Sequential equilibrium

The first model with k-level thinking with Gaussian noise shows that small deviations for common knowledge led to "almost" common knowledge equilibria. The second model demonstrated the semantic economy idea: as agents exchange and adapt beliefs, they create shared informational value by reaching a consensus that reflects network-wide insights rather than mere individual optimization. Higher reasoning depth does not change Nash equilibria but shifts up Kantian beliefs. The shift up in Kantian beliefs suggests a greater alignment toward strategies that maximize collective welfare, rather than purely individualistic or competitive outcomes. This doesn't alter the Nash equilibrium, where players still act independently, but it emphasizes a higher baseline of cooperative or altruistic expectations among players due to more profound belief hierarchies in the reasoning process. In the third model: networked economic context where agents interact in an economic network where competitive advantage depends on the informational value generated across the network, results differ from the second example: Nash beliefs adjust based on others' best responses (shift up), while Kantian beliefs account for mutual benefit, dampening large shifts. Nash and Kantian equilibria differ when only three agents exist versus network economy.

In many situations, such as trade in stock exchanges, agents have many opportunities to act within a short interval of time. The agents in such situations can often coordinate their actions in advance, but coordination during the game consumes too much time. An equilibrium in such situations has to be sequential in order to handle mistakes made by players. In this paper, we present a new solution concept for infinite-horizon dynamic games, which is appropriate for such situations: a sequential normal-form correlated approximate equilibrium. Under additional assumptions, we show that every such game admits this kind of equilibrium.

The so‐called disclosure principle is a ‘puzzle’ in the accounting literature: Game theoretic models of financial markets show that in equilibrium firms should disclose all their private information. Yet, the result is not convincing. Researchers have therefore built sophisticated models in order to demonstrate for which reasons the disclosure principle might fail. This note shows that even in the original model there are multiple equilibria. In those equilibria good types disclose and bad types do not. The commonly known full disclosure equilibrium is a limit point of the equilibrium set.

We introduce new revelation mechanisms for simultaneous common agency games which, although they do not always permit a complete equilibrium characterization, do facilitate the characterization of the equilibrium outcomes that are typically of interest in applications. We then show how these mechanisms can be used in applications such as menu auctions, competition in nonlinear tariffs, and moral hazard settings. Lastly, we show how one can enrich the revelation mechanisms, albeit at a cost of an increase in complexity, to characterize all possible equilibrium outcomes, including those sustained by non-Markov strategies and/or mixed-strategy profiles. (JEL C72, D82, D86)

This study shows that in a two-player infinitely repeated game where one is patient and the other is impatient, Pareto-superior subgame perfect equilibrium can be achieved. An impatient player in this paper is depicted as someone who can truly destroy the possibility of attaining any feasible and individually rational outcome that is supported in equilibrium in repeated games, as asserted by the Folk Theorem. In this scenario, the main ingredient for the restoration of equilibrium is to introduce the notion of tolerant trigger strategy. Consequently, the use of the typical trigger strategy is abandoned since it ceases to be efficient as it only brings automatically the game to its punishment path, therefore eliminating the possibility of extracting other feasible equilibria. I provide a simple characterization of perfect equilibrium payoffs under this scenario and show that cooperative outcome can be approximated. * I am grateful to my advisor Krishnendu Ghosh Dastidar for his guidance and help on this study. I also thank Ramon Torredemer, Mariano Iturbe, and Alfredo Miranda for the many insightful ideas on the topic that came out during our lunch discussions. This research is partly supported by ICCR scholarship. Errors are mine alone.

The network formation model described in this research paper is based on the connection model of Bala and Goyal . We extend this model by introducing two properties: heterogeneity of players and sequentiality of the network formation process. These two characteristics modify the emergence of equilibrium and efficient networks. Our main theoretical finding is that sequentiality is the main determinant of the equilibrium network while heterogeneity is secondary. The equilibrium often requires that the last player of the game incurs a large part of the linking costs. On the contrary, the leader does not create links at the equilibrium. In the following of the paper we build an experimental protocol. Our purpose is to analyse whether our model predicts behaviour in the laboratory and whether individuals are more attracted by the equilibrium or by efficiency. * I thank Marie Claire Villeval and Christophe Bravard for their help for the creation of this research paper. I also wish to thank Yann Bramoullé, Sudipta Sarangi and Jurjen Kamphorst for their precious comments.

