A Model of Behavior in Coordination Game Experiments

This paper constructs a structural model for behavior in expeiments where subjects play a simple coordination game repeatedly under a rotating partner scheme. The model assumes subjects' actions are stochastic best responses to beliefs about opponents' choices, and these beliefs update as subjects observe actual choices during the experiment. The model accounts for heterogeneity across subjects by regarding prior beliefs as random effects and estimating their distribution. Maximum likelihood estimates from experimental data suggest that distributions of initial beliefs vary across games, but in all games studied imply a convergence dynamic toward risk-dominant equilibrium.     

Stated beliefs versus inferred beliefs: A methodological inquiry and experimental test

Belief elicitation in game experiments may be problematic if it changes game play. We experimentally verify that belief elicitation can alter paths of play in a two-player repeated asymmetric matching pennies game. Importantly, this effect occurs only during early periods and only for players with strongly asymmetric payoffs, consistent with a cognitive/affective effect on priors that may serve as a substitute for experience. These effects occur with a common scoring rule elicitation procedure, but not with simpler (unmotivated) statements of expected choices of opponents. Scoring rule belief elicitation improves the goodness of fit of structural models of belief learning, and prior beliefs implied by such models are both stronger and more realistic when beliefs are elicited than when they are not. We also find that “inferred beliefs” (beliefs estimated from past observed actions of opponents) can predict observed actions better than the “stated beliefs” from scoring rule belief elicitation.     

Stated beliefs versus inferred beliefs: A methodological inquiry and experimental test

Belief elicitation in game experiments may be problematic if it changes game play. We experimentally verify that belief elicitation can alter paths of play in a two-player repeated asymmetric matching pennies game. Importantly, this effect occurs only during early periods and only for players with strongly asymmetric payoffs, consistent with a cognitive/affective effect on priors that may serve as a substitute for experience. These effects occur with a common scoring rule elicitation procedure, but not with simpler (unmotivated) statements of expected choices of opponents. Scoring rule belief elicitation improves the goodness of fit of structural models of belief learning, and prior beliefs implied by such models are both stronger and more realistic when beliefs are elicited than when they are not. We also find that “inferred beliefs” (beliefs estimated from past observed actions of opponents) can predict observed actions better than the “stated beliefs” from scoring rule belief elicitation.     

Random belief equilibrium in normal form games

In defining random belief equilibrium (RBE) in finite, normal form games we assume a player's beliefs about others' strategy choices are randomly drawn from a belief distribution that is dispersed around a central strategy profile, the focus. At an RBE: (1) Each chooses a best response relative to her beliefs. (2) Each player's expected choice coincides with the focus of the other players' belief distributions. RBE provides a statistical framework for estimation which we apply to data from three experimental games. We also characterize the limit-RBE as players' beliefs converge to certainty. When atoms in the belief distributions vanish in the limit, not all limit-RBE (called robust equilibria) are trembling hand perfect Nash equilibria and not all perfect equilibria are robust.     

How Much Common Belief Is Necessary for a Convention?

We study Bayesian coordination games in which players choose actions conditional on the realization of their respective signals. Due to differential information, the players do not have common knowledge that a particular game is being played. However, they do have common beliefs with specified probabilities concerning their environment. In our framework, any equilibrium set of rules must be simple enough so that the actions of all players are common belief with probability 1 at every state. Common belief with probability close to 1 will not do.Journal of Economic LiteratureClassification Numbers: C72, D82.     

Preferences and beliefs in a sequential social dilemma: a within-subjects analysis

In empirical analyses of games, preferences and beliefs are typically treated as independent. However, if beliefs and preferences interact, this may have implications for the interpretation of observed behavior. Our sequential social dilemma experiment allows us to separate different interaction channels. When subjects play both roles in such experiments, a positive correlation between first- and second-mover behavior is frequently reported. We find that the observed correlation primarily originates via an indirect channel, where second-mover decisions influence beliefs through a consensus effect, and the first-mover decision is a best response to these beliefs. Specifically, beliefs about second-mover cooperation are biased toward own second-mover behavior, and most subjects best respond to stated beliefs. However, we also find evidence for a direct, preference-based channel. When first movers know the true probability of second-mover cooperation, subjects' own second moves still have predictive power regarding their first moves.     

