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This paper provides a general framework for analyzing self-confirming policies. We study self-confirming equilibria in recurrent decision problems with incomplete information about the true stochastic model. We characterize stationary monetary policies in a linear-quadratic setting.

Evolutionary game theory assumes that players replicate a highly scored player's strategy through genetic inheritance. However, when learning occurs culturally, it is often difficult to recognize someone's strategy just by observing the behaviour. In this work, we consider players with memory-one stochastic strategies in the iterated prisoner's dilemma, with an assumption that they cannot directly access each other's strategy but only observe the actual moves for a certain number of rounds. Based on the observation, the observer has to infer the resident strategy in a Bayesian way and chooses his or her own strategy accordingly. By examining the best-response relations, we argue that players can escape from full defection into a cooperative equilibrium supported by Win-Stay-Lose-Shift in a self-confirming manner, provided that the cost of cooperation is low and the observational learning supplies sufficiently large uncertainty., 16 pages, 3 figures

Equilibrium notions for games with unawareness in the literature cannot be interpreted as steady-states of a learning process because players may discover novel actions during play. In this sense, many games with unawareness are ''self-destroying'' as a player's representation of the game may change after playing it once. We define discovery processes where at each state there is an extensive-form game with unawareness that together with the players' play determines the transition to possibly another extensive-form game with unawareness in which players are now aware of actions that they have discovered. A discovery process is rationalizable if players play extensive-form rationalizable strategies in each game with unawareness. We show that for any game with unawareness there is a rationalizable discovery process that leads to a self-confirming game that possesses a self-confirming equilibrium in extensive-form rationalizable strategies. This notion of equilibrium can be interpreted as steady-state of both a discovery and learning process.

We argue in favor of a departure from the standard equilibrium approach in game theory in favor of the less ambitious goal of describing only the actual behavior of rational players. We investigate the notion of rationality in behavioral models of extensive-form games (allowing for imperfect information), where a state is described in terms of a play of the game instead of a strategy profile. The players' beliefs are specified only at reached decision histories and are modeled as pre-choice beliefs, allowing us to carry out the analysis without the need for (objective or subjective) counterfactuals. The analysis is close in spirit to the literature on self-confirming equilibrium, but it does not rely on the notion of strategy. We also provide a characterization of rational play that is compatible with pure-strategy Nash equilibrium.

We add here another layer to the literature on nonatomic anonymous games started with the 1973 paper by Schmeidler. More specifically, we define a new notion of equilibrium which we call $\varepsilon$-estimated equilibrium and prove its existence for any positive $\varepsilon$. This notion encompasses and brings to nonatomic games recent concepts of equilibrium such as self-confirming, peer-confirming, and Berk--Nash. This augmented scope is our main motivation. At the same time, our approach also resolves some conceptual problems present in Schmeidler (1973), pointed out by Shapley. In that paper\ the existence of pure-strategy Nash equilibria has been proved for any nonatomic game with a continuum of players, endowed with an atomless countably additive probability. But, requiring Borel measurability of strategy profiles may impose some limitation on players' choices and introduce an exogenous dependence among\ players' actions, which clashes with the nature of noncooperative game theory. Our suggested solution is to consider every subset of players as measurable. This leads to a nontrivial purely finitely additive component which might prevent the existence of equilibria and requires a novel mathematical approach to prove the existence of $\varepsilon$-equilibria.

We consider an ambiguity averse, sophisticated decision maker facing a recurrent decision problem where information is generated endogenously. In this context, we study self-confirming actions as the outcome of a process of active experimentation. We provide inter alia a learning foundation for self-confirming equilibrium with model uncertainty ( Battigalli et al., 2015 ), and we analyze the impact of changes in ambiguity attitudes on convergence to self-confirming equilibria. We identify conditions under which the set of self-confirming equilibrium actions is invariant to changes in ambiguity attitudes, and yet ambiguity aversion may affect the dynamics. Indeed, we argue that ambiguity aversion tends to stifle experimentation, increasing the likelihood that the decision maker gets stuck into suboptimal “certainty traps.”

A candidate explanation for the persistence of heterogeneous behavior in a sequential social dilemma played many times is the existence of heterogeneous preferences. Preferences-dependent conjectures about opponents’ behavior are an additional source of heterogeneity. By behaving differently, different preference types acquire different information. Thus, when observing only outcomes of own past interactions heterogeneous and possibly wrong conjectures about opponents’ strategies may endogenously arise and persist. In a Centipede game experiment played for forty rounds, we manipulate the type of ex post information and the method of play. We find that, when the game is played in its reduced normal form and subjects have only access to personal statistics, heterogeneity of behavior across preference types persists in the long run. In this case, behavior resembles a self-confirming equilibrium: selfish subjects take at earlier nodes due to an unjustified lack of trust. When subjects have also access to public statistics, heterogeneity disappears: selfish subjects tend to pass more often and play moves towards Bayes Nash equilibrium.

