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Soulbound Tokens, DAOs, and the Rise of the Decentralized Society: Examining the Path to a New Paradigm

2023-07-11T12:39:03Z

3122188: Created page with "{{DISPLAYTITLE: Soulbound Tokens, DAOs, and the Rise of the Decentralized Society: Examining the Path to a New Paradigm}} Contribution of SIMONE GOZZINI <span id="introd..."

{{DISPLAYTITLE: Soulbound Tokens, DAOs, and the Rise of the Decentralized Society: Examining the Path to a New Paradigm}}

Contribution of [[SIMONE GOZZINI]]


= Introduction =

The concept of Web3 <ref>Web3 refers to the vision and evolution of the internet that aims to shift away from the current centralized model and foster a more open, transparent, and user-centric online ecosystem. Web3 leverages decentralized technologies, such as blockchain, decentralized networks, and cryptographic protocols, to enable greater user control over data, enhance privacy and security, and facilitate peer-to-peer interactions. Its main features are:

* Decentralization: Web3 emphasizes decentralization by dispersing power and data among a network of nodes as opposed to depending on centralized authorities.
* Blockchain technology: By offering a transparent and impenetrable method to record and verify transactions and data, blockchain plays a vital role in Web3.
* User Control and Ownership: The goal of Web3 is to empower people by providing them with more control over their data and digital identities.
* Interoperability: By utilizing open standards and protocols, Web3 encourages interoperability. Users can make use of a variety of services and transfer assets across numerous networks without limitations thanks to the smooth communication and interaction between various decentralized applications (DApps) and platforms built on this new infrastructure.
* Privacy and Security: In order to secure data transmission and storage and ensure privacy in peer-to-peer transactions, it employs cryptographic algorithms.
* Smart Contracts and Decentralized Applications: Web3 leverages smart contracts to automate and enforce rules within decentralized applications. These applications, known as DApps, utilize the decentralized infrastructure provided by Web3 technologies to offer services and functionalities without relying on centralized servers.
* Token economy: Web3 supports the idea of tokenization, in which electronic tokens serve as a representation of ownership or value in decentralized ecosystems. These tokens can be used for a variety of things, including transactions, governance, incentives, and service access.</ref> represents a transformative shift in the internet landscape, aiming to move away from centralized models and foster a more transparent, user-centric online ecosystem. This disruptive innovation has particularly profound implications for the financial world. However, one key aspect that currently lacks in this paradigm is the ability to represent identities, which could unlock numerous new applications.

In their influential papers, Weyl et al. (2022) and Buterin (2022) introduce an intriguing addition to this evolving world: Soulbound tokens (SBTs). These tokens, unlike traditional transferable tokens <ref>A token is a digital representation of value or an asset that exists on a blockchain network. Tokens are created and managed using smart contracts on blockchain platforms such as Ethereum, Binance Smart Chain, or others. Tokens therefore can be utilized to represent full or partial ownership of tangible assets like real estate, works of art, or commodities. Tokenization makes historically illiquid assets more liquid and more transferrable.</ref>, symbolize affiliations, commitments, and social relations. They envision a Web3 where individuals, embodied by their Souls (i.e., digital wallets), can accumulate multiple SBTs to form a comprehensive self-portrait. This innovative approach enables the creation of reputation systems, revolutionizing the way Web3 is conceived. For instance, it opens doors to undercollateralized lending, property rental, and various other possibilities.

Moreover, SBTs have the potential to revolutionize the functioning of decentralized autonomous organizations (DAOs) <ref>DAO stands for Decentralized Autonomous Organization. It refers to an organizational structure that operates through smart contracts on a blockchain network, enabling decentralized decision-making and governance. A DAO is designed to be autonomous, meaning it functions without a central authority or hierarchy. Instead, decision-making power is distributed among its participants, who typically hold tokens representing their ownership or membership in the organization.</ref> by addressing concerns related to colluding accounts and bots in voting processes. By discounting the voting power of colluding accounts and unveiling bots, SBTs can enhance the governance mechanisms of DAOs significantly.

Ultimately, this groundbreaking technology can pave the way for the establishment of a Decentralized Society. Such a society operates on decentralized principles, facilitated by blockchain technology and other Web3 protocols. It empowers individuals with greater control over their data, digital assets, and online interactions, leading to a more democratic and inclusive digital realm.

The structure of the paper is as follows: Section [[#SBTs|2]] delves into the concept of Soulbound tokens (SBTs) and explores why individuals would find it beneficial to have an identity on Web3; Section [[#applications|3]] provides valuable insights into the research conducted by various authors regarding the applications of SBTs; Section [[#DAOs|4]] focuses on the relationship between SBTs and Decentralized Autonomous Organizations (DAOs), with a specific emphasis on how SBTs can enhance voting and funding mechanisms within DAOs; finally, Section [[#DeSoc|5]] presents the concept of a Decentralized Society, which is a social and governance structure enabled by blockchain technology and Web3 protocols. This section explores the potential implications and advantages of a decentralized society in terms of individual control over data, digital assets, and online interactions.


= Soulbound Tokens =


== The Concept ==

Web3 represents a groundbreaking innovation with the capacity to revolutionize numerous industries, ranging from finance and governance to supply chain and social media. Its primary objective is to redefine the internet as a more democratic, secure, and user-centric environment, thereby nurturing innovation, collaboration, and the emergence of new economic models. However, it is noteworthy that developers have predominantly concentrated on exploring Web3’s potential within the realm of finance, leading to the creation of the decentralized finance (DeFi) <ref>DeFi stands for Decentralized Finance. It describes a class of financial applications and platforms that leverage blockchain technology, particularly smart contracts, to provide decentralized and permissionless alternatives to traditional financial services. DeFi seeks to remove the need for intermediaries like banks or other centralized institutions so that people can access financial services like borrowing, lending, trading, and investing directly. Some popular DeFi applications include decentralized exchanges (DEXs), lending and borrowing platforms, stablecoins, and decentralized asset management.</ref> ecosystem. Despite these advancements, certain activities, such as undercollateralized lending, remain unachievable within this ecosystem as of today.

According to Buterin (2022) and Weyl et al. (2022), a notable absence in the current Web3 landscape is the ability to incorporate social identity. Specifically, there is a need for a mechanism to establish a reputation tied to an individual agent, which can be defined based on their past histories and interactions.

The concept of Soulbound tokens draws inspiration from the world of Minecraft, where certain items are designated as soulbound, meaning they cannot be bought, sold, or transferred. Instead, they can only be earned by completing specific tasks. In the context of Web3, Soulbound tokens can be defined as “non-transferable (initially public) tokens representing commitments, credentials, and affiliations. Such tokens would be like an extended resume, issued by other wallets that attest to these social relations” (Weyl et al., 2022, p.1). According to the authors, Soulbound tokens have the potential to enhance Web3 in several ways. They can establish the provenance of assets, allowing for greater transparency and traceability. Additionally, they can facilitate lending opportunities in undercollateralized markets, opening up new avenues for financial inclusion. Furthermore, Soulbound tokens can introduce decomposable shared rights and permissions, creating new markets and opportunities. Lastly, they can be used as a measure of decentralization within the Web3 ecosystem.

The concept of Souls is introduced as “accounts, or wallets, that hold publicly visible, non-transferable (but possibly revocable-by-the-issuer) tokens" (Weyl et al., 2022, p.2). Souls, by storing SBTs, can be regarded as a representation of a user’s history and serve as a form of resume, contributing to the creation of a reputation. It’s important to note that Souls are not necessarily associated with legal names, and there is no explicit protocol-level requirement enforcing a "one Soul per human" rule. Furthermore, the authors do not assume that Souls are non-transferable between humans.

In addition, SBTs play a crucial role in facilitating the formation of communities through unique intersections of Souls. Traditionally, Web3 has relied on airdrops as a means to create communities. Airdrops involve distributing tokens for free to a specific set of wallets that meet certain eligibility criteria, such as holding a minimum amount of a particular token or actively participating in a specific community or platform. However, this approach is susceptible to Sybil attacks and strategic behavior. SBTs offer a solution to these challenges through a process known as "Souldrops." Souldrops allow for a targeted selection of Souls based on their specific characteristics and reputation. For instance, a DAO aiming to gather a community within a particular layer 1 protocol could souldrop to developers who possess a certain number of conference attendance SBTs. By leveraging Souldrops, communities can be formed more effectively, ensuring a higher degree of authenticity and engagement while mitigating the risks associated with sybil attacks and strategic behavior.


=== Community Recovery ===

Preserving one’s Soul is of utmost importance. The authors introduce the concept of community recovery <ref>Community recovery comes from the idea of Social recovery, which pertains to the process of enabling users to regain access to their assets when they encounter situations like losing their private keys or forgetting their passwords. By employing cryptographic safeguards, wallets implementing social recovery offer a reliable and streamlined method to restore users’ identities. Furthermore, social recovery facilitates the restoration of trust between users and their associated networks, while providing a secure means for individuals to store and oversee their personal data and assets. Specifically, social recovery wallets are smart contract wallets that store a user’s assets securely within the confines of the smart contract. These wallets employ a signing key to safeguard access to the smart contract, enabling users to authorize transactions and asset transfers. In the event of a forgotten password, a group of designated trustees, also known as guardians, can collaborate to regain access to the account and facilitate account recovery.</ref>, "where the Soul is the intersectional vote of its social network" (Weyl et al., 2022, p.6). The system involves a group of guardians who possess the authority to modify a user’s wallet keys through a majority vote. However, a challenge arises from the need for users to regularly update and maintain their relationships with these guardians, ensuring that changes can be made when necessary. A stronger approach entails linking Soul recovery to a Soul’s affiliations within various communities. Instead of selectively curating connections, this solution relies on an expansive range of real-time relationships to ensure maximum security. In a community recovery model, the retrieval of a Soul’s private keys would necessitate the consent of a qualified majority of members from a (random subset of) the Soul’s communities. By integrating security into social interactions, a Soul can consistently regenerate their keys through community recovery, effectively deterring Soul theft or sale. This is due to the fact that a potential Seller would have to provide evidence of selling the recovery relationships, making any attempt to sell a Soul lack credibility.

During its early stages, this system encounters a challenge: SBTs are not transferable, which raises the importance of community recovery to enable identity retrieval in case of lost keys. However, community recovery is only feasible when a substantial number of SBTs are already in circulation across various communities. Unfortunately, many individuals may hesitate to participate until a significant number of SBTs are in circulation and community recovery mechanisms are established. To address this issue, the authors propose the introduction of Proto-SBTs. These tokens are both recoverable and transferable, allowing them to be burnt and reissued in a new wallet if needed. This approach ensures the birth of a sufficient number of Souls and enables community recovery before transitioning Proto-SBTs into actual SBTs. By implementing Proto-SBTs as an intermediary step, the system can overcome the initial barrier and facilitate the generation of an ample number of Souls, paving the way for community recovery to become a reality.


== Why a Digital Identity ==

The potential of SBTs is multifaceted, encompassing various aspects, as explained in the previous section. However, it raises a fundamental question: why would individuals adopt a digital identity <ref>The concept of digital identity comes from the principles of Self-sovereign identity (SSI), which is an authentication system designed to decentralize control over personal identity data, empowering individuals to securely, privately, and autonomously manage their own information. SSI grants individuals full ownership and authority over their data. This enables them to freely transfer their information between different service providers without the need for a centralized intermediary to mediate the process. Shuaib et al. (2021) propose a comprehensive framework of principles for Self-Sovereign Identity, encompassing the following key aspects:

* Control: Individuals should have complete authority over their digital identity, personal data, and digital assets.
* Access: Users should enjoy convenient and unrestricted access to their data whenever they require it.
* Transparency: Individuals should be provided with transparent information regarding the usage of their personal data, empowering them to make well-informed decisions.
* Persistence: The connection between an individual’s digital identity and their assets and occurrences should remain persistent throughout their life.
* Portability: It should be effortless for individuals to transfer their assets seamlessly among themselves.
* Interoperability: Digital identities should be compatible and operable across various blockchains and networks.
* Consent: Individuals should retain control over how their data is utilized and who can access it.
* Existence: Measures should be implemented to minimize the risks associated with bots and fake identities, which can be demonstrated through the use of Soul-bound Tokens as proof of existence.
* Minimization: Individuals should only be required to share the minimum necessary information about themselves to prove their identity and access services, thereby prioritizing privacy and data protection.

By adhering to these principles, the SSI framework aims to establish a secure, user-centric approach to digital identity management.</ref> in Web3? To address this query, Chaffer and Goldston (2022) explore the Terror Management Theory (TMT), a model rooted in social and evolutionary psychology that elucidates how individuals safeguard themselves from existential concerns tied to mortality.

In their analysis, they define Web3 identity as a "collection of digital assets" (Chaffer & Goldston, 2022, p.2). A notable characteristic of digital identity is its potential for permanence, allowing individuals to leave behind a long-lasting legacy and thus achieve digital immortality (Allen, 2016). TMT provides insights into why individuals amass digital assets, which contribute to their identity, and why the immortal nature of digital identity holds significance. According to TMT, humans grapple with an existential dread of death, a realization of its inevitability, which conflicts with the innate instinct for self-preservation. To mitigate this existential fear, individuals seek solace in a realm of symbolic existence, such as culture, aiming to live on in the memories and minds of others. Digital identity serves this purpose effectively: given that the Internet does not have a predetermined endpoint, individuals perceive the accumulation of digital assets as a means to validate their existence and etch themselves into the annals of Web3’s history. This is particularly true for those who lack alternative paths to transcendence, such as religious beliefs. However, this assumption relies on the premise that Web3 achieves mainstream adoption; otherwise, insufficient recognition of the digital existence of the deceased would occur. The authors highlight that the sustainability of employing digital identity with this purpose hinges on the self-esteem that individuals derive from the system. In other words, the value people place on their digital identity as a conduit for achieving immortality determines their willingness to combat cyber attacks and identity theft, ensuring the preservation of their online image.

Lastly, the authors underscore the value of SBTs through the lens of logotherapy, a theory emphasizing the pursuit of meaning in one’s life. While holding digital assets in the form of NFTs to construct an identity may lack inherent meaning for some individuals, SBTs hold greater significance. Since SBTs encapsulate the digital history of a user, they enable the preservation not only of assets but also of reputation and transactional history. Consequently, the endeavor to transcend mortality through digital identity becomes more meaningful and valuable.


= Exploring the Diverse Applications of SBTs =

SBTs, although a relatively new concept, have already piqued the interest of authors who are delving into various potential applications of this innovative tool.


== Digital Art ==

Weyl et al. (2022) discover significant implications for the realm of digital art. They highlight that an artist’s identity can be established through a soul possessing multiple STBs, which allows for the development and establishment of their reputation. This, in turn, ensures that the NFTs <ref>Non-Fungible Tokens (NFTs) are a type of digital asset that represent ownership or proof of authenticity of a unique item or piece of content, such as artwork, music, videos, collectibles, virtual real estate, and more. Unlike cryptocurrencies like Bitcoin or Ethereum, which are fungible and interchangeable, NFTs are distinct and unique. Through the blockchain technology, NFTs can be transferred between different individuals or entities, allowing ownership to change hands.</ref>issued by the artist are legitimate and not counterfeit. Furthermore, this approach enables the verification of the authenticity of photographs and videos by not only tracing the time of creation but also capturing their social provenance. Indeed, by utilizing a Soul, it becomes possible to maintain a faithful record of the artist’s history.


== Lending ==

Weyl et al. (2022) find that uncollateralized lending is not widely supported in Web3 due to the absence of systems that can assess credit trustworthiness, challenges in verifying identities, and susceptibility to sybil attacks involving the creation of multiple fraudulent accounts. However, SBTs can address these issues and enable uncollateralized lending by establishing a credit score to evaluate the trustworthiness and reputation of borrowers. SBTs can record the credit transaction history, education, and work background of the associated Soul, providing insights into the likelihood of repayment. Algorithms can then analyze this information to generate a meaningful credit score. Loans can be represented by non-transferable but revocable SBTs, which can be replaced with proof of payment once the loan is fully repaid. This mechanism ensures that borrowers who attempt to evade their obligations by creating new Souls will lack the necessary SBTs to establish a reliable reputation.

However, the challenge lies in initiating this process, as there is no repayment history available initially since Souls have not yet accumulated any debt. To address this, other pertinent information such as education and work history can be considered to enable lending, particularly within social connections. By leveraging these additional factors, lenders can make informed decisions even in the absence of a repayment track record.


== Students ==

Tejashwin et al. (2023) propose a model that utilizes Souls and SBTs as a means to store and authenticate academic achievements. They argue that this approach offers greater privacy and security compared to centralized models, which are susceptible to document forgery and database tampering. The authors demonstrate that their model enables efficient access to student data by universities and recruiters, reducing verification time and ensuring reliable and immutable records.

According to the authors, each student should possess a wallet (a Soul) to store their data. To address privacy concerns regarding school-related information, SBTs can be utilized to store the data off-chain. Students have the flexibility to choose the storage method for their on-chain data, whether it be on their own devices, trusted cloud providers, or decentralized networks like IPFS (Interplanetary File System). Storing data on-chain allows for the utilization of smart contracts <ref>Smart contracts are self-executing agreements or contracts with the terms of the agreement directly written into lines of code. These contracts are stored on a blockchain network, such as Ethereum, and automatically executed when predefined conditions encoded in the contract are met.</ref>, granting permissions to write SBT data while managing different levels of access rights. Students retain the option to selectively disclose the contents of their SBTs or associated data stores as desired.

The process involves the issuer uploading the record onto the blockchain <ref>The blockchain is a distributed digital ledger technology that enables many parties to securely and openly maintain a shared, immutable record of transactions. A blockchain is fundamentally made up of a chain of blocks, each of which contains a list of transactions. A chronological and impenetrable record is produced by grouping these transactions, cryptographically hashing them, and connecting them to the preceding block in the chain. The blockchain is impervious to manipulation thanks to this linking mechanism, which assures that any modifications to a block will need changes to all succeeding blocks. Its key features are:

* Decentralization: A decentralized network of computers known as nodes underlies the operation of a blockchain. The upkeep of the blockchain is split among the participating nodes rather than falling under the control of a single central authority.
* Transparency: Blockchain transactions are frequently accessible to all network users.
* Security: Blockchain uses cryptographic methods to achieve security. Cryptographic hashes are used to link each block in the chain, ensuring that any changes to a block would cause an incorrect hash, alerting the network to tampering efforts. Consensus mechanisms are used to validate and agree upon the state of the blockchain.
* Since the blockchain is distributed and uses cryptographic hashing, any modifications to earlier blocks require the agreement and processing resources of a majority of the network’s nodes, making the blockchain extremely hard to alter.</ref>, where it is encrypted and divided into multiple blocks. These blocks are distributed and stored across nodes within the DCS network. The data undergoes hashing, resulting in hashed values that are utilized for storage and retrieval purposes in decentralized applications.


== Property Rental ==

In their study, Sharma et al. (2023) delve into the role of SBTs in facilitating transactions on property rental websites. They highlight the prevalent issue of scams and fraud in the realm of online real estate, involving both landlords and potential tenants. To address this problem, the authors propose a blockchain-based model that focuses on establishing a robust online reputation through the use of SBTs, thereby instilling trust in the rental process.

To begin, upon user registration, an Entry SBT linked to the user’s personal wallet is created. Landlords, as users, can utilize smart contracts to list their properties on the website. The property information is stored on the InterPlanetary File System (IPFS), a decentralized file sharing peer-to-peer network. Subsequently, when a prospective tenant negotiates terms with a landlord to initiate a rental agreement, they are required to submit a security fee in the form of cryptocurrencies, acting as collateral. In return, the tenant receives a Deal Token SBT, enabling them to pay rent and access other property-related features. Smart contracts replace traditional rental agreements, ensuring the execution of predetermined terms without deviation. Upon the conclusion of the rental contract, the tenant can withdraw the deposit, while the Deal Token remains associated with the Entry Token. The authors propose the use of administrative smart contract "Admin.sol" for issuing Entry Tokens, "PropertyFactory.sol" as a smart contract for uploading property data, and "Property.sol" as a template for property contracts and Deal Token creation. Furthermore, to ensure the credibility of the platform, a comprehensive review system has been implemented. This system allows landlords and tenants to review each other based on a star rating system ranging from 1 to 5. These reviews are permanently linked to their respective Entry Tokens and contracts, providing transparency and accountability. The current reputation at time <math display="inline">t</math> can be obtained through the following equation: <math display="block">R = \frac{\left[1.01 \times \left(R_{\text{max}} - \left(R_{\text{min}} \times \left[1 - \log_5x)\right]\right)\right)\right] + \left[R \times (n - 1)\right]}{n}</math> where <math display="inline">R_{max}</math> is the maximum reputation score possible (set to 100), <math display="inline">R_{min}</math> is the minimum reputation score possible (set to 50), <math display="inline">n</math> is the total number of reviews received and <math display="inline">x</math> is the current rating score. The factor of <math display="inline">1.01</math> is utilized to assign higher importance to the most recent reviews. If, for any reason, the calculated reputation score <math display="inline">R</math> falls below <math display="inline">R_{\text{min}}</math>, the user’s collateral will be liquidated, and they will be prohibited from further transactions on the website. This measure ensures that users with inadequate reputation scores are unable to engage in transactions.

The benefits of this system are manifold. By leveraging blockchain-based payments, the flexibility of transactions is increased, as it is not constrained by commercial bank working hours. Moreover, it facilitates easier cross-border payments and incurs lower transaction fees compared to traditional online rental platforms. The use of smart contracts enables the automation of various rental property operations, such as eviction procedures for tenants in arrears, resulting in time savings. Lastly, the implementation of peer-to-peer reputation systems resolves the vulnerabilities associated with centralized reputation systems, which are susceptible to attacks and manipulation (Dennis & Owen, 2015).


== Inheritance ==

Goldston et al. (2023) present a novel model for digital inheritance, focusing on the Polkadot and Kusama blockchain networks and utilizing SBTs and the social recovery pallet as a practical application. As technology and digital assets like NFTs and cryptocurrencies play an increasingly prominent role in everyday life, the need for effective transfer mechanisms to heirs has become paramount. The model introduces key entities involved in the digital inheritance process. The "testator" is the individual creating their will concerning their assets, while the "executor" is responsible for carrying out the testator’s wishes. The "trustee" manages the assets within a trust, and the "beneficiary" is the recipient of the testator’s wealth. SBTs can be utilized within this framework to offer proof of existence for the entities involved. If the testator, executors, trustees, and beneficiaries have SBTs generated by the testator and transferred to their respective wallets, it would become an integral component of the digital inheritance plan. Such a process has the potential to validate the existence of all users and enhance the security of the testator’s digital assets. To implement their solution, the authors leverage the Substrate framework, a versatile and modular platform that empowers developers to build customized blockchain applications. Within Substrate, "pallets" serve as self-contained units of code, offering various functionalities such as governance, staking, and consensus. Developers can choose from prebuilt pallets or create their own, tailoring their blockchain to specific requirements.

The proposed model incorporates a "Social Recovery" pallet built on the Substrate framework. This pallet addresses the issue of lost credentials by generating a new key and account to recover assets. The process involves the setup of a "createRecovery" mechanism in an individual’s account, where the user designates a group of trustees. Through a multi-signature approach <ref>Multi-signature (multi-sig) is a cryptographic technique employed in blockchain networks, wherein multiple individuals must provide authorized signatures to validate a transaction before it can be executed. This method enhances decentralization by reducing points of failure in the governance model. When initiating a multi-sig transaction, if the required number of signatures from users holding private keys is not obtained, the transaction fails to execute. This ensures that consensus among the authorized parties is established before any action is taken on the blockchain.</ref>, these trustees can approve the recovery process to regain access to the account. In the context of digital inheritance, the testator collaborates with their executor to select friends as trustees. Additionally, determining a "delay period" is crucial. This period represents the time required to pass after the testator’s death before account recovery becomes possible. By incorporating this delay, the model enhances security, mitigating potential fraud attempts while the testator is alive. Upon the testator’s demise, either the digital executor or a designated trustee must initiate the recovery process by submitting a security deposit. This deposit distinguishes the initiating party from others and can be utilized to halt the recovery process if necessary, using the "closeRecovery" function. The executor subsequently contacts all trustees, requesting their participation in a "vouchRecovery" transaction. If the multi-signature approach yields positive results and the delay period has elapsed, the executor gains access to the inheritance via the original account. Assets can then be distributed to beneficiaries through the creation of a new account. After completing this distribution, the "closeRecovery" option is invoked to reclaim the security deposit. Finally, the "removeRecovery" option is called, ensuring the irreversibility of transactions.

In addition to its applicability to Web3 applications, the authors suggest that the proposed process can be expanded using SBTs to enhance interoperability between Web2 and Web3 digital assets. This extension enables the retrieval and inheritance of not only blockchain-based assets but also social media accounts and associated content.


== SBTs, Prediction Markets and AI ==

According to Weyl et al. (2022), SBTs can find applications in various predictive models that utilize user data, such as artificial intelligence (AI) systems and prediction markets. AI models rely on data feeds and employ nonlinear models to generate predictions, while prediction markets serve as platforms enabling individuals to trade contracts based on future events. These markets facilitate the buying and selling of shares or contracts representing potential outcomes, with contract prices reflecting the aggregated beliefs of the market participants regarding the probability of those outcomes.

However, prediction markets face a challenge. Although they claim to aggregate the beliefs of all participants, they often primarily elicit the beliefs of individuals prone to gambling, resulting in enrichment for the winners while impoverishing others and discouraging risk-averse individuals from participating. To address this issue, quadratic rules (see Section [[#Quadratic Mechanisms|4.1]]) can be implemented to elicit precise probability estimates from all participants. Furthermore, SBTs can enhance prediction markets by allowing the computation and synthesis of diverse beliefs, taking into account social context, reputation, credentials, and affiliations, which form the essence of a participant’s Soul.

SBTs also have the potential to transform the field of AI. AI systems heavily rely on data, yet the creators of the surveilled data often remain unaware of their role in shaping these models. They typically retain no residual rights to the data they generate and are considered "incidental" rather than essential participants. Moreover, the process of gathering extensive data divorces models from their social context, concealing their biases and limitations, and hindering our ability to address and compensate for them. By integrating SBTs, AI can be augmented in a natural manner. SBTs enable the programming of economic incentives that reward data creators with rich provenance while empowering them with residual governance rights over their data. Specifically, SBTs allow the implementation of carefully targeted incentives for data (including data quality) tailored to individuals and communities based on their unique characteristics. Concurrently, model-makers can track the characteristics of the collected data and its social context, as reflected by SBTs, and identify contributors who can offset biases and compensate for limitations.


= SBTs and DAOs =

Weyl et al. (2022) study how SBTs can offer several benefits to improve DAOs. Firstly, they can serve as a deterrent against "vampire attacks", which are a type of malicious activity where an attacker exploits the functionality of decentralized finance (DeFi) protocols to drain liquidity and assets from other projects or platforms. To discourage free-riders, one approach is to establish a social norm around souldropping, potentially by vesting SBTs exclusively to Sybil-resistant Souls who have contributed liquidity. Moreover, souldrops can be withheld from Souls involved in vampire attacks by shifting their liquidity.

Secondly, SBTs can facilitate dynamic leadership roles in DAOs, adapting as community compositions change, as reflected by the types of SBTs individuals possess.

Furthermore, SBTs enable DAOs to represent property contracts and decompose property rights that were not feasible with transferable tokens like NFTs. For instance, permissioning access to privately or publicly controlled resources (such as homes, cars, museums) can be accomplished without the risk of the right being transferred to an untrusted user, as SBTs are non-transferable.

DAOs are particularly susceptible to sybil attacks, where a single user gains majority voting power, leading to complete control. SBTs can help mitigate this issue in DAOs through the following features:

* Identifying Souls from bots using the reputation mechanism enabled by SBTs.
* Granting more voting power to Souls with reputable SBTs.
* Checking for correlations between SBTs held by supporting voters and assigning a lower vote weight to highly correlated voters.

The latter element is especially crucial, as a vote from multiple Souls with identical SBTs may indicate a Sybil attack or a group prone to judgment errors, warranting a reduction in vote weight.

Moreover, SBTs facilitate the measurement of decentralization and pluralism in a DAO. This can be achieved by granting voting power only to Sybil-resistant Souls, discounting voting power for correlated Souls with many common SBTs, and assessing correlations between SBTs across different layers of the network stack to gauge decentralization.


== Quadratic Mechanisms, Weighted Voting and Collusion ==

The authors propose the introduction of various weighted voting mechanisms as a solution to the issues discussed in the previous section. Their models address biases and tendencies towards excessive coordination by acknowledging the presence of partial and pre-existing cooperation. This recognition is crucial also in public funding models, which are used to raise funds for blockchain-based projects or initiatives from the public, because it corrects the assumption that agents are purely selfish. In reality, humans are rarely completely selfish or completely cooperative.