Who knows it is a game? On strategic awareness and cognitive ability

We examine strategic awareness in experimental games, that is, the question of whether subjects realize they are playing a game and thus have to form beliefs about others’ actions. We conduct a beauty contest game and elicit measures of cognitive ability and beliefs about others’ cognitive ability. We show that the effect of cognitive ability is highly non-linear. Subjects below a certain threshold choose numbers in the whole interval and their behavior does not correlate with beliefs about others’ ability. In contrast, subjects who exceed the threshold avoid choices above 50 and react very sensitively to beliefs about the cognitive ability of others.     

Belief formation: an experiment with outside observers

In this paper we study the belief formation processes of a group of outside observers making predictions about the actions of a player involved in a repeated game. We document four main results. First, there is substantial heterogeneity in the accuracy of our observers, with average accuracy being quite poor. Second, while there is no difference between the most and the least accurate observer in their initial beliefs, there are striking differences in their belief updating rules. The most accurate observers have a well-formulated model of player behavior, are good at best responding and quickly incorporate new information to their beliefs. The worst observers behave in an opposite manner on all three fronts. Third, when the game does not converge, subjects look beyond historical actions to make predictions and place more emphasis on forgone payoffs. Finally, we document that a “collective wisdom” emerges when our data are pooled across subjects and analyzed. Specifically, the accuracy of the group estimates becomes much higher than that of the average observer.     

Measuring agents’ reaction to private and public information in games with strategic complementarities

In games with strategic complementarities, public information about the state of the world has a larger impact on equilibrium actions than private information of the same precision, because public signals are more informative about the likely behavior of others. We present an experiment in which agents’ optimal actions are a weighted average of the fundamental state and their expectations of other agents’ actions. We measure the responses to public and private signals. We find that, on average, subjects put a larger weight on the public signal. In line with theoretical predictions, as the relative weight of the coordination component in a player’s utility increases, players put more weight on the public signal when making their choices. However, the weight is smaller than in equilibrium, which indicates that subjects underestimate the information contained in public signals about other players’ beliefs.     

Belief and higher-order belief in the centipede games: An experimental investigation

This paper experimentally explores the epistemic conditions behind people's non-equilibrium behaviour in the centipede games. We propose a novel design of laboratory experiment to elicit people's first- and second-order beliefs regarding their opponents' choices and beliefs. The measured beliefs, together with the choice data, help us to estimate people's level of rationality, belief of rationality and second-order belief of rationality. To examine how these epistemic variables are affected by the social-efficiency property of the classic increasing-sum centipede game, we revisit the constant-sum centipede and compare the measured epistemic conditions from the constant-sum with those from the classic centipede. We find that people's non-backward induction behaviour may be attributed to the diffusion of beliefs and higher-order beliefs in the increasing-sum centipede. We consider a behavioural model in which people's preferences for social efficiency are incorporated into the extended utility maximization problem. Our analytical and estimation results indicate that the presence of efficiency-oriented players and people's belief towards the uncertain portion of such type of players may play a part in the non-backward-induction outcomes in experimental centipede games.     

Mindreading and endogenous beliefs in games

We argue that a Bayesian explanation of strategic choices in games requires introducing a psychological theory of belief formation. We highlight that beliefs in epistemic game theory are derived from the actual choice of the players, and cannot therefore explain why Bayesian rational players should play the strategy they actually chose. We introduce the players’ capacity of mindreading in a game theoretical framework with the simulation theory, and characterise the beliefs that Bayes rational players could endogenously form in games. We show in particular that those beliefs need not be ratifiable, and therefore that rational players can form action-dependent beliefs.     

Self-confirming equilibrium and the Lucas critique

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.     

Self-referential thinking and equilibrium as states of mind in games: fMRI evidence

Sixteen subjects' brain activity were scanned using fMRI as they made choices, expressed beliefs, and expressed iterated 2nd-order beliefs (what they think others believe they will do) in eight games. Cingulate cortex and prefrontal areas (active in “theory of mind” and social reasoning) are differentially activated in making choices versus expressing beliefs. Forming self-referential 2nd-order beliefs about what others think you will do seems to be a mixture of processes used to make choices and form beliefs. In equilibrium, there is little difference in neural activity across choice and belief tasks; there is a purely neural definition of equilibrium as a “state of mind.” “Strategic IQ,” actual earnings from choices and accurate beliefs, is negatively correlated with activity in the insula, suggesting poor strategic thinkers are too self-focused, and is positively correlated with ventral striatal activity (suggesting that high IQ subjects are spending more mental energy predicting rewards).     

