The strategy set of every player is compact and convex, the payoff functions are jointly continuous, and each function is linear in a player’s own strategy. Therefore the game admits a Nash equilibrium ¯ p = (¯ p$_{1}$, . . . , ¯ p$_{n}$ ). Because of the linearity in own strategy, the equilibrium property implies that (6) support(¯ p$_{i}$ ) ⊆ arg max s ∈ S F$_{i}$ ( s, ∑ j = i ¯ p$_{j}$ ) for every player i . for every player i . We claim that the strategy profile ¯ a is a 2 mδ -equilibrium. Indeed, for every player i and every strategy s ∈ S , f$_{i}$ ( s, ¯ a$_{−}$$_{i}$ ) = F$_{i}$ ( s, ∑ j = i 1 ¯ s$_{j}$ ) ≤ F$_{i}$ ( s, ∑ j = i ¯ p$_{j}$ ) + mδ ≤ F$_{i}$ ( ¯ s$_{i}$, ∑ j = i ¯ p$_{j}$ ) + mδ ≤ F$_{i}$ ( ¯ s$_{i}$, ∑ j = i 1 ¯ s$_{j}$ ) + 2 mδ = f$_{i}$ (¯ s$_{i}$, ¯ a$_{−}$$_{i}$ ) + 2 mδ. The equalities follow from the definition of F$_{i}$ . The first inequality follows from (5) and (7), the second from the fact that ¯ s$_{i}$ ∈ support(¯ p$_{i}$ ) and (6), and the third again from (5) and (7). □ As a last remark in this section we note that the self-purification argument in the proof of Theorem 2.5 fails when the Lipschitz constant of the game is not sufficiently small, even if the game is anonymous . This is illustrated in the following example. 14 (7) ∥ ∑ j = i 1 ¯ s$_{j}$ − ∑ j = i ¯ p$_{j}$ ∥ 1 ≤ 2 m By the Shapley-Folkman Theorem (e.g., [21]), we can write ∑ i ¯ p$_{i}$ = ∑ i ∈ E q$_{i}$ + ∑ i ∈ E $_{c}$1 ¯ $^{s$_{i}$}$, where E is a set of m − 1 players, q$_{i}$ ∈ Δ$_{1}$( S ) for each i ∈ E and ¯ s$_{i}$ ∈ support(¯ p$_{i}$ ) for each i ∈ E $^{c}$. By replacing each q$_{i}$ , i ∈ E , by some ¯ s$_{i}$ ∈ support(¯ p$_{i}$ ) we get a pure strategy profile ¯ a = (¯ s$_{1}$, . . . , ¯ s$_{n}$ ) such that ‖ ∑ $_{i}$1 ¯ s$_{i}$ − ∑ i ¯ p$_{i}$ ‖ 1 ≤ 2( m − 1). In particular, it follows that