Excessive coordination among agents in public funding models can result in collusion, which can take on different manifestations, such as:

* Token allocation manipulation: colluding parties may coordinate their actions to manipulate the allocation of tokens during token sales or fundraising events. By colluding, they can unfairly influence the distribution of tokens to their advantage.
* Insider collusion: This form of collusion involves individuals or groups working together to gain unfair advantages within the public funding process. They may have privileged access or information that enables them to manipulate the system for personal gain.
* Market manipulation: Colluding parties can engage in market manipulation by deliberately influencing the prices of tokens. This can involve coordinated buying or selling activities to create artificial price movements, exploiting the market for their own benefit.

To mitigate these risks, the authors emphasize the importance of SBTs, which promote cooperation among diverse entities by reducing the cooperative rewards given to correlated Souls (Souls that share some degree of SBTs).

The authors provide an illustrative example using the Quadratic Funding (QF) model, but the same principles can be applied to Quadratic Voting (QV). Quadratic mechanisms, by their nature, encourage collaboration starting from a self-centered perspective, assuming agents are selfish. However, these mechanisms can be vulnerable to groups that already exhibit cooperative behavior.

Quadratic Funding (QF) operates by allocating funds from a community to shared projects in a manner that is proportional to the square of the sum of the square roots of individual contributions. In this model, for a set level of contributions, the matching funds increase exponentially with the number of individual contributors, illustrating increasing returns to collective action. However, individual contributions experience diminishing returns, meaning that concentrated individual action yields less relative impact. In other words, as more individuals contribute to a project, the matching funds available for that project grow quadratically. This encourages collective participation and rewards the collaborative efforts of the community. Conversely, the impact of an individual’s contribution diminishes as more contributors join, emphasizing the need for collective action rather than relying solely on concentrated individual efforts. QF thus fosters an environment where collective engagement and collaboration are incentivized and can yield greater overall results.

Suppose that there are 3 non cooperating individuals who contribute, respectively, with a sum A, B and C. In this case, we have: <math display="block">\text{Simple Match} \sim (\sqrt{A} + \sqrt{B} + \sqrt{C})^2 - (A + B + C)</math>

Suppose now that two of the 3 individuals work in the same place (so they both have the same SBT which attests where they work) and let’s suppose that, for this reason, they coordinate. An approach useful to take into account this would be clustering their contributions (in this example A and B), obtaining: <math display="block">\text{Cluster Match} \sim (\sqrt{A + B} + \sqrt{C})^2 - (A + B + C)</math> Considering the coordination between the co-workers, the two coordinated contributions will be given less weight, while the individual contribution will carry more significance. In the case of perfect coordination, they will evenly divide their joint contribution, resulting in: <math display="block">\text{Cluster Match} \sim (\sqrt{2A} + \sqrt{C})^2 - (2A + C)</math> Another adjustment could be obtained by reducing the weight of the two co-worker’s contribution by a factor of <math display="inline">\sqrt{2}</math> to compensate for their coordination, obtaining: <math display="block">\text{Offset Match} \sim (\frac{\sqrt{A} + \sqrt{B}}{\sqrt{2}} + \sqrt{C})^2 - (A + B + C)</math> In the case of perfect coordination, it remains optimal for both individuals to make the same contribution <math display="inline">A=B</math>, as they effectively act as a single agent.

However, assuming a single common membership is an oversimplification as individuals often have multiple affiliations, some of which may be shared with other users. To address this, the authors propose a general model that considers these complexities. For each individual <math display="inline">i = 1, \ldots, N</math>, let <math display="inline">T_i</math> be the number of affiliations he has. It is also assumed that each affiliation carries equal importance, meaning they have the same weight. Let <math display="inline">\sum</math> represent the set of all "affiliation groups," which are projects formed by a specific set of holders belonging to a given affiliation. These affiliation groups are associated with the set of participants in the match. A typical element in this set can be denoted as <math display="inline">\sigma_j</math>, representing an individual affiliation group within <math display="inline">\sum</math>. Let <math display="inline">T_i = \sum_{j=1}^{|\Sigma|} 1_{i \in \sigma_j}</math>, where 1 is the indicator function. Let <math display="inline">c_i</math> be the individual contribution of <math display="inline">i</math>. The general formula for Cluster Matching will be: <math display="block">ClusterMatch \sim \left( \sum_{j=1}^{|\Sigma|} \sqrt{\sum_{i=1}^{|\sigma_j|} \frac{c_i}{T_i}} \right)^2 - \sum_{i=1}^{N} c_i .</math> Now let <math display="inline">s_{i,k} = \frac{\sum_{j=1}^{|\Sigma|} 1_{i \in \sigma_j} 1_{k \in \sigma_j}} {T_i}</math> be the correlation score between any ordered pair of individuals <math display="inline">i</math> and <math display="inline">k</math>. Let <math display="inline">\alpha_i</math> be the offset coefficient that solves the system of equations (one of each individual) <math display="inline">\alpha_i + \sum\limits_{\begin{array}{c} k \neq i \\ \end{array}}^{N} \alpha_k s_{k,i} = 1.</math> Then we obtain: <math display="block">\text{OffsetMatch} \sim \sum_{i=1}^{N} \left( \sqrt{ \alpha_i c_i }\right)^2 - \sum_{i=1}^{N} c_i .</math> This solution is generally able to achieve optimality (i.e. welfare maximization).

A third mechanism, known as the "Pairwise mechanism" , does not aim for optimality but derives its strength from obtaining coordination information from the contribution values themselves. This approach is particularly relevant in the context of multiple projects and operates under a per-pair matching capacity limit, denoted as <math display="inline">M</math>. <math display="inline">\forall</math> pair of agents <math display="inline">(A, \text{ } B)</math>, if they contribute <math display="inline">x_{A \rightarrow P}</math> and <math display="inline">x_{B \rightarrow P}</math> to the same project <math display="inline">P</math>, they obtain a subsidy <math display="block">\text{Match}_{\text{AB} \rightarrow P} = \frac {2M \sqrt{x_{A \rightarrow P} x_{B \rightarrow P}}} { M + \text{CorrelationScore}_{AB}}</math> where <math display="inline">M</math> is a parameter and <math display="inline">\text{CorrelationScore}_{AB} = \sum_{\text{all projects P }} \sqrt{x_{A \rightarrow P} x_{B \rightarrow P}}</math>, which is designed to indicate the degree of overlap in project contributions between two participants. When participants A and B both contribute <math display="inline">x</math> to a project, the <math display="inline">\text{CorrelationScore}_{AB}</math> increases by <math display="inline">x</math>. However, if they contribute different amounts, the <math display="inline">\text{CorrelationScore}_{AB}</math> increases by the geometric mean of their contributions. A low <math display="inline">\text{CorrelationScore}_{AB}</math> means that agents are independent, and, when they contribute to the same project, the subsidy given will be close to the maximum. If participants A and B contribute frequently and/or in large amounts to the same project, it is assumed that they are highly coordinated and behave more like a unified entity. In such cases, the subsidies to projects they co-fund are discounted, acknowledging their collaborative nature. It is possible to demonstrate that the losses incurred from misidentifying a colluding group as separate, independent agents are limited, while this is not the case for Simple Matching and Cluster Matching. As previously mentioned, Pairwise Matching does not achieve optimality. Colluding actors still maintain an incentive to slightly overstate the value they assign to certain projects, and can even exploit the system by contributing to a fictitious project under their control. Instead, this approach serves as a second-best solution, specifically designed for scenarios where there is limited external information available regarding which actors are genuinely colluding.

Nevertheless, these approaches are not without their risks. When SBTs are employed to quantify and mitigate coordination, there is a possibility that individuals may intentionally evade or avoid using SBTs in order to maximize their influence through their vote or contribution (Hildebrandt, 2022; Weyl et al., 2022).


= SBTs and the Decentralized Society =

According to Weyl et al. (2022), the integration of SBTs within the Web3 environment has the potential to catalyze the emergence of a Decentralized Society (DeSoc). DeSoc is defined as "a co-determined sociality, where Souls and Communities convene bottom-up, as emergent properties of each other to produce plural network goods across different scales" (Weyl et al., 2022, p.17).

Plural network goods are a fundamental aspect of this paradigm, representing goods or resources that are generated, exchanged, and utilized within decentralized networks built on blockchain technology. These goods possess the traits of being non-excludable and non-rivalrous, while their creation, governance, and ownership are primarily driven by the network’s participants through the application of smart contracts, decentralized protocols, and blockchain-based systems. They contribute to the economic growth facilitated by increasing network returns.

DeSoc transforms the competitive race for control and speculation over network value in decentralized finance (DeFi) into a collaborative endeavor focused on building, participating in, and governing these networks from the grassroots level. At the very least, DeSoc’s social foundation can ensure sybil-resistance (enabling community governance) within DeFi, safeguard against vampiric tendencies (internalizing positive externalities to foster open-source networks), and prevent collusion (preserving network decentralization). Through these structural corrections, DeFi can support and expand plural networks that provide broad benefits.

The primary strength of DeSoc lies in its network decomposability, which enables the proliferation and intersection of nested networks. Rather than solely relying on the trustless premise of DeFi, DeSoc incorporates trust networks that underpin the existing real economy. This integration empowers the generation of plural network goods that are resilient against capture, extraction, or dominance. With such an enhanced social fabric, Web3 can move away from short-term hyper-financialization and embrace an expansive future characterized by increasing returns across social distances.


== Challenges ==

DeSoc can give rise to various challenges, among which the following problems stand out:

* Privacy: One of the significant concerns with DeSoc is the potential exposure of sensitive information about an individual’s Soul due to the public nature of the blockchain. One approach to addressing this issue is by adopting multiple identities based on the context, although these identities can still be easily uncovered. Another solution involves allowing SBTs to store data off-chain, such as on personal devices or in cloud services. By doing so, explicit permission would be required to access the data, enabling a Soul to decide when and if they want to disclose the contents of their SBTs. Additionally, the use of "zero-knowledge proofs," a cryptographic technique, can provide a means for individuals to prove statements without revealing any additional information beyond the statement itself.
* Cheating: There is a risk that Souls may engage in fraudulent activities to gain entry into communities and exploit their governance and property rights. Through bribery, both humans and bots could fabricate a counterfeit social network that creates the illusion of an authentic human Soul, complete with (fake) SBTs. To mitigate this issue, various measures can be implemented. One approach involves incentivizing whistleblowers to expose collusions of significant magnitude, making such collusion untenable. Another strategy is to leverage ZK technology, which can cryptographically prevent certain attestations made by a Soul from being verifiable. Consequently, attempts to sell specific types of attestations would lack credibility, as the briber would have no means of determining whether the recipient of the bribe fulfilled their part of the agreement.


== The strengths of DeSoc ==

DeSoc, empowered by Souls and SBTs, represents a significant advancement over other paradigms in Web3, such as the traditional "legacy" identity ecosystem, the pseudonymous economy, and proof of personhood.

The legacy identity system relies on physical documents or ID cards issued and managed by a third party, such as a government, university, or employer. These identities are verified through communication with the respective third parties. However, this system is centralized, inefficient, and lacks composability. In contrast, DeSoc offers a decentralized and horizontal approach, providing a more efficient way to meet the security requirements of government IDs.

The pseudonymous economy involves individuals accumulating transferable zero knowledge (ZK) attestations in their wallets and, in order to evade reputational attacks, they may transfer a subset of attestations to new wallets or split them among multiple wallets, without traceability. However, this approach faces challenges when it comes to initiating a new identity to escape attacks, complicating reputation-staking for lending and provenance. It also doesn’t integrate well with governance mechanisms aiming to correct correlations or address Sybil attacks. In contrast, DeSoc does not heavily rely on identity separation and allows for contextualizing attackers, thereby improving provenance and accountability.

Proof of personhood shares a similar approach to DeSoc by providing tokens with individual uniqueness to represent identities. However, its main limitation is that it only represents individual identities and fails to capture the broader aspects of social identity, including reputation, relationships, and solidarities. DeSoc, on the other hand, offers a more comprehensive framework that encompasses these social aspects, enhancing the representation and understanding of identities.

=Notes=
<references />

=References=

* Allen, A. L. (2016). The duty to protect your own privacy. In A. Moore (Ed.), Privacy, security and accountability: Ethics, law and policy. Rowman Littlefield.
* Buterin, V. (2019). Pairwise coordination subsidies: A new quadratic funding design. [Accessed on June 27, 2023]. Ethereum Research. https://ethresear.ch/t/pairwise-coordination-subsidies-a-new-quadratic-funding-design/5553
* Buterin, V. (2022). Soulbound. [Accessed on June 12, 2023]. Vitalik Buterin’s website. https://vitalik.ca/general/2022/01/26/soulbound.html
* Chaffer, T. J., & Goldston, J. (2022). On the existential basis of self-sovereign identity and soulbound tokens: An examination of the "self" in the age of web3. Journal of Strategic Innovation and Sustainability, 17(3).
* Dennis, R., & Owen, G. (2015). Rep on the block: A next-generation reputation system based on the blockchain. 2015 10th International Conference for Internet Technology and Secured Transactions (ICITST), 131–138.
* Goldston, J., Chaffer, T. J., Osowska, J., & von Goins II, C. (2023). Digital inheritance in web3: A case study of soulbound tokens and the social recovery pallet within the polkadot and kusama ecosystems [Accessed on June 17, 2023]. https://arxiv.org/abs/2301.11074
* Hildebrandt, F. (2022). The future of soulbound tokens and their blockchain accounts. Konferenzband zum Scientific Track der Blockchain Autumn School 2022, (2), 18–24.
* Sharma, S., Kumar, A., Sengar, N., & Kaushik, A. K. (2023). Implementation of property rental website using blockchain with soulbound tokens for reputation and review system [Accessed on June 20, 2023]. https://ceur-ws.org/Vol-3390/Paper3.pdf
* Shuaib, M., Mohd Daud, S., & Alam, S. (2021). Self-sovereign identity framework development in compliance with self-sovereign identity principles using components. International Journal of Modern Agriculture, 10(2), 3277–3296.
* Tejashwin, U., Kennith, S. J., Manivel, R., Shruthi, K. C., & Nirmala, M. (2023). Decentralized society: Student’s soul using soulbound tokens. 2023 International Conference for Advancement in Technology (ICONAT), 1–4.
* Weyl, E. G., Ohlhaver, P., & Buterin, V. (2022). Decentralized society: Finding web3’s soul [Accessed on June 5, 2023]. https://ssrn.com/abstract=4105763

Main Page

2023-07-11T11:57:58Z

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= DeCrypto: the Bocconi Algorand Fintech Lab collaborative wiki =

== Latest==
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*Integrating (Algorand) DLTs with market infrastructures: analysis and proof-of-concept for secure DvP between TIPS and DLT platforms. [https://www.bancaditalia.it/pubblicazioni/mercati-infrastrutture-e-sistemi-di-pagamento/approfondimenti/2022-026/index.html?com.dotmarketing.htmlpage.language=1&dotcache=refresh Paper] and [https://www.bancaditalia.it/media/agenda/2022-09-30_technical-meeting-on-integrating-dlts-with-market-infrastructures/ program of the technical meeting.]
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== Research Insights ==

* [[Crypto Financial Markets and Institutions]].
*[[Blockchain Economics and the Law|Blockchain, the Economy and the Law.]]
* [[Digital Currencies|Digital Currencies with a special Focus on CBDC]].
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* [[Reputation, Trust, and Reputation Games: Exploring the Dynamics of Reputation in Game-Theory Contexts]].
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== Teaching Insights ==
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**[[Blockchain 101 Zero Lab Slides|Slides]]
**[[Blockchain 101 Zero Lab|Videos]]

== Current [[Contributors]] and [[Moderators]] @ Fintech Lab ==
The Lab acknowledges the interaction with the leading Bocconi Student Associations active in the area of [https://bocconiblockchain.com/ Blockchain] and [https://www.bsfintechsociety.com/ Fintech] space and a number of [[Industry adviser|industry advisors]] to foster the interaction with the broader fintech community.

== Why a wiki project: Introduction from the Scientific Director ==
Digital transformation is disrupting the financial sector and more generally industrial organization. This collaborative wiki project offers an efficient medium to cluster and consolidate major achievements in the areas of interdisciplinary research interested by this innovation wave.

The initial focus of the wiki will be on the financial and economic impact of distributed ledger and machine learning technologies.

The supervised contents included in the wiki are produced and made publicly available by experts of the Bocconi community with the unique goal of creating a common knowledge basis. It will hopefully reduce the time to build a solid scientific discipline in the new interdisciplinary areas that are emerging at the boundaries of both social and hard sciences.

Most of the public debate on these interesting topics takes place through social media and is unstructured. On the contrary, wiki contents are supervised in order to disentangle rigorous theoretical and empirical economic analysis from dangerous marketing narratives.

Moderators will accept a contribution only if it satisfies minimum consistency conditions with general financial economic principles. Wiki contributions may leverage on results already established and published in relevant academic sources like top-tier, peer-reviewed journals. Link to external sources is allowed, but the contributor is required to provide a clear assessment about the nature and the quality of the information retrieved from the source. The bibliography section at the end of each contribution provides a complete list of all utilized information sources.

Contributions are not intended to provide any financial advice and are exclusively intended to promote scientific discussion and dissemination.

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Claudio Tebaldi

Reputation, Trust, and Reputation Games: Exploring the Dynamics of Reputation in Game-Theory Contexts

2023-06-11T16:05:29Z

3122188: Created page with "{{DISPLAYTITLE: Reputation, Trust, and Reputation Games: Exploring the Dynamics of Reputation in Game-Theory Contexts}} Contribution of SIMONE GOZZINI <span id="introduc..."

{{DISPLAYTITLE: Reputation, Trust, and Reputation Games: Exploring the Dynamics of Reputation in Game-Theory Contexts}}

Contribution of [[SIMONE GOZZINI]]


= Introduction =

Trust and reputation are different but interrelated concepts. The former can be defined as ''“a particular level of the subjective probability with which an agent assesses that another agent or group of agents will perform a particular action, both before he can monitor such action (or independently of his capacity ever to be able to monitor it) and in a context in which it affects his own action”'' (Gambetta, 2000, p. 5), while the latter can be defined as ''“a situation when agents believe a particular agent to be something"'' (Cabral, 2005, p. 3). Trust is a fundamental human sentiment that plays a crucial role in fostering cooperation among individuals. It serves as a catalyst for positive outcomes in various domains, including stock market participation, firm performance, and efficient market transactions. Trust creates an environment where individuals feel secure and confident in engaging in economic exchanges, leading to smoother and more fluid interactions. In fact, trust is considered a vital ingredient that allows complex modern societies to not only exist but also evolve and thrive (Popitz, 1980). Its influence permeates through different aspects of human interaction, enabling cooperation, economic growth, and societal development. However, trust involves risk and uncertainty about the other party’s behavior, given that perfect monitoring of what the other agent is doing is not possible: trust and trustworthiness are not easy to develop. Throughout history, various solutions have been devised to address this problem, including the use of physical coercion, contract law, and reputation. Among these solutions, reputation holds significant importance in enhancing trust and trustworthiness by reducing uncertainty, given that it ''”establish(es) links between past behavior and expectations of future behavior.”'' (Mailath & Samuelson, 2015, p. 166). By leveraging reputation, individuals can make more informed judgments about the trustworthiness of others, thereby reducing the risks associated with trust. As reputation serves as a bridge between past actions and future expectations, it plays a pivotal role in fostering trust and creating a more conducive environment for cooperative interactions.

Mailath and Samuelson (2006) extensively study the effects of reputation within the environment of repeated games, which has been found effective because those games provide a clear mathematical framework to describe both the short-term incentives that encourage opportunistic behavior and, through well-defined specifications of future actions, rewards, and punishments, the incentives that discourage opportunistic behavior.

The paper is organized as follows: section [[#reputation and trust|2]] offers contextual information on the connection between trust and reputation. It discusses the interplay between these concepts and their significance in various domains; section [[#Definitions|3]] provides a mathematical framework for stage-games (single round interactions) and repeated-games (interactions occurring over multiple rounds), focusing both on the case of perfect and imperfect monitoring; section [[#Reputation Games|4]] deals with the concept of reputation games within the context of both perfect and imperfect monitoring, in the case of one long-lived and one short-lived player; finally, section [[#long-lived players|5]] presents the various specifications of repeated games in the case where the two players are both long-lived.


= Reputation and Trust =

Extensive research in the literature has thoroughly examined the relationship between trust and reputation across various applications and contexts.

Diekmann and Przepiorka (2021) explore the problem of trust in economic transactions, which they define as ''"the uncertainty regarding the trustworthiness and/or competence of the trustee that the truster faces"'' (Diekmann & Przepiorka, 2021, p. 132). Uncertainty can hinder the occurrence of efficient exchanges, as it creates a situation where one party is unsure whether the other party will fulfill their promises. This uncertainty may manifest in various ways, such as doubts regarding whether the other agent will exchange the agreed-upon amount, deliver the promised product of satisfactory quality, or even engage in the exchange at all. Reputation, which creates the possibility to engage in long-lasting relationships, can be a solution given that it is a ''“shadow of the past”'' (Diekmann & Przepiorka, 2021, p. 134) behavior of the trustee. Trustworthiness can arise because reputation is a way to make the past transaction history of the trustee known to the trustor, who, by having access to this information, can gain insights into the characteristics and behavior of his counterpart, thereby reducing uncertainty. This is especially true in online markets, where the seller and the vendor are usually anonymous and are likely to engage in one-time-only transactions: decentralized reputation systems like ratings play an indispensable role in ensuring the viability of these markets, given that buyers are more likely to buy from sellers with good reputation and therefore vendors have the incentive to maintain a good reputation by providing good services. Also according to Einwiller (2003), who conducted an empirical research involving 473 German internet users, reputation systems play a critical role in building trust within online markets. This is especially significant for consumers who are less familiar with engaging in such markets.

Xiong and Liu (2004) study the problem of reputation and trust in peer-to-peer online communities, which are platforms or networks that facilitate direct interaction and collaboration among individuals without relying on a centralized authority or server. To overcome the uncertainty and the threats related to decentralized networks, they develop a reputation-based trust supporting framework called PeerTrust, which aims to assess the trustworthiness of peers within these communities. This system is based on the following features: the feedback a peer receives from other peers, the total number of transactions a peer performs, the credibility of the feedback sources, the transaction context factor, and the community context factor. Once again, it is evident that building a reputation plays a pivotal role in enhancing trust and facilitating efficient exchanges. Similar conclusions are also achieved by Selcuk et al. (2004).

At a more granular level, the reputation of individual firms (and therefore their perceived trustworthiness) holds significant importance. Companies that foster a positive culture, underpinned by a well-defined code of ethics, possess the ability to cultivate a favorable reputation. This, in turn, bolsters the trust vested in them by their stakeholders (Webley, 2004). Consequently, these firms experience enhanced profitability as they attract high-quality employees, cultivate loyal customers, and are perceived as more valuable, affording them the opportunity to command premium pricing for their products (Eccles et al., 2007). Furthermore, reputation enhances the persistence and sustainability of these positive performances over time (Roberts & Dowling, 2002). As such, reputation emerges as an indispensable intangible asset (Dowling, 1993).

= Definitions =

To delve into the realm of game theory, it is essential to establish a clear understanding of the mathematical frameworks underlying stage games and repeated games.


== Stage games with perfect monitoring ==

Repeated games can be defined starting from stage games. There are n players and each of them can choose various actions <math display="inline">a_i</math> from a set of available pure actions <math display="inline">A_i</math>. The set of pure actions profile is <math display="inline">A \equiv \prod_{i}A_i</math>. The payoffs are given by a continuous function <math display="inline">u: \prod_{i} A_i \rightarrow \mathbb{R}^n</math>. The set of mixed actions for player i is <math display="inline">\Delta(A_i)</math>, with a typical element <math display="inline">\alpha_i</math>, while the set of mixed profile is <math display="inline">\prod_{i} \Delta(A_i)</math>. The set of stage-game payoffs generated by pure action profiles in A is <math display="inline">F \equiv \{v \in \mathbb{R}^n : \exists a \in A \text{ such that } v = u(a)\}</math> and the set of feasible payoff is <math display="inline">F^{\dagger} \equiv coF</math>, which is a convex hull of F. A payoff <math display="inline">v \in F^{\dagger}</math> is inefficient if <math display="inline">\exists v^\prime \in F^{\dagger}</math> s.t <math display="inline">v^\prime_i > v_i</math>, otherwise it is efficient. According to Nash (1951), if the stage game is finite, there is a Nash equilibrium, but, because the payoffs are given by a continous function, also infinite stage games have Nash equilibria. Further assumptions are made:

* <math display="inline">A_{i}</math> is either finite, or a compact and convex subset of the Euclidean space <math display="inline">\mathbb{R}^k</math>, for some k.
* If <math display="inline">A_i</math> is a continuum action space, then <math display="inline">u : A \rightarrow \mathbb{R}^n</math> is continuous, and <math display="inline">u_i</math> is quasiconcave in <math display="inline">a_i</math> .

It is important to characterize the worst payoff that an individual can reach while optimizing. The worst outcome for player i, consistent with player i behaving optimally, is reached when the other player choose the action <math display="inline">a_{-i} \in A_{-i} \equiv \prod_{j \neq i} A_j</math> that minimizes the payoffs i gets when he plays the best response to action <math display="inline">a_{-i}</math>. This is called ''minmax payoff'' and mathematically is defined by <math display="inline">v^{p}_i \equiv \min_{\stackrel{a_{-i} \in A_{-i}}{}} \max_{\stackrel{a_i \in A_i}{}} u_i(a_i, a_{-i})</math>. Thus, it is possible to define <math display="inline">\hat{a^i} = (\hat{a^i_i},\hat{a^i_{-i}})</math> as the minmax profile for player i. A payoff vector <math display="inline">v = (v_1, \ldots, v_n)</math> is weakly individually rational if <math display="inline">v_i \geq v^p_i</math> <math display="inline">\forall</math> i, and it is strictly individually rational if <math display="inline">v_i > v^p_i</math> <math display="inline">\forall</math> i.


== Repeated games with perfect monitoring ==

A repeated game is a stage game that is repeated in each period <math display="inline">t \in \{0, 1, \ldots\}</math>. The behavior in this kind of games is called ''strategy'' and it is a collection of actions. The authors firstly deal with ''perfect monitoring'' games, so at the end of each period t each player can observe the actions taken by all the other players. The set of period t histories is given by <math display="inline">A^t</math>, that is the t-fold product of A. A history <math display="inline">h^t \in A^t</math> is thus a list of <math display="inline">t</math> action profiles, identifying the actions played in periods 0 through <math display="inline">t - 1</math>. The set of all possible histories is <math display="inline">H \equiv \bigcup_{t=0}^{\infty} A^t</math>. A pure strategy can be defined as <math display="inline">\sigma_i : H \rightarrow A_i</math>, while a mixed strategy (also called behavior strategy) is <math display="inline">\sigma_i : H \rightarrow \Delta(A_i)</math>. A continuation game is the infinitely repeated game that begins in t, following history <math display="inline">h^t</math>. The continuation strategy induced by <math display="inline">h^t</math> is denoted by <math display="inline">\sigma_{i|h^t}</math>. The continuation game associated with each history is also defined as a subgame identical to the original game. This means that repeated games have a recursive structure. An outcome path in the infinitely repeated game is an infinite sequence of action profiles <math display="inline">a \equiv (a_0, a_1, a_2, \ldots) \in A^\infty</math>. They differ from histories because histories have a finite length. The first <math display="inline">t</math> periods of an outcome are defined as <math display="inline">a^t = (a^0, a^1, \ldots, a^{t-1})</math>. Thus, <math display="inline">a^t</math> is the history in <math display="inline">A^t</math> corresponding to the outcome <math display="inline">a</math>. A strategy profile <math display="inline">\sigma</math> induces an outcome <math display="inline">a(\sigma)</math> and, analogously, a behavior strategy profile <math display="inline">\sigma</math> induces a path of play. For a pure strategy profile, the induced path of play and induced outcome are the same. In period <math display="inline">t</math>, the induced pure strategy profile <math display="inline">a^t(\sigma)</math> yields a payoff <math display="inline">u_i(a^t(\sigma))</math>. An outcome <math display="inline">a(\sigma)</math> induces an infinite stream of stage game payoffs <math display="inline">(u_i(a^0(\sigma)), u_i(a^1(\sigma)), u_i(a^2(\sigma)), \ldots) \in \mathbb{R}^\infty</math> which are discounted with a discount factor <math display="inline">\delta \in [0,1)</math>. The payoff from a pure strategy profile <math display="inline">\sigma</math> is therefore: <math display="block">U_i(\sigma) = (1 - \delta) \sum_{t=0}^{\infty} \delta^t u_i(a^t(\sigma)).</math> The authors assume that long-lived players share a common discount factor <math display="inline">\delta</math>.