We provide a general notion of perfect Bayesian equilibrium which can be applied to arbitrary extensive-form games and is intermediate between subgame-perfect equilibrium and sequential equilibrium. The essential ingredient of the proposed definition is the qualitative notion of AGMconsistency, which has an epistemic justification based on the AGM theory of belief revision.

Many interactions result in a socially suboptimal equilibrium, or in a nonequilibrium state, from which arriving at an equilibrium through simple dynamics can be impossible of too long. Aiming to achieve a certain equilibrium, we persuade, bribe, or coerce a group of participants to make them act in a way that will motivate the rest of the players to act accordingly to the desired equilibrium. Formally, we ask which subset of the players can adopt the goal equilibrium strategies that will make acting according to the desired equilibrium a best response for the other players. We call such a subset a direct control set, prove some connections to strength of equilibrium, and study the hardness to find such lightest sets, even approximately. We then solve important subcases and provide approximation algorithms, assuming monotonicity. Next, we concentrate on potential games and prove that, while the problem of finding such a set is NP-hard, even for constantfactor approximation, we can still solve the problem approximately or even precisely in relevant special cases. We approximately solve this problem for singleton potential games and treat more closely specific potential games, such as symmetric games and coordination games on graphs.

Many interactions result in a socially suboptimal equilibrium, or in a nonequilibrium state, from which arriving at an equilibrium through simple dynamics can be impossible of too long. Aiming to achieve a certain equilibrium, we persuade, bribe, or coerce a group of participants to make them act in a way that will motivate the rest of the players to act accordingly to the desired equilibrium. Formally, we ask which subset of the players can adopt the goal equilibrium strategies that will make acting according to the desired equilibrium a best response for the other players. We call such a subset a direct control set, prove some connections to strength of equilibrium, and study the hardness to find such lightest sets, even approximately. We then solve important subcases and provide approximation algorithms, assuming monotonicity. Next, we concentrate on potential games and prove that, while the problem of finding such a set is NP-hard, even for constantfactor approximation, we can still solve the problem approximately or even precisely in relevant special cases. We approximately solve this problem for singleton potential games and treat more closely specific potential games, such as symmetric games and coordination games on graphs.

Coordination between agents can be modelled using correlated equilibrium and a 'device' allocating roles to players. That coordination takes place within a social context suggests properties that one may expect of a correlated equilibrium. In this paper, given a partition of the player set into (social) groups, we demonstrate the existence of an approximate correlated equilibrium satisfying a within group anonymity property and a group independence property. Within group anonymity assures 'fairness' and equality within groups while group independence rules out correlation of actions between groups. * This paper is a major revision of Cartwright and Wooders (2003a). The main results of that paper were presented at the the 2002 General Equilibrium Conference held in Athens in May 2002 and at Northwestern University in August 2002. We thank the participants for their interest and comments.

Conventional (single-prior) Bayesian games of incomplete information are limited in their ability to capture the extent of informational asymmetry. In particular, they are not capable of representing complete ignorance of an uninformed player about an unknown parameter of the environment. Using a framework of contracting for delegated experimentation, we formulate and analyze a dynamic game of incomplete information that incorporates a multiple-prior belief system. Specifically, we consider a game with a principal contracting with an expert agent for his (observable) effort on a novel experiment – a Poisson process with unknown hazard rate. Although the expert agent has suffi cient knowledge to form a single prior over the hazard rate, the principal initially has complete ignorance and her ambiguous beliefs are represented by the set of all plausible prior distributions over the hazard rate. We propose a new equilibrium concept – Perfect Objectivist Equilibrium – in which the princi...

In an earlier paper [Rational choice and AGM belief revision, Artificial Intelligence, 2009] a correspondence was established between the choice structures of revealed-preference theory (developed in economics) and the syntactic belief revision functions of the AGM theory (developed in philosophy and computer science). In this paper we extend the re-interpretation of (a generalized notion of) choice structure in terms of belief revision by adding: (1) the possibility that an item of "information" might be discarded as not credible (thus dropping the AGM success axiom) and (2) the possibility that an item of information, while not accepted as fully credible, may still be "taken seriously" (we call such items of information "allowable"). We establish a correspondence between generalized choice structures (GCS) and AGM belief revision; furthermore, we provide a syntactic characterization of the proposed notion of belief revision, which we call filtered belief revision.

