The Lipschitz constant of perturbed anonymous games

Ron Peretz, Amnon Schreiber, Ernst Schulte-Geers

Research output: Contribution to journalArticlepeer-review


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→ ∞.

Original languageEnglish
Pages (from-to)293-306
Number of pages14
JournalInternational Journal of Game Theory
Issue number2
StatePublished - Jun 2022

Bibliographical note

Funding Information:
We are grateful to MathOverflow for facilitating this collaboration. We thank the members of the Facebook group 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.

Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.


  • Anonymous games
  • Approximate Nash equilibrium
  • Large games
  • Perturbed games


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