## Equilibrium

An n-tuple […] such that each player's mixed strategy maximizes his payoff if the strategies of the others are held fixed. Thus each player's strategy is optimal against those of the others."

Nash Equilibrium (NE) must exist for all finite games with any number of players. Until Nash, this had only been proved for 2-player zero-sum games by von Neumann and Morgestern.

If wea allow mixed strategies, then every n-player game in which every player can choose from finitely many strategies admits at least one NE.

It is a kind of optimal collective strategy in a game involving two or more players, where no player has anything to gain by changing only his or her own strategy. If each player has chosen a strategy and no player can get benefit by changing his or her strategy while the other players keep theirs unchanged, then the current set of strategy choices and the corresponding payoffs constitute a NE.

## Aumann - Cooperative Games

The basic difficulty in n-person game theory is due to the lack of a clear meaning as to what is the purpose of the game. Certainly, the purpose is not just to get the maximum amount of profits, because if every will demand the maximum he can get in a coalition, no agreement will be reached. Thus, one decides that the purpose of the game is to reach some kind of stability, to which the players would or sould agree if they want any agreement to be enforced.

## Bishop-Cannings Theorem

All member of a mixed evolutionary stable strategy have the same payoff, and that that none of these can be a pure evolutionary stable strategy.

D.T. Bishop, C. Cannings, 1978. A generalized war of attrition, J. Theroretical Biology 70:85-124