6.3. Example. Consider a town with 2 n + 1 individuals (players) and one restaurant. Each player should decide between going out for dinner in the restaurant (strategy R ) or cook her own dinner at home ( H ). Players like the restaurant to be populated but not crowded. Specifically, the payoff to a player that chooses R when k out of her 2 n opponents also choose R is given by First, notice that this is an anonymous game (actually this game is even symmetric in the sense that all the players have the same payoff function). Second, the Lipschitz constant of this game is δ . Third, the strategy profile µ in which all players mix between R and H with equal probabilities is an equilibrium. The function g was chosen such that, when n is large, E ( g ( X )) is approximately 1/2; this follows from the central limit theorem. Thus, each player can guarantee a payoff close to 1/2 by staying at home. It follows that, for ϵ = 1 / 4, a strategy profile is an ϵ -equilibrium if and only if the number of players k that choose R is either zero or in the interval where g ( k ) is close to 1/2. But when the equilibrium µ is played the probability of this event diminishes when n increases. Thus, the random realized profile will typically not be a 1 / 4-equilibrium. Appendix A. From pure ϵ -equilibrium to exact mixed equilibrium Theorem 2.5 states that if the Lipschitz constant of a game is small enough then pure ϵ - equilibrium exists. This result is an assertion about relationships between linear inequalities: It says that if a certain finite number of linear inequalities about the entries of a payoff matrix (inequalities that assert that the Lipschitz constant is smaller than δ ) are satisfied, then there 15 g ( k ) =    $^{1}$$^{,}$ if | k − n | ≤ 0 . $_{477}$√$_{n}$ ( 1 − δ ( | k − n | − 0 . $^{$_{477}$√$_{n}$}$) ) $^{+}$, otherwise, where δ > 0 is a small number. A player that chooses to stay home ( H ) gets a fixed payoff of E ( g ( X )) (independently of the choices of other players), where X is a random variable with binomial distribution X ∼ Bin (2 n, 1 / 2).