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Encyclopedia > Purification theorems

In game theory, the purification theorem was contributed by Nobel laurate John Harsanyi in 1973[1]. The theorem aims to justify a puzzling aspect of mixed strategy Nash equilibria: that each player is wholly indifferent amongst each of the actions he puts non-zero weight on, yet he mixes them so as to make every other player also indifferent. Game theory is a branch of applied mathematics that studies strategic situations where players choose different actions in an attempt to maximize their returns. ... John Charles Harsanyi (May 29, 1920 – August 9, 2000) was a Hungarian-American business and economics professor who contributed to the study of game theory in mathematics by developing the analysis of games of incomplete information. ... A mixed strategy is used in game theory economics to describe a strategy comprising possible moves and a probability distribution which corresponds to how frequently each move is chosen. ... In game theory, the Nash equilibrium (named after John Nash) is a kind of optimal strategy for games involving two or more players, whereby the players reach an outcome to mutual advantage. ...


The mixed strategy equilibria are explained as being the limit of pure strategy for a perturbed game of imperfect information in which the payoffs of each player are known to themselves but not their opponents. The idea is that the predicted mixed strategy of the original game emerge as ever improving approximations of a game that is not observed by the theorist who designed the original, idealized game. A pure strategy is a term used to refer to strategies in Game theory. ... Perfect information is a term used in economics and game theory to describe a state of complete knowledge about the actions of other players that is instantaneously updated as new information arises. ...


The apparently mixed nature of the strategy is actually just the result of each player playing a pure strategy with threshold values that depend on the ex-ante distribution over the continuum of payoffs that a player can have. As that continuum shrinks to zero, the players strategies converge to the predicted nash equilibria of the original, unperturbed, perfect information game. Perfect information is a term used in economics and game theory to describe a state of complete knowledge about the actions of other players that is instantaneously updated as new information arises. ...


The result is also an important aspect of modern day inquiries in evolutionary game theory where the perturbed values are interpreted as distributions over types of players randomly paired in a population to play games. Evolutionary game theory (EGT) is the application of game theory in evolutionary biology. ...


References

  1. ^ J.C. Harsanyi. 1973. "Games with randomly disturbed payoffs: a new rationale for mixed-strategy equilibrium points. Int. J. Game Theory 2 (1973), pp. 1–23.
Topics in game theory
Definitions Normal form game - Extensive form game - Cooperative game - Information set - Strategy - Mixed strategy - Preference
Equilibrium concepts Dominant strategy equilibrium - Nash equilibrium - Subgame-perfect Nash equilibrium - Bayes-Nash equilibrium - Perfect Bayes-Nash equilibrium - Sequential equilibrium - Evolutionarily stable strategy - Other equilibria
Classes of games Symmetric game - Perfect information - Dynamic game - Repeated game - Signaling game - Cheap talk - Zero-sum game - Mechanism design - Win-win game
Games Prisoner's dilemma - Chicken - Stag hunt - Ultimatum game - Matching pennies - Minority Game - Rock, Paper, Scissors - Dictator game -...
Theorems Revelation principle - Minimax theorem - Purification theorems - Folk theorem of repeated games - Bishop-Cannings theorem
Related topics Mathematics - Economics - Behavioral economics - Evolutionary biology - Evolutionary game theory - Population genetics - Behavioral ecology - List of game theorists
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