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We introduce a new model of repeated games in large populations with random matching, overlapping generations, and limited records of past play. We prove that steady-state equilibria exist under general conditions on records. When the updating of a player’s record can depend on the actions of both players in a match, any strictly individually rational action can be supported in a steady-state equilibrium. When record updates can depend only on a player’s own actions, fewer actions can be supported. Here, we focus on the prisoner’s dilemma and restrict attention to strict equilibria that are coordination-proof, meaning that matched partners never play a Pareto-dominated Nash equilibrium in the one-shot game induced by their records and expected continuation payoffs. Such equilibria can support full cooperation if the stage game is either “strictly supermodular and mild” or “strongly supermodular,” and otherwise permit no cooperation at all. The presence of “supercooperator” records, where a player cooperates against any opponent, is crucial for supporting any cooperation when the stage game is “severe.”

We study how an agent learns from endogenous data when their prior belief is misspecified. We show that only uniform Berk–Nash equilibria can be long‐run outcomes, and that all uniformly strict Berk–Nash equilibria have an arbitrarily high probability of being the long‐run outcome for some initial beliefs. When the agent believes the outcome distribution is exogenous, every uniformly strict Berk–Nash equilibrium has positive probability of being the long‐run outcome for any initial belief. We generalize these results to settings where the agent observes a signal before acting.

We summarize our recent work that uses machine learning techniques as a complement to theoretical modeling, rather than a substitute for it. The key concepts are those of the completeness and restrictiveness of a model. A theory's completeness is how much it improves predictions over a naive baseline, relative to how much improvement is possible. When a theory is relatively incomplete, machine learning algorithms can help reveal regularities that the theory doesn't capture, and thus lead to the construction of theories that make more accurate predictions. Restrictiveness measures a theory's ability to match arbitrary hypothetical data: A very unrestrictive theory will be complete on almost any data, so the fact that it is complete on the actual data is not very instructive. We algorithmically quantify restrictiveness by measuring how well the theory approximates randomly generated behaviors. Finally, we propose "algorithmic experimental design" as a method to help select which experiments to run.

We add the assumption that players know their opponents' payoff functions and rationality to a model of non-equilibrium learning in signaling games. Agents are born into player roles and play against random opponents every period. Inexperienced agents are uncertain about the prevailing distribution of opponents' play, but believe that opponents never choose conditionally dominated strategies. Agents engage in active learning and update beliefs based on personal observations. Payoff information can refine or expand learning predictions, since patient young senders' experimentation incentives depend on which receiver responses they deem plausible. We show that with payoff knowledge, the limiting set of long-run learning outcomes is bounded above by rationality-compatible equilibria (RCE), and bounded below by uniform RCE. RCE refine the Intuitive Criterion (Cho and Kreps, 1987) and include all divine equilibria (Banks and Sobel, 1987). Uniform RCE sometimes but not always exists, and implies universally divine equilibrium., Comment: This material was previously part of a larger paper titled "Type-Compatible Equilibria in Signalling Games," which split into two smaller papers: "Learning and Type Compatibility in Signaling Games" and "Payoff Information and Learning in Signaling Games."

We model the joint distribution of choice probabilities and decision times in binary decisions as the solution to a problem of optimal sequential sampling, where the agent is uncertain of the utility of each action and pays a constant cost per unit time for gathering information. We show that choices are more likely to be correct when the agent chooses to decide quickly, provided the agent’s prior beliefs are correct. This better matches the observed correlation between decision time and choice probability than does the classical drift-diffusion model (DDM), where the agent knows the utility difference between the choices. (JEL C41, D11, D12, D83)

Which equilibria will arise in signaling games depends on how the receiver interprets deviations from the path of play. We develop a micro-foundation for these off-path beliefs, and an associated equilibrium refinement, in a model where equilibrium arises through non-equilibrium learning by populations of patient and long-lived senders and receivers. In our model, young senders are uncertain about the prevailing distribution of play, so they rationally send out-of-equilibrium signals as experiments to learn about the behavior of the population of receivers. Differences in the payoff functions of the types of senders generate different incentives for these experiments. Using the Gittins index (Gittins (1979)), we characterize which sender types use each signal more often, leading to a constraint on the receiver's off-path beliefs based on “type compatibility” and hence a learning-based equilibrium selection. ©2018 The Econometric Society, NSF grant (SES 1643517)

Game theory has been a huge success in economics. Many important questions have been answered, and game theoretic methods are now central to much economic investigation. We suggest areas where further advances are important, and argue that models of learning are a promising route for improving and widening game theory's predictive power while preserving the successes of game theory where it already works well. We emphasize in particular the need for better understanding of the speed with which learning takes place.

