Mixed strategy

A mixed strategy is used in game theory economics to describe a strategy compromising of possible moves and a 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.

Illustration

Suppose the following payoff matrix (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 1 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 equilbria 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.

See also

• Pure strategy
• Nash equilibrium
• Trembling hand perfect equilibrium