Approximating Nash Equilibria in Normal-Form Games via Stochastic Optimization

For example, the first column indicates the payoff when all background players play action 0. The second column indicates all background players play action 0 except for one which plays action 1, and so on. The last column indicates all background players play action 1. These 2n scalars uniquely define the payoffs of a symmetric game. Given that this game only has two actions, we represent a mixed strategy by a single scalar p [0, 1], i.e., the probability of the first action. Furthermore, this game is symmetric and we seek a symmetric equilibrium, so we can represent a full Nash equilibrium by this single scalar p. This reduces our search space from 7 2 = 14 variables to 1 variable (and obviates any need for a map s from the unit hypercube to the simplex--see Lemma 25).