We characterize a generalization of discounted logistic choice that incorporates a parameter to capture different views the agent might have about the costs and benefits of larger choice sets. The discounted logit model used in the empirical literature is the special case that displays a “preference for flexibility†in the sense that the agent always prefers to add additional items to a menu. Other cases display varying levels of “choice aversion,†where the agent prefers to remove items from a menu if their ex ante value is below a threshold. We show that higher choice aversion, as measured by dislike of bigger menus, also corresponds to an increased preference for putting off decisions as late as possible.

Player-Compatible Equilibrium (PCE) imposes cross-player restrictions on the magnitudes of the players' “trembles” onto different strategies. These restrictions capture the idea that trembles correspond to deliberate experiments by agents who are unsure of the prevailing distribution of play. PCE selects intuitive equilibria in a number of examples where trembling-hand perfect equilibrium ( Selten, 1975 ) and proper equilibrium ( Myerson, 1978 ) have no bite. We show that rational learning and weighted fictitious play imply our compatibility restrictions in a steady-state setting.

We let participants indicate their intended action in a repeated game experiment where actions are implemented with errors. Even though communication is cheap talk, we find that the majority of messages were honest (although the majority of participants lied at least occasionally). As a result, communication has a positive effect on cooperation when the payoff matrix makes the returns to cooperation high; when the payoff matrix gives a lower return to cooperation, communication reduces overall cooperation. These results suggest that cheap talk communication can promote cooperation in repeated games, but only when there is already a self-interested motivation to cooperate. ©2017, National Science Foundation Grant (no. SES- 1258665)

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)

We use machine learning to uncover regularities in the initial play of matrix games. We first train a prediction algorithm on data from past experiments. Examining the games where our algorithm predicts correctly, but existing economic models don’t, leads us to add a parameter to the best performing model that improves predictive accuracy. We obtain even better predictions with a hybrid model that uses a decision tree to decide game-by-game which of two economic models to use for prediction. Finally, we explore the usefulness of crowd-sourced predictions for making better predictions, and for discovering additional relevant game features.

We study the design of careers by a principal who trains a cash-constrained agent, or apprentice, who is free to walk away at any time. The principal specifies time paths of knowledge transfer, effort provision, and task allocation, subject to the apprentice's continued participation. In the optimal contract, the apprentice pays for training by working for low or no wages and working inefficiently hard. The apprentice can work on both "skilled" (knowledge-complementary) and "unskilled" (knowledge-independent) tasks. If the principal specifies inefficiently much skilled effort at any time, she shortens the apprenticeship compared to its length when skilled effort is efficient. Otherwise, she specifies inefficiently much unskilled effort throughout and leaves the apprenticeship length unchanged. We then consider the effect of regulations that limit how hard the apprentice can work and how long the apprenticeship can last.

We reexamine the effects of the anchoring manipulation of Ariely, Loewenstein, and Prelec (2003) on the evaluation of common market goods and find very weak anchoring effects. We perform the same manipulation on the evaluation of binary lotteries, and find no anchoring effects at all. This suggests limits on the robustness of anchoring effects. (JEL C91, D12, D44)

We study the perfect type-contingently public ex-post equilibrium (PTXE) of repeated games with incomplete information where players observe imperfect public signals of the actions and the map from actions to signal distributions is itself unknown. The PTXE payoffs when players are patient are determined by the intersection of the maximal half spaces in various directions; we focus on the “cross-state” directions that consider payoffs in two or more states. We develop conditions under which the maximal half spaces in these directions impose no constraints on the equilibrium set, so that equilibrium play can be as if the players have learned the state. We use these conditions to provide a sufficient condition for the folk theorem, and a characterization of the PTXE payoffs in games with a known monitoring structure.

We develop a dual-self model of self-control that is compatible with modern dynamic macroeconomic theory and evidence. We show that a convex cost of self-control explains a wide range of behavioral anomalies concerning risk, including the Allais paradox, and also explains the observed interaction between risk and delay. We calibrate the model to obtain a quantitative fit. We find that most of the data can be explained with subjective interest rates in the range of 1–7 percent, short-run relative risk aversion of about two, and a time horizon of one day for the short-run self. (JEL D11, D44, D81)

This paper studies repeated games with imperfect public monitoring where the play- ers are uncertain both about the payoff functions and about the relationship between the distribution of signals and the actions played. We introduce the concept of perfect public ex post equilibrium (PPXE), and show that it can be characterized with an exten- sion of the techniques used to study perfect public equilibria. We develop identifiability conditions that are sufficient for a folk theorem; these conditions imply that there are PPXE in which the payoffs are approximately the same as if the monitoring structure and payoff functions were known. Finally, we define perfect type-contingently public ex post equilibria (PTXE), which allows players to condition their actions on their initial private information, and we provide its linear programming characterization.

