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

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

Contribution of [[SIMONE GOZZINI]]

<span id="introduction"></span>
= 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&quot;'' (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.

<span id="reputation and trust"></span>
= 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 ''&quot;the uncertainty regarding the trustworthiness and/or competence of the trustee that the truster faces&quot;'' (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.

<span id="Stage games with perfect monitoring"></span>
== 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.

<span id="RGPM"></span>
== 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.

<span id="equilibria"></span>
=== 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.

<span id="simple-strategy-and-penal-code"></span>
=== 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.

<span id="long-lived-and-short-lived-players"></span>
=== 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>.

<span id="games-with-imperfect-monitoring"></span>
== 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.

<span id="stage-games"></span>
=== 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.

<span id="repeated-games"></span>
=== 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.

<span id="Reputation Games"></span>
= Reputation games =

Reputation can be considered as a specification of the concept of trust. It can be defined as &quot;a link between past behavior and expectations of future behavior&quot; (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.

<span id="commitment-types"></span>
== 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 &quot;Stackelberg action&quot; 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>.

<span id="perfect-monitoring-games"></span>
== 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.

<span id="example-product-choice-game"></span>
== 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).

<span id="Product-choice game" label="Product-choice game"></span>

<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>

<span id="imperfect-monitoring-games"></span>
== 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>

<span id="example-product-choice-game-with-public-monitoring"></span>
=== 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 < p < 1</math>. 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.

<span id="tr"></span>
=== 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>

<span id="long-lived players"></span>
= 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.

<span id="perfect-monitoring"></span>
== 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.

<span id="other-actions"></span>
=== 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>.

<span id="imperfect-public-monitoring"></span>
== 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 &quot;one-block&quot; 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>

<span id="commitment-types-who-punish"></span>
=== 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.

<span id="temporary-reputations-with-two-long-lived-players"></span>
== 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.

<span id="References"></span>
= References =

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