A Nash equilibrium is a strategy profile in which each player is optimally responding to the strategies of the others. Formally, <math display="inline">\sigma</math> is a Nash equilibrium if <math display="inline">\forall</math> players <math display="inline">i</math> and strategies <math display="inline">\sigma_i^{\prime}</math>, we have <math display="inline">U_i(\sigma) \geq U_i(\sigma_i^{\prime}, \sigma_{-i}).</math> Furthermore, a strategy profile <math display="inline">\sigma</math> is a subgame-perfect equilibrium if <math display="inline">\forall</math> histories <math display="inline">h^t \in H</math>, <math display="inline">\sigma_{i|h^t}</math> is a Nash equilibrium of the repeated game.

The authors further define the concept of one shot deviation: for player i and from strategy <math display="inline">\sigma_i</math>, a one shot deviation is a strategy <math display="inline">\hat{\sigma_i} \neq \sigma_i</math> with the property that <math display="inline">\exists</math> a unique history <math display="inline">h^t</math> <math display="inline">\in</math> H such that <math display="inline">\forall h^\tau \neq \tilde{h^t}</math>, <math display="inline">\sigma_i(h^\tau) = \hat{\sigma}_i(h^\tau)</math>. A one shot deviation <math display="inline">\hat{\sigma_i}</math> is profitable if, fixed an opponent strategy <math display="inline">\sigma_{-i}</math>, at the history <math display="inline">\tilde{h^t}</math> for which <math display="inline">\hat{\sigma}_i(\tilde{h}^t) \neq \sigma_i(\tilde{h}^t)</math>, <math display="inline">U_i(\hat{\sigma}_i \mid \tilde{h}^t, \sigma_{-i} \mid \tilde{h}^t) > U_i(\sigma \mid \tilde{h}^t)</math>. A strategy profile is a subgame perfect IFF there are no profitable one shot deviations. However, along the equilibrium path, the absence of one-shot deviations is not sufficient to establish a Nash equilibrium, even though in the equilibrium path there cannot be profitable one-shot deviations. The one-shot principle is useful because it simplifies the strategies to check when evaluating subgame perfection. Another way to simplify is by grouping repeated-game strategies in automata. An automaton <math display="inline">(W, w^0, f, \tau)</math> is a set of states W, an initial state <math display="inline">w^0 \in W</math>, a decision function f : <math display="inline">W \rightarrow \prod_{i} \Delta(A_i)</math> associating mixed action profiles with states, and a transition function <math display="inline">\tau : W\times A \rightarrow W</math>, which identifies the next state of the automaton, given its current state and the realized stage-game pure action profile. Any strategy profile can be represented by an automaton. It is possible to represent a single strategy <math display="inline">\sigma_i</math> by the automaton <math display="inline">(W_i, w^0_i, f_i, \tau_i)</math>, the strategy profile <math display="inline">\sigma</math> as the automaton <math display="inline">(W, w^0, f, \tau)</math> and a continuation strategy as the automaton <math display="inline">(W, \tau(w^0, h^t),f, \tau)</math>. The strategy profile with an automaton <math display="inline">(W, w^0, f, \tau)</math> is a subgame perfect equilibrium IFF <math display="inline">\forall</math> w <math display="inline">\in</math> W accessible from <math display="inline">w^0</math>, the strategy profile induced by <math display="inline">(W, w, f, \tau)</math> is a Nash equilibrium for the repeated game.


=== Equilibria ===

To discuss the notion of equilibrium, the authors describe the notions of enforceability and pure-action decomposability:

* Enforceability: A pure action profile <math display="inline">a^*</math> is enforceable on W if <math display="inline">\exists</math> some specification of continuation promises (i.e. a commitment made by a player in a sequential game to take a specific action in future stages of the game) <math display="inline">\gamma : A \rightarrow W</math> such that, <math display="inline">\forall</math> players i and <math display="inline">a_i \in A_i</math>, <math display="inline">(1 - \delta) u_i(a^*) + \delta \gamma_i(a^*) \geq (1 - \delta) u_i(a_i, a_{-i}^*) + \delta \gamma_i(a_i, a_{-i}^*)</math>.
* Decomposability: A payoff <math display="inline">v \in F^{\dagger}</math> is pure action decomposable on W if <math display="inline">\exists</math> a pure action profile <math display="inline">a^*</math> enforceable on W such that <math display="inline">v_i = (1 - \delta)u_i(a^*) + \delta\gamma_i(a^*)</math> where <math display="inline">\gamma</math> is a function enforcing <math display="inline">a^*</math>.

Any set of payoff <math display="inline">W \subset F^{\dagger}</math> with the property that every payoff in W is pure-action decomposable on W is a set of pure-strategy subgame-perfect equilibrium payoffs. Moreover, a set W is pure-action self-generating if every payoff in W is pure-action decomposable on W.


=== Simple strategy and penal code ===

Simple strategy profile: given (n + 1) outcomes <math display="inline">\{a(0), a(1), \ldots, a(n)\}</math>, the associated simple strategy profile <math display="inline">\sigma(a(0), a(1), \ldots, a(n))</math> is given by the automaton: <math display="inline">W = \{0, 1, \ldots, n\} \times \{0, 1, 2, \ldots\}</math>, <math display="inline">w_0 = (0, 0)</math>, <math display="inline">f(j, t) = a^{t}(j)</math> and <math display="block">\tau((j, t), a) = \begin{cases}
(i, 0) & \text{if } a_i \neq a^{t}_i(j) \text{ and } a_{-i} = a^{t}_{-i}(j) \\
(j, t + 1) & \text{otherwise}
\end{cases}</math> Therefore, a simple strategy consist in a prescribed outcome a(0) and a punishment outcome a(i) <math display="inline">\forall</math> player i. The games continues according to the outcome a(0), but, if there is any deviation by player i, the other players respond with the player i outcome path a(i). it is important to underline that the punishment for a deviation is independent of when the deviation occurs and the nature of it. The simple strategy profile <math display="inline">\sigma(a(0), a(1), \ldots, a(n))</math> is a subgame-perfect equilibrium IFF: <math display="block">U_t^i(a(j)) \geq \max_{a_i \in A_i} \left[(1 - \delta)u_i(a_i, a_{-i}^t(j)) + \delta U^0_i(a(i))\right],</math> <math display="inline">\forall</math> <math display="inline">i = 1, \ldots, n</math>, <math display="inline">j = 0, 1, \ldots, n</math>, and <math display="inline">t = 0, 1, \ldots</math>.

Then, a definition of an optimal penal code is given. Let <math display="inline">\{a(i) : i = 1, \ldots, n\}</math> be <math display="inline">n</math> outcome paths satisfying <math display="inline">U^0_i(a(i)) = v^*_i</math>, for <math display="inline">i = 1, \ldots, n</math>. The collection <math display="inline">\sigma(i) = \sigma(a(i), a(1), \ldots, a(n))</math> is an optimal penal code if <math display="inline">\sigma(i) \in \mathcal{E}^p, \quad i = 1, \ldots, n</math>, being <math display="inline">\mathcal{E}^p</math> the set of pure strategy subgame-equilibrium payoffs.


=== Long-lived and short-lived players ===

The authors introduce short-lived players, who differ from long-lived players since the latter live throughout the game. On the contrary, short-lived players are concerned only with the current period payoffs (therefore they do not discount) and they are called myopic. There are two interpretations for them:

* In each period a collection of short-lived players enter the games and, after one period, they leave.
* Each of them represents a continuum of long-lived agents.

Small players are assumed to be anonymous (i.e a change in the behavior of a member of the continuum does not affect the ditribution of play, so it does not affect the behavior of other players) and each of them observes observes only the history <math display="inline">h^t \in A^t</math>. The notion of one-shot deviation applies also in games with long and short lived players. Moreover, being myopic, the short-lived players play a Nash equilibrium of the induced stage game, given the actions of the long-lived players.

It is possible to generalize the concept of minmax payoff with short-lived players. Let B: <math display="inline">\prod_{i=1}^{n} \Delta(A_i) \rightarrow \prod_{i=n+1}^{N} \Delta(A_i)</math> be the correspondence that maps any mixed-action profile for the long-lived players to the corresponding set of static Nash equilibria for the short-lived players. For each long-lived player i, the payoff <math display="inline">v_i = \min_{\stackrel{\alpha \in B}{}}
\max_{\stackrel{a_i \in A_i}{}}
u_i(a_i, a_{-i})</math> is player i’s (mixed-action) minmax payoff with short-lived players and it is a lower bound on the payoff that player i can obtain in an equilibrium of the repeated game.

Finally, repeated games with short-lived players have some features that must be underlined:

* There are some restrictions on the payoffs that can be achieved by the long-lived players. In particular, short-lived players impose restrictions on the set of equilibrium payoffs that go beyond the specification of minmax payoffs.
* There are restrictions on the structure of the equilibrium.

Let i being a long-lived player and <math display="block">\overline{v}_i = \sup_{\alpha \in B} \min_{a_i \in \text{supp}(\alpha_i)} u_i(a_i, \alpha_{-i})</math> be the minimum payoff that it is possible to construct by adjusting player i’s behavior within the support of his action <math display="inline">\alpha_i</math>. In every subgame-perfect equilibrium for player i, the payoff for player i will be less or equal to <math display="inline">\overline{v}_i</math>.


== Games with imperfect monitoring ==

The focus before was on perfect monitoring games, where deviations from the equilibrium can be easily detected and punished, thus providing incentives for players not to myopically optimize. Now the authors focus on games with imperfect monitoring, so games where players have only noisy information about past play, thus making deviations more difficult to discover. However, it is still possible to have players that do not myopically optimize, since punishments are still possible. When the noisy signals are observed by all players we are in the case of imperfect public monitoring, while if some signals are observed by some players but not others, we are in the case of private monitoring. The following sections focus on imperfect public monitoring, but the results can be extended for the case of private monitoring.


=== Stage games ===

The specification of this games is very similar to the stage games with perfect monitoring, seen in section [[#Stage games with perfect monitoring|3.1]]. The main difference relies in the presence of a public signal <math display="inline">y</math>, from the finite signal space <math display="inline">Y</math>. <math display="inline">\rho(y | a)</math> will be the probability that signal <math display="inline">y</math> is realized, given <math display="inline">a \in A \equiv \prod_{i} A_i</math>. The function <math display="inline">\rho: Y \times A \rightarrow [0,1]</math> is continuous. <math display="inline">\rho</math> has full support if <math display="inline">\forall y \text{ and } a, \rho(y | a)>0</math>. Player i’s payoff at the end of each period, given the realization <math display="inline">(y, a)</math> is <math display="inline">u^*_i(y, a_i)</math>, while ex-ante stage game payoffs are <math display="inline">u_i(a) = \sum_{y \in Y} u^*_i(y, a_i) \rho(y | a)</math>. The other assumptions are maintained.


=== Repeated games ===

Repeated games with imperfect monitoring have a similar structure to the games presented in section [[#RGPM|3.2]]. The only public information available in t is the t-period history of public signals <math display="inline">h^t \equiv (y^0, y^1, \ldots, y^{t-1})</math>. The set of public histories is <math display="inline">H \equiv \bigcup_{t=0}^{\infty} Y^t</math>. A history for a long-lived player will include the public history and the history of the actions he has taken. It is assumed that each short lived-player in period t only observes the public history <math display="inline">h^t</math>. A pure strategy profile does not induce a deterministic outcome path, since public signals may be random. Public monitoring games include also the special case of perfect monitoring games, where Y=A and <math display="inline">\rho(y | a) = 1</math> if y=a, 0 otherwise.

Long-lived players have private information (the knowledge of their past actions), so information sets are isomorphic to private histories, not to public histories. A behavior strategy <math display="inline">\sigma_i</math> is public if, in every period <math display="inline">t</math>, it depends only on the public history <math display="inline">h^t \in Y^t</math> and not on <math display="inline">i</math>’s private history: <math display="inline">\forall</math> <math display="inline">h^t_i, \hat{h}^t_i \in H_i</math> satisfying <math display="inline">y^\tau = \hat{y}^\tau</math> <math display="inline">\forall</math> <math display="inline">\tau \leq t - 1</math>, <math display="inline">\sigma_i(h^t_i) = \sigma_i(\hat{h}^t_i)</math>. Otherwise, it is private. If all players other than i are playing a public strategy, then player i has a public strategy as a best reply. <math display="inline">\sigma_i</math> and <math display="inline">\hat{\sigma_i}</math> are said to be realization equivalent if, for all strategies for the other players, <math display="inline">\sigma_{-i}</math>, the distributions over outcomes induced by <math display="inline">(\sigma_i, \sigma_{-i})</math> and <math display="inline">(\hat{\sigma}_i, \sigma_{-i})</math> are the same. Thus, it is possible to demonstrate that every pure strategy in a public monitoring game is realization equivalent to a public pure strategy.

Finally, it is possible to define a perfect public equilibrium (PPE) as a profile of public strategies <math display="inline">\sigma</math> that, <math display="inline">\forall</math> <math display="inline">h^t</math>, specifies a Nash equilibrium for the repeated game. That is, <math display="inline">\forall</math> <math display="inline">t</math> and <math display="inline">h^t \in Y^t</math>, <math display="inline">\sigma \vert h^t</math> is a Nash equilibrium. <math display="inline">\sigma</math> is a PPE IFF there are no profitable one shot deviations.


= Reputation games =

Reputation can be considered as a specification of the concept of trust. It can be defined as "a link between past behavior and expectations of future behavior" (Mailath & Samuelson, 2006, p.459) that arise after multiple interactions among agents. Repeated games are therefore a good environment to study reputation.

There are two kind of approaches:

* In the first, an equilibrium of the repeated game is selected and it involves an equilibrium path that is not a Nash equilibrium of the stage game. Players who choose the equilibrium are said to maintain a reputation, triggering a punishment if they deviate, consisting in loosing their reputation. The link between past and future behavior, i.e. reputation, is an equilibrium phenomenon.
* In the second (adverse selection approach), each player is uncertain about the characteristics of his opponent. Therefore, this incomplete information introduces a link between past and future behavior, thus making the concept of reputation arise. Here, the reputation approach does not describe a possible equilibrium, but places constraints on the equilibria available.


== Commitment types ==

The authors firstly consider game with one long-lived player and one short-lived player. Player 2 does not know the type of player 1, but he has a prior belief <math display="inline">\mu</math> about player 1’s type <math display="inline">\xi</math>, which comes from a countable set <math display="inline">\Xi</math>. There are two possible categories of types for player 1:

* the payoff type <math display="inline">\Xi_1</math>, which maximizes the average discounted value of his payoffs. Of particular interest, in this category, there is the normal type <math display="inline">\xi_0 \in \Xi_1</math>, who has a stationary payoff function.
* the commitment type <math display="inline">\Xi_2</math>, who does not have payoffs but play a particular game strategy. If the strategy played consist in the same stage-game action in every period, regardless of the past history, the player will be a simple commitment type.

Usually, the probability <math display="inline">\mu(\xi_0)</math> is large.

Player 1’s pure-action Stackelberg payoff can be defined as <math display="block">v^*_{1} = \sup_{a_1 \in A_1} \min_{\alpha_2 \in B(a_1)} u_1(a_1, \alpha_2)</math> where <math display="inline">B(a_1)</math> is the set of player 2 myopic best replies to <math display="inline">a_1</math>. If the supremum is achieved by some action <math display="inline">a^*_1</math>, that action will be a Stackelberg action <math display="block">a^*_{1} \in \arg\max_{a_1 \in A_1} \min_{\alpha_2 \in B(a_1)} u_1(a_1, \alpha_2).</math> This is a pure action, known as the Stackelberg action, that player 1 would commit to if given the opportunity to do so. The name "Stackelberg action" arises because such a commitment elicits a best response from player 2. The Stackelberg type of player 1 will play <math display="inline">a^*_1</math> and is denoted by <math display="inline">\xi(a^*_1) \equiv \xi^*</math>.


== Perfect monitoring games ==

The authors firstly focus on repeated games with perfect monitoring. The action set <math display="inline">A_2</math> of player 2 is assumed to be finite and each player only play a pure strategy. The basic reputation result establishes a minimum threshold for the equilibrium payoffs of the normal long-lived player. H is the set of public histories in the complete information game and also in the incomplete information game, and an history for player 1 will be <math display="inline">\Xi \times H</math>, which specifies player 1’s type and the public history. A behavior strategy for player 1 will be <math display="inline">\sigma_1 : H \times \Xi \rightarrow \Delta(A_1)</math> such that, for all commitment types <math display="inline">\xi(\hat{\sigma_1}) \in \Xi_2</math>, <math display="inline">\sigma_1(h^t, \xi(\hat{\sigma}_1)) = \hat{\sigma}_1(h^t) \quad \forall h^t \in H</math>. A behavior strategy for player 2 will be <math display="inline">\sigma_2 : H \rightarrow \Delta(A_2)</math>.

Let <math display="inline">U_1(\sigma, \xi)</math> be the type <math display="inline">\xi</math> long-lived player’s payoff. A strategy profile <math display="inline">(\tilde{\sigma}_1, \tilde{\sigma}_2)</math> is a Nash equilibrium of the reputation game with perfect monitoring if <math display="inline">\forall</math> <math display="inline">\xi \in \Xi_1</math>, <math display="inline">\tilde{\sigma}_1</math> maximizes <math display="inline">U_1(\sigma_1, \tilde{\sigma}_2, \xi)</math> over player 1’s repeated game strategies, and if <math display="inline">\forall</math> <math display="inline">t</math> and <math display="inline">h_t \in H</math> that have positive probability under <math display="inline">(\tilde{\sigma}_1, \tilde{\sigma}_2)</math> and <math display="inline">\mu</math>, <math display="block">E[u_2(\tilde{\sigma}_1(h^t, \xi), \tilde{\sigma}_2(h^t)) | h^t] = \max_{a_2 \in A_2} E[u_2(\tilde{\sigma}_1(h^t, \xi), a_2) | h^t].</math>

The first goal is to demonstrate that, when player 2 assign some probability to 1 being the simple type <math display="inline">\xi(a^\prime_1)=\xi^\prime</math>, if the normal player 1 repeatedly plays <math display="inline">a^\prime_1</math>, then player 2 will place an high probability on the fact that such action will be played in the future. Obviously, the reputation for playing <math display="inline">a^\prime_1</math> will not arise instantaneously, and it can also be costly. However, the cost will be negligible if player 1 is sufficiently patient. If this action is the Stackelberg action <math display="inline">a^{*}_1</math> , when player 1 is sufficiently patient, the resulting lower bound on player 1’s payoff is close to his Stackelberg payoff <math display="inline">v^{*}_1</math>. Let <math display="inline">\Omega \equiv \Xi \times (A1 \times A2) ^ \infty</math> be the space of outcomes, being <math display="inline">\omega</math> a specific outcome. A profile of strategy <math display="inline">(\sigma_1, \sigma_2)</math>, along with <math display="inline">\mu</math>, induces a probability measure on the set of outcomes <math display="inline">\Omega</math>, <math display="inline">P \in \Delta(\Omega)</math>. <math display="inline">\Omega^\prime</math> is the event that action <math display="inline">a^\prime_1</math> is chosen in every period. <math display="inline">q^t</math> will be the probability that <math display="inline">a^\prime_1</math> is chosen in period t conditional on <math display="inline">h^t</math>, that is <math display="inline">q^t \equiv P(a^t_1 = a^\prime_1 | h^t)</math>, and it is a random variable. The normal player 1 receives a payoff of at least <math display="inline">\min_{a_2 \in B(a^\prime_1)} u_1(a^\prime_1, a_2)</math> in any period t in which <math display="inline">q^t</math> is large enough to permit player 2 to choose the best response to <math display="inline">a^\prime_1</math> and player 1 effectively plays <math display="inline">a^\prime_1</math>. Since player 1 can always play <math display="inline">a^\prime_1</math>, the payoff generated by always playing <math display="inline">a^\prime_1</math> must be the lower bound of the payoff in any Nash equilibrium.

Let <math display="inline">n_\zeta : \Omega \rightarrow N_0 \cup {\infty}</math> be the number of random variables <math display="inline">q^t</math> for which <math display="inline">q^t \leq \zeta</math> and <math display="inline">\xi^\prime</math> the event that player 1 is type <math display="inline">\xi^\prime</math>. Fix <math display="inline">\zeta \in [0,1)</math>. Suppose <math display="inline">\mu(\xi(a^\prime_1)) \in [\mu^\prime, 1)</math> for some <math display="inline">\mu^\prime > 0</math> and <math display="inline">a^\prime_1 \in A_1</math>. <math display="inline">\forall</math> profile <math display="inline">(\sigma_1, \sigma_2)</math>, <math display="block">P\left(n_\zeta > \frac{\ln \mu^\prime}{\ln \xi} \; \Bigg| \; \Omega^\prime\right) = 0</math> and <math display="inline">\forall</math> <math display="inline">\omega \in \Omega^\prime</math> such that all histories <math display="inline">(h^t(\omega))_{t=0}^\infty</math> have positive probability under P, <math display="inline">P(\xi(a^\prime_1) | h^t(\omega))</math> is non decreasing in t. Therefore, the more player 2 observes player 1 playing <math display="inline">a^\prime_1</math>, the more he will expect <math display="inline">a^\prime_1</math> to be played with an higher and higher probability. The bound <math display="inline">n_\zeta</math> is independent of P, meaning that this result does not say that the posterior probability attached to be the simple type converges to 1 as t increases, leaving the possibility that player 1 is a normal type playing like the simple type.

By committing to action <math display="inline">a_1</math>, player 1 can guarantee the payoff <math display="block">v^*_1(a_1) \equiv \min_{a_2 \in B(a_1)} u_1(a_1, a_2)</math> which is the one-shot bound from <math display="inline">a_1</math>. Let <math display="inline">v_1(\xi_0, \mu, \delta)</math> the infimum over the set of normal player 1’s payoff in any Nash Equilibrium. The reputation result establishes a lower bound on the equilibrium payoff of player 1. In particular, let <math display="inline">A_2</math> be finite and <math display="inline">\mu(\xi_0) > 0</math>. Suppose <math display="inline">A^\prime_1</math> is a finite subset of <math display="inline">A_1</math> with <math display="inline">\mu(\xi(a_1)) > 0 \text{ }\forall a_1 \in A^\prime_1</math>. Then, <math display="inline">\exists</math> k such that:<math display="block">v_1(\xi_0, \mu, \delta) \geq \delta^k \max_{a_1 \in A^\prime_1} v^*_1(a_1) + (1 - \delta^k) \min_{a \in A} u_1(a).</math> If the set of commitment types is sufficiently rich, the lower bound of player’s 1 payoff is the Stackelberg payoff, if he is the normal type. Indeed, if there is a Stackelberg action and the Stackelberg type has <math display="inline">\mu > 0</math>, the normal player 1 effectively builds a reputation for playing like the Stackelberg type, thus receiving a payoff no less than <math display="inline">v^*_1</math> despite the fact that there are other possible commitment types. However, the result does not tell if it is optimal for player 1 to play the Stackelberg action in each period. Reputation effects resulting from pure-action commitment types in perfect monitoring games impose a minimum threshold on player 1’s equilibrium payoffs, which can be considerably high. However, unlike mixed-action commitment types or imperfect monitoring games in general, they do not introduce the possibility of new payoffs. In this context, discounting plays a dual role:

* It makes future payoffs relatively more important.
* it diminishes the significance of the initial sequence of periods in which player 1 may incur in costs to imitate the commitment type.


== Example: product-choice game ==

Let’s consider the following product-choice game (i.e. a game where the first agent is a firm that decides whether to exert high (H) or low (L) effort in producing its output, while the second agent is a consumer that can buy a high-priced (h) or low-priced (l) product).



<div id="Product-choice game">

<center>
{| class="wikitable"
|-
! !! !! colspan="2" | '''Player 2''' ||
|-
! !! | !! '''h''' !! '''l''' ||
|-
| rowspan="2" | '''Player 1''' || '''H''' || 2,3 || 0,2 ||
|-
|| '''L''' || 3,0 || 1,1 ||
|}
<center>

</div>
Player 1 is a long lived player, while player 2 is a short lived player. Player 1 can build a reputation of playing H by persistently doing so. This at the beginning could be costly for player 1, since it may take time for player 2 to be convinced (and in the meanwhile he will play l), but the subsequent payoffs could make the initial investment rewarding, if player 1 is sufficiently patient. However, nothing in the repeated game structure captures this feature. Therefore, it is necessary to introduce the probability <math display="inline">\hat{\mu} > 0</math>, i.e. the probability that player 2 assigns to the possibility that player 1 is a commitment type (so it has some hidden characteristics that ensure that he will exert high effort) and not a normal type (so a player without the hidden characteristics previously mentioned). This introduces a link between past and expected future play of H: player 2 can decide the action to play after having seen the behavior of player 1. Therefore, (L,l), which would have been the unique Nash equilibrium in the perfect monitoring game of complete information, is not the necessary outcome anymore. For example, if the product choice game is played twice in this setting, the response of player 2 will depend on what player 1 does, assuming that he is normal. If player 1 plays L, player 2 will subsequently play l, believing that he is facing the normal type. On the other hand, if player 1 plays H, player 2 will possibly conclude that he is facing the commitment type, thus best responding with h. Player 1, who has behaved like the commitment type, can sacrifice current payoff to get more in the following period.

Continuing with the example, the pure Stackelberg type of player 1 chooses H, with a payoff of 2. Suppose <math display="inline">\Xi = \left\{\xi_0, \xi^*, \xi(L)\right\}</math>. If <math display="inline">\delta \geq 1/2</math>, always playing Hh is a subgame perfect equilibrium of the complete information game. Adapting the profile for incomplete information games we have that <math display="block">\sigma_1(h^t, \xi) =
\begin{cases}
H, & \text{if } \xi = \xi^* \text{ or } (\xi = \xi_0 \text{ and } a^\tau = Hh \text{ }\forall\text{ } \tau < t), \\
L, & \text{otherwise},
\end{cases}</math>

<math display="block">\sigma_2(h^t) =
\begin{cases}
h, & \text{if } a^\tau = Hh \text{ }\forall\text{ } \tau < t, \\
\text{l}, & \text{otherwise}.
\end{cases}</math> This will be a Nash equilibrium for <math display="inline">\delta \geq 1/2</math> and <math display="inline">\mu(\xi(L)) < 1/2</math>. Player 2 will find optimal to play h in period 0. If he observes L, he will place probability 1 on type <math display="inline">\xi(L)</math> or normal type, thus optimally punishing player 1. This ensures that player 1 will find optimal to play H. After observing <math display="inline">a^0_1</math> in period 0, player 2 will assign 0 probability to <math display="inline">\xi(L)</math>. It is also possible to demonstrate that it is impossible to obtain Nash equilibria with a low payoff for normal player 1. If <math display="inline">\mu(\xi^*) < 1/3</math> and <math display="inline">\mu(\xi(L)) < 1/3</math>, the normal player 1’s payoff in any pure strategy Nash equilibrium is bounded below by <math display="inline">2\delta</math> and above by 2.