We study repeated games with frequent actions and frequent imperfect public signals, where the signals are the aggregate of many discrete events, such as sales or tasks. The high-frequency limit of the equilibrium set depends on both the probability law governing the discrete events and on how many events are aggregated into a single signal. When the underlying events have a binomial distribution, the limit equilibria correspond to the equilibria of the associated continuous-time game with diffusion signals, but other event processes that aggregate to a diffusion limit can have a different set of limit equilibria. Thus the continuous-time game need not be a good approximation of the high-frequency limit whenever the underlying events have three or more possible values.

We provide a characterization of the limit set of perfect public equilibrium payoffs of repeated games with imperfect public monitoring as the discount factor goes to one. Our result covers general stage games including those that fail a “full-dimensionality†condition that had been imposed in past work. It also provides a characterization of the limit set when the strategies are restricted in a way that endogenously makes the full-dimensionality condition fail, as in the strongly symmetric equilibrium studied by Abreu [Abreu, D., 1986. Extremal equilibria of oligopolistic supergames. J. Econ. Theory 39, 191–228] and Abreu et al. [Abreu, D., Pearce, D., Stacchetti, E., 1986. Optimal cartel equilibria with imperfect monitoring. J. Econ. Theory 39, 251–269]. Finally, we use our characterization to give a sufficient condition for the exact achievability of first-best outcomes. Equilibria of this type, for which all continuation payoffs lie on the Pareto frontier, have a strong renegotiation-proofness property: regardless of the history, players can never unanimously prefer another equilibrium.

We show that the use of communications to coordinate equilibria generates a Nash-threats folk theorem in two-player games with “almost public” information. The results generalize to the n-person case. However, the two-person case is more difficult because it is not possible to sustain equilibria by comparing the reports of different players, and using these “third parties” to effectively enforce contracts.

Evolutionary models in which N players are repeatedly matched to play a game have “fast convergence” to a set A if the models both reach A quickly and leave A slowly, where “quickly” and “slowly” refer to whether the expected hitting and exit times remain bounded when N tends to infinity. We provide simple and general Lyapunov criteria which are sufficient for reaching quickly and leaving slowly. We use these criteria to determine aspects of learning models that promote fast convergence.

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.

This paper studies whether agents must agglomerate at a single location in a class of models of two-sided interaction. In these models there is an increasing returns effect that favors agglomeration, but also a crowding or market-impact effect that makes agents prefer to be in a market with fewer agents of their own type. We show that such models do not tip in the way the term is commonly used. Instead, they have a broad plateau of equilibria with two active markets, and tipping occurs only when one market is below a critical size threshold. Our assumptions are fairly weak, and are satisfied in Krugman's model of labor market pooling, a heterogeneous-agent version of Pagano's asset market model, and Ellison, Fudenberg, and Mobius' model of competing auctions. I. INTRODUCTION Many economic activities are agglomerated: people are crowded into a small fraction of the Earth's land mass; individual industries are geographically concentrated; trading is concen- trated in a few marketplaces. The standard way to account for concentration (and the arbitrariness of where activity concen- trates) has been to propose "tipping models" with three equilibria: one with most activity at location A; one with most activity at B; and an unstable "knife-edge" equilibrium with exactly half of the activity in each market. At the core of most models of agglomeration is some type of increasing returns or "scale effect" that favors the emergence of a single dominant site. In some of these models, all agents are ex ante identical, and they all prefer to be part of the larger market. In such cases, it is clear that an equilibrium where all markets are exactly the same size is an unstable knife-edge; every agent would rather be in a market with 51 percent of the agents than in a market with 49 percent, so any departure from exactly equal sizes leads to "tipping" to a single site. In other models, there are differences between the agents, and while all agents prefer larger markets, they also prefer markets where the other agents are less like themselves. For example, firms like markets with an excess of workers, upstream firms like markets with many downstream

We model the joint distribution of choice probabilities and decision times in binary choice tasks as the solution to a problem of optimal sequential sampling, where the agent is uncertain of the utility of each action and pays a constant cost per unit time for gathering information. In the resulting optimal policy, the agent's choices are more likely to be correct when the agent chooses to decide quickly, provided that the agent's prior beliefs are correct. For this reason it better matches the observed correlation between decision time and choice probability than does the classical drift-diffusion model, where the agent is uncertain which of two actions is best but knows the utility difference between them.