We study the dynamical system of expectations generated by a simple general equilibrium model of an exchange economy in which each agent considers a finite collection of models, each of which specifies a relationship between payoff-relevant information and equilibrium prices. One of the models under consideration is a correct description of the rational expectations equilibrium. We find that under a Bayesian type of learning process the rational expectations equilibrium is locally stable, but that nonrational equilibria may also be locally stable. Journal of Economic Literature Classification Numbers: 021, 026. *The authors gratefully acknowledge the comments offered by Richard Erikson, John Ledyard, and Roy Radner. Discussions with Jim Jordan have been especially insightful. Also helpful were discussions in seminars at Stonybrook, Washington University, and at the Midwestern Conference on Mathematical Economics.

We study behavior and equilibrium selection in experimental network games. We vary two important factors: (a) actions are either strategic substitutes or strategic complements, and (b) subjects have either complete or incomplete information about the structure of a random network. Play conforms strongly to the theoretical predictions, providing an impressive behavioral confirmation of the Galeotti, Goyal, Jackson, Vega-Redondo, and Yariv (2010) model. The degree of equilibrium play is striking, even with incomplete information. We find that under complete information, subjects typically play the stochastically-stable (inefficient) equilibrium when the game involves strategic substitutes, but play the efficient one with strategic complements. Our results suggest that equilibrium multiplicity may not be a major concern. Subjects' actions and realized outcomes under incomplete information depend strongly on both the degree and the connectivity. When there are multiple equilibria, subjects begin by playing the efficient equilibrium, but eventually converge to the inefficient one.

We study behavior and equilibrium selection in experimental network games. We vary two important factors: (a) actions are either strategic substitutes or strategic complements, and (b) subjects have either complete or incomplete information about the structure of a random network. Play conforms strongly to the theoretical predictions, providing an impressive behavioral confirmation of the Galeotti, Goyal, Jackson, Vega-Redondo, and Yariv (2010) model. The degree of equilibrium play is striking, even with incomplete information. We find that under complete information, subjects typically play the stochastically-stable (inefficient) equilibrium when the game involves strategic substitutes, but play the efficient one with strategic complements. Our results suggest that equilibrium multiplicity may not be a major concern. Subjects' actions and realized outcomes under incomplete information depend strongly on both the degree and the connectivity. When there are multiple equilibria, subjects begin by playing the efficient equilibrium, but eventually converge to the inefficient one.

Dynamic contracts with multiple agents is a classical decentralized decision-making problem with asymmetric information. In this paper, we extend the single-agent dynamic incentive contract model in continuous-time to a multi-agent scheme in finite horizon and allow the terminal reward to be dependent on the history of actions and incentives. We first derive a set of sufficient conditions for the existence of optimal contracts in the most general setting and conditions under which they form a Nash equilibrium. Then we show that the principal's problem can be converted to solving Hamilton-Jacobi-Bellman (HJB) equation requiring a static Nash equilibrium. Finally, we provide a framework to solve this problem by solving partial differential equations (PDE) derived from backward stochastic differential equations (BSDE).

This study shows that in a two-player in…nitely repeated game where one is impatient, Pareto-superior subgame perfect equilibria can still be achieved. An impatient player in this paper is depicted as someone who can truly destroy the possibility of attaining any feasible and individually rational outcome that is supported in equilibrium in repeated games, as asserted by the Folk Theorem. In this scenario, the main ingredient for the restoration of equilibrium is to introduce the notion of tolerant trigger strategy. Consequently, the use of the typical trigger strategy is abandoned since it ceases to be e¢cient as it only brings automatically the game to its punishment path, therefore eliminating the possibility of extracting other feasible equilibria. I provide a simple characterization of perfect equilibrium payo¤s under this scenario and show that cooperative outcome can be approximated.

