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A mixed strategy should be understood in contrast to a Pure Strategy where a player plays a single strategy with probability 1. ILLUSTRATION Suppose the Payoff Matrix pictured to the right (known as a Coordination Game ). Here one player chooses the row and the other chooses a column. The row player receives the first payoff, the column the second. If row opts to play A with probability 1 (i.e. play A for sure), then he is said to be playing a pure strategy. If column opts to flip a coin and play A if the coin lands heads and '''B''' if the coin lands tails, then she is said to be playing a mixed strategy not a pure strategy. SIGNIFICANCE In his famous paper John Forbes Nash proved that there is a Nash Equilibrium (not his term) for every finite game. One can divide Nash equilibria into two types. ''Pure strategy Nash equilibria'' are Nash equilibria where all players are playing pure strategies. ''Mixed strategy Nash equilibria'' are equilibria where at least one player is playing a mixed strategy. While Nash proved that every finite game has a Nash equilibrium, not all have pure strategy Nash equilibria. For an example of a game that does not have a Nash equilibrium in pure strategies see Rock Paper Scissors . A DISPUTED MEANING During the 1980s, the concept of mixed strategies came under heavy fire for being "intuitively problematic" (. Game theorist Ariel Rubinstein points out two alternative ways of understanding the concept (Rubinstein 1991Rubinstein, A. "Comments on the interpretation of Game Theory", ''Econometrica'', July, 1991 (Vol. 59, n°4)): One is to imagine that the game players stand for a large population of agents. Each of the agents chooses a pure strategy, and the payoff depends on the fraction of agents choosing each strategy. The mixed strategy hence represents the distribution of pure strategies chosen by each population. However, this does not provide any justification for the case when players are individual agents. The other, called ''purification'', is to suppose that the mixed strategies interpretation merely reflects our lack of knowledge of the agent's information and decision-making process. Apparently random choices are then seen as consequences of non-specified, payoff-irrelevant exogeneous factors, such as Keynes ' "animal spirits". However, it is unsatisfying to have results that hang on unspecified factors, and this dismisses the possibility of a mixed-strategies analysis to have any predictive power. Arguing that those factors are simply other players' beliefs about a player's strategy (hence, adopting a mixed strategy is the best response to a player playing mixed strategies) gives a credible interpretation, but does not restore predictive power to the concept of mixed equilibria. Ever since, economists' attitude towards mixed strategies-based results have been ambivalent. Mixed strategies are still widely used for their capacity to provide Nash equilibria in any game, but the model shall specify why and how players randomize their decisions. REFERENCES SEE ALSO |
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