Strategic thinking, best-response, and mutual consistency (equilibrium) are three key modeling principles in noncooperative game theory. This paper relaxes mutual consistency to predict how players are likely to behave in one-shot games before they can learn to equilibrate. We introduce a one-parameter cognitive hierarchy (CH) model to predict behavior in one-shot games, and initial conditions in repeated games. The CH approach assumes that players use k steps of reasoning with frequency f(k). Zero-step players randomize. Players using k ( ≥ 1) steps best respond given partially rational expectations about what players doing 0 through k − 1 steps actually choose. A simple axiom which expresses the intuition that steps of thinking are increasingly constrained by working memory, implies that f(k) has a Poisson distribution (characterized by a mean number of thinking steps τ). The CH model converges to dominance-solvable equilibria when τ is large, predicts monotonic entry in binary entry games for τ < 1.25, and predicts effects of group size which are not predicted by Nash equilibrium. Best-fitting values of τ have an interquartile range of (.98,2.21) and a median of 1.55 across 60 experimental samples of matrix games, entry games and mixed-equilibrium games. The CH model also has economic value because subjects would have raised their earnings substantially if they had best-responded to model forecasts instead of making the choices they did. 1

The existence of a beauty premium in the labor market and the malefemale wage gap suggests that appearance can matter in the real world. We explore beauty and gender in a public goods experiment and find similar effects. We find a beauty premium, even though beautiful people contribute, on average, no more or less than others. The beauty premium, however, disappears when we provide information on individual contributions, and becomes a beauty penalty. Players seem to expect beautiful people to be more cooperative. Relative to these expectations, they appear more selfish, which in turn results in less cooperation by others. These appear to be clear examples of stereotyping. We also find a substantial benefit tobeing male, especially with information. This is primarily due to men being better “leaders.” Men tend to make large contributions, and people follow their example and give more in later rounds.

A One-parameter Theory of Learning in Games Functional experience weighted attraction (fEWA) is a one-parameter theory of learning in games. It approximates the free parameters in an earlier model (EWA) with functions of experience. The theory was originally tested on seven different games and compared to four other learning and equilibrium theories, then four more games were added. Generally fEWA or parameterized EWA predict best out-of-sample, but one kind of reinforcement learning predicts well in games with mixed-strategy equilibrium. Of the learning models, belief learning models fit worst but fit better than noisy (quantal response) equilibrium models. The economic value of a theory is measured by how much more subjects would have earned if they followed the theory's recommendations. Most learning theories add value (though equilibrium theories often subtract value) and fEWA and EWA usually add the most value.

Using data on one-shot games, we investigate the assumption that players respond to underlying expectations about their opponent’s behavior. In our laboratory experiments, subjects play a set of 14 two-person 3x3 games, and state first order beliefs about their opponent’s behavior. The sets of responses in the two tasks are largely inconsistent. Rather, we find evidence that the subjects perceive the games differently when they (i) choose actions, and (ii) state beliefs – they appear to pay more attention to the opponent’s incentives when they state beliefs than when they play the games. On average, they fail to best respond to their own stated beliefs in almost half of the games. The inconsistency is confirmed by estimates of a unified statistical model that jointly uses the actions and the belief statements. There, we can control for noise, and formulate a statistical test that rejects consistency. Effects of the belief elicitation procedure on subsequent actions are mostly insignificant.