Since the late 1960s, the efforts of general equilibrium theorists have been directed towards overcoming the evident limitation of the Arrow-Debreu model, i.e. the assumption that the transactions associated with the future activities of agents are all regulated at the initial date on a complete system of forward markets. Research has thus focused on 'sequential economies', in which spot markets are active in each period, and has developed along two paths, both inspired by Hicks's Value and Capital and stressing the dependence of agents' choices on their expectations of future prices. The first is temporary equilibrium theory, in which expectations are assumed to be subjective. The second postulates that all agents exactly predict the future prices (sequential economies with perfect foresight). This paper examines the analytical problems that the inclusion of expectations among the determinants of equilibrium originates within each approach. In the light of the studies of the 1970s and 1980s, it first illustrates the difficulties that arise in temporary equilibrium theory due to the subjective nature of individual forecasts. Then it moves on to examine sequential economies with perfect foresight. After illustrating the equilibrium notion on which the analysis of those economies relies, i.e. the 'equilibrium of plans, prices and price expectations' introduced by Radner (1972), it indicates, on the basis of recent contributions, that for plausible configurations of the economy the perfect foresight associated with Radner equilibria proves not only unrealistic but also theoretically dubious.

Treating games of incomplete information, we demonstrate that the existence of an ex post stable strategy vector implies the existence of an approximate Bayesian equilibrium in pure strategies that is also ex post stable. Through examples we demonstrate the 'bounds obtained on the approximation' are tight.

In this paper the new concept of coalition Nash equilibrium of a strategic game is introduced, and it is shown that a communication among the players in a coalition leads to the equilibrium through messages. A coalition Nash equilibrium for a strategic game consists of (1) a subset S of players, (2) independent mixed strategies for each member of S, (3) the conjecture of the actions for the other players not in S with the condition that each member of S maximises his/her expected payoff according to the product of all mixed strategies for S and the other players' conjecture. However, this paper stands on the Bayesian point of view as follows: The players start with the same prior distribution on a state-space. In addition they have private information which is given by a reflexive and transitive binary relation on the state space. Each player in a coalition S predicts the other players' actions as the posterior of the others' actions given his/her information. He/she communicates privately their beliefs about the other players' actions through messages among all members in S according to the communication network in S, which message is information about his/her individual conjecture about the others' actions. The recipients update their belief by the messages. Precisely, at every stage each player communicates privately not only his/her belief about the others' actions but also his/her rationality as messages according to a protocol and then the recipient updates their private information and revises her/his prediction. In this circumstance, we show that the conjectures of the players in a coalition S regarding the future beliefs converge in the long run communication, which lead to a coalition Nash equilibrium for the strategic game.

We ask whether communication can directly substitute for memory in dynastic repeated games in which short lived individuals care about the utility of their offspring who replace them in an infinitely repeated game. Each individual is unable to observe what happens before his entry in the game. Past information is therefore conveyed from one cohort to the next by means of communication. When communication is costless and messages are sent simultaneously, communication mechanisms or protocols exist that sustain the same set of equilibrium payoffs as in the standard repeated game. When communication is costless but sequential, the incentives to "whitewash" the unobservable past history of play become pervasive. These incentives to whitewash can only be countered if some player serves as a "neutral historian" who verifies the truthfulness of others' reports while remaining indifferent in the process. By contrast, when communication is sequential and (lexicographically) costly, all protocols admit only equilibria that sustain stage Nash equilibrium payoffs. We also analyze a centralized communication protocol in which history leaves a "footprint" that can only hidden by the current cohort by a unanimous "coverup." We show that in this case the set of payoffs that are sustainable in equilibrium coincides with the weakly renegotiation proof payoffs of the standard repeated game.

We include endogenous differential information in a model with sequential trade and incomplete financial participation. Agents update information through market signals given by commodity prices and asset deliveries. Information acts over admissible strategies and consumption tastes, allowing discontinuities in preferences and choice sets. Therefore, equilibrium may cease to exist. However, internalizing the compatibility between information and consumption through preferences, and without requiring either financial survival assumptions or fully revealing prices, equilibrium existence can be ensured. We would like to thank Jean-Marc Bonnisseau, Bernard Cornet, and João Correia-da-Silva for helpful discussions. We specially thank the detailed comments and suggestions of an anonymous referee. S. Cea-Echenique acknowledges financial support from Conicyt and University of Chile; C. Hervés-Beloso acknowledges financial support from Research Grants ECO2012-38860-C02-02 (Ministerio de Economia y Competitividad) and RGEA (Xunta de Galicia and FEDER); J. P. Torres-Martínez acknowledges financial support from Conicyt through Fondecyt Projects 1120294 and 1150207.

