function on the simplex, use Brower’s theorem to obtain a fixed point of this function, and then use Hall’s marriage lemma for the “purification” procedure. Proof of Theorem 6.1. We denote by S the common set of strategies, so that | S | = m and A = S $^{n}$. We consider the vector space R S equipped with the standard basis { 1 $^{s}$| s ∈ S } and the L$_{1}$ norm ‖ · ‖ $_{1}$. Let Δ$_{n}$ ( S ) be the set of all elements { x [ s ] } s ∈ S ∈ R S such that x [ s ] ≥ 0 for every s ∈ S and ∑ $_{s}$x [ s ] = n . Thus, Δ$_{n}$( S ) is an m − 1 dimensional simplex in R $^{S}$, whose vertices are n 1 $^{s}$. Elements of Δ$_{n}$ ( S ) with integral entries are called distributions . The set of all distributions in Δ$_{n}$( S ) is denoted D$_{n}$ ( S ). One can think about D$_{n}$ ( S ) as the set of possible ways to distribute n identical balls in m cells. Fix a strategy profile a = ( s$_{1}$, . . . , s$_{n}$ ) ∈ A and a player i . Let d ( a$_{−}$$_{i}$ ) = ∑ j = $_{i}$1 s$_{j}$ ∈ D$_{n}$$_{−}$$_{1}$ ( S ) be the distribution induced by a$_{−}$$_{i}$ . Since the game is anonymous, if d ( a$_{−}$$_{i}$ ) = d ( a ′ − $_{i}$) then player i gets the same payoff against a$_{−}$$_{i}$ and a ′ − $_{i}$. Thus, we can describe the game by payoff functions { F$_{i}$ : S × D$_{n}$$_{−}$$_{1}$ ( S ) → R } n i $_{=1}$, where F$_{i}$ ( s, d ) is the payoff for player i under a strategy profile in which player i plays s and the distribution of her opponent’s strategies is d . If the Lipschitz constant of an anonymous game is δ then the payoff functions satisfy the following Lipschitz property: (5) | F$_{i}$ ( s, d ) − F$_{i}$ ( s, d $^{′}$) | ≤ δ ‖ d − d $^{′}$‖ 1 2 for every s ∈ S and d, d ′ ∈ D$_{n}$$_{−}$$_{1}$ ( S ). We extend the domain of each F$_{i}$ ( s, · ) from D$_{n}$$_{−}$$_{1}$ ( S ) to Δ$_{n}$$_{−}$$_{1}$( S ), so that the Lipschitz property is preserved, for example by defining F$_{i}$ ( s, x ) = max d ∈ D$_{n}$$_{−}$$_{1}$ ( S ) { F$_{i}$ ( s, d ) − δ ‖ d − x ‖ $_{1}$/ 2 } for every x ∈ Δ$_{n}$$_{−}$$_{1}$( S ). Consider now the auxiliary n -player game in normal form where the strategy set of each player is Δ$_{1}$( S ) and the payoff for player i under strategy profile p = ( p$_{1}$, . . . , p$_{n}$ ) is given by R$_{i}$ ( p ) = ∑ s ∈ S p$_{i}$ [ s ] F$_{i}$ ( s, ∑ j = i p$_{j}$ ) .