Now, let <math display="inline">a^\prime_1 = H = a^*_1</math>, the Stackeblerg action. Let <math display="inline">\tilde{\xi_t}</math> a commitment type who: plays H if <math display="inline">\tau < t</math> and L afterwards. <math display="inline">\tilde{\xi_0}</math> will be <math display="inline">\xi(L)</math>, the simple commitment type who always plays L and <math display="inline">\tilde{\xi_t}</math> the nonsimple commitment type for t <math display="inline">\geq</math> 1. Finally, let <math display="inline">\hat{\xi}</math> be the type that plays H in period 0 and in every other period if 2 plays h in time 0. Consider the strategy profile where normal player 1 always play H and player 2 always play h. <math display="inline">\Omega^\prime</math> is the set of outcomes in which player 1 always plays H. <math display="inline">q^0 = 1 - \mu(\tilde{\xi}_0)</math>. In period 1 Hh is the only history consistent with <math display="inline">\Omega^\prime</math> that has a non zero probability. With Bayes rule, we obtain that <math display="inline">q^1(Hh) = \frac{1 - \mu(\tilde{\xi}_0) - \mu(\tilde{\xi}_1)}
{1 - \mu(\tilde{\xi}_0)}.</math> If instead player 1 still always plays H, but player 2 plays h with probability 1/2 and l with probability 1/2 in the first period and then always plays h, the calculation for <math display="inline">q^0</math> does not change, but now Hl and Hh are both consistent with <math display="inline">\Omega^\prime</math>. So, <math display="inline">q^1(Hh) = \frac{1 - \mu(\tilde{\xi}_0) - \mu(\tilde{\xi}_1)}
{1 - \mu(\tilde{\xi}_0)}.</math> and <math display="inline">q^1(Hl) = \frac{1 - \mu(\hat{\xi}) - \mu(\tilde{\xi}_0) - \mu(\tilde{\xi}_1)}
{1 - \mu(\tilde{\xi}_0)}.</math>


== Imperfect monitoring games ==

The authors now study imperfect monitoring games, focusing firstly on the case with one long-lived and one short-lived players. <math display="inline">A_i</math> will be the action space and <math display="inline">Z_i</math> will be the finite signal space. In each stage of the repeated game, each player only learns the realized value of this (private) signal. Let <math display="inline">\pi(z | a)</math> be the distribution over private signals <math display="inline">z = (z_1, z_2)</math> for each action profile <math display="inline">a</math> and let <math display="inline">u^*_i(z_i, a_i)</math> be the ex-post payoff of normal player 1 and player 2 after the realization of z and a. Ex-ante stage game payoffs are <math display="inline">u_i(a) \equiv \sum_{z} u^*_i(z_i, a_i) \pi(z | a)</math>. <math display="inline">\Xi</math> is the same as the previous section. The set of private histories for player 1, excluding his type, is <math display="inline">H_1 \equiv \bigcup_{t=0}^{\infty} (A_1 \times Z_1)^t</math>, while a behavior strategy for player 1 can be defined as <math display="inline">\sigma_1: H_1 \times \Xi \rightarrow \Delta(A_1)</math>, such that <math display="inline">\forall \xi(\hat{\sigma_1}) \in \Xi_2</math>, <math display="inline">\sigma_1(h^t_1, \xi(\hat{\sigma}_1)) = \hat{\sigma}_1(h^t_1) \quad \forall h^t_1 \in H_1</math>. The set of private histories for player 2 is <math display="inline">H_2 \equiv \bigcup_{t=0}^{\infty} (A_2 \times Z_2)^t</math>, while a behavior strategy is <math display="inline">\sigma_2: H_2 \rightarrow \Delta(A_2)</math>. A strategy profile <math display="inline">(\tilde{\sigma}_1, \tilde{\sigma}_2)</math> is a Nash equilibrium of the reputation game with imperfect monitoring if <math display="inline">\forall</math> <math display="inline">\xi \in \Xi_1</math>, <math display="inline">\tilde{\sigma}_1</math> maximizes <math display="inline">U_1(\sigma_1, \tilde{\sigma}_2, \xi)</math> (type <math display="inline">\xi</math> long lived player’s payoff) over player 1’s repeated game strategies, and if <math display="inline">\forall</math> <math display="inline">t</math> and <math display="inline">h^t_2 \in H_2</math> that have positive probability under <math display="inline">(\tilde{\sigma}_1, \tilde{\sigma}_2)</math> and <math display="inline">\mu</math>, <math display="block">E[u_2(\tilde{\sigma}_1(h^t_1, \xi), \tilde{\sigma}_2(h^t_2)) | h^t_2] = \max_{a_2 \in A_2} E[u_2(\tilde{\sigma}_1(h^t_1, \xi ), a_2) | h^t_2].</math>

Since in this environment monitoring is not perfect, the best responses obtainable by playing <math display="inline">a_1</math> are not simply the actions in <math display="inline">B(a_1)</math>. Consider the set of possible best responses of player 2 to player 1’s mixed action <math display="inline">\alpha_1</math>. An action <math display="inline">\alpha_2</math> is an <math display="inline">\varepsilon-confirmed</math> best response to <math display="inline">\alpha_1</math> if <math display="inline">\exists \alpha^\prime_1</math> such that <math display="inline">\alpha_2(a_2) > 0 \Rightarrow a_2 \in \text{arg} \max_{a^\prime_2} u_2(\alpha^\prime_1, a^\prime_2)</math> and <math display="inline">|\pi_2(\cdot | \alpha_1, \alpha_2) - \pi_2(\cdot | \alpha^\prime_1, \alpha_2)| \leq \varepsilon</math>. <math display="inline">B_\varepsilon(\alpha_1)</math> is the set of <math display="inline">\varepsilon</math>-confirmed best responses to <math display="inline">\alpha_1</math>. For private monitoring games, let <math display="inline">B^*_\epsilon(\hat{\alpha}_1) \equiv \{\alpha_2 : \text{supp}(\alpha_2) \subset B_\epsilon(\hat{\alpha}_1)\}</math>. The results of the authors show that if player 2 assign positive probability to a simple type <math display="inline">\xi(\alpha^\prime_1)</math>, a patient normal player 1’s payoff in every Nash equilibrium (with an <math display="inline">\varepsilon > 0</math> approximation) can be no lower than <math display="inline">\bar{v}_1(\alpha^\prime_1) \equiv \min_{\alpha_2 \in B^*_0(\alpha^\prime_1)} u_1(\alpha^\prime_1, \alpha_2)</math>. Taking the supremum over <math display="inline">\alpha^\prime_1</math>, the payoff will be <math display="inline">v^{**}_1 \equiv \sup_{\alpha^\prime_1} \min_{\alpha_2 \in B^*_0(\alpha^\prime_1)} u_1(\alpha^\prime_1, \alpha_2)</math>.

Further assumptions are necessary. In particular, it must be that <math display="inline">\forall a_2 \in A_2</math>, (and <math display="inline">\forall \alpha_2 \in \Delta(A_2)</math>), <math display="inline">\{ \pi_2(\cdot | (a_1, a_2)) : a_1 \in A_1 \}</math> is linearly independent, that is for any action of player 2 no two actions for player 1 should generate the same distribution of signals. For the private-monitoring game, this means that <math display="inline">B(\alpha_1) = B^*_0(\alpha_1)</math> and <math display="inline">v^{**}_1</math> equals the mixed action Stackelberg payoff.

It is possible to demonstrate the following proposition. Let <math display="inline">\hat{\xi}</math> be the simple commitment type that plays <math display="inline">\hat{\alpha_1} \in \Delta(A_1)</math> (or <math display="inline">\hat{\alpha_1} \in A_1</math>). Suppose <math display="inline">\mu(\xi_0)</math>, <math display="inline">\mu(\hat{\xi}) > 0</math>. In the private monitoring game (or canonical public monitoring game), <math display="inline">\forall \varepsilon > 0</math>, <math display="inline">\exists</math> K such that <math display="inline">\forall \delta</math>:<math display="block">\bar{v_1}(\xi_0,\mu,\delta) \geq (1 - \varepsilon)\delta^K \inf_{\alpha_2 \in B^*_\varepsilon(\hat{\alpha}_1)} u_1(\hat{\alpha}_1, \alpha_2) + (1 - (1 - \varepsilon)\delta^K) \min_{a \in A} u_1(a)</math> This means that, as for perfect monitoring games, the normal player 1 manages to establish a reputation for consistently playing like a simple type. This phenomenon persists even when there are numerous other potential commitment types present. This guarantees that even if it is not possible to be certain about player 2’s ultimate beliefs regarding player 1, we can be confident that player 2’s beliefs will converge towards something.

The same results can be obtained when the short-lived player is interpreted as a continuum of small and anonymous long lived players. Each small player receives a private signal from the finite set <math display="inline">Z_2</math>, while the large player observes a private signal <math display="inline">z_1 \in Z_1</math> of the aggregate behavior of the small players. If the private signal received by the small players is common across all small players, the model is identical to what previously described. If each small player observes different realization of the private signal (idiosyncratic signals), the model is different. In this case, let <math display="inline">\hat{\xi}</math> denote the simple commitment type that always plays <math display="inline">\hat{\alpha}_1 \in \Delta(A_1)</math> (if <math display="inline">A_1</math> is finite) or <math display="inline">\hat{\alpha}_1 \in A_1</math> (if <math display="inline">A_1</math> is infinite). Suppose <math display="inline">\xi_0, \hat{\xi} \in \Xi</math>. <math display="inline">\forall</math> <math display="inline">\varepsilon > 0</math>, <math display="inline">\exists</math> <math display="inline">K</math> such that <math display="inline">\forall</math> <math display="inline">\delta</math>, <math display="block">v_1(\xi_0, \mu, \delta) \geq (1 - \varepsilon)\delta^K \inf_{\alpha_2 \in B^*_\varepsilon(\hat{\alpha}_1)} u_1(\hat{\alpha}_1, \alpha_2) + (1 - (1 - \varepsilon)\delta^K) \min_{a \in A} u_1(a).</math>


=== Example: product-choice game with public monitoring ===

Player 1’s actions are not public. There is a public signal that can take values <math display="inline">\bar{y}</math> and <math display="inline">\underline{y}</math>, with distribution <math display="inline">\rho(\bar{y} | a) =
\begin{cases}
p, & \text{if } a_1 = H, \\
q, & \text{if } a_1 = L,
\end{cases}</math>, with <math display="inline">0 < q . Player 2’s actions are public and <math display="inline">\hat{\alpha_1}</math> is player 1’s mixed action that randomize equally between his possible actions. <math display="inline">\forall \varepsilon \geq 0</math>, every pure or mixed action for player 2 <math display="inline">\in B_\varepsilon(\hat{\alpha_1})</math>. Therefore, <math display="inline">\min_{\alpha_2 \in B_0(\hat{\alpha}_1)} u_1(\hat{\alpha}_1, \alpha_2) = 1/2</math> and <math display="inline">v^{**}_1 = 5/2</math>. The resulting payoff is the mixed action Stackelberg payoff and it is higher than <math display="inline">2 - \frac{1 - p}{p - q} < 2</math>, which it is possible to demonstrate that it is the upper bound player 1’s payoff in the public monitoring game with complete information.


=== Temporary reputations ===

Under imperfect monitoring, the authors show that player 2 must eventually learn the type of player 1. While the normal and commitment types may exhibit similar behavior over an extended duration, the normal type will inevitably find motivation to deviate slightly from the commitment strategy. This contradicts the belief held by player 2 that player 1 will consistently demonstrate commitment. Consequently, player 2 will eventually acquire knowledge about player 1’s true type.

The environment in which they operate is that of incomplete information private monitoring games. The authors assume that <math display="inline">\forall i = 1, 2, a \in A, z_i \in Z_i, \pi_i(z_i | a) > 0</math> and <math display="inline">\forall a_1 \in A_1</math>, the collection of probability distributions <math display="inline">\{\pi_1(\cdot, (a_1, a_2)) : a_2 \in A_2\} \text{ is linearly independent.}</math> This assumptions implies that player i is able to correctly identify any fixed-stage game action of player j. The focus will be on one simple commitment type for player 1. For the the normal type, a strategy will be denoted as <math display="inline">\tilde{\sigma_1}</math>, while for the commitment type <math display="inline">\hat{\sigma_1}</math>. Let <math display="inline">P \in \Delta(\Omega)</math> be the unconditional probability measure induced by <math display="inline">\mu</math> and the strategy profile, while <math display="inline">\hat{P}</math> will be the measure induced by conditioning on <math display="inline">\hat{\xi}</math>. For player 2, <math display="inline">\hat{\sigma_2}</math> will be the unique stage game best response to <math display="inline">\hat{\sigma_1}</math>. <math display="inline">(\hat{\sigma_1}, \hat{\sigma_2})</math> will not be a stage-game Nash equilibrium. Suppose that there is a Nash equilibrium of the incomplete information game in which it is possible that player 1 can be either the normal or the commitment type. Player 2 will not distinguish between the signals generated by the two types, thus believing that both are playing the same strategy. Player 2 will play a best response to this strategy, that is a best response to the commitment type. Since it is not a best response also for player 1, he will find optimal to deviate, contradicting player 2’s beliefs. This means that in any Nash equilibrium of the game with imcomplete information, <math display="inline">\hat{\mu^t} \equiv P(\{\hat{\xi}\} | G^t_2) \rightarrow 0</math>, being <math display="inline">G^t_2</math> the filtration on <math display="inline">\Omega</math> generated by player 2’s histories.

It is also possible to demonstrate that there cannot be equilibria in which uncertainty about player 1’s type survives after T, for any period T. Indeed <math display="inline">\forall \varepsilon > 0, \exists \text{ } T</math> such that for any Nash equilibrium of the game with incomplete information, <math display="inline">\tilde{P}(\hat{\mu}^t < \varepsilon, \forall t > T) > 1 - \varepsilon.</math>


= Reputation with long-lived players =

This section deals with perfect monitoring games with two long-lived players. The goal is to demonstrate that when player 1 consistently plays the Stackelberg action and there exists a type of player 1 committed to that action, player 2 will eventually assign a high probability to the occurrence of the Stackelberg action in future rounds. However, being now player 2 long-lived, he may not play a best response to the Stackelberg type, but something else. When dealing with two long-lived players, a crucial step is to determine the conditions in which player 2, as his conviction regarding the appearance of the Stackelberg action grows, will ultimately choose to play a best response to that action. The following considerations applies in this setting: it is possible to think that as long as player 2 considers future benefits (thus discounting), any losses incurred by not playing a current best response must be compensated within a finite duration. However, if player 2 holds a strong conviction that the Stackelberg action will be played not only in the present but also in numerous subsequent periods, there will be no chance to accumulate future gains. Consequently, player 2 may find advantageous to simply opt for a stage-game best response. If this is the case, player 1 will receive almost the Stackelberg payoff in each period, thus putting a lower bound on his payoff if he is sufficiently patient. However, here lies the difference with the setting with short-lived players: player 2 might select an option other than a best response to the Stackelberg action due to concerns about triggering a future punishment by playing a current best response. This punishment would not occur if player 2 were facing the Stackelberg type, but player 2 can only be certain that he is facing the Stackelberg action, not the Stackelberg type. Short-lived players have the same fear, but this uncertainty does not affect their behavior given their time horizon.


== Perfect monitoring ==

Consider a perfect monitoring repeated game with two long-lived players. The two players have different discount factors <math display="inline">\delta_i</math> and the characteristics of player 2 are known. The remaining environment is similar to the case with a short-lived player 2.

The focus is now on a commitment type that minmaxes player 2. It is possible to demonstrate that if <math display="inline">\exists</math> a pure action <math display="inline">a^\prime_1</math> that mixed-action minmaxes player 2 and there is a positive probability that player 1 is the simple type <math display="inline">\xi(a^\prime_1)</math>, a sufficiently patient normal player 1 gets a payoff arbitrarily close to <math display="inline">v^*_1(a^\prime_1)</math>, which is the one-shot bound on player 1’s payoff when he commits to <math display="inline">a^\prime_1</math>. The following definition applies: the stage game has conflicting interests if a pure Stackelberg action <math display="inline">a^*_1</math> mixed action minmaxes player 2. The highest reputation bound is obtained when the game has conflicting interests. Let <math display="inline">v1(\xi_0, \mu, \delta_1, \delta_2)</math> be the infimum of the normal player 1’s payoffs. Suppose <math display="inline">\mu(\xi(a^\prime_1)) > 0 \text{ for some pure action } a^\prime_1 \text{ that mixed-action minmaxes player 2.}</math> <math display="inline">\exists</math> a value k, independent of <math display="inline">\delta_1</math> (but depending on <math display="inline">\delta_2</math>), such that <math display="inline">v_1(\xi_0, \mu, \delta_1, \delta_2) \geq \delta^k v_1^*(a^\prime_1) + (1 - \delta^k_1) \min_{a} u_1(a).</math> Therefore, only in some periods k player 2 can play something other than the best response to <math display="inline">a^\prime_1</math> and if player 1 is patient enough, these k periods have a small effect on his payoffs. Eventually, player 2 will play the best response to <math display="inline">a^\prime_1</math>. Moreover, <math display="inline">\forall \varepsilon >0, \exists \underline{\delta_1} \in (0,1)</math> such that <math display="inline">\forall \delta_1 \in (\underline{\delta_1},1)</math>, <math display="inline">v1(\xi_0,\mu, \delta_1, \delta_2) > v_1^*(a^\prime_1) - \varepsilon.</math> If the equilibrium strategy of player 2 results in a payoff lower than his minmax payoff, given that <math display="inline">a^\prime_1</math> is always played, then it implies that player 2 does not anticipate <math display="inline">a^\prime_1</math> to be played consistently.

Example: The product choice game (same structure of section [[#Product-choice game|1]]). This game is not one of conflicting interests. When player 1 takes the Stackelberg action H, it elicits a best response h from player 2, resulting in a payoff of 3 for player 2, surpassing his minmax payoff of 1. In contrast to games with conflicting interests, both normal player 1 and player 2 benefit more when player 1 chooses the Stackelberg action (with player 2 best responding) compared to the Nash equilibrium in the stage-game.

It is also possible that not only player 1’s type is unknown, but also player 2’s type. Each player’s type comes from a countable set before the game begins. <math display="inline">\lambda^0 > 0</math> is the probability that player 2 is normal. Also in this setting there is a maximum number of periods in which normal player 2 can play something other than the best response to <math display="inline">a^\prime_1</math>. Suppose <math display="inline">\mu(\xi(a_1)) > 0</math> for some action <math display="inline">a^\prime_1</math> minmaxing player 2. <math display="inline">\exists</math> a constant k, independent of player 1’s discount factor, such that the normal player 1’s payoff in any Nash equilibrium of the repeated game is at least <math display="inline">\lambda^0 \delta^k_1 v_1^*(a^\prime_1) + (1 - \lambda^0 \delta^k_1) \min_a u_1(a).</math> All in all, to establish a reputation, it is not important that incomplete information is only one sided, but that player 1 is sufficiently patient and that <math display="inline">\mu(\xi(a^\prime_1)) > 0.</math> Therefore, this ensures that player 2 will eventually play an optimal response to <math display="inline">a^\prime_1</math> and not in a very far away period.


=== Other actions ===

If action <math display="inline">a^\prime_1</math> does not minmax player 2, the number of periods in which player 2 is not optimally responding cannot be bound anymore. However, it is possible to bound the number of times in which player 2 can expect a continuation payoff lower than his minmax value. Player 1 payoff bound is <math display="inline">{v}_{1}^{\text{†}}(a^\prime_{1}) \equiv \min_{\alpha_{2} \in D(a^\prime_{1})} u_{1}(a^\prime_{1}, \alpha_{2})</math> where <math display="inline">D(a^\prime_{1}) = \{\alpha_{2} \in \Delta(A_{2}) \mid u_{2}(a^\prime_{1}, \alpha_{2}) \geq v_{2}\}</math> is the set of player 2 actions that imply his minmax utility. It is possible to demonstrate that, fixing <math display="inline">\delta_{2} \in [0, 1)</math> and <math display="inline">a^\prime_{1} \in A_{1}</math> with <math display="inline">\mu(\xi(a^\prime_{1})) > 0</math>, <math display="inline">\forall \text{ }\varepsilon > 0</math>, <math display="inline">\exists \underline{\delta_1} <1</math> such that <math display="inline">\forall \delta_{1} \in (\underline{\delta_{1}}, 1), v_{1}(\xi_{0}, \mu, \delta_{1}, \delta_{2}) \geq v_{1}^{\text{†}}(a^\prime_{1}) - \varepsilon.</math> If <math display="inline">a^\prime_{1}</math> does not minmax player 2, then <math display="inline">v_{1}^{\text{†}}(a^\prime_{1}) < v^*_1(a^\prime_1)</math>, the one shot bound. Moreover, it is not necessary that the Stackelberg action maximizes <math display="inline">v_{1}^{\text{†}}(a_{1})</math> and this bound is verified for all actions. If <math display="inline">\mu >0</math> for all simple pure commitment types, then a normal player 1 will get a payoff close to <math display="inline">\max_{a_1 \in A_1} v^{\dagger}_1(a_1)</math>.


== Imperfect Public Monitoring ==

In this section, there are 2 long-lived players who play an imperfect public monitoring game. <math display="inline">A_1, A_2</math> are finite action sets. <math display="inline">\rho</math> is the public monitoring distribution and it has full support, so <math display="inline">\forall y \in Y \text{and } a \in A_1 \times A_2</math>, <math display="inline">\rho(y | a) > 0.</math> Player 2’s actions are imperfectly monitored by player 1 and player 2 is able to update his belief based on what he observes about player 1. Therefore, it is also assumed that <math display="inline">\forall</math> mixed action <math display="inline">\alpha_2 \in \Delta(A_2)</math>, <math display="inline">\rho(\cdot | (\alpha_1, \alpha_2)) = \rho(\cdot | (\alpha^\prime_1, \alpha_2)) \Rightarrow \alpha_1 = \alpha^\prime_1.</math> As usual, <math display="inline">\delta_i</math> is the discount factor for player i, <math display="inline">\mu</math> is the prior with a support <math display="inline">\Xi</math>. Player 1’s set of history is <math display="inline">H_1 = \bigcup_{t=0}^{\infty} (A_1 \times Y)^t</math> and a strategy behavior is <math display="inline">\sigma_1 : H_1 \times \Xi \rightarrow \Delta(A_1)</math>, while a set of histories for player 2 is <math display="inline">H_2 =\bigcup_{t=0}^{\infty} (A_2 \times Y)^t</math>, with a behavior strategy <math display="inline">\sigma_2 : H_2 \times \Xi \rightarrow \Delta(A_2)</math>. A Nash equilibrium <math display="inline">\sigma = (\sigma_1, \sigma_2)</math> and <math display="inline">\mu</math> induce a measure P over the set of outcomes <math display="inline">\Omega \equiv \Xi \times (A_1 \times A_2 \times Y)^{\infty}</math>. It is possible that player 1 is committed to a non simple strategy. Being <math display="inline">G^N(\delta_2)</math> the complete information finitely repeated game that plays the complete information stage game N times, it is possible to define the payoff as follow: for player 1, <math display="inline">\frac{1}{N} \sum_{t=0}^{N-1} u_1(a^t)</math>; for player 2, <math display="inline">\frac{1 - \delta_2}{1 - \delta_2^N} \sum_{t=0}^{N-1} \delta_2^t u_2(a^t)</math>. <math display="inline">\sigma^N_1</math> is a strategy in a infinitely repeated game or in a finite repeated game of length N or integer multiple of N. The target for player 1’s payoff is the maximum payoff achievable by the strategies <math display="inline">\Xi</math>, the support of <math display="inline">\mu</math>, within the corresponding finitely repeated game, when player 1 exhibits arbitrary patience. The set of player 1 payoffs can be defined as <math display="block">V_1(\delta_2, \Xi) \equiv \left\{ v_1 : \forall \varepsilon > 0, \exists N, \xi(\sigma^N_1) \in \Xi \text{ s.t. } \forall \sigma^N_2 \in B^N(\sigma^N_1; \delta_2), U^N_1(\sigma^N_1, \sigma^N_2) \geq v_1 - \varepsilon \right\}</math> <math display="inline">\text{and set } v^{\ddagger}_1(\delta_2, \Xi) = \sup V_1(\delta_2, \Xi)</math>. If <math display="inline">\xi(\alpha_1) \in \Xi</math>, then <math display="inline">v^\ddagger_1(\delta_2, \xi) \geq v^*_1(\alpha_1)</math>. If <math display="inline">\xi</math> contains only simple commitment types, then <math display="inline">v^\ddagger_1(\delta_2, \xi) = v^{**}_1</math>. <math display="inline">v^\ddagger_1</math> may be much higher than <math display="inline">v^{**}_1</math> It is possible to demonstrate that <math display="inline">\forall \eta > 0</math> and <math display="inline">\delta_2</math>, <math display="inline">\exists \underline{\delta_1} < 1</math> such that <math display="inline">\forall \delta_1 \in (\underline{\delta_1},1)</math>, <math display="inline">v_1(\xi_0, \mu, \delta_1, \delta_2) \geq v^\ddagger_1(\delta_2, \Xi) - \eta</math>. Moreover, <math display="inline">\forall \eta > 0 \text{ and } \delta_2 > 0, \exists N^\prime, \delta_1^\prime, \varepsilon^\prime, \text{ and a strategy } \sigma^{N^\prime}_1 \text{ for } G^{N^\prime}(\delta_2)</math>, with <math display="inline">\xi(\sigma^{N^\prime}_1) \in \Xi</math>, such that if player 2 plays an <math display="inline">\varepsilon^\prime</math>-best response to <math display="inline">\sigma^{N^\prime}_1</math> in <math display="inline">G^{N^\prime}(\delta_2)</math>, then player 1’s <math display="inline">\delta_1</math>-discounted payoff in <math display="inline">G^{N^\prime}(\delta_2)</math>is at least <math display="inline">v^\ddagger_1(\delta_2, \Xi) - \frac{\eta}{2}.</math> It can also be shown that <math display="inline">\forall \varepsilon > 0, N \in \mathbb{N}, \text{ and } \sigma^N_1, \exists \gamma > 0</math> such that <math display="inline">\forall \tilde{\sigma}^N_1</math>, if <math display="inline">|\rho^N(\cdot | (\sigma^N_1, \sigma^N_2)) - \rho_N(\cdot | (\tilde{\sigma}^N_1, \sigma^N_2))| < \gamma \text{ for } \sigma^N_2</math> an <math display="inline">\varepsilon\text{-best response to } \sigma^N_1 \text{ in } G^N(\delta_2)</math>, then <math display="inline">\sigma^N_2</math> is a <math display="inline">2\varepsilon\text{-best response}</math> to <math display="inline">\tilde{\sigma}^N_1.</math>

Given <math display="inline">G^N(\delta_2)</math>, let us divide the infinitely repeated game into blocks of length N. Incomplete information repeated games have a prior <math display="inline">\mu \in \Delta(\Xi)</math> and a posteriors <math display="inline">\mu^\prime \in \Delta(\Xi)</math>. <math display="inline">\text{Given a strategy profile } \sigma \text{ and a prior } \mu^\prime, \text{ let } \rho^N_{\sigma,\mu^\prime}(\cdot | h^{Nk}_2)</math> be player 2’s "one-block" ahead prediction of the distribution over signals in block <math display="inline">G^{N,k}</math> (the <math display="inline">k^{th}</math> block of periods of length N) for any private history <math display="inline">h^{N,k}_2 \in H^{N,k}_2.</math> <math display="inline">P^{(\sigma^N_1, \sigma_2)}</math> is the probability measure over <math display="inline">\Omega</math> implied by <math display="inline">\sigma_2</math> conditioning on the event <math display="inline">\xi(\sigma^N_1) \in \Xi</math>. It can be shown that, fixing <math display="inline">\lambda, \mu^{\text{†}} \in (0, 1)</math> and <math display="inline">\gamma > 0</math>, integer N, and a strategy <math display="inline">\sigma^N_1</math>, there exists an integer L such that <math display="inline">\forall (\sigma_1, \sigma_2)</math> and all <math display="inline">\mu^\prime \in (\Delta(\Xi))</math> with <math display="inline">\mu^\prime(\xi(\sigma^N_1)) \geq \mu^{\text{†}}</math>, <math display="block">P^{(\sigma^N_1, \sigma_2)} (| \left\{ k \geq 0:
| \rho^N_{(\sigma^N_1, \sigma_2), \mu^\prime} (\cdot | h^{Nk}_2) - \rho^N_{(\sigma_1, \sigma_2), \mu^\prime} (\cdot | h^{Nk}_2) | \geq \gamma \right\} | \leq L) \geq 1 - \lambda.</math>


=== Commitment types who punish ===

A similar result can be obtained by adding, in the environment of perfect monitoring, commitment types who punish player 2 for not behaving properly. Player 2 will thus know the features of player 1, which previously remained hidden. This uncertainty was the reason why player 2 did not play a best response to the Stackelberg type in the simple environment of perfect monitoring.