Peer-prediction is a mechanism which elicits privately-held, non-variable information from self-interested agents---formally, truth-telling is a strict Bayes Nash equilibrium of the mechanism. The original Peer-prediction mechanism suffers from two main limitations: (1) the mechanism must know the "common prior" of agents' signals; (2) additional undesirable and non-truthful equilibria exist which often have a greater expected payoff than the truth-telling equilibrium. A series of results has successfully weakened the known common prior assumption. However, the equilibrium multiplicity issue remains a challenge. In this paper, we address the above two problems. In the setting where a common prior exists but is not known to the mechanism we show (1) a general negative result applying to a large class of mechanisms showing truth-telling can never pay strictly more in expectation than a particular set of equilibria where agents collude to "relabel" the signals a...

We consider generalized sender-receiver games in which the sender also has a decision to make, but this decision does not directly a¤ect the receiver. We introduce speci…c perfect Bayesian equilibria, in which the players agree on a joint decision after that a message has been sent ("talk and cooperate equilibrium," TCE). We establish that a TCE exists provided that the receiver has a "uniform punishment decision" (UPD) against the sender. This paper is part of a more general research project entitled "Cheap talk and commitment" and was completed while the …rst author was visiting Humboldt University. Financial support by Deutsche Forschungsgemeinschaft through CRC TRR 190 is gratefully acknowledged. We thank the participants of the Berlin micro theory seminar, the economic theory seminar of Toulouse School of Economics and the workshop "Incentives for Con ‡ict and Cooperation" (Bristol, May 2016). We also wish to thank Gorkem Celik, Johannes Hörner and Andrés Salamanca, for helpful comments. We are specially grateful to Roland Strausz for many insightful conversations and extensive comments on an earlier draft. .

We consider a general framework of optimal mechanism design under adverse selection and ambiguity about the type distribution of agents. We prove the existence of optimal mechanisms under minimal assumptions on the contract space and prove that centralized contracting implemented via mechanisms is equivalent to delegated contracting implemented via a contract menu under these assumptions. Our abstract existence results are applied to a series of applications that include models of optimal risk sharing and of optimal portfolio delegation. Keywords robust contracts • nonmetrizable contract spaces • ambiguity • financial markets Horst and Beissner gratefully acknowledge financial support by the CRC TRR 190 Rationality and competition-the economic performance of individuals and firms

We consider the existence of Partition Equilibrium in Resource Selection Games. Super-strong equilibrium, where no subset of players has an incentive to change their strategies collectively, does not always exist in such games. We show, however, that partition equilibrium (introduced in [4] to model coalitions arising in a social context) always exists in general resource selection games, as well as how to compute it efficiently. In a partition equilibrium, the set of players has a fixed partition into coalitions, and the only deviations considered are by coalitions that are sets in this partition. Our algorithm to compute a partition equilibrium in any resource selection game (i.e., load balancing game) settles the open question from [4] about existence of partition equilibrium in general resource selection games. Moreover, we show how to always find a partition equilibrium which is also a Nash equilibrium. This implies that in resource selection games, we do not need to sacrifice the stability of individual players when forming solutions stable against coalitional deviations. In addition, while super-strong equilibrium may not exist in resource selection games, we show that its existence can be decided efficiently, and how to find one if it exists.

We present a decentralization result which is useful for practical and theoretical work with sequential equilibrium, perfect Bayesian equilibrium, and related equilibrium concepts for extensive form games. A weak consistency condition is sufficient to obtain an analogy to the well known One-Stage-Deviation Principle for subgame perfect equilibrium. Journal of Economic Literature Classification Number: C72.

We formulate an evolutionary learning process with trembles for static games of incomplete information. For many games, if the amount of trembling is small, play will be in accordance with the games' (strict) Bayesian equilibria most of the time supporting the notion of Bayesian equilibrium. Often the process will select a speciÞc equilibrium. We study an extension to incomplete information of the prototype conßict known as "Chicken" and Þnd that the equilibrium selection by evolutionary learning may well be in favor of inefficient Bayesian equilibria where some types of players fail to coordinate.

