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Decision theory

A **pure strategy** is a term used to refer to strategies in Game theory. Each player is given a set of strategies, if a player chooses to take one action with probability 1 then that player is playing a pure strategy. This is in contrast to a mixed strategy where individual players choose a probability distribution over several actions.

## Illustration Edit

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.

## Significance Edit

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. However, many games do have pure strategy Nash equilibria (e.g. the Coordination game, the Prisoner's dilemma, the Stag hunt). Further, games can have both pure strategy and mixed strategy equilibria.

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