We consider a game in extensive form recurrently played by agents who are randomly drawn from large populations and matched. We assume that preferences over actions at any information set admit a smooth-ambiguity representation in the sense of Klibanoff et al. (2005) , which may induce dynamic inconsistencies. We take this into account in our analysis of self-confirming equilibrium (SCE) given players' feedback about the path of play. Battigalli et al. (2015) show that the set of SCE's of a simultaneous-move game with feedback expands as ambiguity aversion increases. We show by example that SCE in a sequential game is not equivalent to SCE applied to the strategic form of such game, and that the previous monotonicity result does not extend to general sequential games. Still, we provide sufficient conditions under which the monotonicity result holds for SCE.

Rationalizable partition-conrme d equilibrium (RPCE) describes the steady state outcomes of rational learning in extensive-form games, when rationality is common knowledge and players observe a partition of the terminal nodes. RPCE allows players to make inferences about unobserved play by others; We discuss the implications of this using numerous examples, and discuss the relationship of RPCE to other solution concepts in the literature.

Motivated by the fact that people’s perceptions of their societies are routinely incorrect, we study the possibility and implications of misperception in social interactions. We focus on coordination games with assortative interactions, where agents with higher types (e.g., wealth, political attitudes) are more likely than lower types to interact with other high types. Assortativity creates scope for misperception, because what agents observe in their local interactions need not be representative of society as a whole. To model this, we define a tractable solution concept, “local perception equilibrium” (LPE), that describes possible behavior and perceptions when agents’ beliefs are derived only from their local interactions. We show that there is a unique form of misperception that can persist in any environment: This is assortativity neglect, where all agents believe the people they interact with to be a representative sample of society as a whole. Relative to the case with correct perceptions, assortativity neglect generates two mutually reinforcing departures: A “false consensus effect,” whereby agents’ perceptions of average characteristics in the population are increasing in their own type; and more “dispersed” behavior in society, which adversely affects welfare due to increased miscoordination. Finally, we propose a comparative notion of when one society is more assortative than another and show that more assortative societies are characterized precisely by greater action dispersion and a more severe false consensus effect, and as a result, greater assortativity has an ambiguous effect on welfare.

Equilibrium notions for games with unawareness in the literature cannot be interpreted as steady-states of a learning process because players may discover novel actions during play. In this sense, many games with unawareness are "self-destroying" as a player's representation of the game may change after playing it once. We define discovery processes where at each state there is an extensive-form game with unawareness that together with the players' play determines the transition to possibly another extensive-form game with unawareness in which players are now aware of actions that they have discovered. A discovery process is rationalizable if players play extensive-form rationalizable strategies in each game with unawareness. We show that for any game with unawareness there is a rationalizable discovery process that leads to a self-confirming game that possesses a self-confirming equilibrium in extensive-form rationalizable conjectures. This notion of equilibrium can be interpreted as steady-state of both a discovery and learning process., Comment: arXiv admin note: substantial text overlap with arXiv:1707.08761

Investigating interaction of the lumpy nature of educational investments and informational frictions on returns to and costs of education, I show that pessimistic beliefs can be self-confirmed in equilibrium. Among some of its consequences, I show that the commonly pursued research methods may not always identify the true underlying skill distributions.

I introduce a new framework to study environments with both structural and strategic uncertainty, different from Harsanyi’s ( 1967–1968) “Bayesian games”, that allows a researcher to test the robustness of Nash predictions while maintaining certain desirable restrictions on players’ beliefs. The solution concept applied to this environment is rationalizable conjectural equilibrium (RCE), which integrates both learning from feedback (in the spirit of self-confirming equilibrium) and from introspection (in the spirit of rationalizability). I provide an epistemic definition of RCE and obtain a characterization in terms of a procedure that generalizes iterated deletion of strategies that are not a best response.