Let <math display="inline">a^\prime_1 \in A_1</math> be an action for player 1, <math display="inline">a^\prime_2</math> the best response for player 2 for which <math display="inline">u_2(a^\prime_1, a^\prime_2) > v^p_2</math> and <math display="inline">\hat{a^2_1}</math> the action for player 1 that minmaxes plaer 2. Player 1 is a commitment type who plays strategy <math display="inline">\hat{\sigma_1}</math> which consist in phase k, where he plays <math display="inline">\hat{a^2_1}</math> and then he plays <math display="inline">a^\prime_1</math>. If player 2 does not play <math display="inline">a^\prime_2</math>, he will punish him. It is possible to demonstrate that fixing an integer <math display="inline">K > 0</math> and <math display="inline">\eta > 0</math>, there exists an integer <math display="inline">T(K,\eta,\hat{\mu}^0)</math> such that <math display="inline">\forall</math> pure strategy <math display="inline">\sigma_2</math> and any <math display="inline">\omega \in \hat{\Omega}</math>, there are no more than <math display="inline">T(K,\eta,\hat{\mu}^0)</math> periods <math display="inline">t</math> in which player 2 attaches probability no greater than <math display="inline">1 - \eta</math> to the event that player 1 plays as <math display="inline">\hat{\sigma}_1</math> in periods <math display="inline">t, \ldots, t + K</math>, given that player 2 plays as <math display="inline">\sigma_2</math>. Moreover, fixing <math display="inline">\varepsilon > 0</math> and letting <math display="inline">\Xi</math> contain <math display="inline">\hat{\sigma}_1</math>, for some action profile <math display="inline">a^\prime</math> with <math display="inline">u_2(a^\prime) > v^p_2</math>, <math display="inline">\exists</math> a <math display="inline">\underline{\delta_2} < 1</math> such that <math display="inline">\forall \delta_2 \in (\underline{\delta_2}, 1)</math>, <math display="inline">\exists</math> a <math display="inline">\underline{\delta_1}</math> such that <math display="inline">\forall \delta_1 \in (\underline{\delta_1}, 1)</math>, <math display="inline">v(\xi_0, \mu, \delta_1, \delta_2) \geq u_1(a^\prime) - \varepsilon</math>. If <math display="inline">a^\prime_1</math> is player 1’s Stackelberg action, then this result gives player 1’s Stackelberg payoff as a lower bound of his equilibrium payoff in the game of incomplete information. The bound on the payoff for the normal player 1 is determined by demonstrating that player 2 will face only a finite number of punishments resulting from player 1’s commitment type.


== Temporary reputations with two long-lived players ==

The results obtained in section [[#tr|4.4.2]] can be generalized also in the case of two long-lived players. Considering the case in which player 1’s type is unknown, it is possible to show some conditions under which player 2 is effectively able to learn about player 1’s type. The authors assume a commitment type who plays <math display="inline">\hat{\sigma_1}</math>, a strategy with no long-run credibility for which:

* <math display="inline">\hat{\sigma_2}</math> is the best response for player 2 and it is unique on the equilibrium path.
* <math display="inline">\exists \text{ }T^0</math> such that <math display="inline">\forall t > T^0</math>, normal player 1 is likely to deviate from <math display="inline">\hat{\sigma_1}</math>, given <math display="inline">\hat{\sigma_2}</math>.

The set of best responses to <math display="inline">\sigma_1</math> for player 2 in the game of complete information is <math display="inline">B(\sigma_1) \equiv \{\sigma_2 : U_2(\sigma_1, \sigma_2) \geq U_2(\sigma_1, \sigma_2^\prime) \text{ }\forall\text{ }\sigma_2^\prime\}</math>, with <math display="inline">U^t_i</math> being player i’s continuation value in period t.

Let <math display="inline">\pi</math> be the monitoring distribution and let the commitment type’s strategy <math display="inline">\hat{\sigma}_1</math> be public and with no long-run credibility. Then in any Nash equilibrium of the game with incomplete information, <math display="inline">\hat{\mu}^t \rightarrow 0</math> <math display="inline">\tilde{P}</math>-almost surely. <math display="inline">\hat{\sigma_1}</math> is public so that player 1 can anticipate player 2’s optimal response to <math display="inline">\hat{\sigma_1}</math>. It is important to underline that a long-lived player 2 will best respond to the commitment type once he is convinced that he is almost certainly facing the commitment strategy. Moreover, the normal type deviations from the commitment strategy will occur only in a finite number of periods.


= References =

* Cabral, L. M. B. (2005). The economics of trust and reputation: A primer [Last access June 8, 2023]. [https://pages.stern.nyu.edu/~lcabral/reputation/ Reputation June05.pdf]
* Diekmann, A., & Przepiorka, W. (2021). Trust and reputation in historical markets and contemporary online markets. In A. Maurer (Ed.), Handbook of economic sociology for the 21st century: New theoretical approaches, empirical studies and developments (pp. 131–145). Springer International Publishing.
* Dowling, G. (1993). Developing your company image into a corporate asset. Long Range Planning, 26, 101–109.
* Eccles, R., Newquist, S., & R., S. (2007). Reputation and its risks [Last access June 10, 2023]. [https://hbr.org/2007/02/reputation-and-its-risks]
* Einwiller, S. (2003). When reputation engenders trust: An empirical investigation in business-to-consumer electronic commerce. Electronic Markets, 13 (3), 196–209.
* Gambetta, D. (2000). Can we trust trust? Trust: Making and Breaking Cooperative Relations, electronic edition, Department of Sociology, University of Oxford, 213–237.
* Mailath, G. J., & Samuelson, L. (2006). Repeated Games and Reputations: Long-Run Relationships. Oxford: Oxford University Press.
* Mailath, G. J., & Samuelson, L. (2015). Chapter 4 - reputations in repeated games. In H. P. Young & S. Zamir (Eds.), Handbook of Game Theory with Economic Applications. Elsevier. 165-238.
* Nash, J. (1951). Non-cooperative games. Annals of Mathematics, 54 (2), 286–295.
* Popitz, H. (1980). Die normative konstruktion von gesellschaft. Tubinga: Mohr Siebeck.
* Roberts, P. W., & Dowling, G. R. (2002). Corporate reputation and sustained superior financial performance. Strategic Management Journal, 23 (12), 1077–1093.
* Selcuk, A., Uzun, E., & Pariente, M. (2004). A reputation-based trust management system for p2p networks. IEEE International Symposium on Cluster Computing and the Grid, 2004. CCGrid 2004., 251–258.
* Webley, S. (2004). Risk, reputation and trust. Journal of Communication Management, 8, 9–12.
* Xiong, L., & Liu, L. (2004). Peertrust: Supporting reputation-based trust for peer-to-peer electronic communities. IEEE Transactions on Knowledge and Data Engineering, 16 (7), 843–857.

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The role of trust in economics and finance

2023-06-09T15:31:29Z

3122188:

{{DISPLAYTITLE:The role of trust in economics and finance}}

Contribution of [[SIMONE GOZZINI]]


= Introduction =

Trust is a fundamental sentiment, the binding force behind modern societies: without it, no progress would have been possible. Trust is what permitted the birth of modern finance with the Buttonwood Agreement of 1792, it is what makes people able to rely on another person or organization without the continuous need to assess what the other party is doing and it is ultimately what permits the existence of modern democracies (Warren, 2018) . Without it, no meaningful relationship would be possible. However, in the last few years, a general disbelief about trust is permeating the civil society. Edelman is a global communication firm that conducts a comprehensive survey about trust every year. For 2022, the results picture a general sentiment of distrust across all segments of the population: 60% of the interviewed say that their default tendency is to distrust others, media are seen as a divisive and untrustworthy institution by around 50% of the people and trust in government is significantly dropping, year after year. The problem is particularly accentuated regarding governments, which are seen as unable to fix societies’ problems (Edelman, 2022). Distrust affects modern societies as a whole, impacting not only social relationships and the economy, but also human health: for example, lower trust in government has led to lower vaccinations against COVID-19, threatening society as a whole (Bajos et al., 2022).

This paper highlights the importance of trust in modern economies and in the financial world. Section [[#Concept|2]] describes the concept of trust, differentiating it from other human sentiments like cooperation and confidence. In general, the concept of risk is concerned, given that trust involves a sort of faith in someone or something. Section [[#Measure|3]] describes the various methodologies used in the literature to measure trust: trust games, surveys and the frontier of neuroscience. Section [[#Comparative|4]] presents trust as a source of comparative advantage in world trade patterns: societies with more trust have bigger and more productive firms. Section [[#stock|5]] studies how trust affects stock market participation: people with a higher tendency to trust are more likely to participate in the stock market and, conditional on participating, they invest a higher fraction of their wealth. Section [[#Money|6]] describes a general equilibrium model where money is seen as a substitute of trust: the allocation of resources in a trustworthy society can be reached also in a trust-less society which employs money. Section [[#Institution|7]] describes a stylized model of trust between individuals and an institution: the exchange of information among individuals is found to be a tool that improves the assessment of the true trustworthiness of an institution. Section [[#Trust_blockchain|8]] presents the blockchain technology as a new architecture of trust, describing also how trust can be enhanced to reach a higher diffusion and application of this technology. Section [[#Games|9]] presents various papers regarding trust games in the blockchain technology, considering in particular how to reach and improve the consensus process. Section [[#algorithms|10]] describes how algorithms, which are becoming more and more important in modern life, can be trusted: in particular, the author highlights transparency and accessibility as fundamental characteristics to enhance trust. Section [[#conclusion|11]] concludes.


= The concept of Trust =

According to the definition of Gambetta (2000), trust is ''“a particular level of the subjective probability with which an agent assesses that another agent or group of agents will perform a particular action, both before he can monitor such action (or independently of his capacity ever to be able to monitor it) and in a context in which it affects his own action”'' (Gambetta, 2000, p. 5).

This definition highlights important concepts:

* Trust is a probability ''p'', a threshold, but subjective: people engage in a trust relationship if they believe that the probability that the person will perform the particular action mentioned in the definition is higher than a certain level, which depends on the individual predisposition to trust and the circumstances under which the relationship is being created, like the cost of misplacing trust.
* Trust is related to uncertainty: the underlying assumption is that the agent is not able to fully monitor the other agent while he performs the particular action (this is usually the case in practice), otherwise trust would not be necessary since the first could keep track of the second while performing the action.
* It has an impact on the trustor, otherwise the actions of the second agent would not matter to the first, and engaging in a relationship would not be necessary.

Trust is relevant when the other agents (trustees) are free to betray the trustor, otherwise, if coercion intervenes, the outcome of the trust relationship is known ex-ante. Assessing a probability would not be needed anymore and uncertainty would not be involved: the resulting interaction would not be a trust relationship, given that it lacks of its fundamental characteristics. Furthermore, also the trustor needs to be free to choose whether to engage in this relationship or to escape from it, otherwise also in this case assessing ''p'' would not be needed since he would have no choice.

Therefore, trust is fundamentally a free choice between two individuals who seek mutual benefits and it involves a level of risk for the trustor, given that it affects his personal sphere of action.


== Cooperation and trust ==

Trust, however, must not be confused with other human actions, sentiments, or beliefs. Trust is different from friendship, from passion, from loyalty and from cooperation. In particular, the relationship between the latter and trust is investigated extensively by the author.

Trust can be seen as:

* a precondition for cooperation, which, together with sane competition is beneficial to foster human progress. However, although being probably the most efficient way to achieve cooperation, it is not a necessary condition: people have used surrogates in history to overcome the problem of the lack of trust, like coercion, contracts and promises. All of them have the objective of diminishing the possible alternatives that the trustor and the trustee can face, thus reducing the risk for both parties in engaging in this relationship. A higher level of trust increases the probability of cooperating, but it is possible that, even though the level of ''p'' is low, the result is cooperation anyway. This is because an agent takes also into consideration the cost and the benefit of engaging (or not) in such a relationship, the other alternatives he has and the specific situation.
* an outcome of cooperation. Societies may engage in cooperation thanks to ''“a set of fortunate practices”'' (Gambetta, 2000, p. 10), particular circumstances and the need to satisfy mutual interests, for which the cost of not engaging in cooperation is higher than the risk of engaging. Trust is therefore the outcome of these practices and there is no need for prior beliefs about the trustworthiness of the other party, since trust will arise only after the beginning of the relationship, when information is collected. This statement is reinforced by the fact that cooperation exists in animals, that are unlikely to experience trust.

However, according to the author, there is no reason for saying that cooperation is a spontaneous equilibrium in human interaction: cooperation is just as likely as non cooperation. A predisposition to trust may be rational for humans in order to achieve their objectives, since trust is fundamentally an efficient way to achieve cooperation, but it is not necessary to wait for trust to evolve in order to initiate cooperation. Common interests and constraints can be enough and they can be beneficial especially in underdeveloped countries which present a low level of trust. In fact, although trust and trustworthiness can be advantageous for an individual’s purposes, they cannot be artificially induced in a rational person. Moreover, the author argues that rational trustors and trustees may seek and present evidence to trust and be trustworthy. However, more information cannot fully solve the problem of trust. People, once they trust, do not try to find evidence to corroborate their belief, but rather they change their mind only if they find contrary evidence, which is not easy.


== The concept of trust in organizations ==

Mayer et al. (1995) start from the studies of Gambetta to further develop the understanding of trust in the context of organizations. In particular, they focus on the trust relationship between two individuals: a trustor who trusts or not an individual to perform a particular action and the trustee who receives the trustor’s trust, deciding then if fulfilling or not that action. The flow of trust is unidirectional: mutual trust between two parties is not developed in the paper, nor is trust in a social system. In particular, according to the authors, the concept of vulnerability is what is missing in the definition of Gambetta, given that ''“Trust is not taking risk per se, but rather it is a willingness to take risk.”'' (Mayer et al., 1995, p. 712). Then, trust is differentiated from different constructs such as:

* Cooperation, which is intensively studied also by Gambetta. The authors highlight the fact that trust is not a conditio sine qua non for cooperation, since it is possible to cooperate with someone not trustworthy (for example when there are external controls and constraints like those discussed in the previous section).
* Confidence: the main difference relies on the fact that, with trust, risk must be assumed, while in the second it is not necessary. Moreover, when a person chooses to trust, he will consider a set of possible alternatives, while that is not the case with confidence.
* Predictability: trust and predictability are a way to cope with uncertainty but, if a person is predictable, it does not necessarily mean that it is worth putting trust in him. This is because it is possible to predict that the other person will consistently behave in negative ways (and the uncertainty is reduced), but no rational individual would put trust in him.

Then, the characteristics of the trustor and the trustee are analyzed, which can together initiate a trust relationship between the two agents. The most important feature of the trustor is his propensity to trust another person, which is a personal trait constant over time and across situations. It is a general willingness to trust and it is not related to another party, since it is measurable before any interaction with the other agent. However, each trustor has different levels of trust for various trustees, which arise after the relationship is initiated. Therefore, they depend on the characteristics and actions of the trustee, i.e. his trustworthiness. According to the authors, there are 3 main characteristics of the trustee that are able to explain trustworthiness:

* Ability: it is defined as the set of skills and competencies of the trustee over a specific domain. It is possible to trust another person to perform a particular action if the other agent is competent in that field otherwise he should not be trusted, even though he may be committed to completing the task. Therefore, trust should not be intended in absolute terms, but over a specific field of knowledge.
* Benevolence: it is a personal trait of the trustee towards the trustor which is related to how much the former wants good for the latter. More benevolence leads to higher trust because the trustor can be more sure that the trustee will perform the action taking into account also his benefit and not only the trustee’s egoistic motives.
* Integrity: it is defined as ''“the trustor’s perception that the trustee adheres to a set of principles that the trustor finds acceptable”'' (Mayer et al., 1995, p. 719). A trustee’s integrity therefore depends on what the trustor’s set of beliefs are. If the trustor thinks that the integrity of the trustee is not sufficient, he will not engage in a trust relationship with him.

In particular, integrity will be central in the early stages of the relationship, before gaining any insights; then, benevolence will become important over time, as the trustor retrieves information during the course of the relationship; ability, instead, continues to stay important from the beginning to the end. After engaging in the trust relationship, the trustor will be able to gain new data and information, through which he can update his beliefs about these three characteristics of the trustor, eventually deciding whether the placement of trust is still reasonable. While a trustor tries to assess these characteristics of the trustee, the role of context becomes important because it affects ability (for example because a change in a situation may change the skills needed to complete a certain task), the level of benevolence (for example if the trustee changes his behavior during time) and integrity (for example because a certain action of the trustee is not interpreted as coherent with the trustor’s set of values only because it was obliged to do so by the specific situation).

Finally, the authors deal with the risk involved in trusting. In particular, they highlight the fact that there is no risk taking in the propensity to trust, but risk arises only when an agent effectively engages in a trust relationship. However, the form and the level of risk assumed by the trustor will depend on the level of trust involved in a relationship: the more the trustor trusts the trustee, the more risk he will be willing to take. So, before initiating a trust relationship, the agent has to assess whether the level of trust is higher or lower than the perceived level of risk, so that he can decide whether it makes sense to engage in such a relationship.


= How to measure trust =

Trust is therefore a fundamental device in human society and it is also important in economics and finance, as this paper will later explain. A natural question arises: ''how it is possible to measure trust?'' The question is not easy to answer, since trust is a human sentiment, therefore subjective and emotional, and which is also interwoven with other human sentiments and beliefs. review the major methods used by the literature to measure trust, highlighting the main limitations of each model.


== Trust Games and Game Theory ==

Experimental economics has intensively relied on game theory to quantify trust. The games mostly used nowadays are various versions of the TRUST GAME, which was invented by Berg et al. (1995). Alós-Ferrer and Farolfi (2019) describe it as follows: ''“A first agent, called the trustor, is given a monetary endowment X, and can choose which fraction p of it (zero being an option) will be sent to the second agent, called the trustee. The transfer p · X is then gone, and there is nothing the trustor can do to ensure a return of any kind. Before the transfer arrives into the trustee’s hands, the transfer is magnified by a factor K <math display="inline">></math> 1. The trustee is free to keep the whole amount without repercussion. Crucially, however, the trustee has the option to send a fraction q of the received transfer back to the trustor, hence honoring the trustor’s initial sacrifice”'' (Alós- Ferrer & Farolfi, 2019, p. 1). The transfer of the trustor can become a measure of trust, while the subsequent transfer of the trustee is a measure of trustworthiness. These games underline some important features of trust as described by Gambetta (2000): trustor and trustee decisions are free and voluntary, uncertainty and risk are involved and there are possible repercussions for the trustor (a loss in utility).

However, despite the popularity of this method, there are various limitations that need to be addressed. In the agents’ behavior, there might be possible motivational confounds that affect the measurement of trust and trustworthiness, like selfish or altruistic tendencies, efficiency reasons, or prior personal preferences (like inequity aversion). To address this problem, the authors suggest taking as a measure the difference between the transfers in the trust game and those in a game called ''the Dictator Game'' (i.e. a game where the proposer’s decisions are implemented without the possibility for the responder to do something).

Then, the question of the risk attitudes of the agents is addressed. Trust involves a certain risk (given that the trustor cannot monitor the response of the trustee) so the aptitudes towards risk may affect the monetary transfers. The evidence is mixed, with early studies (like Houser et al., 2010) finding no relationship between risk attitudes and trust and with others finding a correlation. The lack of agreement might be due to the concept of risk involved in the trust games themselves, which is not a pure financial risk, but a ''betrayal aversion'', i.e. the risk and fear of being betrayed by another human being. Taking into account this, the authors mention the study of Bohnet and Zeckhauser (2004), where a “betrayal aversion” was found in the decisions of the agents, different from the standard risk aversion. Therefore, to disentangle the trust component and the risk component of the agents’ transfers, standard measures of risk might not fit properly. The authors, however, criticize also the use of game variants to address this issue, since new measures may capture other undesired effects.

Another problem can arise when there are changes in the parameters, implementation and description of the trust game: the responses of the agents might not be consistent in all contexts, thus creating an impossibility of comparability between different experiments. For instance, increasing the multiplier K will likely increase the trustor’s transfer and also the fraction returned by the trustee according to Lenton and Mosley (2011). Moreover, also the way that the game is framed can have an impact: Burnham et al. (2000) show that the responses of the agents involved depend on whether, in the instructions of the game, the other agent was called partner or opponent. In the former case, the trustor trusted more the trustee than in the latter case. However, if the game is not framed at all the participants might create their own frame, thus interpreting the play in different and unpredictable ways, conducting to biased results.


== Surveys ==

Another possible measure of trust relies on the use of surveys. The most important example is the General Social Survey (GSS) of the U.S. National Opinion Research Center. The question asked is: ''“Generally speaking, would you say that most people can be trusted or that you can’t be too careful in dealing with people?”''. The possible answers are: “Most people can be trusted” or “Can’t be too careful” or “I don’t know”. This question is used also in other important surveys, like the EVS (European Values Survey), the WVS (World Values Survey), the BHPS (British household panel study) and the ANES (American National Election Studies).

This method is not immune from problems. For example, the interpretation of each individual might play a role in the response, as seen in the Trust Game. Moreover, the relationship between these two methods should be taken into account. Ideally, if both were valid and consistent, the responses should be highly correlated. However, the evidence is mixed. Glaeser et al. (2000) find no correlation between the two measures while Fehr et al. (2003) find evidence of the contrary. An explanation could be that surveys test a general propensity to trust, while Trust Games measure a specific strategic situation of the agents’ behavior. The concept of trust is therefore not uniquely determined and different methodologies might capture different aspects of this complex human attitude.

Moreover, the authors suggest that, if surveys are used as a measure, one must take into account various controls (like culture, geography and age) to interpret and therefore compare the responses.


== Neuroscience ==

The new frontier in the measurement of trust is represented by neuroscience, which tries to give more objective and biological methods.

Firstly, the relationship between oxytocin (OT) and trust is investigated, in particular to link OT levels with the behavior in the Trust Game. Zak et al. (2005) find that OT levels can predict trustee trustworthiness but not trustors’ transfers. However, when the change in OT levels is endogenous (i.e. natural, like in the paper mentioned above), the studies cannot establish causality. Hence, another set of studies, where the level of OT was exogenously determined, is examined. Kosfeld et al. (2005) find that the treatment group in their experiment (i.e. the people whom OT was administered) presents larger trustors’ transfers compared to the control group, but no significant differences in the trustees’ transfers. Moreover, their results suggest that OT causally increases trust through a reduction of betrayal aversion and it does not increase risk-taking behavior or prosocial aptitudes in general. The two methods of investigation lead therefore to inconsistent result with each other. Therefore, no conclusion can be reached: the relationship of OT with trust and trustworthiness is not simple as previously thought.

Finally, the authors introduce the latest studies about the use of brain imaging to understand where trust comes from and how it forms. This might be useful to develop more reliable measures of trust in the future.


= Trust as a source of comparative advantage =

Cingano and Pinotti (2016) study the effect of trust on firm organization and on comparative advantage. The authors argue that interpersonal trust means more delegation of decisions within a firm, resulting in a larger firm size and in the expansion of more productive units. If trust is established, it is possible to expand the firm outside familiar and friendly relationships, thus using the firm’s own productivity advantage over a larger amount of input, given that the firm is bigger and has more factors of production. The principal-agent problem (that comes with delegation and prevents a higher level of it) can be partially solved by this human device. In particular, higher delegation causes higher productivity through:

* higher exploitation of the informational advantage of the managers and of specific skills of some workers.
* the reduction of information costs.
* more resiliency and ability to cope with changes in profit and growth opportunities.

Studying a sample of Italian and European companies, the authors find that trust, together with human capital and intangible intensity, is associated with greater delegation, which, in turn, is associated with larger firm size. Their findings suggest that high-trust countries present a higher value-added per worker and higher exports in industries where delegation is needed, thus making trust a source of comparative advantage in trade patterns. This effect is the result of a reduction of smaller size firms towards bigger size firms.

The authors test their hypotheses through empirical data obtained with surveys. They retrieve data from:

* The INVIND survey from the Bank of Italy, which provides information about inputs, outputs, internal organization and governance of a sample of more than 6500 firms. These data are used to test trust differences across Italian regions.
* The World Values Survey (WVS) and the European Social Survey (ESS) to measure interpersonal trust and delegation.
* The OECD Structural Analysis Database (STAN) and the OECD Business Demographic Statistics, which provide information about value added per worker, organization and the number of workers of European firms.

The analysis starts with the following regression:

<math display="block">Y_{jr} = \alpha + \beta(Trust_r \times Delegation_j) + \delta X_{jr} + \mu_r + \mu_j + \varepsilon_{jr}</math>

Where <math display="inline">Y_{jr}</math> is industry specialization (measured through value added per worker or exports), <math display="inline">Trust_r</math> is the average level of trust, <math display="inline">Delegation_j</math> is a measure of the need for delegation in each industry and <math display="inline">X_{jr}</math>, <math display="inline">\mu_r</math> and <math display="inline">\mu_j</math> are controls respectively for other determinants of specialization and geographical factors.

Then, the authors estimate <math display="inline">Delegation_j</math> through the following regression:

<math display="block">Centers_{jr} = \eta + \theta lnL_{ijr} + \mathit{f_j} + \mathit{f_r} + \mathit{v_{ijr}}</math>

Where <math display="inline">Centers_{jr}</math> is the number of responsibility centers (which is a measure of delegation inside firms), <math display="inline">lnL_{ijr}</math> is the log of the number of workers (which is kept fixed) and <math display="inline">\mathit{f_j}</math> and <math display="inline">\mathit{f_r}</math> are firm’s controls.

In particular, the analysis shows that, for the Italian sample, higher trust leads to an increase in the production of delegation intensive industries. Starting with the log of value added per worker as the dependent variable, the authors add a series of controls. Introducing human capital, the calculations show that it remains the main source of the pattern of specialization but, despite being correlated with delegation (which in turn has an effect on trust), the latter variable remains statistically significant. Then, two other controls are introduced: financial development and judicial quality. However, they do not affect the coefficient of trust, thus making the estimation more robust and consistent. The results are similar when the dependent variable is export. 
For the international sample, the analysis is more complicated because different countries present different institutional dimensions, like labor market regulations and property protections. The results, however, are very similar, making their thesis consistent also at the international level.


= Trust and the stock market =

Guiso et al. (2008) study the effect of trust on stock market participation across individuals and across countries. Starting from Gambetta (2000), they define trust as ''“the subjective probability individuals attribute to the possibility of being cheated”'' (Guiso et al., 2008, p. 2557), which depends on the characteristics of the financial system and the individual priors and predisposition to trust.

Firstly, they develop a theoretical model in which they reproduce the effect of trust on portfolio decisions, starting with a two asset model (one safe asset and one stock). They assume that investors know the distribution of returns but they are worried, with a level of subjective probability ''p'', about other bad events, like the possibility of fraud perpetrated by their broker, which will lead to 0 return in the stock. They also assume 0 participation cost. Given a level ''W'' of wealth, being <math display="inline">\tilde{r}</math> the return in the stock investment and <math display="inline">r_f</math> the risk free rate, each of the agents chooses a share <math display="inline">\alpha</math> of their wealth to invest in the risky asset so that they can maximize their expected utility <math display="block">Max (1-p)EU(\alpha \tilde{r} W +(1-\alpha)r_f W) + pU((1-\alpha)r_f W).</math>

They also calculate that a risk averse individual will invest in the stock market if his subjective probability p > <math display="inline">\bar{p}</math>, where <math display="inline">\bar{p}</math> is <math display="inline">\bar{p}</math> = (<math display="inline">\bar{r}</math> - <math display="inline">r_f</math>)/<math display="inline">\bar{r}</math> and <math display="inline">\bar{r}</math> is the mean of the true distribution of the returns of the stock. This last relationship comes from the fact that an investor invests in a risky asset if the expected return of investing is higher than the risk free rate, i.e. (1 - ''p'') <math display="inline">\times</math> <math display="inline">\bar{r}</math> + ''p'' <math display="inline">\times</math> 0.

An important result of this model is that the decision to participate or not in the stock market depends on the subjective probability ''p'' of being cheated (since it reduces the expected return of the investment) and it does not depend on the level of ''W''. Since ''W'' is not significantly correlated with trust (as calculated through the survey data they use in their empirical analysis), this can explain why also the wealthy might not engage in stock trading. Moreover, <math display="inline">\alpha</math> itself depends on the level of trust: more trust means more wealth invested in risky assets and vice-versa.

Then, participation costs are introduced in the theoretical model. To enter the market, the investor now has to pay a fixed cost ''f'' (thus reducing the allocable wealth to ''W'' - ''f''). As ''f'' increases, in order to invest in stocks, a higher level of trust is necessary (<math display="inline">\bar{p}</math> decreases). In particular, less trust reduces the return on stock investment (thus making the participation less attractive) because it reduces the share of wealth invested in stocks and it reduces the expected utility from participating.

Finally, the authors demonstrate that risk tolerance and trust are two different things by looking at the optimal amount of stocks: this number increases with trust and it increases also with risk aversion (for the benefits of diversification). Therefore, since risk tolerance reduces the optimal number of stocks and the contrary is true for trust, the latter cannot be a proxy of the former. As the empirical analysis will demonstrate, this is consistent with the data. This result is also reinforced by the fact that the authors find that individuals with high levels of trust buy more insurance, while risk tolerant individuals buy less.

