Lecture Notes on Game Theory Set 3 – Mixed Strategy Equilibria

A game in strategic form does not always have a Nash equilibrium in which each player deterministically chooses one of his strategies. However, players may instead randomly select from among these pure strategies with certain probabilities. Randomizing one’s own choice in this way is called a mixed strategy. A profile of mixed strategies is called a mixed equilibrium if no player can gain on average by unilateral deviation. Nash showed in 1951 that any finite strategic-form game has a mixed equilibrium (J. F. Nash (1951), Non-cooperative games. Annals of Mathematics 54, pp. 286–295). We will show how Nash proved this theorem in Section 3.7 below. Average (that is, expected) payoffs must be considered because the outcome of the game may be random. This requires that each payoff in the game represents an “expected utility”, in the sense that the payoffs can be weighted with probabilities in order to represent the player’s preference for a random outcome.