Strategy (game theory)#Mixed strategy

The concept of mixed strategy is used in game theory as a generalization of the notion of ( pure ) strategy. A strategy is a carried out before a game definition of a complete action plan. In a mixed strategy of player makes no direct decision, but he picks a random mechanism that determines a pure strategy. The decision taken in a game decision for a concrete action plan is thus purely coincidental and is subject only indirectly strategic considerations, unless the player has to take into account when choosing the random mechanism.

Mixed strategies were first described by Émile Borel (1921) and John von Neumann (1928 ) is used.

Existence of a Nash equilibrium under mixed strategies

In some normal form games there is no Nash equilibrium in the field of pure strategies. That is, there is no strategy combination, starting from which no single player can gain an advantage for himself by changing his own strategy. However, every finite game has a Nash equilibrium in mixed strategies. A Nash equilibrium in mixed strategies therefore consists of a mixed strategy for each player, with the property that each player's mixed strategy is the best response to the mixed strategies of the other players.

Example

2 players each have one black and one white marble. The rules are as follows: Player A wins if the colors of the marbles in drawing are the same ( black - white or black - white). Player B wins if the colors of the marbles are different ( white-black or black - white). How could the strategy of player A look like? If he chooses the black marble, Player B will always choose the whiteness and A player loses. Even if player A changes his strategy and chooses to use the white marble, Player B changes its strategy and also chooses this time in response black - A loses again.

Starts Player B, Player A will also adjust its strategy. It follows that no player can gain an advantage through the right combination of marbles. If the opponent guesses the strategy, he can always choose an appropriate counter-strategy that guarantees the win and vice versa.

In this game that was described there can be no Nash equilibrium in which both players choose a pure strategy. Remedy can only be a randomized selection, so a game by having a random selection of procedures. Only if both players with a probability of 50 % take by chance the white or black marble, there would be no incentive for this random strategy departing and it inevitably results in a Nash equilibrium.

The proof:

In practice, the problem can be solved in the example described above, so that both players the marbles drawn from a darkened vessel ( urn drawing ).