The authors use survey data to test their model. In particular, they employ the DNB Household Survey (to which they have directly contributed), which maps about 1990 individuals and tries to capture their level of generalized trust, their risk and ambiguity aversion and their optimism. It also reports some statistics about households’ assets, distinguishing in particular between listed and unlisted stocks and securities held directly or through financial intermediaries. To measure generalized trust, this survey uses the same question as the World Values Survey (see section [[#sec:surveys|3.2]] for explanation); to measure risk aversion and ambiguity aversion, the authors ask the interviewed their willingness to pay for some lotteries; to measure optimism, they ask to quantify their agreement (on a scale from 1 to 5) with the following statement: ''“I expect more good things to happen to me than bad things”''. Then, the Italian Bank customers survey is used to capture the ''personalized trust'', i.e. the trust that an individual has towards its financial intermediary, which could be different from the general propensity to trust. This data set contains information about the financial assets the interviewed have and their demographic characteristics. More importantly, to measure personalized trust, the survey asks the following question: ''“How much do you trust your bank official or broker as financial advisor for your investment decisions?”''.

The empirical analysis confirms their hypothesis. Starting from the study of the relationship between generalized trust (i.e the level of trust measured in the survey) on stock market participation, the authors find that trust has a positive and highly significant coefficient (so more trust means more participation), even after controlling for a number of variables (like age, sex and wealth). In particular, ''"Trusting others increases the probability of direct participation in the stock market by 6.5 percentage points”'' (Guiso et al., 2008, p. 2578). Risk aversion and ambiguity aversion do not seem significant, as well as optimism, since the coefficient of trust remains unchanged. Moreover, when studying the effect of wealth, the authors find that the coefficient of trust remains significant even after controlling for this variable, thus providing a proof for their previous statement: the lack of trust may be an explanation for the fact that rich people do not invest in stocks even though they should not be affected by the participation costs. Then, the relationship between trust and the amount invested in risky assets is studied. The result confirms, again, the hypothesis: ''"Individuals who trust have a 3.4 percentage points higher share in stocks, or about 15.5% of the sample mean"'' (Guiso et al., 2008, p. 2580). The same results hold for risky assets in general: risk and ambiguity aversion are not statistically significant also in this case. However, a significant control is represented by the level of education. The authors find that trust increases the holding of risky securities for everyone, but less in more educated people, since they know better how the market works with respect to the less educated and they are less affected by priors and cultural stereotypes.

Considering now the Italian Banks costumers survey, the results confirm the previous ones: trust in one’s own financial intermediary increases the probability of investing in stock and the share of the wealth allocated in this type of security.

Finally, the authors investigate the implication of the level of trust on market participation across countries. The analysis is based on the following statement: less trust should mean that agents are less willing to invest and, in turn, firms will be less willing to float their equity given that it is less rewarding. Therefore, countries with lower levels of trust should have lower participation in the market. The empirical analysis confirms the previous claims: trust has a positive and significant effect on stock ownership among individuals and it has also a positive effect on stock market capitalization.


= Money as a substitute for trust =

Gale (1978) develops a theoretical model to study the effect of the introduction of money in an economy characterized by a lack of trust between its agents. The author starts from the Arrow-Debreu model of Walrasian equilibrium. This model is characterized by a finite number of consumers (who have an initial endowment of resources) and commodities, perfect competition in all markets and constant return to scale. Moreover, markets are complete, which means that all transactions in the economy can be arranged at one time. This is made possible because transactions that involve the delivery of a commodity in a different time period (i.e. in t=0 a commodity is sold but the delivery will be arranged in t=1) can be concluded through contracts at time t=0. The contract specifies that the delivery will occur in t=1, even though the transaction itself is completed in t=0. Therefore, the contract is seen as the commodity being traded. This mechanism operates under the assumption that there is no uncertainty in the market. In such an environment, agents can trust each other to fulfill the contracts they have agreed upon. As a result, there is no need to distinguish between the contracts and their execution. Nevertheless, if for some reason agents start not to trust each other, and therefore uncertainty arises, some agents may prefer not to fulfill their promise and other agents, anticipating that, might not engage in a transaction in the very first place. If trust were to vanish, therefore, the allocation process would break down if no other substitutes were found. The scholar demonstrates that money can be a substitute for trust and it can permit the allocation and redistribution of resources even in the absence of trust.

To illustrate that formally, the author employs the concept of core that is ''“the set of attainable allocations such that (a) neither agent can make himself better off by remaining self-sufficient and (b) two agents cannot both be made better off by any feasible redistribution of their joint endowment.”'' (Gale, 1978, p. 459), through which he develops the concept of sequential core to integrate time periods and uncertainty about the outcome of a contract. An allocation of commodities is trustworthy if the sequential core applies to it, that is if it cannot be improved by any redistribution of resources in any time period. If this were not the case, an agent would have the incentive to break the contract in later periods. Therefore, any exchange of commodities without trust would not form a sequential core, because agents would have the incentive to deviate from equilibrium to increase their own utility.

To resolve this issue, the author introduces money in the model. In particular, each agent is given an endowment of money at time t=0 and it is assumed that at the end of time t=1 (the second and last period) the same amount of money must be returned as a tax. Implicitly, the model introduces a social institution (for example a government) that issues fiat money, which has no intrinsic value but it is guaranteed by the imposition of the government itself (this is the case in modern economies). In between the two periods, the agents can exchange money among themselves. This solves the issue: the agents who were reluctant to keep their promises in the model without money and without trust now have an incentive to fulfill the contract, given that they need the money to pay their taxes. Money does not restore trust among agents, but they act as a substitute, a way to enforce previous contracts and agreements. The possibility for the government to directly intervene in the fulfillment of contracts should be discarded, since it is not plausible that a human institution could be so almighty that it can oversee every transaction in a complex economy. Therefore, money can create the conditions for trustworthy transactions (without trust) in a decentralized way. However, the institution must be able to credibly impose the payment of taxes, otherwise agents would face the same problem as before. To do that, penalties for those who do not want to pay taxes should be sufficiently ''gruesome'', but the author does not quantify the penalty. Moreover, the author argues that, despite money can substitute trust, there could be a loss in overall utility with respect to the case with trust. In the model, the social institution is introduced without any explicit cost, but this is unlikely to be the case in reality, since introducing a government that is able to enforce tax payment and issue securities is certainly not free.


== The gruesome penalty ==

Grimes (1990) continues the work of Gale (1978) in analyzing the role of money in the same theoretical framework studied by the previous author. The results of Gale are confirmed: without money, the outcome of an economy without trust would be autarky, since no transaction can effectively occur. With the introduction of money, however, it is possible to replicate the allocation of the economy with trust. The contribution of his work with respect to the research of his predecessor is about a quantification of the ''gruesome'' penalty that agents face when they do not respect their tax obligations.

In particular, the author shows that the simple introduction of money does not necessarily replicate the outcomes of an economy without trust, because a sufficient incentive (i.e. a penalty higher than a certain threshold) to make inefficient for the agents not to fulfill their promises must be introduced. Under this threshold, the increase in utility derived from reneging the contract is higher than the reduction in utility due to the penalty. Therefore, the optimal choice is not to fulfill the agreement. On the contrary, above that threshold, the optimal choice is to fulfill the contracts (therefore replicating the allocation with trust). It is worth noting that the intensity of the penalty has no effect on the final allocation of goods, since they are already Pareto-efficiently allocated, but the author shows that this has an impact on prices.

To calculate the threshold, the author considers a world with two agents, two periods, no uncertainty and one good in each period. Each agent’s endowment in each period is defined as <math display="inline">(1-\lambda, \lambda)</math> and <math display="inline">(\lambda, 1- \lambda)</math>, where <math display="inline">\lambda</math> is a small positive number. The other features of this world are the same described in the previous paragraph. His calculations show that, to replicate trust, the maximum penalty should be: <math display="block">q(0) > (1-\lambda)/\lambda,</math> where <math display="inline">q(0)</math> is the maximum penalty and <math display="inline">\lambda</math> is the small positive number described above.


= Trusting an institution =

Meylahn et al. (2023) study the dynamics of the trust between individuals and institutions using a stylized model of social network learning. Firstly, the authors define a model to describe the relationship between only one individual and the institution, in which the agent has repeated opportunities to place trust. The institution’s behavior is modeled by a parameter <math display="inline">\theta</math> that represents its trustworthiness, i.e. the probability that the institution honors the trust placed by the individual. So, in each round the institution honors the trust that has been placed by the agent with probability <math display="inline">\theta</math> and abuses it with probability <math display="inline">1 - \theta</math>. Similarly, the agent, in each round, can decide whether or not to place trust in the institution. The decisions taken by the two are independent in each round and the agent observes the actions of the institution only when he places trust. If trust is honored, he gains ''r'', while if it is abused he loses <math display="inline">-c</math>. Therefore, his expected utility is <math display="inline">r\theta - c(1-\theta)</math>. The agent behaves with myopic rationality, so he maximizes the expected utility in each round without taking into consideration future rounds. Moreover, the agent starts the interaction with the institution having a prior belief ''P0'', which is a function of <math display="inline">\alpha</math> and <math display="inline">\beta</math>, which can be considered the number of times trust was honored and betrayed in a past setting, before the beginning of the experiment. The variables of interest are <math display="inline">\tau</math>, the number of rounds after which the agent decides not to place trust anymore, through which determining the probability of quitting, and ''q'', the expected time spent playing before quitting. In each round, the agent updates his knowledge by taking into consideration the actions taken by the institution and, therefore, he updates its estimation of <math display="inline">\theta</math>. If the agent quits, he will never trust the institution again, given that there is no possibility to update his estimation of the trustworthiness of the institution.

Then, the authors define another model where another agent is added: the relationship between the two plays an important role in determining the relationship with the institution. The agents’ behavior and the institution’s behavior share the same characteristics as the model with one agent: the agents choose in each round whether to place trust or not, they have a prior belief and the institution decide whether to honor or betray the agents’ trust. The authors further assume that both agents share the same prior. The key feature of this model is that each agent, in each round, receives information from the other, through which he can update his information. Two cases are analyzed:

* Agents fully communicate with each other the interactions they have with the institution. Given that the agents have the same prior and the same information available, they will have the same estimate of <math display="inline">\theta</math>.
* Agents do not communicate explicitly, but they only observe the actions of the other agent. Therefore, the information received from the other agent will be incorporated only a round later.

They run their model 4000 times for the single agent model and 2000 times for the dual agents model, for a maximum of 500 rounds. They find that the probability of quitting in most of the settings (i.e. in various calibrations of the parameters) is higher in the single agent model. When considering only the two agents model, the probability is higher when the agents can only observe the actions of the other but they are not able to fully communicate. However, there are some exceptions and in some simulations the observable actions setting outperforms the full communication model, thus having a lower probability of quitting. The expected time to quit is lower in the two agents model with respect to the case where there is one agent only, in particular in the model when they fully communicate (in which therefore they receive more information). This is due to the fact that having more information will make their estimations more precise: they either quit quickly or they do not, since they need less time to have a good estimation of <math display="inline">\theta</math> and, if the estimation is not high enough, they will quit after fewer round, otherwise they are likely to place trust indefinitely.

Overall, the authors find that communication is always helpful since it increases the probability of continuing to trust a reliable institution and decreases the expected time of quitting an untrustworthy institution. Moreover, they find that more optimistic priors increase the possibility of trusting a trustworthy institution. Finally, they highlight that it is not possible to say which of the two agents model is better, since it depends on the parameters setting and which criterion taking into consideration.


= Trust and the blockchain =

As highlighted before, trust, with its dynamics, is fundamental in every aspect of a society and it is what permits societies in themselves to evolve and transform. Without trust, each individual would have the burden of verifying the reliability of every other agent he encounters, which would be impossible. Trust is also what permitted the birth of modern finance, with the Buttonwood agreements of 1792 that led to the creation of the stock market. In recent years, however, trust within modern societies is decreasing, putting at risk the way the society in itself operates. People not only do not trust each other anymore, but they also do not trust the government, or the media, or any other authority that once was considered credible and reliable. It is in this framework that ''“a new architecture of trust”'' was developed, leading to the birth of bitcoin and the blockchain technology in 2009. Werbach (2018) analyzes the relationship between trust and the blockchain in his book ''“The blockchain and the new architecture of trust”''.


== What are the blockchain and Bitcoin ==

The blockchain is a distributed and decentralized digital ledger (i.e. a record of accounts) that records transactions across a network of computers in a secure, transparent, and tamper-proof manner. In a blockchain, transactions are grouped into blocks, which are linked together in a chronological and linear order, forming a chain of blocks. Each block contains a list of transactions, a timestamp, and a reference to the previous block in the chain, creating a verifiable record of all transactions that have ever occurred on the network. One of the key features of a blockchain is its consensus mechanism, which ensures that all participants in the network agree on the state of the ledger. Once a block is added to the blockchain, it is considered immutable, meaning that the data in the block cannot be altered or deleted without the consensus of the majority of the network. This makes blockchains secure and resistant to tampering or manipulation. The transactions registered on the blockchain are performed through smart contracts, which are pieces of code that execute a predetermined function, like transferring a bitcoin, with no possibility to alter the agreement. Finally, a cryptocurrency is a digital currency that runs on the blockchain network.

Bitcoin, introduced by Nakamoto (2009), was the first digital currency and the first example of the blockchain. It relies on 3 elements: cryptography, digital cash and distributed systems. Cryptography can be considered as the science of secure communications and it is employed for this purpose in the blockchain technology. Each agent that interacts with Bitcoin is identified with a private key associated with a public key through the mechanism of cryptography, so that each transaction can be verified and associated with a user without the need to disclose his private key. What is called coin is in reality a chain of signatures of verified transactions. Bitcoin comes from the unspent output of previous transactions, all register on the blockchain. Each transaction is verified by a network of nodes (i.e. a participant in a distributed network that maintains a copy of the blockchain ledger and participates in the consensus process). All the agents need to trust the state of the ledger: this is achieved by the consensus mechanism. Consensus comes from a process called mining, in which agents compete to verify the transactions and create a new block of the blockchain, in exchange for a reward (transaction fees and newly mined bitcoins). The winner is randomly decided, but all the other agents verify independently that the new block is legitimate. Being untrustworthy is not profitable: mining is an expensive activity, because miners engage in a proof of work system, where they have to solve a cryptographic puzzle to have the right to validate the transaction. This requires energy and money and the more energy and money an agent put into mining, the more chances he will have to win. The benefits of cheating are much lower than the costs, so in this way each agent can trust the state of the ledger because there are no incentives to deviate. Finally, the consensus mechanism has also the objective to make the ledger immutable because each transaction is recorded from the hash of the previous block. Changing a past block would mean forking the chain, and this would be rejected by the majority of users. Only in the case that an agent has more than 50% of the computing power (which is almost impossible) this change would be viable.


== A new form of trust ==

The innovation of the blockchain is connected to the fact that every participant can trust the information recorded on the ledger without necessarily trusting another agent to validate it. There is no need for a central authority to validate the transactions and trust is reinforced by the fact that there are mechanisms that make impossible to alter the transactions already recorded on the ledger. The idea of Satoshi Nakamoto was to design a system that, through incentives, made the needs and objectives of every participant aligned with each other, so that what is recorded on the ledger can be trusted without trusting (or knowing) the other agents. Nakamoto claimed to have eliminated the need for trust but, according to Werbach (2018), that would be impossible. What Nakamoto created is trust in ''“a new architecture of trust”'', where independent agents run this technology, validating the transactions so that they can be recorded on the ledger. This is reinforced by the fact that distributed ledger networks make people work together in a way that otherwise would not have been possible since they would not have trusted each other sufficiently.

To better understand what he means by a ''“new architecture”'', the author firstly outlines the various architectures (which define as "the ways the components of a system interact with one another" (Werbach, 2018, p. 25) ) of trust that humans have developed over time. The main architectures are:

* Peer to peer (P2P): here, trust is based on a face to face relationship that arises because the agents share ethical norms and mutual commitment. The downside of this architecture is that this is possible for only a few people and small communities, given that the knowledge of each other is pivotal in creating trust.
* Leviathan: this vision starts from the belief that humans are not fully trustable and therefore a powerful third party, the state/government, is needed to enforce private contracts and property rights. This is achieved through the monopoly of violence held by the state: people can now trust each other because, if something goes wrong, the leviathan can punish the guilty and enforce previous commitments.
* Intermediaries: transactions are guaranteed by a third party (different from the government), which is trusted to perform certain actions. They create the possibility to perform certain transactions that in a peer to peer network would have been difficult: the other agent is trusted because there is an intermediary that makes the transaction happen. Examples are e-commerce platforms such as Amazon, or financial services companies.

The new architecture of trust created by the blockchain is defined as a “trustless trust”. Without trust it would fail since no engagement between individuals is possible without a form of trust, but if it relied on old trust structures it would not be a revolution and would fail its primary object. On the blockchain network, no agent is assumed to be trustworthy, but the output of the network is. Generally speaking, in every transaction, the counterpart, the intermediary and the dispute resolution mechanism must be trusted, but the blockchain substitutes these elements with code. There is no possibility to assess the other party’s trustworthiness, since all agents are represented by private\public keys in the network which allow for their anonymity; there is no central intermediary, since the platform is a distributed machine operated by all the participants; the disputes are solved through pieces of codes called smart contracts, that perform a certain action with no possibility to stop them. Transactions are verified through cryptographic proofs that other agents can verify mathematically. Therefore, it is not possible to frame this system within the common architectures: it is not a P2P since the other parties are unknown, there is no central authority and also there is no central intermediary since the platform is operated in a decentralized way. Each agent needs to trust the network and not each agent with whom he is engaging in a transaction. The blockchain (and Bitcoin) seems the perfect solution for the lack of trust in the modern society and for the problems that the previous architectures of trust presented. The fact that Bitcoin was born after the Great Financial Crisis is not random. P2P relationships were not sufficient in a world so deeply interconnected, intermediaries were considered the cause of the crisis itself and the Leviathan, i.e. the government, was not able to foresee the crisis and prevent it.

Blockchain trust relies also on the immutability of the information recorded, through the mechanisms beforehand explained. However, immutability must be understood in a probabilistic way. The more blocks are added, the more the previous transactions will be immutable because it would require an infinite amount of power to alter the transactions. Each agent can decide after how long they trust the state of the ledger. Therefore, blockchain trust is not instantaneous. Moreover, the transparency of the ledger, meaning that the record of every transaction is publicly available and the software itself through which the blockchain operates is open source, is an important characteristic that increases trust. Finally, blockchain’s trust is algorithmic, meaning that it relies on algorithms to maintain the system: what must be trusted are not the people operating on it, but the software and the math behind the consensus process.

Satoshi’s error was to believe that in his architecture trust was absent, while in reality it reduced the need of trusting some part of the system. Trust is needed and the blockchain could not function without it. Firstly, engaging in a transaction in a system without central control and with immutability means that no one is able to oversee the transaction and amend it if something is wrong. Agents can be confident that the transaction will be correctly registered, but a distributed ledger will not be able to verify if the content is legitimate and, if something is wrong with the transaction itself, there is no possibility to reverse it: smart contracts are unstoppable. Moreover, humans are not out of the system entirely, which means that errors and misunderstandings can occur. And the cryptographic techniques are still vulnerable to attacks: they may be difficult to perform, but users that engage with a blockchain need to trust that this will not happen.

The author argues that the success of the blockchain as an architecture of trust will depend on its governance. The blockchain is a way to enforce some rules, but it is also a product of some rules designed by humans, which therefore would need a governance to continue to operate and to decide the next rules of the game. Moreover, the law should regulate the blockchain framework: without legal rules, the blockchain could be used as an instrument by criminals and terrorists (for recycling money for example), and this would reduce the trust that normal people put in this system. Crypto enthusiasts argue that the role of law would be replaced by smart contracts, but codes cannot fully formulate human intentions, which are an important part behind private contracts, and this could create misunderstandings between the parties. The law can intervene where smart contracts are not able to. Finally, also regulation can play an important role in developing the future of the blockchain and fostering its trustworthiness, as it does with other financial instruments and institutions.


== Trust and blockchain in practice ==

Some scholars have started to think about how trust between users can be enhanced in real blockchain applications. You et al. (2022) find the main challenge to be the fact that there is no consensus about how to measure trust in the blockchain environment. Therefore, they develop a framework to do that, creating a system based on subjective ratings of trustworthiness. The authors start by identifying six different blockchain applications, considering which factors can be used to measure trust in each specific domain. Identifying the key factors behind trustworthiness is essential for creating a system to enhance trust. In particular:

* Supply chain: it is possible to measure how trustworthy the supplier is by the average order arrival time and the defect rate, and how trustworthy is the buyer by the number of days for payment.
* Healthcare industry: to assess the trustworthiness of these firms, regulatory compliance proof, claim approval rate and drug prescription regularity can be the starting point.
* E-commerce: to assess the trustworthiness of those firms, the accuracy of ratings provided by the users and the security of payments represent the most important features.
* IoT devices: system security data and the reliability of the data provided by these devices are the most important features.
* Finance: pivotal factors are the security of transactions and data and the efficiency and quality of communications.
* Social media: news and reputation credit represent the most important characteristics to assess trustworthiness.

The problem of the blockchain is that, although the information recorded cannot be modified easily, the data may not always be true: the need for accountability arises because of this fact.

The system presented by the authors is based on trust scores given by agents that interact with other agents on the blockchain applications. Initially, there would be no score, since no transaction has occurred yet. Then, the two parties start to interact and they begin to collect trust factors about each other. The specific factors, which are described above, will depend on which application is under consideration. Then, each actor will give his score, which will be recorded on the blockchain and will be available for other users, who are now better informed regarding the other users of the blockchain application and can decide to interact with them or not. The validity of the scores will be ensured by the fact that each user will have followed the KYC validation procedures before interacting on the application and it will possible to identify the particular participant from the outside through verifiable credentials. Therefore, no rating will be anonymous.

This system may increase the trust between users because they are incentivized to adhere to the common organizational norms of each sector, because otherwise they would damage their reputation by having a low score permanently recorded on the blockchain. Therefore, this model may create a set of incentives to align the two sides of each transaction.


= Trust games in the blockchain =

As explained before, the blockchain system employs game theory and incentives to make the agents act honestly on the network. After the work of Satoshi Nakamoto, several papers were developed to study the incentive structures and the games behind the blockchain and its consensus mechanism.


== ==

Breiki (2022) studies how trust among players evolves over time when they perform trust evolution games. To do that, he defines the features of its abstract game. Firstly, the author identifies the parameters of the model: there are various miners and each of them has the possibility to cooperate (acting honestly) or defect (cheating); there is a vector of probabilities that defines the likelihood of each player to succeed in solving the puzzle, which is proportional to their computational power; there are the costs and rewards of mining, taking into account also the propagation delay (i.e. the time needed to validate a transaction); there is the market value. Moreover, the author uses two learning algorithms: fictitious play, where prior beliefs are defined; satisficing learning, where aspiration level of payoff and learning rates are defined. All in all, the author finds that the players learn to cooperate in the game to get a better payoff and, for satisfice players, lower learning rates increase the final payoffs.


== ==

J. Zhang and Wu (2021) study evolutionary game theory applied to the blockchain network to understand the strategies and incentives of the participants and their cooperative behavior. The authors explain that the blockchain is a perfect environment for evolutionary game theory because:

* There is information symmetry, since all individuals have and share the same information on the network and each participant has complete transactions data.
* All the participants are equal so no party has a dominant advantage when the game begins.
* Participants are prone to trust each other and engage in the game because of the cryptographic mechanisms, which make the environment credible and immutable.
* The process of adding new blocks can be seen as a form of repeated games.

Agents have bounded rationality, since they cannot get global information because the network is complex, and therefore they are not fully able to maximize their payoffs. Each participant can have two possible behaviors, cooperation or defection, and they update their strategy considering the maximum payoff. Indeed, during the generation of new blocks, each agent is able to learn from his actions and the actions of the winners.

The model developed comprises two groups of miners: group A, with the inclination for cooperation, and group B, inclined to cheating. Participating in each group has a cost <math display="inline">C_a</math> and <math display="inline">C_b</math> and each game brings a revenue R, which will be rewarded to the participants. Each group will have different benefits (for group A, transactions fees and mining rewards, for group B illegal revenue). Finally, there are also punitive measures, denominated P.

Each player, in both groups, can decide which strategy to adopt. Their payoffs are: <math display="block">\begin{aligned}
E_{h,a} &= y(K_a + \lambda R - C_a) + (1-y)(K_a - C_{a}) \\
E_{m,a} &= y(K_a - C_a - P) + (1-y)(K_a - P) \\
E_{h,b} &= x(K_b + (1-\lambda)R - C_{b}) + (1-x)(K_b - C_b - P) \\
E_{m,b} &= x(K_b - C_{b}) + (1-x)(K_b - P)
\end{aligned}</math> Where y and x are the probabilities of winning the game, ''h'' means the honest strategy, ''m'' means the dishonest strategy and <math display="inline">\lambda</math> is the portion of R won.

The authors assume that, at the beginning, each participant has a decided strategy to start with. The goal is to examine the change in the population of honest and dishonest agents after several rounds, taking also into account changes in the parameters of the game. The authors argue that, unlike classic evolution games, the relationship among agents in the blockchain is random. Moreover, the size of the network matters: with small networks, the emergence of cooperative behaviors is easier. Finally, a definition of the evolutionary stable strategy (ESS) is given. The ESS is ''“a strategy that other strategies cannot invade”'' (J. Zhang & Wu, 2021, p. 5).

The authors run various simulation. Firstly, 67% of group A are honest agents and 20% of group B are betrayers. Here, the honest strategy is an ESS, since the number of betrayers tends to 0 as the rounds increase. However, as the expected payoff of group B augments, the honest strategy is still an ESS but weak, because higher payoffs with the dishonest strategy tend to tempt agents to cheat. Therefore, future revenue expectations influence the behavior of participants in the blockchain. Then, the influence of the network structure is analyzed within the group A population. They use a Watts-Strogatz (WS) small-world model and a Barabasi-Albert (BA) scale-free network model. The authors find that it takes quite a lots of rounds for honest agents to establish a trusting cooperative relationship and this may represent an opportunity for cheating agents in the blockchain. Therefore, security is a relevant topic especially in the initial stage of the blockchain.


== ==

L. Zhang and Tian (2023) develop a Byzantine consensus protocol (i.e. a consensus protocol where there are faulty and malicious agents) on the blockchain, shaping it as a dynamic game. Their main contribution to the current literature relies on the fact that the agents in their model have bounded rationality and can learn from the historical observations. In particular, this means that the participants can choose among a limited set of strategies (honest or dishonest), they are able to learn from historical observations and they choose their strategy accordingly, taking also into account the current state of information, but they are not able to forecast the future. Moreover, they are allowed to have inconsistent subjective beliefs about the probability of meeting agents with their same strategy: each agent believes that, for a portion ''m'' of rounds, he will meet a proposer with the same strategy, ranging from ''m''=1 (meaning that he and the proposer will always have the same strategy in every round) to ''m''=0 (meaning that he and the proposer will have the same strategy only by chance). Their model, based on a BFT (Byzantine Fault Tolerance) consensus protocol, consists of the following features:

* The agents, before the game, are selected to form different parallel committees, which compete in a ''n'' round mining game and which will not change until the end of the game.
* Each agent has one vote in the mining game.
* In each round, an agent is randomly selected to make a proposal about the validity of a block. The other agents become validators and they vote if the block in the proposal is valid. The block will be validated if the number of votes is higher than ''v'', a majority threshold.
* There is a reward R for validating the block, a cost <math display="inline">c_{check}</math> for verifying a transaction, a cost <math display="inline">c_{sent}</math> for voting for a transaction and a penalty ''k'' that validators encounter if they misbehave.

Before each round, each validator checks if their congeners (the other nodes in the network) have pivotality, i.e. the ability to control the consensus outcome because they have the majority. As specified before, the participants have two strategies: the honest strategy, where miners achieve the consensus protocol and the Byzantine (dishonest) strategy, where miners damage the consensus protocol. The authors assume that participants are fixed and no one would quit. Initially, a number <math display="inline">x_1</math> of miners choose the honest strategy in the first turn.

The authors specify the concept of stable equilibrium, which can be defined as the situation when <math display="inline">x_t=x_{t-1}</math>, so when the portion of miners which perform an honest strategy remains stable (so the number of agents that changes their strategy from honest to dishonest is equal to the number of agents that changes from dishonest to honest). There are 3 possible stable equilibria:

* The honest stable equilibrium, where no agent is cheating, so <math display="inline">x_t=x_{t-1}=1</math>.
* The Byzantine stable equilibrium, where all agents are cheating, so <math display="inline">x_t=x_{t-1}=0</math>.
* The pooling stable equilibrium, where both strategies exist, so <math display="inline">x_t=x_{t-1} \in (0,1)</math>.

Each equilibrium can be reached depending on the number of initial cheater/honest miners, their belief ''m'', the cost-reward mechanism and the pivotality rate (i.e. the minimum percentage of nodes that must agree to reach consensus and add a block). They find that only the honest stable equilibrium can support the safety, the liveness (so the fact that all non faulty agents should have output) and the validity (the fact that all participants have the same valid output) of the blockchain. Moreover, they find that if the reward-punishment increases, the blockchain will become safer and the honest stable equilibrium will be easier to achieve, while if the cost-punishment ratio increases, the safety and the liveness of the ledger are threatened and the honest equilibrium is more difficult to achieve. Finally, if the pivotality rate increases, every stable equilibrium is harder to achieve.