Many models of learning in games implicitly or explicitly assume there are many agents in the role of each player. In principle this allows different agents in the same player role to have different beliefs and play differently, and this is known to occur in laboratory experiments. To explore the impact of this heterogeneity, along with the idea that subjects use their information about other players' payoffs, we define rationalizable partition-confirmed equilibrium (RPCE). We provide several examples to highlight the impact of heterogeneous beliefs, and show how mixed strategies can correspond to heterogeneous play in a large population. We also show that every heterogeneous-belief RPCE can be approximated by a RPCE in a model where every agent in a large pool is a separate player. Keywords: Rationalizability; Extensive-form games; Self-confirming equilibrium; Heterogeneous beliefs; Purification; Random matching, National Science Foundation (Grant SES-0646816), National Science Foundation (Grant SES-0951462), National Science Foundation (Grant SES-1258665)

This article argues that process tracing is a viable and suitable methodological alternative to probe the implications of formal models specifying how the dynamics of belief formation may systematically cause bargaining failures under uncertainty. I illustrate the argument with a brief case study of the failure of the European Defence Community in postwar Europe.

Many claims about political behavior are based on implicit assumptions about how people think. One such assumption, that political actors use identical conjectures when assessing others' strategies, is nested within applications of widely used game-theoretic equilibrium concepts. When empirical findings call this assumption into question, the self-confirming equilibrium (SCE) concept provides an alternate criterion for theoretical claims. We examine applications of SCE to political science. Our main example focuses on the claim of Feddersen and Pesendorfer that unanimity rule can lead juries to convict innocent defendants (1998. Convicting the innocent: The inferiority of unanimous jury verdicts under strategic voting.American Political Science Review92:23–35). We show that the claim depends on the assumption that jurors have identical beliefs about one another's types and identical conjectures about one another's strategies. When jurors' beliefs and conjectures vary in ways documented by empirical jury research, fewer false convictions can occur in equilibrium. The SCE concept can confer inferential advantages when actors have different beliefs and conjectures about one another.

Fudenberg and Levine (1993a) introduced the notion of self-confirming equilibrium, which is generally less restrictive than Nash equilibrium. Fudenberg and Levine also defined a concept of consistency, and claimed in their Theorem 4 that with consistency and other conditions on beliefs, a self-confirming equilibrium has a Nash equilibrium outcome. We provide a counterexample that disproves Theorem 4 and prove an alternative by replacing consistency with a more restrictive concept, which we call strong consistency. In games with observed deviators, self-confirming equilibria are strongly consistent self-confirming equilibria. Hence, our alternative theorem ensures that despite the counterexample, the corollary of Theorem 4 is still valid.

This study explores theoretical and macroeconomic implications of the self-confirming equilibrium in a standard growth model. When rational expectations are replaced by adaptive expectations, we prove that the self-confirming equilibrium is the same as the steady state rational expectations equilibrium, but that dynamics around the steady state are substantially different between the two equilibria. We show that, in contrast to Williams (2003), the differences are driven mainly by the lack of the wealth effect and the strengthening of the intertemporal substitution effect, not by escapes. As a result, adaptive expectations substantially alter the amplification and propagation mechanisms and allow technology shocks to exert much more impact on macroeconomic variables than do rational expectations.

We offer a game-theoretic framework that enables the analysis of situations whereby rational individuals with different beliefs and views of the world agree to a shared course of action. We introduce a new solution concept: a mutually acceptable course of action, which can be viewed as an (incomplete) contract or a social norm that free rational individuals would be willing to follow for their own diverse reasons. We show that by varying the degree of completeness of the underlying course of action, our concept can be related to commonly used solutions, such as perfect equilibrium, perfect Bayesian equilibrium, (rationalizable) self-confirming equilibrium, and rationalizable outcomes.

We study experimentally whether heterogeneity of behavior in the Centipede game can be interpreted as the result of a learning process of individuals with different preference types (more and less pro-social) and coarse information regarding the opponent's past behavior. We manipulate the quality of information feedbacks provided after each play. If subjects rely only on their personal database, long run behavior resembles a Self-confirming equilibrium whereby less pro-social types take at earlier nodes due to prediction errors. Aggregate information release decreases heterogeneity of behavior by increasing the passing rates of pro-selfs and play moves towards Bayesian Nash equilibrium.

Recent papers have analyzed how economies with adaptive agents may converge to and escape from self-confirming equilibria. These papers have imputed to agents a particular prior about drifting coefficients. In the context of a model of monetary policy, this paper analyzes dynamics that govern both convergence and escape under a more general class of priors for the government. We characterize how the shape of the prior influences possible cycles, convergence, and escapes. There are priors for which the E -stability condition is not enough to assure local convergence to a self-confirming equilibrium. Our analysis also isolates the source of differences in the sustainability of Ramsey inflation encountered in the analyses of Sims [Revista de Analisis Economico 3 (1988) 3] and Chung [PhD Thesis, University of Minnesota, 1990], on the one hand, and Cho, Williams, and Sargent [Rev. Econ. Stud. 69 (2002) 1], on the other.