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.

We examine games played by a single large player and a large number of opponents who are small, but not anonymous. If the play of the small players is observed with noise, and if the number of actions the large player controls is bounded as the number of small players grows, the equilibrium set converges to that of the game where there is a continuum of small players. This paper extends previous work on the negligibility of small players by dropping the assumption that small players' actions are “anonymous.” That is, we allow each small player's actions to be observed separately, instead of supposing that the small players' actions are only observed through their effect on an aggregate statistic.Journal of Economic LiteratureClassification Numbers: C72, C73.

We examine the long-term implication of two models of learning with recency bias: recursive weights and limited memory. We show that both models generate similar beliefs and that both have a weighted universal consistency property. Using the limited-memory model we produce learning procedures that both are weighted universally consistent and converge with probability one to strict Nash equilibrium. We are grateful to the National Science Foundation (Grants SES-08-51315 and 1258665) for financial support.

We extend the folk theorem of repeated games to two settings in which players' information about others' play arrives with stochastic lags. In our first model, signals are almost-perfect if and when they do arrive, that is, each player either observes an almost-perfect signal of period-t play with some lag or else never sees a signal of period-t play. The second model has the same lag structure, but the information structure corresponds to a lagged form of imperfect public monitoring, and players are allowed to communicate via cheap-talk messages at the end of each period. In each case, we construct equilibria in “delayed-response strategies,†which ensure that players wait long enough to respond to signals that with high probability all relevant signals are received before players respond. To do so, we extend past work on private monitoring to obtain folk theorems despite the small residual amount of private information.

In some experiments rational players who understand the structure of the game could improve their payoff. We hound the size of the observed losses in several such experiments, lb do this, we suppose that observed play resembles an equilibrium because players learn about their opponents' play. Consequently, in an extensive-form game, some actions that are not optimal given the true distribution of opponents' play could be optimal given available information. We find that average losses are small: $0.03 to $0.64 per player with stakes between $2 and $30. In one of the three experiments we examine, this also implies a narrow range of outcome.

Different discrete time triangular arrays representing a noisy signal of players' activities can lead to the same limiting diffusion process yet lead to different limit equilibria. Whether the limit equilibria are equilibria of the limiting continuous time game depends on the limit properties of test statistics for whether a player has deviated. We provide an estimate of the tail probabilities along these arrays that allows us to determine the asymptotic behavior of the best test and thus of the best equilibrium.

We consider the robustness of extensive form mechanisms to deviations from common knowledge about the state of nature, which we refer to as information perturbations. First, we show that even under arbitrarily small information perturbations the Moore-Repullo mechanism does not yield (even approximately) truthful revelation and that in addition the mechanism has sequential equilibria with undesirable outcomes. More generally, we prove that any extensive form mechanism is fragile in the sense that if a non-Maskin monotonic social objective can be implemented with this mechanism, then there are arbitrarily small information perturbations under which an undesirable sequential equilibrium also exists. Finally, we argue that outside options can help improve efficiency in asymmetric information environments, and that these options can be thought of as reflecting ownership of an asset. JEL Codes: C72, D23, D78, D82 Copyright 2012, Oxford University Press.

The dual self-model of self-control with one-period lived short-run selves is excessively sensitive to the timing of shocks and to the interpolation of additional “noaction†time periods in between the dates when decisions are made. We show that when short-run selves have a random length of time this excess sensitivity goes away. We consider both linear and convex cost of self-control models, illustrating the theory through a series of examples. We examine when opportunities to consume will be avoided or delayed; we consider the way in which the marginal interest declines with delay, and we examine how preference “reversals†depend on the timing of information. To accommodate the combination of short time periods and convex costs of self control we extend the model to treat willpower as a cognitive resource that is limited in the short run.

We study the experimental play of the repeated prisoner's dilemma when intended actions are implemented with noise. In treatments where cooperation is an equilibrium, subjects cooperate substantially more than in treatments without cooperative equilibria. In all settings there was considerable strategic diversity, indicating that subjects had not fully learned the distribution of play. Furthermore, cooperative strategies yielded higher payoffs than uncooperative strategies in the treatments with cooperative equilibria. In these treatments successful strategies were “lenient” in not retaliating for the first defection, and many were “forgiving” in trying to return to cooperation after inflicting a punishment. (JEL C72, C73, D81)

We consider a game between a patient player 1 and a nonmyopic but less patient opponent, player 2. As usual in reputation models, we suppose that the patient player's type is private information, and that he may be a commitment type who is locked into playing a particular strategy. We investigate the extent to which an uncommitted or normal type of patient player can exploit his less patient opponent's uncertainty to maintain a reputation for playing like a commitment type.

