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game theory Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...
, the purification theorem was contributed by
Nobel laureate The Nobel Prizes ( sv, Nobelpriset, no, Nobelprisen) are awarded annually by the Royal Swedish Academy of Sciences, the Swedish Academy, the Karolinska Institutet, and the Norwegian Nobel Committee to individuals and organizations who make out ...
John Harsanyi John Charles Harsanyi ( hu, Harsányi János Károly; May 29, 1920 – August 9, 2000) was a Hungarian-American economist and the recipient of the Nobel Memorial Prize in Economic Sciences in 1994. He is best known for his contributions to the ...
in 1973. The theorem aims to justify a puzzling aspect of
mixed strategy In game theory, a player's strategy is any of the options which they choose in a setting where the outcome depends ''not only'' on their own actions ''but'' on the actions of others. The discipline mainly concerns the action of a player in a game ...
Nash equilibria In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equili ...
: 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. The mixed strategy equilibria are explained as being the limit of
pure strategy In game theory, a player's strategy is any of the options which they choose in a setting where the outcome depends ''not only'' on their own actions ''but'' on the actions of others. The discipline mainly concerns the action of a player in a game ...
equilibria for a disturbed game of
incomplete information In economics and game theory, complete information is an economic situation or game in which knowledge about other market participants or players is available to all participants. The utility functions (including risk aversion), payoffs, strategies ...
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. 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 Continuum may refer to: * Continuum (measurement), theories or models that explain gradual transitions from one condition to another without abrupt changes Mathematics * Continuum (set theory), the real line or the corresponding cardinal number ...
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,
complete information In economics and game theory, complete information is an economic situation or game in which knowledge about other market participants or players is available to all participants. The utility functions (including risk aversion), payoffs, strategies ...
game. The result is also an important aspect of modern-day inquiries in
evolutionary game theory Evolutionary game theory (EGT) is the application of game theory to evolving populations in biology. It defines a framework of contests, strategies, and analytics into which Darwinian competition can be modelled. It originated in 1973 with John Ma ...
where the perturbed values are interpreted as distributions over types of players randomly paired in a population to play games.


Example

Consider the Hawk–Dove game shown here. The game has two
pure strategy In game theory, a player's strategy is any of the options which they choose in a setting where the outcome depends ''not only'' on their own actions ''but'' on the actions of others. The discipline mainly concerns the action of a player in a game ...
equilibria (Defect, Cooperate) and (Cooperate, Defect). It also has a mixed equilibrium in which each player plays Cooperate with probability 2/3. Suppose that each player ''i'' bears an extra cost ''ai'' from playing Cooperate, which is uniformly distributed on ˆ’''A'', ''A'' Players only know their own value of this cost. So this is a game of
incomplete information In economics and game theory, complete information is an economic situation or game in which knowledge about other market participants or players is available to all participants. The utility functions (including risk aversion), payoffs, strategies ...
which we can solve using Bayesian Nash equilibrium. The probability that ''ai'' ≤ . If player 2 Cooperates when , then player 1's expected utility from Cooperating is ; his expected utility from Defecting is . He should therefore himself Cooperate when . Seeking a symmetric equilibrium where both players cooperate if , we solve this for ). Now we have worked out ''a*'', we can calculate the probability of each player playing Cooperate as :\Pr(a_i \le a^*) = \frac = \frac+\frac. As ''A'' → 0, this approaches 2/3 – the same probability as in the mixed strategy in the complete information game. Thus, we can think of the mixed strategy equilibrium as the outcome of pure strategies followed by players who have a small amount of private information about their payoffs.


Technical details

Harsanyi's proof involves the strong assumption that the perturbations for each player are independent of the other players. However, further refinements to make the theorem more general have been attempted. The main result of the theorem is that all the mixed strategy equilibria of a given game can be purified using the same sequence of perturbed games. However, in addition to independence of the perturbations, it relies on the set of payoffs for this sequence of games being of full measure. There are games, of a pathological nature, for which this condition fails to hold. The main problem with these games falls into one of two categories: (1) various mixed strategies of the game are purified by different sequences of perturbed games and (2) some mixed strategies of the game involve weakly dominated strategies. No mixed strategy involving a weakly dominated strategy can be purified using this method because if there is ever any non-negative probability that the opponent will play a strategy for which the weakly dominated strategy is not a best response, then one will never wish to play the weakly dominated strategy. Hence, the limit fails to hold because it involves a discontinuity.Fudenberg, Drew and Jean Tirole: ''Game Theory'', MIT Press, 1991, pp. 233–234


References

{{game theory Game theory Theorems