Mixed strategy
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In game theory a mixed strategy is a strategy which chooses randomly between possible moves. The strategy has some probability distribution which corresponds to how frequently each move is chosen. A totally mixed strategy is a mixed strategy in which the player assigns strictly positive probability to every pure strategy. (Totally mixed strategies are important for the equilibrium refinement Trembling hand perfect equilibrium.)
A mixed strategy should be understood in contrast to a pure strategy where a player plays a single strategy with probability 1.
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[edit] Illustration
| A | B | |
|---|---|---|
| A | 1, 1 | 0, 0 |
| B | 0, 0 | 1, 1 |
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.
[edit] 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 equilibria, 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.
Mixed strategies are important in game theory because they can allow players to reach a Nash Equilibrium when one would not normally exist. Sometimes, a mixed strategy can allow a player to attain a higher expected payoff than choosing any available pure strategy. Finally, mixed strategies are useful because an opponent is less likely to correctly guess your move, which can also help a player gain a higher than average payoff. (Dutta:103-115)
[edit] A disputed meaning
During the 1980's, the concept of mixed strategies came under heavy fire for being "intuitively problematic" (Aumann 1985). Randomization, central in mixed stategies, lack behavioral support. Seldom do people make their choices following a lottery. This behavioral problem is compounded by the cognitive difficulty that people are unable to generate random outcomes without the aid of a random or pseudo-random generator.
Game theorist Ariel Rubinstein points out two alternative ways of understanding the concept (Rubinstein 1991: 909-924).
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 unsatistafiyng 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, adpoting 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.
[edit] References
- Aumann, R. "What is Game Theory Trying to accomplish?", Frontiers of Economics, edited by K. Arrow and S. Honkapohja, Basil Blackwell, Oxford, 1985.
- Dutta, Prajit K. (April 1999). Strategies and Games: Theory and Practise, Cambridge: The MIT Press. ISBN 1899235744
- Rubinstein, A. "Comments on the interpretation of Game Theory", Econometrica, July, 1991 (Vol. 59, n°4).