We examine the role of off-path “superstitions†in macro-economics, and show how a false belief about off-path play is the key element underlying both the Lucas Critique and the game-theoretic concept of self-confirming equilibrium. However, the impact of false beliefs in these two cases is different: In the Lucas case, a policy maker's incorrect beliefs about off-path play can lead to the adoption of mistaken policy innovation. However, the consequences of such an innovation provide evidence that the belief that motivated them was wrong. In contrast, play may never escape an undesirable self-confirming equilibrium, as the action implied by the mistaken belief does not generate data that contradicts it; escape from the self-confirming equilibrium requires that players do a sufficient amount of experimentation with off-path actions.

In a repeated game with imperfect public information, the set of equilibria depends on the way that the distribution of public signals varies with the players' actions. Recent research has focused on the case of "frequent monitoring," where the time interval between periods becomes small. Here we study a simple example of a commitment game with a long-run and short-run player in order to examine different specifications of how the signal distribution depends upon period length. We give a simple criterion for the existence of efficient equilibrium, and show that the efficiency of the equilibria that can be supported depends in an important way on the effect of the player's actions on the variance of the signals, and whether extreme values of the signals are "bad news" of "cheating" behavior, or "good news" of "cooperative" behavior.

We analyze a class of imitation dynamics with mutations for games with any finite number of actions, and give conditions for the selection of a unique equilibrium as the mutation rate becomes small and the population becomes large. Our results cover the multiple-action extensions of the aspiration-and-imitation process of Binmore and Samuelson [Muddling through: noisy equilibrium selection, J. Econ. Theory 74 (1997) 235–265] and the related processes proposed by Benai¨m and Weibull [Deterministic approximation of stochastic evolution in games, Econometrica 71 (2003) 873–903] and Traulsen et al. [Coevolutionary dynamics: from finite to infinite populations, Phys. Rev. Lett. 95 (2005) 238701], as well as the frequency-dependent Moran process studied by Fudenberg et al. [Evolutionary game dynamics in finite populations with strong selection and weak mutation, Theoretical Population Biol. 70 (2006) 352–363]. We illustrate our results by considering the effect of the number of periods of repetition on the selected equilibrium in repeated play of the prisoner's dilemma when players are restricted to a small set of simple strategies.

We show the existence of a pure strategy, symmetric, increasing equilibrium in double auction markets with correlated, conditionally independent private values and many participants. The equilibrium we find is arbitrarily close to fully revealing as the market size grows. Our results provide strategic foundations for price-taking behavior in large markets.

Each period, a player must choose an action without knowing the outcome that will be chosen by "Nature," according to an unknown and possibly history-dependent stochastic rule. We discuss have a class of procedures that assign observations to categories, and prescribe a simple randomized variation of fictitious play within each category. These procedures are "conditionally consistent," in the sense of yielding almost as high a time-average payoff as could be obtained if the player chose knowing the conditional distributions of actions given categories. Moreover given any alternative procedure, there is a conditionally consistent procedure whose performance is no more than epsilon worse regardless of the discount factor. Cycles can persist if all players classify histories in the same way; however in an example, where players classify histories differently, the system converges to a Nash equilibrium. We also argue that in the long run the time-average of play should resemble a correlated equilibrium.

Forecasts are said to be calibrated if the frequency predictions are approximately correct. This is a refinement of an idea first introduced by David Blackwell in 1955. We show that “ K -initialized myopic strategies” are approximately calibrated when K is large. These strategies first “initialize” by making each forecast exactly K times, and thereafter play, in each period t , the minmax strategy in a static game. Journal of Economic Literature Classification Numbers: C72, C73, C63, D83.

This paper shows that larger auctions are more efficient than smaller ones, but that despite this scale effect, two competing and otherwise identical markets or auction sites of different sizes can coexist in equilibrium. We find that the range of equilibrium market sizes depends on the aggregate buyer—seller ratio, and also whether the markets are especially thin.

We study the evolution of the continuous-time replicator dynamics when payoffs are subject to aggregate shocks that take the form of a Wiener process. In the absence of “mutation”, the system need not have an ergodic distribution. With mutation, the system does have an ergodic distribution. In the limit as the mutation rate and the variance of the shocks converge to zero, this distribution concentrates on the risk-dominant equilibrium. This result is not, however, robust to changes in the underlying deterministic dynamics.