The Lipschitz constant of perturbed anonymous games

Abstract

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 $\delta$-perturbed game, $\lambda (n,k,\delta )$, is given an explicit probabilistic description. In the case of $k\ge 3$, it is identified with the passage probability of a certain symmetric random walk on ${\mathbb {Z}}$. In the case of $k=2$ and n even, $\lambda (n,2,\delta )$ 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 $\lambda (n,k,\delta )$ as $\delta n /k\rightarrow \infty$.