A Cognitive Hierarchy Theory of One-shot Games

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, mixed-equilibrium games and entry 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.