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 δ n /k→∞ .