A Tutorial on the Proof of the Existence of Nash Equilibria

Often we do not need the notion of an outcome as distinct from a strategy profile. In such cases the game has the simpler formN,A, u, and in the remainder of the article we will adopt this form. We have so far defined the actions available to each player in a game, but not yet his set of strategies, or his available choices. Certainly one kind of strategy is to select a single action and play it. We call such a strategy a pure strategy. We call a choice of pure strategy pure strategy for each agent a pure strategy profile. Players could also follow another, less obvious type of strategy: randomizing over the set of available actions according to some probability distribution. Such a strategy is called a mixed strategy. Although it may not be immediately obvious why a player should introduce randomness into his choice of action, in fact in a multiagent setting the role of mixed strategies is critical. We define a mixed strategy for a normal form game as follows.