The Lipschitz Constant of Perturbed Anonymous Games

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 $k\geq 3$, $λ(n,k,δ)$ is identified with the passage probability of a certain symmetric random walk on $\mathbb Z$. In the case of $k=2$ and $n$ even, $λ(n,2,δ)$ is identified with the probability that two 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 characterisation implies a sharp closed form asymptotic estimate of $λ(n,k,δ)$ as $δn /k\to\infty$.

💡 Research Summary

The paper studies the worst‑case Lipschitz constant of δ‑perturbed anonymous games, denoted λ(n,k,δ), where n is the number of players, k the number of actions, and δ∈(0,1) the perturbation level. An anonymous game is one in which each player’s payoff depends only on his own action and the distribution of actions chosen by the other players, not on their identities. The Lipschitz constant of a game g measures the maximal change in any player’s payoff when a single opponent changes his pure strategy while all other strategies remain the same; formally λ(g)=max_{i,a,b}|g_i(a)−g_i(b)| where a and b differ only in the action of one opponent.

A δ‑perturbation replaces each pure action a_i by the mixture (1−δ)·a_i+δ·Uniform(