A cognitive hierarchy theory of one-shot games: Some preliminary results

Strategic thinking, best-response, and mutual consistency (equilibrium) are three key modelling principles in noncooperative game theory. This paper relaxes mutual consistency to predict how players are likely to behave in 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 (.98,2.40) and a median of 1.65 across 80 experimental samples of matrix games, entry games, mixed-equilibrium games, and dominance-solvable p-beauty contests. 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]  J. Robinson AN ITERATIVE METHOD OF SOLVING A GAME , 1951, Classics in Game Theory.

[2]  O. H. Brownlee,et al.  ACTIVITY ANALYSIS OF PRODUCTION AND ALLOCATION , 1952 .

[3]  N. Dalkey EQUIVALENCE OF INFORMATION PATTERNS AND ESSENTIALLY DETERMINATE GAMES , 1952 .

[4]  A. W. Tucker,et al.  Advances in game theory , 1964 .

[5]  D M Messick,et al.  Interdependent decision strategies in zero-sum games: a computer-controlled study. , 1967, Behavioral science.

[6]  S. Vajda Some topics in two-person games , 1971 .

[7]  J. Harsanyi The tracing procedure: A Bayesian approach to defining a solution forn-person noncooperative games , 1975 .

[8]  Patrick D. Larkey,et al.  Subjective Probability and the Theory of Games , 1982 .

[9]  B. Bernheim Rationalizable Strategic Behavior , 1984 .

[10]  R. Roll Orange Juice and Weather , 1984 .

[11]  David Pearce Rationalizable Strategic Behavior and the Problem of Perfection , 1984 .

[12]  A. Shaked,et al.  Testing non-cooperative bargaining theory: a preliminary study , 1985 .

[13]  J. Tirole ASSET BUBBLES AND OVERLAPPING GENERATIONS , 1985 .

[14]  Eugene B. Brody,et al.  A Psychological Perspective , 1986 .

[15]  Robert E. Lucas,et al.  Adaptive Behavior and Economic Theory , 1986 .

[16]  B. O'Neill Nonmetric test of the minimax theory of two-person zerosum games. , 1987, Proceedings of the National Academy of Sciences of the United States of America.

[17]  Ken Binmore,et al.  Modeling Rational Players: Part II , 1987, Economics and Philosophy.

[18]  V. Smith,et al.  Bubbles, Crashes, and Endogenous Expectations in Experimental Spot Asset Markets , 1988 .

[19]  Daniel Kahneman,et al.  EXPERIMENTAL ECONOMICS: A PSYCHOLOGICAL PERSPECTIVE , 1988 .

[20]  John C. Harsanyi,et al.  Общая теория выбора равновесия в играх / A General Theory of Equilibrium Selection in Games , 1989 .

[21]  Robert W. Rosenthal,et al.  A bounded-rationality approach to the study of noncooperative games , 1989 .

[22]  Colin Camerer Behavioral Game Theory , 1990 .

[23]  John Nachbar “Evolutionary” selection dynamics in games: Convergence and limit properties , 1990 .

[24]  Andrei Shleifer,et al.  Equilibrium Short Horizons of Investors and Firms , 1990 .

[25]  Andrew Schotter,et al.  A LABORATORY INVESTIGATION OF MULTI-PERSON RATIONALITY AND PRESENTATION EFFECTS , 1994 .

[26]  D. Stahl Evolution of Smart n Players , 1991 .

[27]  Johanna Helena Kerstholt,et al.  Insights in decision making: A tribute to Hillel J. Einhorn , 1991 .

[28]  Jordi Brandts,et al.  An Experimental Test of Equilibrium Dominance in Signaling Games , 1992 .

[29]  A. Rapoport,et al.  Mixed strategies in strictly competitive games: A further test of the minimax hypothesis , 1992 .

[30]  Ken Binmore,et al.  Frontiers of game theory , 1993 .

[31]  Keith Weigelt,et al.  Convergence in experimental double auctions for stochastically live assets , 1993 .

[32]  Colin Camerer,et al.  Cognition and framing in sequential bargaining for gains and losses , 1993 .

[33]  Dale O. Stahl,et al.  Evolution of Smartn Players , 1993 .

[34]  R. Bloomfield Learning a mixed strategy equilibrium in the laboratory , 1994 .

[35]  Jörgen W. Weibull,et al.  Evolutionary Game Theory , 1996 .

[36]  J. Banks,et al.  An Experimental Analysis of Nash Refinements in Signaling Games , 1994 .

[37]  Dilip Mookherjee,et al.  Learning behavior in an experimental matching pennies game , 1994 .

[38]  Ken Binmore,et al.  Learning to be imperfect: The ultimatum game , 1995 .

[39]  George Loewenstein,et al.  Biased Judgments of Fairness in Bargaining , 1995 .

[40]  D. Stahl,et al.  On Players' Models of Other Players: Theory and Experimental Evidence , 1995 .

[41]  A. Roth,et al.  Learning in Extensive-Form Games: Experimental Data and Simple Dynamic Models in the Intermediate Term* , 1995 .

[42]  J. Ochs Games with Unique, Mixed Strategy Equilibria: An Experimental Study , 1995 .