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The game theorists who defined the equilibrium concepts that we now think of as various forms of backwards induction, namely subgame perfect equilibrium (Selten, 1965), perfect equilibrium (Selten, 1975), sequential equilibrium (Kreps and Wilson, 1982), and quasi-perfect equilibrium (van Damme, 1984), explicitly restricted their analysis to games with perfect recall. In spite of this the concepts are well defined, exactly as they defined them, even in games without perfect recall. There is now a small literature examining the behaviour of these concepts in games without perfect recall. Jeff Kline (2005) and Joe Halpern and Rafael Pass (2008) look at what happens in games without perfect recall to solutions defined in exactly the same way as they were defined in games with perfect recall. The Nature of Our Project What we want to do is to give definitions of the "backward induction" solutions-in this paper sequential equilibrium and quasi-perfect equilibrium-that 1. "coincides" with the original definitions in games with perfect recall, and

This paper examines the question of the extent to which it is true that any equilibrium that is quasi-perfect in any extensive form game having a given normal form is necessarily proper. If one fixes not only the equilibrium in question but also a a sequence of completely mixed strategies converging to that equilibrium then indeed the notions are equivalent.

Many conditions have been introduced to ensure equilibrium existence in games with discontinuous payoff functions. This paper introduces a new condition, called regularity, that is simple and easy to verify. Regularity requires that if there is a sequence of strategies converging to s * such that the players' payoffs along the sequence converge to the best-reply payoffs at s * , then s * is an equilibrium. We show that regularity is implied both by Reny's better-reply security and Simon and Zame's endogenous sharing rule approach. This allows us to explore a link between these two distinct methods. Although regularity implies that the limits of-equilibria are equilibria, it is in general too weak for implying equilibrium existence. However, we are able to identify extra conditions that, together with regularity, are sufficient for equilibrium existence. In particular, we show how regularity allows the technique of approximating games both by payoff functions and space of strategies.

We investigate the effect of introducing costs of complexity in the n-person unanimity bargaining game. As is well-known, in this game every individually rational allocation is Ž sustainable as a Nash equilibrium also as a subgame perfect equilibrium if players are. sufficiently patient and if n) 2. Moreover, delays in agreement are also possible in such equilibria. By limiting ourselves to a plausible notion of complexity that captures length of Ž memory, we find that the introduction of complexity costs lexicographically with the. standard payoffs does not reduce the range of possible allocations but does limit the amount of delay that can occur in any agreement. In particular, we show that in any n-player game, for any allocation z, an agreement on z at any period t can be sustained as a Nash equilibrium of the game with complexity costs if and only if t F n. We use the limit on delay result to establish that, in equilibrium, the strategies implement stationary behavior. Finally, we also show that ''noisy Nash equilibrium'' with complexity costs sustains only the unique stationary subgame perfect equilibrium allocation.

Two of the most important refinements of the Nash equilibrium concept for extensive form games with perfect recall are Selten's (1975) perfect equilibrium and Kreps and Wilson's (1982) more inclusive sequential equilibrium. These two equilibrium refinements are motivated in very different ways. Nonetheless, as Kreps and Wilson (1982, Section 7) point out, the two concepts lead to similar prescriptions for equilibrium play. For each particular game form, every perfect equilibrium is sequential. Moreover, for almost all assignments of payoffs to outcomes, almost all sequential equilibrium strategy profiles are perfect equilibrium profiles, and all sequential equilibrium outcomes are perfect equilibrium outcomes. We establish a stronger result: For almost all assignments of payoffs to outcomes, the sets of sequential and perfect equilibrium strategy profiles are identical. In other words, for almost all games each strategy profile which can be supported by beliefs satisfying the rationality requirement of sequential equilibrium can actually be supported by beliefs satisfying the stronger rationality requirement of perfect equilibrium. We obtain this result by exploiting the algebraic/geometric structure of these equilibrium correspondences, following from the fact that they are semi-algebraic sets; i.e., they are defined by finite systems of polynomial inequalities. That the perfect and sequential equilibrium correspondences have this semi-algebraic structure follows from a deep result from mathematical logic, the Tarski-Seidenberg Theorem; that this structure has important game-theoretic consequences follows from deep properties of semi-algebraic sets.

and participants in seminars at Queen's university, the universities of Melbourne and Birmingham, the Econometric Society World meetings in Tokyo 1995 and the ESRC game theory workshop, Kenilworth 1997. We thank two anonymous referees of this journal for detailed comments which helped to improve the final version. We

