# Purification theorem

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In game theory, the purification theorem was contributed by Nobel laureate 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 the study of mathematical models of strategic interaction in between rational decision-makers. It has applications in all fields of social science, as well as in logic and computer science. Originally, it addressed zero-sum games, in which each participant's gains or losses are exactly balanced by those of the other participants. Today, game theory applies to a wide range of behavioral relations, and is now an umbrella term for the science of logical decision making in humans, animals, and computers.

John Charles Harsanyi was a Hungarian-American economist.

## Contents

The mixed strategy equilibria are explained as being the limit of pure strategy equilibria for a disturbed game of incomplete 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.

Idealization is the process by which scientific models assume facts about the phenomenon being modeled that are strictly false but make models easier to understand or solve. That is, it is determined whether the phenomenon approximates an "ideal case," then the model is applied to make a prediction based on that ideal case.

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, complete information game.

The term ex-ante is a phrase meaning "before the event". Ex-ante or notional demand refers to the desire for goods and services which is not backed by the ability to pay for those goods and services. This is also termed as ‘wants of people’. Ex-ante is used most commonly in the commercial world, where results of a particular action, or series of actions, are forecast in advance. The opposite of ex-ante is ex-post (actual). Buying a lottery ticket loses you money ex ante, but if you win, it was the right decision ex post.

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, payoffs, strategies and "types" of players are thus common knowledge.

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 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 Maynard Smith and George R. Price's formalisation of contests, analysed as strategies, and the mathematical criteria that can be used to predict the results of competing strategies.

## Example

 C D C 3, 3 2, 4 D 4, 2 0, 0 Fig. 1: a Hawk–Dove game

Consider the Hawk–Dove game shown here. The game has two pure strategy equilibria (Defect, Cooperate) and (Cooperate, Defect). It also has a mixed equilibrium in which each player plays Cooperate with probability 2/3.

The game of chicken, also known as the hawk–dove game or snowdrift game, is a model of conflict for two players in game theory. The principle of the game is that while it is to both players’ benefit if one player yields, the other player's optimal choice depends on what their opponent is doing: if the player opponent yields, they should not, but if the opponent fails to yield, the player should.

Suppose that each player i bears an extra cost ai from playing Cooperate, which is uniformly distributed on [−A, A]. Players only know their own value of this cost. So this is a game of incomplete information which we can solve using Bayesian Nash equilibrium. The probability that aia* is (a* + A)/2A. If player 2 Cooperates when a2a*, then player 1's expected utility from Cooperating is a1 + 3(a* + A)/2A + 2(1 − (a* + A)/2A); his expected utility from Defecting is 4(a* + A)/2A. He should therefore himself Cooperate when a1 ≤ 2 - 3(a*+A)/2A. Seeking a symmetric equilibrium where both players cooperate if aia*, we solve this for a* = 1/(2 + 3/A). Now we have worked out a*, we can calculate the probability of each player playing Cooperate as

${\displaystyle \Pr(a_{i}\leq a^{*})={\frac {{\frac {1}{2+3/A}}+A}{2A}}={\frac {A}{4A^{2}+6A}}+{\frac {1}{2}}.}$

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. [2] [3]

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. [4]

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## 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. doi : 10.1007/BF01737554
2. R. Aumann, et al. 1983. "Approximate Purificaton of Mixed Strategies. Mathematics of Operations Research 8 (1983), pp. 327–341.
3. Govindan, S., Reny, P.J. and Robson, A.J. 2003. "A Short Proof of Harsanyi's Purification Theorem. Games and Economic Behavior v45,n2 (2003), pp. 369–374. doi : 10.1016/S0899-8256(03)00149-0
4. Fudenberg, Drew and Jean Tirole: Game Theory, MIT Press, 1991, pp. 233–234