content_image/303451.jpg content_image/303455.jpg content_image/303457.jpg content_image/303459.jpg
one has 

<img src='content_image/303451.jpg'>

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 

<img src='content_image/303455.jpg'>

The payoff for male j is given by K ( v$_{j}$ ) where the untruncated payoff of male j is given by 

<img src='content_image/303457.jpg'>

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 

<img src='content_image/303459.jpg'>

for every profile (¯ x, ¯ y ). 