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We ask whether communication can directly substitute for memory in dynastic repeated games in which short lived individuals care about the utility of their offspring who replace them in an infinitely repeated game. Each individual is unable to observe what happens before his entry in the game. Past information is therefore conveyed from one cohort to the next by means of communication. When communication is costless and messages are sent simultaneously, communication mechanisms or protocols exist that sustain the same set of equilibrium payoffs as in the standard repeated game. When communication is costless but sequential, the incentives to "whitewash" the unobservable past history of play become pervasive. These incentives to whitewash can only be countered if some player serves as a "neutral historian" who verifies the truthfulness of others' reports while remaining indifferent in the process. By contrast, when communication is sequential and (lexicographically) costly, all protocols admit only equilibria that sustain stage Nash equilibrium payoffs. We also analyze a centralized communication protocol in which history leaves a "footprint" that can only hidden by the current cohort by a unanimous "coverup." We show that in this case the set of payoffs that are sustainable in equilibrium coincides with the weakly renegotiation proof payoffs of the standard repeated game.

In this report, the fundamental limits of decentralized information transmission in the K-user Gaussian multiple access channel (G-MAC), with K 2, are fully characterized. Two scenarios are considered. First, a game in which only the transmitters are players is studied. In this game, the transmitters autonomously and independently tune their own transmit configurations seeking to maximize their own information transmission rates, R 1 , R 2 ,. .. , R K , respectively. On the other hand, the receiver adopts a fixed receive configuration that is known a priori to the transmitters. The main result consists of the full characterization of the set of rate tuples (R 1 , R 2 ,. .. , R K) that are achievable and stable in the G-MAC when stability is considered in the sense of the η-Nash equilibrium (NE), with η 0 arbitrarily small. Second, a sequential game in which the two categories of players (the transmitters and the receiver) play in a given order is presented. For this sequential game, the main result consists of the full characterization of the set of rate tuples (R 1 , R 2 ,. .. , R K) that are stable in the sense of an η-sequential equilibrium, with η 0.

We introduce a tractable model of cheap talk among players located on networks. In our model, a player can send a message to another player if and only if he is linked to him. We derive a sharp equilibrium and welfare characterization which reveals two basic insights. In equilibrium, the willingness of a player to communicate with a neighbor decreases with the number of opponents who communicate with the neighbor. The ex-ante equilibrium welfare of every player increases not only with the number of truthful reports transmitted in the network, but also when truthful reports are more evenly distributed across players. We apply our findings to the analysis of homophily in communities, to organization design, and to the study of endogenous network formation. Communication across communities decreases as communities become larger, and communication may be asymmetric: From large communities to small ones. In our set up, fully decentralized organizations maximize all players' welfare. Further, decentralized networks, where information may flow asymmetrically, endogenously form in equilibrium. Finally, we introduce the possibility of public communication in networks, and identify conditions under which public communication Pareto dominates private communication.

Game theory lacks an explanation of how players' beliefs arc formed and why they are in equilibrium. This is the reason why it has failed to make significant advances with the problem of equilibrium selection even for quite simple games, as 2 x 2 games with two strict Nash equilibria. Our paper models the introspection process by which the selected equilibrium is achieved in this class of games. Players begin their analysis with imprecise priors, obtained under weak restrictions formulated as Axioms. For a large class of reasoning dynamics we obtain as the solution the risk dominant Nash equilibrium.

Game theory lacks an explanation of how players' beliefs arc formed and why they are in equilibrium. This is the reason why it has failed to make significant advances with the problem of equilibrium selection even for quite simple games, as 2 x 2 games with two strict Nash equilibria. Our paper models the introspection process by which the selected equilibrium is achieved in this class of games. Players begin their analysis with imprecise priors, obtained under weak restrictions formulated as Axioms. For a large class of reasoning dynamics we obtain as the solution the risk dominant Nash equilibrium.

We provide a characterization of quasi-perfect equilibria in n-player games, showing that any quasiperfect equilibrium can be obtained as limit point of a sequence of Nash equilibria of a certain class of perturbed games in sequence form, and any limit point of a sequence of Nash equilibria of these perturbed games is a quasi-perfect equilibrium. We prove that, in games with three or more players, we need trembles defined as rational functions of the perturbation magnitude ε, whereas, in two-player games with nature, trembles expressed in terms of polynomial functions of ε suffice. Exploiting the relationship between sequence form and extensive form, we also provide a similar characterization in terms of perturbed games in extensive form, though not compliant with Selten's definition of perturbed game.