[43]  R. McKelvey,et al.  Quantal Response Equilibria for Normal Form Games , 1995 .

[44]  Colin Camerer Individual Decision Making , 2020, The Handbook of Experimental Economics.

[45]  Teck-Hua Ho,et al.  Task complexity, equilibrium selection, and learning: an experimental study , 1996 .

[46]  G. Loewenstein,et al.  Behavioral Law and Economics: Explaining Bargaining Impasse: The Role of Self-serving Biases , 1997 .

[47]  Dilip Mookherjee,et al.  Learning and Decision Costs in Experimental Constant Sum Games , 1997 .

[48]  J. Friedman,et al.  Boundedly Rational Nash Equilibrium: A Probabilistic Choice Approach☆☆☆ , 1997 .

[49]  R. Selten Features of experimentally observed bounded rationality , 1998 .

[50]  Vijay Krishna,et al.  On the Convergence of Fictitious Play , 1998, Math. Oper. Res..

[51]  D. Stahl Is step-j thinking an arbitrary modelling restriction or a fact of human nature? , 1998 .

[52]  R. McKelvey,et al.  Quantal Response Equilibria for Extensive Form Games , 1998 .

[53]  D. Fudenberg,et al.  The Theory of Learning in Games , 1998 .

[54]  Colin Camerer,et al.  Iterated Dominance and Iterated Best-Response in Experimental P-Beauty Contests , 1998 .

[55]  Colin Camerer,et al.  Overconfidence and Excess Entry: An Experimental Approach , 1999 .

[56]  Canice Prendergast The Provision of Incentives in Firms , 1999 .

[57]  Miguel A. Costa-Gomes,et al.  Cognition and Behavior in Normal-Form Games: An Experimental Study , 1998 .

[58]  J K Goeree,et al.  Stochastic game theory: for playing games, not just for doing theory. , 1999, Proceedings of the National Academy of Sciences of the United States of America.

[59]  Colin F. Camerer,et al.  Strategic Learning and Teaching , 2000 .

[60]  Roberto A. Weber,et al.  The effects of payoff magnitude and heterogeneity on behavior in 2 x 2 games with unique mixed strategy equilibria , 2000 .

[61]  Colin Camerer,et al.  Sophisticated EWA Learning and Strategic Teaching in Repeated Games , 2000 .

[62]  A. Rapoport,et al.  Mixed strategies and iterative elimination of strongly dominated strategies: an experimental investi , 2000 .

[63]  Kevin McCabe,et al.  The Impact of Exchange Context on the Activation of Equity in Ultimatum Games , 2000 .

[64]  Rachel Croson,et al.  Thinking like a game theorist: factors affecting the frequency of equilibrium play , 2000 .

[65]  K. Binmore,et al.  Does Minimax Work? An Experimental Study , 2001 .

[66]  Ariel Rubinstein,et al.  Correct Belief, Wrong Action and a Puzzling Gender Difference , 2001 .

[67]  Fang-Fang Tang,et al.  Anticipatory learning in two-person games: some experimental results , 2001 .

[68]  Georg Weizsäcker Ignoring the Rationality of Others: Evidence from Experimental Normal-form Games , 2001 .

[69]  C. Plott,et al.  NONSPECULATIVE BUBBLES IN EXPERIMENTAL ASSET MARKETS: LACK OF COMMON KNOWLEDGE OF RATIONALITY VS. ACTUAL IRRATIONALITY , 2001 .

[70]  Eric J. Johnson,et al.  Detecting Failures of Backward Induction: Monitoring Information Search in Sequential Bargaining , 2002, J. Econ. Theory.

[71]  Teck-Hua Ho,et al.  Sophisticated Experience-Weighted Attraction Learning and Strategic Teaching in Repeated Games , 2002, J. Econ. Theory.

[72]  J. Tirole,et al.  Self-Confidence and Personal Motivation , 2002 .

[73]  Colin Camerer Behavioral Game Theory: Experiments in Strategic Interaction , 2003 .

[74]  P. Glimcher Decisions, Uncertainty, and the Brain: The Science of Neuroeconomics , 2003 .

[75]  Colin Camerer,et al.  Strategic teaching and equilibrium models of repeated trust and entry games , 2003 .

[76]  C. Monica Capra,et al.  The Effects of Common Advice on One-shot Traveler’s Dilemma Games: Explaining Behavior through an Introspective Model with Errors , 2003 .

[77]  David J. Cooper,et al.  Evidence on the equivalence of the strategic and extensive form representation of games , 2003, J. Econ. Theory.

[78]  Jacob K. Goeree,et al.  A model of noisy introspection , 2004, Games Econ. Behav..

[79]  Vincent P. Crawford,et al.  Theory and experiment in the analysis of strategic interaction , 2004 .

[80]  Colin Camerer,et al.  Neuroeconomics: How Neuroscience Can Inform Economics , 2005 .

[81]  Colin Camerer,et al.  “Economic man” in cross-cultural perspective: Behavioral experiments in 15 small-scale societies , 2005, Behavioral and Brain Sciences.

[82]  David I. Laibson,et al.  Bounded Rationality and Directed Cognition , 2005 .