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 \(\delta n /k\rightarrow \infty \).