This paper discusses the implications of learning theory for the analysis of games with a move by Nature. One goal is to illuminate the issues that arise when modeling situations where players are learning about the distribution of Nature's move as well as learning about the opponents' strategies. A second goal is to argue that quite restrictive assumptions are necessary to justify the concept of Nash equilibrium without a common prior as a steady state of a learning process.

Rationalizable partition-confirmed equilibrium (RPCE) describes the steady state outcomes of rational learning in extensive-form games, when rationality is common knowledge and players observe a partition of the terminal nodes. RPCE allows players to make inferences about unobserved play by others. We discuss the implications of this using numerous examples, and discuss the relationship of RPCE to other solution concepts in the literature.

We study experimentally whether heterogeneity of behavior in the Centipede game can be interpreted as the result of a learning process of individuals with different preference types (more and less pro-social) and coarse information regarding the opponent's past behavior. We manipulate the quality of information feedbacks provided after each play. If subjects rely only on their personal database, long run behavior resembles a Self-confirming equi- librium whereby less pro-social types take at earlier nodes due to prediction errors. Aggregate information release decreases heterogeneity of behavior by increasing the passing rates of pro-selfs and play moves towards Bayesian Nash equilibrium.

In order to model the subjective uncertainty of a player over the behavior strategies of an opponent, one must consider the player's beliefs about the opponent's play at information sets that the player thinks have probability zero. This corregendum uses “trembles” to provide a definition of the convex hull of a set of behavior strategies. This corrects a definition we gave in [E. Dekel, D. Fudenberg, and D. K. Levine, 1999, J. Econ. Theory 89 , 165–185], which led to two of the solution concepts we defined there not having the properties we intended. Journal of Economic Literature Classification Numbers: C72, D82, C610.

In a self-confirming equilibrium, each player correctly forecasts the actions that opponents will take along the equilibrium path, but may be mistaken about the way that opponents would respond to deviations. This paper develops a refinement of self-confirming equilibrium in which players use information about opponents' payoffs in forming beliefs about the way that opponents play off of the equilibrium path. We show that this concept is robust to payoff uncertainty. We also discuss its relationship to other concepts and show that it is closely related to assuming almost common certainty of payoffs in an epistemic model with independent beliefs. Journal of Economic Literature Classification Numbers: C72, D84.

How do individuals' spatial decisions affect the institutions for public goods provision over time? This paper describes a dynamic model in which the provision mechanism for a public project is itself the object of locational choice of individuals. Individuals in an ongoing society must choose between a location with a Majority Rule mechanism and one with a Voluntary Contribution mechanism. Each mechanism determines a funding decision for a local public project which is repeated over time. Generations of individuals asynchronously supercede their ‘parents’, creating an entry/exit process that allows individuals with possibly different beliefs to enter society. A self-confirming equilibrium (SCE) belief process describes an evolution of beliefs in this society consistent with a self-confirming equilibrium (Fudenberg and Levine, 1993) of the repeated location/provision game. It is shown that the process with belief mutation as new individuals enter society results in a globally absorbing state in which the Majority Rule mechanism is the unique survivor of the two.

In the paper, I examine free entry in homogeneous product markets and its social efficiency. Previous research on free entry in homogeneous product markets has shown that under Cournot oligopoly with fixed setup costs the free entry equilibrium always delivers excessive entry. In contrast, I demonstrate in this paper that free entry along with excessive entry might also lead to a socially insufficient number of firms when a demand parameter uncertainty is considered. My findings support the validity of the traditional wisdom in industrial organization that free entry is desirable for social efficiency and call for revision of restrictive entry regulation practices which been based on previous research findings.

I introduce a new framework to study environments with both structural and strategic uncertainty, different from Harsanyi's (1967-8) `Bayesian games', that allows a researcher to test the robustness of Nash predictions while maintaining certain desirable restrictions on players' beliefs. The solution concept applied to this environment is rationalizable conjectural equilibrium (RCE), which integrates both learning from feedback (in the spirit of self-confirming equilibrium) and from introspection (in the spirit of rationalizability). I provide an epistemic definition of RCE and obtain a characterization in terms of a procedure that generalizes iterated deletion of strategies that are not a best response.

Several studies show that evolution favors non-selfish preferences only if preference types are observable. We present a new evolutionary scenario applied to the Centipede Game, where we adopt self-confirming equilibrium to capture behavior. We show that altruism may be evolutionarily successful even if preferences are unobservable.