= Trust in algorithms =

Algorithms are becoming more and more important in everyday life, from health care to criminal justice systems, contributing to decision making processes in many fields. Therefore, the natural question of whether it is possible to trust algorithms arises.

According to Spiegelhalter (2020), the trustworthiness of an algorithm comes from the claims made about the system, including how its developers describe how the system works and what it can do, as well as the claims made by the system itself, which refer to the algorithm’s responses and output regarding specific cases. Therefore, he proposes a model to assess and boost trustworthiness in algorithms.

Regarding the first kind of claims, developers should clearly state what the benefits and drawbacks of using their algorithms are. To assess that, the author proposes an evaluation structure of 4 phases:

* Digital testing: the algorithm accuracy should be tested on digital datasets.
* Laboratory testing: the algorithm’s results should be compared with human experts in their field. An independent committee should evaluate which response is better.
* Field testing: the system should be tested on field, to decide whether it does more harm or good, considering also the effects that it can have on the overall population.
* Routine use: if the algorithm passes the 3 previous phases, it should be monitored continuously, in order to solve problems that can eventually arise.

Having explicit positive evidence in all these phases would boost the trustworthiness of the claims made about the system by developers.

Considering the second type of claims, to reach a higher degree of trustworthiness, it is necessary that the algorithm specifies the chain of reasoning behind its claims, which are the most important factors that led to its output and which is the uncertainty around the claim. Moreover, also a counterfactual analysis should be performed (i.e. what would be the output if the input changed). Overall, algorithms should be made clearer and more explainable and transparency can play an important role in that. To increase trustworthiness, an algorithm should be accessible and intelligible by people, it should be useable, so have an effective utility, and it should be assessable, so the process behind every claim should be available. Ultimately, it should show how it works. And more importantly, it should also clearly state its own limitations, so that trustworthiness does not become blind trust.


= Conclusion =

Trust plays a pivotal role in ensuring the existence and development of modern societies. This paper provides a comprehensive summary of the current literature on how trust relates to the world of economics and finance. To begin with, the concept of trust is clearly defined and differentiated from other human sentiments such as cooperation and confidence. The notion of risk is also discussed, as a complete assessment of the other party’s actions is antithetical to a trust relationship. Furthermore, various methods for measuring trust are explored, with trust games being the most commonly used tool by researchers. The paper goes on to explain how trust is a source of comparative advantage, which determines trade patterns. Additionally, trust is linked to stock market participation, with more trusting individuals being more likely to invest in risky assets and, conditional on participating, they allocate a larger portion of their wealth. While complete trust among individuals would potentially be beneficial, it is not possible in the real world. However, money can act as a substitute that replicates the allocations of a trustworthy economy. The paper also emphasizes the importance of trusting institutions, especially in a world where trust is lacking: effective communication among individuals is critical in assessing the true trustworthiness of an institution. The paper then delves into the relationship between trust and blockchain technology, which can be seen as a new architecture of trust. Moreover, various authors have developed blockchain trust games to better understand the best consensus mechanism process. Finally, the importance of trusting algorithms is highlighted, given their widespread use in everyday technology, healthcare, and the justice system.

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The role of trust in economics and finance

2023-05-12T10:04:44Z

3122188: Created page with "{{DISPLAYTITLE:The role of trust in economics and finance}} Contribution of SIMONE GOZZINI =1. Introduction= Trust is a fundamental sentiment, the binding force behind..."

{{DISPLAYTITLE:The role of trust in economics and finance}}

Contribution of [[SIMONE GOZZINI]]

=1. Introduction=

Trust is a fundamental sentiment, the binding force behind modern societies: without it, no progress would have been possible. Trust is what permitted the birth of modern finance with the Buttonwood agreement of 1792, it is what makes people able to rely on another person or organization without the continuous need to assess what the other party is doing and it is ultimately what permits the existence of modern democ- racies (Warren, 2018). Without it, no meaningful relationship would be possible. However, it the last few years, a general disbelief about trust is permeating the civil society. Edelman is a global communication firm that every year makes a comprehen- sive survey about trust. For 2022, the results picture a general sentiment of distrust across all segments of the population: 60% of the interviewed say that their default tendency is to distrust others, media are seen as a divisive and untrustworthy institu- tion by around 50% of the people and trust in government is significantly dropping, year after year. The problem is particularly accentuated regarding governments, which are seen unable to fix societies’ problems (Edelman, 2022).
Distrust affects modern societies as a whole, impacting not only social relationships and the economy, but also human health: for example, lower trust in government has lead to lower vaccinations against COVID-19, threatening the society as a whole (Bajos et al., 2022).

This paper highlights the importance of trust in modern economies and in the finan- cial world.
Section 2 describes the concept of trust, differentiating it from other human sentiments like cooperation and confidence. In general, the concept of risk is concerned, given that trust involves a sort of faith in someone or something. Section 3 describes the various methodologies used in the literature to measure trust: trust games, surveys and the frontier of neuroscience. Section 4 presents trust as a source of comparative advantage in world trade patterns: societies with more trust have bigger and more productive firms. Section 5 studies how trust affects stock market participation: peo- ple with an higher tendency to trust are more likely to participate in the stock market and, conditional on participating, they invest an higher fraction of their wealth. Sec- tion 6 describes a general equilibrium model where money are seen as a substitute of trust: the allocation of resources in a trustworthy society can be reached also in a trust-less society which employs money. Section 7 describes a stylized model of trust between individuals and an institution: the exchange of information among individ- uals is found to be a tool that improves the assessment of the true trustworthiness of an institution. Section 8 presents the blockchain technology as a new architecture of trust, describing also how trust can be enhanced to reach an higher diffusion and application of this technology. Section 9 presents various paper regarding trust games in the blockchain technology, considering in particular how to reach and improve the consensus process. Section 10 describes how algorithms, which are becoming more and more import in modern life, can be trusted: in particular, the author highlights transparency and accessibility as fundamental characteristics to enhance trust. Sec- tion 11 concludes.

=2. The concept of trust=

According to the definition of Gambetta (2000), trust is “a particular level of the subjective probability with which an agent assesses that another agent or group of agents will perform a particular action, both before he can monitor such action (or independently of his capacity ever to be able to monitor it) and in a context in which it affects his own action” (Gambetta, 2000, p. 5)
This definition highlights important concepts:
* Trust is a probability p, a threshold, but subjective: people engage in a trust relationship if they believe that the probability that the person will perform the particular action mentioned in the definition is higher than a certain level, which depends on the individual predisposition to trust and the circumstances under which the relationship is being created, like the cost of misplacing trust.
* Trust is related to uncertainty: the underlying assumption is that the agent is not able to fully monitor the other agent while he performs the particular action (this is usually the case in practice), otherwise trust will not be necessary since the first could keep track of the second while performing the action.
* It has an impact on the trustor, otherwise the actions of the second agent would not matter to the first and engaging in a relationship would not be necessary.

Trust is relevant when the other agents (trustees) are free to betray the trustor, otherwise, if coercion intervenes, the outcome of the trust relationships is known ex ante. Assessing a probability would not be needed anymore and uncertainty would not be involved: the resulting interaction would not be a trust relationship, given that it lacks of its fundamental characteristics. Furthermore, also the trustor needs to be free to choose whether to engage in this relationship or to escape from it, otherwise also in this case assessing p would not be needed since he would have no choice.
Therefore, trust is fundamentally a free choice between two individuals who seek mutual benefits and it involves a level of risk for the trustor, given that it affects his personal sphere of action.

==2.1 Cooperation and trust==

Trust, however, must not be confused with other human actions, sentiments, or beliefs. Trust is different from friendship, from passion, from loyalty and from cooperation. In particular, the relationship between the latter and trust is investigated extensively by the author.

Trust can be seen as:
* a precondition for cooperation, which, together with a sane competition is ben- eficial to foster human progress. However, although being probably the most efficient way to achieve cooperation, it is not a necessary condition: people have used surrogates in history to overcome the problem of the lack of trust, like coercion, contracts and promises. All of them have the objective of diminishing the possible alternatives that the trustor and the trustee can face, thus reducing the risk for both parties in engaging in this relationship. A higher level of trust increases the probability of cooperating, but it is possible that, even though the level of p is low, the result is cooperation anyway. This is because an agent takes also into consideration the cost and the benefit of engaging (or not) in such a relationship, the other alternatives he has and the specific situation.
* an outcome of cooperation. Societies may engage in cooperation thanks to “a set of fortunate practices”, particular circumstances and the need to satisfy mutual interests, for which the cost of not engaging in cooperation is higher than the risk of engaging. Trust is therefore the outcome of these practices and there is no need of prior beliefs about the trustworthiness of the other party, since trust will arise only after the beginning of the relationship, when information are collected. This statement is reinforced by the fact that cooperation exists in animals, that are unlikely to experience trust.

However, according to the author, there is no reason for saying that cooperation is a spontaneous equilibrium in human interaction: cooperation is just as likely as non cooperation. A predisposition to trust may be rational for humans in order to achieve their objectives, since trust is fundamentally an efficient way to achieve cooperation, but it is not necessary to wait for trust to evolve in order to initiate cooperation. Common interests and constraints can be enough and they can be beneficial especially in underdeveloped countries which present low level of trust. In fact, although trust and trustworthiness can be advantageous for an individual’s purposes, they cannot be artificially induced in a rational person. Moreover, the author argues that rational trustors and trustees may seek and present evidence to trust and being trustworthy. However, more information cannot fully solve the problem of trust. People, once they trust, do not try to find evidences to corroborate their belief, but rather they change their mind only if they find contrary evidence, which is not easy.

==2.2 The concept of trust in organizations==

Mayer et al. (1995) starts from the studies of Gambetta to further develop the un- derstanding of trust in the context of organizations. In particular, they focus on the trust relationship between two individuals: a trustor who trusts or not an individ- ual to perform a particular action and the trustee who receives the trustor’s trust, deciding then if fulfilling or not that action. The flow of trust is unidirectional: mu- tual trust between two parties is not developed in the paper, nor trust in a social system. In particular, according to the authors, the concept of vulnerability is what is missing in the definition of Gambetta, given that “Trust is not taking risk per se, but rather it is a willingness to take risk.” (Mayer et al., 1995, p. 712). Then, trust is differentiated from different constructs like:
* Cooperation, which is intensively studied also by Gambetta. The authors high- light the fact trust is not a conditio sine qua non for cooperation, since it is possible to cooperate with someone not trustworthy (for example when there are external controls and constraints like discussed in the previous section).
* Confidence: the main difference relies in the fact that, with trust, risk must be assumed, while in the second it is not necessary. Moreover, when a person chooses to trust, he will consider a set of possible alternatives, while that is not the case with confidence.
* Predictability: trust and predictability are way to cope with uncertainty but, if a person is predictable it does not necessary mean that it is worth putting trust in him. This is because it is possible to predict that the other person will consistently behave in negative ways (and the uncertainty is reduced), but no rational individual would put trust in him.

Then, the characteristics of the trustor and the trustee are analyzed, which together can initiate a trust relationship between the two agents. The most important feature of the trustor is his propensity to trust another person, which is a personal trait con- stant over time and across situations. It is a general willingness to trust and it is not related to another party, since it is measurable before any interaction with the other agent. However, each trustor has different levels of trust for various trustee, which arise after the relationship is initiated. Therefore, they depend on the characteristics and actions of the trustee, i.e. his trustworthiness. According to the authors, there are 3 main characteristics of the trustee that are able to explain trustworthiness:
* Ability: it is defined as the set of skills and competencies of the trustee over a specific domain. It is possible to trust another person to perform a particular action if the other agent is competent in that field otherwise he should not be trusted, even though he may be committed to complete the task. Therefore, trust should not be intended in absolute terms, but over a specific field of knowledge.
* Benevolence: it is a personal trait of the trustee towards the trustor which is related to how much the former wants good for the latter. More benevolence leads to higher trust because the trustor can be more sure that the trustee will perform the action taking into account also his benefit and not only the trustee’s egoistic motives.
* Integrity: it is defined as “the trustor’s perception that the trustee adheres to a set of principles that the trustor finds acceptable” (Mayer et al., 1995, p. 719). A trustee integrity therefore depends on what the trustor’s set of beliefs are. If the trustor thinks that the integrity of the trustee is not sufficient, he will not engage in a trust relationship with him.

In particular, integrity will be central in the early stages of the relationship, before gaining any insights; then, benevolence will become important over time, as the trustor retrieves information during the course of the relationship; ability, instead, continues to stay important from the beginning to the end. After engaging in the trust relationship, the trustor will be able to gain new data and information, through which he can update his beliefs about these three characteristics of the trustor, eventually deciding whether the placement of trust is still reasonable. While a trustor tries to assess these characteristics of the trustee, the role of context becomes important because it affects ability (for example because a change in a situation may change the skills needed to complete a certain task), the level of benevolence (for example if the trustee changes its behavior during time) and integrity (for example because a certain action of the trustee is not interpreted as coherent with the trustor’s set of values only because it was obliged to do so by the specific situation).

Finally, the authors deal with the risk involved in trusting. In particular, they high-
light the fact that there is no risk taking in the propensity to trust, but risk arises only when an agent effectively engage in a trust relationship. However, the form and the level of risk assumed by the trustor will depend on the level of trust involved in a relationship: the more the trustor trusts the trustee, the more risk he will be willing to take. So, before initiating a trust relationship, the agent has to assess whether the level of trust is higher or lower than the perceived level of risk, so that he can decide whether it makes sense to engage in such relationship.

=3. How to measure trust=

Trust in therefore a fundamental device in human society and it is also important in economics and finance, as this paper will later explain. A natural question arises: how it is possible to measure trust? The question is not easy to answer, since trust is a hu- man sentiment, therefore subjective and emotional, and which is also interwoven with other human sentiments and beliefs. Al ́os-Ferrer and Farolfi (2019) review the major methods used by the literature to measure trust, underlying the main limitations of each model.

==3.1 Trust Games and Game Theory==

Experimental economics has intensively relied on game theory to quantify trust. The games mostly used nowadays are various version of the TRUST GAME, which was invented by Berg et al. (1995). Al ́os-Ferrer and Farolfi (2019) describe it as follows: “A first agent, called the trustor, is given a monetary endowment X, and can choose which fraction p of it (zero being an option) will be sent to the second agent, called the trustee. The transfer p · X is then gone, and there is nothing the trustor can do to ensure a return of any kind. Before the transfer arrives into the trustee’s hands, the transfer is magnified by a factor K > 1. The trustee is free to keep the whole amount without repercussion. Crucially, however, the trustee has the option to send a fraction q of the received transfer back to the trustor, hence honoring the trustor’s initial sacrifice” (Alo ́s-Ferrer & Farolfi, 2019, p. 1). The transfer of the trustor can become a measure of trust, while the subsequent transfer of the trustee a measure of trustworthiness. These games underline some important features of trust as described by Gambetta (2000): trustor and trustee decisions are free and voluntarily, uncertainty and risk are involved and there are possible repercussions for the trustor (a loss in utility).

However, despite the popularity of this method, there are various limitations that need to be addressed. In the agents behavior, there might be possible motivational confounds that affect the measurement of trust and trustworthiness, like selfish or altruistic tendencies, efficiency reasons, or prior personal preferences (like inequity aversion). To address this problem, the authors suggest to take as a measure the difference between the transfers in the trust game and those in a game called the Dictator Game (i.e. a game where the proposer’s decisions are implemented without the possibility for the responder to do something).

Then, the question of risk attitudes of the agents is addressed. Trust involves a certain risk (given that the trustor cannot monitor the response of the trustee) so the aptitudes towards risk may affect the monetary transfers. The evidence is mixed, with early studies (like Houser et al., 2010) finding no relationship between risk attitudes and trust and with others finding a correlation. The lack of agreement might be due to the concept of risk involved in the trust games themselves, which is not a pure financial risk, but a betrayal aversion, i.e. the risk and fear of being betrayed by another human being. Taking into account this, the authors mention the study of Bohnet and Zeckhauser (2004), where a “betrayal aversion” was found in the decisions of the agents, different from the standard risk aversion. Therefore, to disentangle the trust component and the risk component of the agents’ transfers, standard measures of risk might not fit properly. The authors, however, criticize also the use of game variants to address this issue, since new measures may capture other undesired effects.

Another problem can arise when there are changes in the parameters, implementation and description of the trust game: the response of the agents might not be consistent in all context, thus creating an impossibility of comparability between different ex- periments. For instance, increasing the multiplier K will likely increase the trustor’s transfer and also the fraction returned by the trustee according to Lenton and Mosley (2011). Moreover, also the way that the game is framed can have an impact: Burn- ham et al. (2000) show that the responses of the agents involved depend on whether, in the instructions of the game, the other agent was called partner or opponent. In the former case, the trustor trusted more the trustee than in the second case. How- ever, if the game is not framed at all the participants might create their own frame, thus interpreting the play in different and unpredictable ways, conducting to biased results.

==3.2 Surveys==

Another possible measure of trust relies in the use of surveys. The most import example is the General Social Survey (GSS) of the U.S. National Opinion Research Center. The question asked is: “Generally speaking, would you say that most people can be trusted or that you can’t be too careful in dealing with people?”. The possible answers are: “Most people can be trusted” or “Can’t be too careful” or “I don’t know”. This question is used also in other important surveys, like the EVS (European Values Survey), the WVS (World values Survey), the BHPS (British household panel study) and the ANES (American National Election studies).

This method is not immune from problems. For example, the interpretation of each individual might play a role in the response, as seen for the Trust Game. Moreover, the relationship between these two methods must be taken into account. Ideally, if both were valid and consistent, the responses should be highly correlated. However, the evidence is mixed. Glaeser et al. (2000) find no correlation between the two measures while Fehr et al. (2002) find evidences of the contrary. An explanation could be that surveys test a general propensity to trust, while Trust Games measure a specific strategic situation of the agents’ behavior. The concept of trust is therefore not uniquely determined and different methodologies might capture different aspects of this complex human attitude.

Moreover, the authors suggest that, if surveys are used as a measure, one must take into account various controls (like culture, geography and age) to interpret and there- fore compare the responses.

==3.3 Neuroscience==

The new frontier in the measurement of trust is represented by neuroscience, which tries to give more objective and biological methods.

Firstly, the relationship between of oxytocin (OT) and trust is investigated, in par- ticular to link OT levels with the behavior in the Trust Game. Zak et al. (2005) find that OT levels can predict trustee trustworthiness but not trustors’ transfers. However, when the change in OT levels in endogenous (i.e. natural, like in the paper mentioned above), the studies cannot establish causality. Hence, another set of stud- ies, where the level of OT was exogenously determined, is examined. Kosfeld et al. (2005) find that the treatment group in their experiment (i.e. the people who OT was administered) presents larger trustors’ transfers compared to the control group, but no significant differences in the trustees’ transfers. Moreover, their results suggest that OT causally increases trust through a reduction of betrayal aversion and it does not increase risk-taking behavior or prosocial aptitudes in general. The two methods of investigation leads therefore to inconsistent result with each other. Therefore, no conclusion can be reached: the relationship of OT with trust and trustworthiness is not simple as previously thought.

Finally, the authors introduce the latest study about the use of brain imaging to understand where trust come from and how it forms. This might be useful to develop more reliable measures of trust in the future.

=4. Trust as a source of comparative advantage=

Cingano and Pinotti (2016) study the effect of trust on firm organization and on comparative advantage. The authors argue that interpersonal trust means more dele- gation of decisions within a firm, resulting in a larger firm size and in the expansion of more productive units. If trust is established, it is possible to expand the firm outside familiar and friendly relationships, thus using the firm own productivity advantage over a larger amount of input, given that the firm is bigger and has more factors of production. The principal-agent problem (that comes with delegation and prevent higher level of it) can be partially solved by this human device. In particular, higher delegation causes higher productivity through:
* higher exploitation of the informational advantage of the managers and of spe- cific skills of some workers.
* the reduction of information costs.
* more resiliency and ability to cope with changes in profit and growth opportu-
nities.

Studying a sample of Italian and European companies, the authors find that trust, together with human capital and intangible intensity, is associated with greater dele- gation, which, in turn, is associated with larger firm size. Their findings suggest that high-trust countries present an higher value added per worker and higher exports in industry where delegation is needed, thus making trust a source of comparative advantage in trade patterns. This effect is the result of a reduction of smaller size firms towards bigger size firms.

The authors test their hypotheses through an empirical data obtained with surveys. They retrieve data from:
* The INVIND survey from the Bank of Italy, which provides information about inputs, outputs, internal organization and governance of a sample of more than 6500 firms. These data are used to test trust differences across Italian regions.
* The World Values Survey (WVS) and the European Social Survey (ESS) to measure interpersonal trust and delegation.
* The OECD Structural Analysis Database (STAN) and the OECD Business De- mographic Statistics, which provide information about value added per worker, organization and number of workers of European firms.

The analysis starts from the following regression:
<math>Y_{jr} = \alpha + \beta(Trust_r \times Delegation_j) + \delta X_{jr} + \mu_r + \mu_j + \epsilon_{jr}</math>

Where $Y_{jr}$ is industry specialization (measured through value added per worker or exports), $Trust_r$ is the average level of trust, $Delegation_j$ is a measure of the need of delegation in each industry and $X_{jr}$, $\mu_r$ and $\mu_j$ are controls respectively for other determinants of specialization and geographical factors.

Then, the authors estimate $Delegation_j$ through the following regression:

<math>Centers_{jr} = \eta + \theta \ln L_{ijr} + f_j + f_r + v_{ijr}</math>

Where $Centers_{jr}$ is the number of responsibility centers (which is a measure of delegation inside firms), $lnL_{ijr}$ is the log of the number of workers (which is kept fixed) and $\mathit{f_j}$ and $\mathit{f_r}$ are firm's controls.

In particular, the analysis show that, for the Italian sample, higher trust leads to an increase in the production of delegation intensive industries. Starting with the log of value added per worker as the dependent variable, the authors add a series of controls. Introducing human capital, the calculations show that it remain the main source of the pattern of specialization but, despite being correlated with delegation (which in turn has an effect on trust), the latter variable remain statistically significant. Then, two other controls are introduced: financial development and judicial quality. How- ever, they do not affect the coefficient of trust, thus making the estimation more robust and consistent. The results are similar when the dependent variable is export. For the international sample, the analysis is more complicated because different coun- tries present different institutional dimensions, like labor market regulations and prop- erty protections. The results, however, are very similar, making their thesis consistent also at the international level.

=5. Trust and the stock market=

Guiso et al. (2008) study the effect of trust on stock market participation across individuals and across countries. Starting from Gambetta (2000), they define trust as “the subjective probability individuals attribute to the possibility of being cheated” (Guiso et al., 2008, p. 2557), which depends on the characteristics of the financial system and the individual priors and predisposition to trust.

Firstly, they develop a theoretical model in which they reproduce the effect of trust on portfolio decisions, starting with a two asset model (one safe asset and one stock). They assume that investors know the distribution of returns but they are worried, with a level of subjective probability p, about other bed events, like the possibility of fraud perpetrated by their broker, which will lead to 0 return in the stock. They also assume 0 participation cost. Given a level W of wealth, being r ̃ the return in the stock investment and rf the risk free rate, each agents choose a share α of their wealth to invest in the risky asset so that they can maximize their expected utility

<math>\max\{(1-p)\text{EU}(\alpha \tilde{r} W +(1-\alpha)r_f W) + p\text{U}((1-\alpha)r_f W)\}</math>

They also calculate that a risk averse individual will invest in the stock market if their subjective probability p \textgreater\ $\bar{p}$, where $\bar{p}$ is $\bar{p}$ = ($\bar{r}$ - $r_f$)/$\bar{r}$ and $\bar{r}$ is the mean of the true distribution of the returns of the stock. This last relationship comes from the fact that an investor invests in a risky asset if the expected return of investing is higher than the risk free rate, i.e. (1 - \textit{p}) $\times$ $\bar{r}$ + \textit{p} $\times$ 0 .

An important result of this model is that the decision to participate or not in the stock market depends on the subjective probability p of being cheated (since it reduces the expected return of the investment) and it does not depend on the level of W. Since W is not significantly correlated with trust (as calculated through the survey data they use in their empirical analysis), this can explain why also the wealthy might not engage in stock trading. Moreover, $\alpha$ depends itself on the level of trust: more trust means more wealth invested in risky assets and vice-versa.

Then, participation costs are introduced in the theoretical model. To enter the mar-
ket, the investor now has to pay a fixed cost f (thus reducing the allocable wealth to W - f ). As f increases, in order to invest in stock, an higher level of trust is nec- essary ($\bar{p}$ decreases). In particular, less trust reduces the return on stock investment (thus making the participation less attractive) because it reduces the share of wealth invested in stock and it reduces the expected utility from participating.

Finally, the authors demonstrate that risk tolerance and trust are two different things by looking at the optimal amount of stocks: this number increases with trust and it increases also with risk aversion (for the benefits of diversification). Therefore, since risk tolerance reduces the optimal number of stocks and the contrary is true for trust, the latter cannot be a proxy of the former. As the empirical analysis will demonstrate, this is consistent with the data. This result is also reinforced by the fact that the authors find that individuals with high level of trust buy more insurance, while risk tolerant individuals buy less.

The authors use survey data to test their model. In particular, they employ the DNB Household Survey (at which they have directly contributed), which maps about 1990 individuals and tries to capture their level of generalized trust, their risk and ambi- guity aversion and their optimism. It also reports some statistics about households’ assets, distinguishing in particular between listed and unlisted stocks and securities held directly or through financial intermediaries. To measure generalized trust, this survey uses the same question of the World Values Survey (see section 3.2 for ex- planation); to measure risk aversion and ambiguity aversion, the authors ask the interviewed their willingness to pay for some lotteries; to measure optimism, they ask to quantify their agreement (on a scale from 1 to 5) with the following statement: “I expect more good things to happen to me than bad things”. Then, the Italian Bank costumers survey is used to capture the personalized trust, i.e. the trust that an individual has towards its financial intermediary, which could be different from the general propensity to trust. This data set contains information about the financial assets the interviewed have and their demographic characteristics. More importantly, to measure personalize trust, the survey ask the following question: “How much do you trust your bank official or broker as financial advisor for your investment deci- sions?”.
The empirical analysis confirms their hypothesis. Starting from the study of the relationship between generalized trust (i.e the level of trust measured in the survey) on stock market participation, the authors find that trust has a positive and highly significant coefficient (so more trust means more participation), even after controlling for a number of variables (like age, sex and wealth). In particular, ”Trusting others increases the probability of direct participation in the stock market by 6.5 percentage points” (Guiso et al., 2008, p. 2578). Risk aversion and ambiguity aversion do not seem significant, as well as optimism, since the coefficient of trust remains unchanged. Moreover, when studying the effect of wealth, the authors find that the coefficient of trust remain significant even after controlling for this variable, thus providing a proof for their previous statement: the lack of trust may be an explanation for the fact that rich people do not invest in stocks even though they should not be affected by the participation costs. Then, the relationship between trust and the amount invested in risky asset is studied. The result confirm, again, the hypothesis: ”Individuals who trust have a 3.4 percentage points higher share in stocks, or about 15.5% of the sample mean” (Guiso et al., 2008, p. 2580). The same results hold for risky assets in general:
risk and ambiguity aversion are not statistically significant also in this case. However, a significant control is represented by the level of education. The authors find that trust increases the holding of risky securities for everyone, but less in more educated people, since they know better how the market works respect to the less educated and they are less affected by priors and cultural stereotypes.

Considering now the Italian Banks costumers survey, the results confirm the previous ones: trust in one’s own financial intermediary increases the probability of investing in stock and the share of the wealth allocated in this type of security.

Finally, the authors investigate the implication of the level of trust on market par- ticipation across countries. The analysis is based on the following statement: less trust should mean that agents are less willing to invest and, in turn, firms will be less willing to float their equity given that it is less rewarding. Therefore, countries with lower levels of trust should have lower participation in the market. The empirical analysis confirms the previous claims: trust has a positive and significant effect on stock ownership among individuals and it has also a positive effect of stock market capitalization.

=6. Money as a substitute of trust=

Gale (1978) develops a theoretical model to study the effect of the introduction of money in an economy characterized by a lack of trust between its agents. The authors starts from the Arrow-Debreu model of Walrasian equilibrium. This model is charac- terized by a finite number of consumers (who have an initial endowment of resources) and commodities, perfect competitions in all markets and constant return to scale.

