Hierarchy of beliefs

Construction by Jean-François Mertens and Zamir implementing with John Harsanyi's proposal to model games with incomplete information by supposing that each player is characterized by a privately known type that describes his feasible strategies and payoffs as well as a probability distribution over other players' types.^[1]

Such probability distribution at the first level can be interpreted as a low level belief of a player. One level up the probability on the belief of other players is interpreted as beliefs on beliefs. A recursive universal construct is built—in which player have beliefs on their beliefs at different level—this construct is called the hierarchy of beliefs.

The result is a universal space of types in which, subject to specified consistency conditions, each type corresponds to the infinite hierarchy of his probabilistic beliefs about others' probabilistic beliefs. They also showed that any subspace can be approximated arbitrarily closely by a finite subspace.

Another popular example of the usage of the construction is the Prisoners and hats puzzle. And so is Robert Aumann's construction of Common knowledge (logic).^[2]

References

↑ Jean -François Mertens and Shmuel Zamir (1985-03-01). "Formulation of Bayesian analysis for games with incomplete information". INTERNATIONAL JOURNAL OF GAME THEORY. 14 (1): 1–29. doi:10.1007/BF01770224.
↑ Herbert Gintis (16 March 2009). The bounds of reason: game theory and the unification of the behavioral sciences. Princeton University Press. p. 158. ISBN 978-0-691-14052-0. Retrieved 3 March 2012.

Topics in game theory

Definitions	Normal-form game Extensive-form game Escalation of commitment Graphical game Cooperative game Succinct game Information set Hierarchy of beliefs Preference

Equilibrium concepts	Nash equilibrium Subgame perfection Mertens-stable equilibrium Bayesian Nash equilibrium Perfect Bayesian equilibrium Trembling hand Proper equilibrium Epsilon-equilibrium Correlated equilibrium Sequential equilibrium Quasi-perfect equilibrium Evolutionarily stable strategy Risk dominance Core Shapley value Pareto efficiency Gibbs equilibrium Quantal response equilibrium Self-confirming equilibrium Strong Nash equilibrium Markov perfect equilibrium

Strategies	Dominant strategies Pure strategy Mixed strategy Tit for tat Grim trigger Collusion Backward induction Forward induction Markov strategy

Classes of games	Symmetric game Perfect information Simultaneous game Sequential game Repeated game Signaling game Screening game Cheap talk Zero-sum game Mechanism design Bargaining problem Stochastic game n-player game Large Poisson game Nontransitive game Global games Strictly determined game Potential game

Games	Prisoner's dilemma Traveler's dilemma Coordination game Chicken Centipede game Volunteer's dilemma Dollar auction Battle of the sexes Stag hunt Matching pennies Ultimatum game Rock-paper-scissors Pirate game Dictator game Public goods game Blotto games War of attrition El Farol Bar problem Fair division Fair cake-cutting Cournot game Deadlock Diner's dilemma Guess 2/3 of the average Kuhn poker Nash bargaining game Prisoners and hats puzzle Trust game Princess and monster game Monty Hall problem Rendezvous problem

Theorems	Minimax theorem Nash's theorem Purification theorem Folk theorem Revelation principle Arrow's impossibility theorem

Key figures	Albert W. Tucker Amos Tversky Ariel Rubinstein Daniel Kahneman David K. Levine David M. Kreps Donald B. Gillies Drew Fudenberg Eric Maskin Harold W. Kuhn Herbert Simon Hervé Moulin Jean Tirole Jean-François Mertens John Harsanyi John Maynard Smith John Nash John von Neumann Kenneth Arrow Kenneth Binmore Leonid Hurwicz Lloyd Shapley Melvin Dresher Merrill M. Flood Oskar Morgenstern Paul Milgrom Peyton Young Reinhard Selten Robert Aumann Robert B. Wilson Roger Myerson Samuel Bowles Thomas Schelling William Vickrey

See also	All-pay auction Alpha–beta pruning Bertrand paradox Bounded rationality Combinatorial game theory Confrontation analysis Coopetition List of game theorists List of games in game theory No-win situation Topological game Tragedy of the commons Tyranny of small decisions

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