Lipschitz Games

TL;DR

This paper investigates how the Lipschitz constant of finite normal-form games influences the existence of pure epsilon-equilibria, providing bounds and proofs using probabilistic methods.

Contribution

It establishes a relationship between the Lipschitz constant and the existence of pure epsilon-equilibria, including bounds depending on game size and strategies.

Findings

Small Lipschitz constants guarantee pure epsilon-equilibria.

Derived bounds for Lipschitz constants ensuring equilibrium existence.

Used probabilistic methods to prove key results.

Abstract

The Lipschitz constant of a finite normal-form game is the maximal change in some player's payoff when a single opponent changes his strategy. We prove that games with small Lipschitz constant admit pure {\epsilon}-equilibria, and pinpoint the maximal Lipschitz constant that is sufficient to imply existence of pure {\epsilon}-equilibrium as a function of the number of players in the game and the number of strategies of each player. Our proofs use the probabilistic method.