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 δn/ k→ ∞.
Bibliographical noteFunding Information:
We are grateful to MathOverflow for facilitating this collaboration. We thank the members of the Facebook group https://www.facebook.com/groups/305092099620459/ for pointing out references related to the reflection principle. We thank anonymous editor and referee for many corrections and ideas for improvement that helped to shape the final version of the paper. Amnon Schreiber is supported in part by the Israel Science Foundation Grant 2897/20. Ron Peretz is supported in part by the Israel Science Foundation Grant 2566/20.
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.
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