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.
[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.
[edit] See also
| Topics in game theory | |
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Definitions |
Normal form game · Extensive form game · Cooperative game · Information set · Preference |
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Nash equilibrium · Subgame perfection · Bayes-Nash · Trembling hand · Correlated equilibrium · Sequential equilibrium · Quasi-perfect equilibrium · Evolutionarily stable strategy |
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Dominant strategies · Mixed strategy · Grim trigger · Tit for Tat |
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Classes of games |
Symmetric game · Perfect information · Dynamic game · Repeated game · Signaling game · Cheap talk · Zero-sum game · Mechanism design |
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Prisoner's dilemma · Coordination game · Chicken · Battle of the sexes · Stag hunt · Matching pennies · Ultimatum game · Minority game · Rock, Paper, Scissors · Pirate game · Dictator game |
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Theorems |
Minimax theorem · Purification theorems · Folk theorem · Revelation principle · Arrow's Theorem |
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Related topics |
Mathematics · Economics · Behavioral economics · Evolutionary game theory · Population genetics · Behavioral ecology · Adaptive dynamics · List of game theorists |
