The Lipschitz constant of perturbed anonymous games

The Lipschitz constant of a game measures the maximal amount of influence that one player has on the payoff of some other player. The worst-case Lipschitz constant of an n-player k-action δ-perturbed game, λ(n, k, δ) , is given an explicit probabilistic description. In the case of k≥ 3 , it is identified with the passage probability of a certain symmetric random walk on Z. In the case of k= 2 and n even, λ(n, 2 , δ) is identified with the probability that two i.i.d. binomial random variables are equal. The remaining case, k= 2 and n odd, is bounded through the adjacent (even) values of n. Our characterization implies a sharp closed-form asymptotic estimate of λ(n, k, δ) as δn/ k→ ∞.