purpose, it suffices to show that This simply states that each player gains no benefit by unilaterally changing their strategy, which is exactly the necessary condition for a Nash equilibrium. Now assume that the gains are not all zero. Therefore, and such that . Note then that So let Also we shall denote as the gain vector indexed by actions in . Since is the fixed point we have: Since we have that is some positive scaling of the vector . Now we claim that To see this, we first note that if then this is true by definition of the gain function. Now assume that . By our previous statements we have that and so the left term is zero, giving us that the entire expression is as needed. So we finally have that where the last inequality follows since is a non-zero vector. But this is a clear contradiction, so all the gains must indeed be zero. Therefore, is a Nash equilibrium for as needed. Computing Nash equilibria If a player A has a dominant strategy then there exists a Nash equilibrium in which A plays . In the case of two players A and B, there exists a Nash equilibrium in which A plays and B plays a best response to . If is a strictly dominant strategy, A plays in all Nash equilibria. If both A and B have strictly dominant strategies, there exists a unique Nash equilibrium in which each plays their strictly dominant strategy. In games with mixed-strategy Nash equilibria, the probability of a player choosing any particular (so pure) strategy can be computed by assigning a variable to each strategy that represents a fixed probability for choosing that strategy. In order for a player to be willing to randomize, their expected payoff for each (pure) strategy should be the same. In addition, the sum of the probabilities for each strategy of a particular player should be 1. This creates a system of equations from which the probabilities of choosing each strategy can be derived. Examples In the matching pennies game, player A loses a point to B if A and B play the same strategy and wins a point from B if they play different strategies. To compute the mixed-strategy Nash equilibrium, assign A the probability of playing H and of playing T, and assign B the probability of playing H and of