one has f$_{i}$ ( d, a ∗ − $_{i}$) ≤ ∫ f$_{i}$ ( d, τ ) µ$_{−}$$_{i}$ (d τ ) + ϵ/ 2 ≤ ∫ f$_{i}$ ( a ∗ $_{i}$, τ ) µ$_{−}$$_{i}$ (d τ ) + ϵ/ 2 ≤ f$_{i}$ ( a ∗ $_{i}$, a ∗ − $_{i}$) + ϵ where the first inequality follows from the fact that a ∗ ∈ E$_{i,d}$ , the second from (1) and the third from the fact that a ∗ ∈ E$_{i,a}$ ∗ $_{i}$. 4. Proof of Theorem 2.8 Let the number of players be n = 2 k and let the set of strategies for each player be { +1 , − 1 } . We divide the players into two groups of k players, females and males, and denote their strategy profiles by ¯ x = ( x$_{1}$, . . . , x$_{k}$ ) and ¯ y = ( y$_{1}$, . . . , y$_{k}$ ) respectively, viewed as row vectors. Fix some constant δ > 0. We consider games that can be described by a k × k matrix M = { m$_{ij}$ } with entries ± 1. The payoff for female i is K ( u$_{i}$ ) where K ( t ) = t for | t | ≤ 1 and K ( t ) = t/ | t | for | t | > 1, and u$_{i}$ , the untruncated payoff of female i , is given by u$_{i}$ (¯ x, ¯ y ) = δx$_{i}$ ∑ j m$_{ij}$y$_{j}$ = δx$_{i}$ · ( M ¯ y $^{T}$)$_{i}$ . The payoff for male j is given by K ( v$_{j}$ ) where the untruncated payoff of male j is given by v$_{j}$ (¯ x, ¯ y ) = − δy$_{j}$ ∑ i m$_{ij}$x$_{i}$ = − δy$_{j}$ · (¯ xM )$_{j}$ . Thus, up to truncation (which ensures that payoffs are bounded in [ − 1 , 1]), every player plays matching pennies against each player of the opposite gender. Each player uses the same coin in all their matching pennies games, and the parameter m$_{ij}$ dictates which player wants to match in the game between female i and male j . Notice that the Lipschitz constant in every such game is at most 2 δ . Also, from the definitions of the untruncated payoff it follows that (2) ∑ i u$_{i}$ (¯ x, ¯ y ) = − ∑ j v$_{j}$ (¯ x, ¯ y ) for every profile (¯ x, ¯ y ). 8