a Nash equilibrium, let be the best response of player i to the strategies of all other players. Here, , where , is a mixed-strategy profile in the set of all mixed strategies and is the payoff function for player i. Define a set-valued function such that . The existence of a Nash equilibrium is equivalent to having a fixed point. Kakutani's fixed point theorem guarantees the existence of a fixed point if the following four conditions are satisfied. is compact, convex, and nonempty. is nonempty. is upper hemicontinuous is convex. Condition 1. is satisfied from the fact that is a simplex and thus compact. Convexity follows from players' ability to mix strategies. is nonempty as long as players have strategies. Condition 2. and 3. are satisfied by way of Berge's maximum theorem. Because is continuous and compact, is non-empty and upper hemicontinuous. Condition 4. is satisfied as a result of mixed strategies. Suppose , then . i.e. if two strategies maximize payoffs, then a mix between the two strategies will yield the same payoff. Therefore, there exists a fixed point in and a Nash equilibrium. When Nash made this point to John von Neumann in 1949, von Neumann famously dismissed it with the words, "That's trivial, you know. That's just a fixed-point theorem." (See Nasar, 1998, p. 94.) Alternate proof using the Brouwer fixed-point theorem We have a game where is the number of players and is the action set for the players. All of the action sets are finite. Let denote the set of mixed strategies for the players. The finiteness of the s ensures the compactness of . We can now define the gain functions. For a mixed strategy , we let the gain for player on action be The gain function represents the benefit a player gets by unilaterally changing their strategy. We now define where for . We see that Next we define: It is easy to see that each is a valid mixed strategy in . It is also easy to check that each is a continuous function of , and hence is a continuous function. As the cross product of a finite number of compact convex sets, is also compact and convex. Applying the Brouwer fixed point theorem to and we conclude that has a fixed point in , call it . We claim that is a Nash equilibrium in . For this