Moreover, the markets are complete, so it means that all transaction in the economy can be arranged at one time. This is made possible because transactions that involve the delivery of a commodity in a different time period (i.e in t=0 a commodity is sold but the delivery will be arranged in t=1) can be concluded through contracts at time t=0. The contract specifies that the delivery will occur in t=1, even though the transaction itself is completed in t=0. Therefore, the contract is seen as the com- modity being traded. This mechanism operates under the assumption that there is no uncertainty in the market. In such an environment, agents can trust each other to fulfill the contracts they have agreed upon. As a result, there is no need to dis- tinguish between the contracts and their execution. Nevertheless, if for some reasons agents start not to trust each other, and therefore uncertainty arises, some agents may prefer not to fulfill their promise and other agents, anticipating that, might not engage in a transaction in the very first place. If trust were to vanish, therefore, the allocation process would break down if no other substitutes were found. The scholar demonstrates that money can be a substitute of trust and it can permit the allocation and redistribution of resources even in the absence of trust.

To illustrate that formally, the author employs the concept of core that is “the set of attainable allocations such that (a) neither agent can make himself better off by remaining self-sufficient and(b) two agents cannot both be made better off by any feasible redistribution of their joint endowment.” (Gale, 1978, p. 459), through which he develops the concept of sequential core to integrate time periods and uncertainty about the outcome of a contract. An allocation of commodities is trustworthy if the sequential core applies to it, that is if it cannot be improved by any redistribution of resources in any time period. If this were not the case, an agent would have the incentive to break the contract in later periods. Therefore, any exchange of commodities without trust would not form a sequential core, because agents would have the incentive to deviate from equilibrium to increase their own utility.

To resolve this issue, the author introduces money in the model. In particular, each agent is given an endowment of money at time t=0 and it is assumed that at the end of time t=1 (the second and last period) the same amount of money must be returned as a tax. Implicitly, the model introduces a social institution (for example a government) that issues fiat money, which has no intrinsic value but it guaranteed by the imposition of the government itself (this is the case in modern economies). In between the two periods, the agents can exchange money among themselves. This solves the issue: the agents who were reluctant to keep their promises in the model without money and without trust now have an incentive to fulfill the contract, given that they need the money to pay their taxes. Money do not restore trust among agents, but they act as a substitute, a way to enforce previous contracts and agreements. The possibility for the government to directly intervene in the fulfillment of contracts should be discarded, since it is not plausible that a human institution could be so almighty that it can oversee every transaction in a complex economy. Therefore, money can create the conditions for trustworthy transactions (without trust) in a decentralized way. However, the institution must be able to credibly impose the payment of taxes, otherwise agents would face the same problem as before. To do that, penalties for those who do not want to pay taxes should be sufficiently gruesome, but the author does not quantify the penalty. Moreover, the author argues that, despite money can substitute trust, there could be a loss in overall utility respect to the case with trust. In the model, the social institution is introduced without any explicit cost, but this is unlikely to be the case in reality, since introducing a government that is able to enforce tax payment and issue securities is certainly not free.

==6.1 The gruesome penalty==

Grimes (1990) continues the work of Gale (1978) in analyzing the role of money in the same theoretical framework studied by the previous author. The results of Gale are confirmed: without money, the outcome of an economy without trust is autarky, since no transaction can effectively occur. With the introduction of money, however, it is possible to replicate the allocation of the economy with trust. The contribution of his work respect to the research of his predecessor is about a quantification of the grue- some penalty that agents face when they do not respect their tax obligations.

In particular, the author shows that the simple introduction of money does not nec- essarily replicate the outcomes of an economy without trust, because a sufficient incentive (i.e. a penalty higher than a certain threshold) to make inefficient for the agents not to fulfill their promises must be introduced. Under this threshold, the increase in utility derived from reneging the contract his higher than the reduction in utility due to the penalty. Therefore, the optimal choice is not to fulfill the agree- ment. On the contrary, above that threshold, the optimal choice is to fulfill the contracts (therefore replicating the allocation with trust). It is worth noting that the intensity of the penalty has no effect on the final allocation of goods, since they are already Pareto-efficiently allocated, but the author shows that this has an impact on prices.

To calculate the threshold, the author considers a world with two agents, two periods,
no uncertainty and one good in each period. Each agent's endowment in each period is defined as $(1-\lambda, \lambda)$ and $(\lambda, 1- \lambda)$, where $\lambda$ is a small positive number. The other features of this world are the same described in the previous paragraph. His calculations show that, to replicate trust, the maximum penalty should be:

<math>q(0) > \frac{1 - \lambda}{\lambda}</math>

Where $q(0)$ is the maximum penalty and $\lambda$ is the small positive number described above.

=7. Trusting an institution=

Meylahn et al. (2023) study the dynamics of the trust between individuals and institutions using a stylized model of social network learning.
Firstly, the authors define a model to describe the relationship between only one individual and the institution, in which the agent has repeated opportunities to place trust. The institution's behavior is modeled by a parameter $\theta$ that represents its trustworthiness, i.e. the probability that the institution honors the trust placed by the individual. So, in each round the institution honors the trust that has been placed by the agent with probability $\theta$ and abuses it with probability $1 - \theta$. Similarly, the agent, in each round, can decide whether or not placing trust in the institution. The decisions taken by the two are independent in each round and the agent observes the actions of the institution only when he places trust. If trust is honored, he gains r, while if it is abused he looses $-c$. Therefore, its expected utility is $r\theta - c(1-\theta)$.
The agent behave with myopic rationality, so he maximizes the expected utility in each round without taking into consideration the future rounds. Moreover, the agent starts the interaction with the institution having a prior belief P0, which is a function of $\alpha$ and $\beta$, which can be considered the number of times trust was honored and betrayed in a past setting, before the beginning of the experiment.
The variables of interest are $\tau$, the number of rounds after which the agent decide not to place trust anymore, through which determining the probability of quitting, and q, the expected time spent playing before quitting. In each round, the agent updates its knowledge by taking into consideration the actions taken by the institution and, therefore, he updates its estimation of $\theta$. If the agent quits, he will never trust the institution again, given that there is no possibility to update his estimation of the trustworthiness of the institution.

Then, the authors define another model where another agent is added: the relation- ship between the two plays an important role in determining the relationship with the institution. The agents’ behavior and the institution’s behavior share the same characteristics as the model with one agent: the agents choose in each round whether to place trust or not, they have a prior belief and the institution decide whether to honor or betray the agents’ trust. The authors further assume that both agents share the same prior. The key feature of this model is that each agent, in each round, receives information from the other, through which he can update his information. Two cases are analyzed:
* Agent fully communicate with each other the interactions they have with the institution. Given that the agents have the same prior and the same information available, they will have the same estimate of $\theta$.
* Agents do not communicate explicitly, but they only observe the actions of the other agent. Therefore, the information received from the other agent will be incorporated only a round later.

They run their model 4000 times for the single agent model and 2000 times for the dual agent models, for a maximum of 500 rounds.
They find that the probability of quitting in most of the settings (i.e. in various calibrations of the parameters) is higher in the single agent model. When considering only the two agents model, the probability is higher when the agents can only observe the actions of the other but they are not able to fully communicate. However, there are some exceptions and in some simulations the observable actions setting outperforms the full communication model, thus having a lower probability of quitting.
The expected time to quit is lower in the two agents model respect to the case where there is one agent only, in particular in the model when they fully communicate (in which therefore they receive more information). This is due to the fact that having more information will make their estimations more precise: they either quit quickly or they do not, since they need less time to have a good estimation of $\theta$ and, if the estimation is not high enough, they will quit after fewer round, otherwise they are likely to place trust indefinitely.

Overall, the authors find that communication is always helpful since it increases the probability of continuing to trust a reliable institution and decrease the expected time of quitting an untrustworthy institution. Moreover, they find that more optimistic priors increase the possibility of trusting a trustworthy institution. Finally, they highlight that it is not possible to say which of the two agents model is better, since it depends on the parameters setting and which criterion taking into consideration.

=8. Trust and the blockchain=

As highlighted before, trust, with its dynamics, is fundamental in every aspect of a society and it is what permits societies in themselves to evolve and transform. Without trust, each individual would have the burden of verifying the reliably of every other agent he encounters, which would be impossible. Trust is also what permitted the birth of modern finance, with the Buttonwood agreements of 1792 that led to the creation of the stock market. In recent years, however, trust within modern societies is decreasing, putting at risk the way the society in itself operates. People not only do not trust each other anymore, but they also do not trust the government, or the media, or any other authority that once was considered credible and reliable. It is in this framework that “a new architecture of trust” was developed, leading to the birth of bitcoin and the blockchain technology in 2009. Werbach (2018) analyzes the relationship between trust and the blockchain in his book “The blockchain and the new architecture of trust".

==8.1 What are the blockchain and Bitcoin==

The blockchain is a distributed and decentralized digital ledger (i.e. a record of accounts) that records transactions across a network of computers in a secure, trans- parent, and tamper-proof manner. In a blockchain, transactions are grouped into blocks, which are linked together in a chronological and linear order, forming a chain of blocks. Each block contains a list of transactions, a timestamp, and a reference to the previous block in the chain, creating a verifiable record of all transactions that have ever occurred on the network. One of the key features of a blockchain is its consensus mechanism, which ensures that all participants in the network agree on the state of the ledger. Once a block is added to the blockchain, it is consid- ered immutable, meaning that the data in the block cannot be altered or deleted without the consensus of the majority of the network. This makes blockchains se- cure and resistant to tampering or manipulation. The transactions registered on the blockchain are performed through smart contracts, which are pieces of codes that execute a predetermined function, like transferring a bitcoin, with no possibility to alter the agreement. Finally, a cryptocurrency is a digital currency that runs on the blockchain network.

Bitcoin, introduced by Nakamoto (2009), was the first digital currency and the first example of the blockchain. It relies on 3 elements: cryptography, digital cash and distributed systems. Cryptography can be considered as the science of secure commu- nications and it is employed for this purpose in the blockchain technology. Each agent that interacts with Bitcoin is identified with a private key associated with a public key thorough the mechanism of cryptography, so that each transaction can be verified and associated with an user without the need to disclose his private key. What is called coin is in reality a chain of signatures of verified transaction. Bitcoin comes from the unspent output of previous transactions, all register on the blockchain. Each transaction is verified by a a network of nodes (i.e. a participant in a distributed net- work that maintains a copy of the blockchain ledger and participates in the consensus process). All the agents need to trust the state of the ledger: this is achieved by the consensus mechanism. Consensus comes from a process called mining, in which agents compete to verify the transactions and create a new block of the blockchain, in exchange for a reward (transaction fees and newly mined bitcoins). The winner is randomly decided, but all the other agent verify independently that the new block is legitimate. Being untrustworthy is not profitable: mining is an expensive activity, because miners engage in a proof of work system, where they have to solve a crypto- graphic puzzle to have the right to validate the transaction. This requires energy and money and the more energy and money a gent put in mining, the more chances he will have to win. The benefits of cheating are much lower than the costs, so in this way each agent can trust the state of the ledger because there are no incentives to deviate. Finally, the consensus mechanism has also the objective to make the ledger immutable because each transaction is recorded from the hash of the previous block. Changing a past block would mean forking the chain, and this would be rejected by the majority of users. Only in the case that an agent has more than 50% of the computing power (which is almost impossible) this change would be viable.

==8.2 A new form of trust==

The innovation of the blockchain is connected to the fact that every participants can trust the information recorded on the ledger without necessarily trusting another agent to validate it. There is no need for a central authority to validate the trans- actions and trust is reinforced by the fact that there are mechanisms that make im- possible to alter the transactions already recorded on the ledger. The idea of Satoshi Nakamoto was to design a system that, through incentives, made the needs and ob- jectives of every participant aligned with each other, so that what is recorded on the ledger can be trusted without trusting (or knowing) the other agents. Nakamoto claimed to have eliminated the need for trust but, according to Werbach (2018), that would be impossible. What Nakamoto created is trust in “a new architecture of trust”, where independent agents run this technology, validating the transactions so that they can be recorded on the ledger. This is reinforced by the fact that distributed ledgers networks make people work together is a way that otherwise would not have been possible since they would not trust each other sufficiently.

To better understand what he means by a “new architecture”, the author firstly outline the various architectures (which define as ”the ways the components of a system interact with one another” (Werbach, 2018, p. 25) ) of trust that humans have developed over time. The main architectures are:
* Peer to peer (P2P): here, trust is based on a face to face relationship that arise because the agents share ethical norms and mutual commitment. The downside of this architecture is that is this possible for only few people and small communities, given that the knowledge of each other is pivotal in creating trust.
* Leviathan: this vision start from the belief that humans are not fully trustable and therefore a powerful third party, the state/government, is needed to enforce private contracts and property rights. This is achieved through the monopoly of violence held by the state: people can now trust each other because, if some- thing goes wrong, the leviathan can punish the guilty and enforce previous commitments.
* Intermediaries: transactions are guaranteed by a third party (different from the government), which is trusted to perform certain actions. They create the possibility to perform certain transactions that in a peer to peer network would have been difficult: the other agent is trusted because there is an intermediary that makes the transaction happen. Example are e-commerce platforms such as Amazon, or financial services companies.

The new architecture of trust created by the blockchain is defined as a “trustless trust”. Without trust it would fail since no engagement between individuals is possible without a form of trust, but if it relied on old trust structures it would not be a revolution and would fail its primary object. On the blockchain network, no agent is assumed to be trustworthy, but the output of the network is. Generally speaking, in every transaction, the counterpart, the intermediary and the dispute resolution mechanism must be trusted, but the blockchain substitutes these elements with code. There is no possibility to assess the other party trustworthiness, since all agents are represented by private\public keys in the network which allow for their anonymity; there is no central intermediary, since the platform is a distributed machine operated by all the participants; the disputes are solved through pieces of codes called smart contracts, that perform a certain action with no possibility to stop them. Transactions are verified through cryptographic proofs that other agents can verify mathematically. Therefore, it is not possible to frame this system within the common architectures: it is not a P2P since the other parties are unknown, there is no central authority and also there is no central intermediary since the platform is operated in a decentralized way. Each agent needs to trust the network and not each agent with whom he is engaging in a transaction. The blockchain (and Bitcoin) seems the perfect solution for the lack of trust of the modern society and for the problems that the previous architectures of trust presented. The fact that Bitcoin was born after the Great Financial Crisis is not random. P2P relationships were not sufficient in a world so deeply interconnected, intermediaries were considered the cause of the crisis itself and the Leviathan, i.e. the government, was not able to foresee the crisis and prevent it.

Blockchain trust relies also on the immutability of the information recorded, through the mechanisms beforehand explained. However, immutability must be understood in a probabilistic way. The more blocks are added, the more the previous transactions will be immutable because it would require an infinite amount of power to alter the transactions. Each agent can decide after how long they trust the state of the ledger. Therefore, blockchain trust is not instantaneous. Moreover, the transparency of the ledger, meaning that the record of every transaction is publicly available and the software itself through which the blockchain operates is open source, is an important characteristic that increases trust. Finally, blockchain’s trust is algorithmic, meaning that it relies on algorithms to maintain the system: what must be trusted are not the people operating on it, but the software and the math behind the consensus process.

Satoshi’s error was to believe that in his architecture trust was absent, while in reality it reduced the need of trusting some part of the system. Trust is needed and the blockchain could not function without it. Firstly, engaging in a transaction in a system without central control and with immutability means that no one is able to oversee the transaction and amend it if something is wrong. Agents can be confident that the transaction will be correctly registered, but a distributed ledger will not be able to verify if the content is legitimate and, if something is wrong with the transaction itself, there is no possibility to reverse it: smart contracts are unstoppable. Moreover, humans are not out of the system entirely, which means that error and misunderstandings can occur. And the cryptographic techniques are still vulnerable to attacks: they may be difficult to perform, but users that engage with a blockchain need to trust that this will not happen.

The author argues that the success of the blockchain as an architecture of trust will depend on its governance. The blockchain is a way to enforce some rules, but it is also a product of some rules designed by humans, which therefore would need a governance to continue to operate and to decide the next rules of the game. Moreover, law should regulate the blockchain framework: without legal rules, the blockchain could be used as an instrument by criminals and terrorists (for recycling money for example), and this would reduce the trust that normal people put in this system. Crypto enthusiasts argue that the role of law would be replaced by smart contracts, but codes cannot fully formulate human intentions, which are an important part behind private contracts, and this could create misunderstandings between the parties. Law can intervene where smart contracts are not able to. Finally, also regulation can play an important role in developing the future of the blockchain and foster its trustworthiness, as it does with other financial instruments and institutions.

==8.3 Trust and the blockchain in practice==

Some scholars have started to think about how trust between users can be enhanced in real blockchain applications. You et al. (2022) find the main challenge to be the fact that there is no consensus about how to measure trust in the blockchain environment. Therefore, they develop a framework to do that, creating a system based on subjective ratings of trustworthiness. The authors start by identifying six different blockchain pplication, considering which factors can be used to measure trust in each specific domain. Identifying the key factors behind trustworthiness is essential for creating a system to enhance trust. In particular:
* Supply chain: it is possible to measure how trustworthy the supplier is by the average order arrival time and the defect rate, and how trustworthy is the buyer by the number of days for payment.
* Healthcare industry: to assess the trustworthiness of these firms, regulatory compliance proof, claim approval rate, drug prescription regularity can be the starting point.
* E-commerce: to assess the trustworthiness of those firms, the accuracy of ratings provided by the users and the security of payments represent the most important features.
* IoT devices: system security data and reliability of the data provided by these devices are the most important features.
* Finance: pivotal factors are the security of transactions and data and the effi- ciency and quality of the communications.
* Social media: news and reputation credit represent the most important charac- teristics to assess trustworthiness.
The problem of the blockchain is that, although the information recorded cannot be modified easily, the data may not always be true: the need for accountability arises because of this fact.

The system presented by the authors is based on trust scores given by agents that interacts with other agents on the blockchain applications. Initially, there would be no score, since no transaction has occurred yet. Then, the two parties start to interact and they begin to collect trust factors about each other. The specific factors, which are described above, will depend on which application is under consideration. Then, each actor will give its score, which will be recorded on the blockchain and will be available for other users, who are now better informed regarding the other users of the blockchain application and can decide to interact with them or not. The validity of the scores will be ensured by the fact that each user will have followed the KYC validation procedures before interacting on the application and it will possible to identify the particular participant from the outside through verifiable credentials. Therefore, no rating will be anonymous.

This system may increase the trust between users because they are incentivized to adhere to the common organizational norms of each sector, because otherwise they would damage their reputation by having a low score permanently recorded on the blockchain. Therefore, this model may create a set of incentives to align the two sides of each transaction.

=9. Trust games in the blockchain=

As explained before, the blockchain system employs game theory and incentives to make the agents act honestly on the network. After the work of Satoshi Nakamoto, several papers were developed to study the incentives structures and the games behind the blockchain and its consensus mechanism.

==9.1 ==

Breiki (2022) studies how trust among players evolve over time when they perform trust evolution games. To do that, he defines the features of its abstract game. Firstly, the author defines the parameters of the model: there are various miners and each of them has the possibility to cooperate (acting honestly) or defect (cheating); there is a vector of probabilities that define the likelihood of each player to succeed in solving the puzzle, which is proportional to their computational power; there are the costs and rewards of mining, taking into account also the propagation delay (i.e. the time needed to validate a transaction); there is the market value. Moreover, the author uses two learning algorithms: fictitious play, where prior believes are defined; satisficing learning, where aspiration level of payoff and leaning rates are defined. All in all, the author finds that the player learn to cooperate in the game to get a better payoff and, for satisfice players, lower learning rates increased the final payoff.

==9.2 ==

J. Zhang and Wu (2021) study evolutionary game theory applied on the blockchain network to understand the strategies and incentives of the participants and their co- operative behavior. The authors explain that the blockchain is a perfect environment for evolutionary game theory because:
* There is information symmetry, since all individual have and share the same information on the network and each participant have complete transactions data.
* All the participants are equal so no party has a dominant advantage when the game begins.
* Participants are prone to trust each other and engage in the game because of the cryptographic mechanisms, which make the environment credible and immutable.
* The process of adding new blocks can be seen as a form of repeated games.

Agents have bounded rationality, since they cannot get global information because the network is complex, and therefore they are not fully able to maximize their payoffs. Each participant can have two possible behaviors, cooperation or defection, and they update their strategy considering the maximum payoff. Indeed, during the generation of new blocks, each agent is able to learn from its actions and the actions of the winners.

The model developed comprises two groups of miners: group A, with the inclination for cooperation, and group B, inclined to cheating. Participating in both groups has a cost Ca and Cb and each game bring a revenue R, which will be rewarded to the participants. Each group will have different benefits (for group A, transactions fees and mining rewards, for group B illegal revenue). Finally, there are also punitive measures, denominated P.

Each player, in both groups, can decide which strategy to adopt. Their payoff are:
\begin{align*}
E_{h,a} &= y(K_a + \lambda R - C_a) + (1-y)(K_a - C_{a}) \\
E_{m,a} &= y(K_a - C_a - P) + (1-y)(K_a - P) \\
E_{h,b} &= x(K_b + (1-\lambda)R - C_{b}) + (1-x)(K_b - C_b - P) \\
E_{m,b} &= x(K_b - C_{b}) + (1-x)(K_b - P)
\end{align*}

Where y and x are the probabilities of winning the game, h means the honest strategy, m means the dishonest strategy and λ is the portion of R won.

The authors assume that, at the beginning, each participant has a decided strategy to start with. The goal is to examine the change in population of honest and dishonest after several rounds, taking also into account changes in the parameters of the game. The authors argue that, unlike classic evolution games, the relationship among agents in the blockchain is random. Moreover, the size of the network matters: with small networks, the emergence of cooperative behaviors is easier. Finally, a definition of the evolutionary stable strategy (ESS) is given. The ESS is “a strategy that other strategies cannot invade” (J. Zhang & Wu, 2021, p. 5).

The authors runs various simulation. Firstly, 67% of group A are honest agents and 20% of group B are betrayers. Here, the honest strategy is an ESS, since the number of betrayers tends to 0 as the rounds increase. However, as the expected payoff of group B augment, the honest strategy is still an ESS but weak, because higher payoffs with the dishonest strategy tend to tempt agents to cheat. Therefore, future revenue expectations influence the behavior of participants in the blockchain. Then, the influence of the network structure is analyzed within the group A population. They use a Watts-Strogatz (WS) small-world model and a Barabasi-Albert (BA) scale-free network model. The authors find that it takes quite lots of rounds for honest agents to establish a trusting cooperative relationship and this may represent an opportunity for cheating agents in the blockchain. Therefore, security is a relevant topic expecially in the initial stage of the blockchain.

==9.3 ==

L. Zhang and Tian (2023) develop a Byzantine consensus protocol (i.e. a consensus protocol where there are faulty and malicious agents) on the blockchain, shaping it as a dynamic game. Their main contribution to the current literature relies in the fact that the agents in their model have bounded rationality and can learn from the historical observations. In particular, this means that the participants can choose among a limited set or strategy (honest or dishonest), they are able to learn from historical observations and they choose their strategy accordingly, taking also into account the current state of information, but they are not able to forecast the future. Moreover, they are allowed to have inconsistent subjective believes about the probability of meeting agents with their same strategy: each agent believes that, for a portion m of rounds, he will meet a proposer with the same strategy, ranging from m=1 (meaning that he and the proposer will always have the same strategy in every round) to m=0 (meaning that he and the proposer will have the same strategy only by chance). Their model, based on a BFT (Byzantine Fault Tolerance) consensus protocol, consists in the following features:
* The agents, before the game, are selected to form different parallel committees,
which compete in a n round mining game and which will not change until the end of the game.
* Each agent has one vote in the mining game.
* In each round, an agent is randomly selected to make a proposal about the validity of a block. The other agents become validators and they vote if the block in the proposal is valid. The block will be validated if the number of votes is higher than v, a majority threshold.
* There is a reward R for validating the block, a cost $c_{check}$ for verifying a transaction, a cost $c_{sent}$ for voting for a transaction and a penalty k that validators encounter if they misbehave.

Before each round, each validator check if their congeners (the other nodes in the network) have pivotality, i.e. the ability to control the consensus outcome because they have the majority. As specified before, the participant have two strategy: honest strategy, where miners achieve the consensus protocol and the Byzantine (dishonest) strategy, where miners damage the consensus protocol. The authors assume that participants are fixed and no one would quit. Initially, a number x1 of miners choose the honest strategy in the first turn.

The authors define the concept of stable equilibrium, which can be defined as the situation when $x_t=x_{t-1}$, so when the portion of miners which perform an honest strategy remains stable (so the number of agents that changes their strategy from honest to dishonest is equal to the number of agents that changes from dishonest to honest). There are 3 possible stable equilibria:
* The honest stable equilibrium, where no agent is cheating, so $x_t=x_{t-1}=1$.
* The Byzantine stable equilibrium, where all agents are cheating, so $x_t=x_{t-1}=0$.
* The pooling stable equilibrium, where both strategy exists, so $x_t=x_{t-1} \in (0,1)$.

Each equilibrium can be reached depending on the number of initial cheaters/honest miners, their belief m, the cost-reward mechanism and the pivotality rate (i.e. the minimum percentage of nodes that must agree to reach consensus and add a block). They find that only the honest stable equilibrium can support the safety, the liveness (so the fact that all non faulty agents should have output) and the validity (the fact that all participants have the same valid output) of the blockchain. Moreover, they find that if the reward-punishment increases, the blockchain will become safer and the honest stable equilibrium will be easier to achieve, while if the cost punishment ratio increases, the safety and the liveness of the ledger are threatened and the honest equilibrium is more difficult to achieve. Finally, if the pivotality rate increases, every stable equilibria is harder to achieve.

=10. Trust in algorithms=

Algorithms are becoming more and more important in everyday life, from health care to criminal justice systems, contributing in decision making processes in many fields. Therefore, the natural question on whether it is possible to trust algorithms arises.

According to Spiegelhalter (2020), algorithm’ trustworthiness come from the claims made about the system (how its developers say that the system works and what it can do) and claims by the system (the algorithm’s responses and output about a specific case). Therefore, he proposes a model to assess and boost trustworthiness in algorithms.
Regarding the first kind of claims, developers should clearly state what are the benefits and downside of using their algorithms. To assess that, the authors proposes an evaluation structure of 4 phases:
* Digital testing: the algorithm accuracy should be tested on digital datasets.
* Laboratory testing: the algorithm results should be compared with human ex- perts in this field. An independent committee should evaluate which response is better.
* Field testing: the system should be tested on field, to decide whether is does more harm or good, considering also the effects that it can have on the overall population.
* Routine use: if the algorithms passes the 3 previous phases, it should be moni- tored continuously, in order to solve problems that can eventually arise.
Having explicit positive evidence in all these phases would boost the trustworthiness of the claims made about the system by developers.
Considering the second type of claims, to reach an higher degree of trustworthiness, it is necessary that the algorithms specifies the chain of reasoning behind its claims, which are the most important factor that led to its output and which is the uncertainty around the claim. Moreover, also a counterfactual analysis should be performed (i.e what would be the output If the input changed). Overall, the algorithms should be made clearer and more explainable and transparency can play an important role for that. To increase trustworthiness, an algorithm should be accessible and intelligible by people, it should be useable, so have an effective utility, and it should be assessable, so the process behind every claim should be available. Ultimately, it should show how it works. And more importantly, it should also clearly state its own limitations, so that trustworthiness do not become blind trust.

=11. Conclusion=

Trust plays a pivotal role in ensuring the existence and development of modern soci- eties.
This paper provides a comprehensive summary of the current literature on how trust relates to the world of economics and finance. To begin with, the concept of trust is clearly defined and differentiated from other human sentiments such as cooperation and confidence. The notion of risk is also discussed, as a complete assessment of the other party’s actions is antithetical to a trust relationship. Furthermore, various methods for measuring trust are explored, with trust games being the most commonly used tool by researchers. The paper goes on to explain how trust is a source of com- parative advantage, which determines trade patterns. Additionally, trust is linked to stock market participation, with more trusting individuals being more likely to invest in risky assets and, conditional on participating, they allocate a larger portion of their wealth. While complete trust among individuals would potentially be ben- eficial, it is not possible in the real world. However, money can act as a substitute that replicates the allocations of a trustworthy economy. The paper also emphasizes the importance of trusting institutions, especially in a world where trust is lacking. Effective communication among individuals is critical in assessing the true trustwor- thiness of an institution. The paper then delves into the relationship between trust and blockchain technology, which can be seen as a new architecture of trust. More- over, various authors have developed blockchain trust games to better understand the best consensus mechanism process. Finally, the importance of trusting algorithms is highlighted, given their widespread use in everyday technology, healthcare, and the justice system.

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