Between-game rule learning in dissimilar symmetric normal-form games

Rule learning posits that decision makers, rather than choosing over actions, choose over behavioral rules with different levels of sophistication. Rules are reinforced over time based on their historically observed payoffs in a given game. Past works on rule learning have shown that when playing a single game over a number of rounds, players can learn to form sophisticated beliefs about others. Here we are interested in learning that occurs between games where the set of actions is not directly comparable from one game to the next. We study a sequence of ten thrice-played dissimilar games. Using experimental data, we find that our rule learning model captures the ability of players to learn to reason across games. However, this learning appears different from within-game rule learning as previously documented. The main adjustment in sophistication occurs by switching from non-belief-based strategies to belief-based strategies. The sophistication of the beliefs themselves increases only slightly over time.     

Beliefs and (in)stability in normal-form games

Experimental Economics - In this paper, we use experimental data to study players’ stability in normal-form games where subjects have to report beliefs and choose actions. Subjects saw each...     

Knightian games and robustness to ambiguity

This paper introduces a notion of robustness to ambiguous beliefs for Bayesian Nash equilibria. An equilibrium is robust if the corresponding strategies remain approximately optimal for a class of games with ambiguous beliefs that results from an appropriately defined perturbation of the belief structure of the original non-ambiguous belief game. The robustness definition is based on a novel definition of equilibrium for games with ambiguous beliefs that requires equilibrium strategies to be approximate best responses for all measures that define a player's belief. Conditions are derived under which robustness is characterized by a newly defined strategic continuity property, which can be verified without reference to perturbations and corresponding ambiguous belief games.     

Out of your mind: Eliciting individual reasoning in one shot games

We experimentally investigate the fundamental element of the level-k model of reasoning, the level-0 actions and beliefs. We use data from a novel experimental design that allows us to obtain incentivised written accounts of individuals' reasoning. In particular, these accounts allow to infer level-0 beliefs. Level-0 beliefs are not significantly different from 50, and almost 60% of higher level players start their reasoning from a level-0 belief of exactly 50. We also estimate that around one third of the participants play non-strategically. The non-strategic level-0 actions are not uniformly distributed.     

Learning to alternate

The Individual Evolutionary Learning (IEL) model explains human subjects’ behavior in a wide range of repeated games which have unique Nash equilibria. Using a variation of ‘better response’ strategies, IEL agents quickly learn to play Nash equilibrium strategies and their dynamic behavior is like that of humans subjects. In this paper we study whether IEL can also explain behavior in games with gains from coordination. We focus on the simplest such game: the 2 person repeated Battle of Sexes game. In laboratory experiments, two patterns of behavior often emerge: players either converge rapidly to one of the stage game Nash equilibria and stay there or learn to coordinate their actions and alternate between the two Nash equilibria every other round. We show that IEL explains this behavior if the human subjects are truly in the dark and do not know or believe they know their opponent’s payoffs. To explain the behavior when agents are not in the dark, we need to modify the basic IEL model and allow some agents to begin with a good idea about how to play. We show that if the proportion of inspired agents with good ideas is chosen judiciously, the behavior of IEL agents looks remarkably similar to that of human subjects in laboratory experiments.     

Identification of firms’ beliefs in structural models of market competition

Firms make decisions under uncertainty and differ in their ability to collect and process information. As a result, in changing environments, firms have heterogeneous beliefs on the behaviour of other firms. This heterogeneity in beliefs can have important implications on market outcomes, efficiency and welfare. This paper studies the identification of firms’ beliefs using their observed actions—a revealed preference and beliefs approach. I consider a general structural model of market competition where firms have incomplete information and their beliefs and profits are nonparametric functions of decisions and state variables. Beliefs may be out of equilibrium. The framework applies both to continuous and discrete choice games and includes as particular cases models of competition in prices or quantities, auction models, entry games and dynamic games of investment decisions. I focus on identification results that exploit an exclusion restriction that naturally appears in models of competition: an observable variable that affects a firm's cost (or revenue) but does not have a direct effect on other firms’ profits. I present identification results under three scenarios—common in empirical industrial organization—on the data available to the researcher.     

Strategic sophistication of individuals and teams. Experimental evidence

Many important decisions require strategic sophistication. We examine experimentally whether teams act more strategically than individuals. We let individuals and teams make choices in simple games, and also elicit first- and second-order beliefs. We find that teams play the Nash equilibrium strategy significantly more often, and their choices are more often a best response to stated first order beliefs. Distributional preferences make equilibrium play less likely. Using a mixture model, the estimated probability to play strategically is 62% for teams, but only 40% for individuals. A model of noisy introspection reveals that teams differ from individuals in higher order beliefs.