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 $δn /k\to\infty$.