Lipschitz Games

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.

💡 Research Summary

The paper introduces a quantitative measure of a game’s sensitivity to unilateral deviations, called the Lipschitz constant, and investigates how small this constant must be to guarantee the existence of pure ε‑equilibria in finite normal‑form games.

Model and Definitions
Consider an n‑player normal‑form game where each player i has a finite strategy set A_i (|A_i| ≤ m) and payoff function f_i : A → ℝ, with A = Π_i A_i. The Lipschitz constant δ(G) of the game G is defined as the maximum absolute change in any player’s payoff when a single opponent switches his strategy while all other opponents keep theirs fixed. Formally, δ(G) = max_{i, a_i, a_{−i}, a’{−i}: ρ(a{−i}, a’{−i})=1} |f_i(a_i, a{−i}) – f_i(a_i, a’_{−i})|, where ρ is the Hamming distance on opponent profiles. This definition captures the idea that each player’s payoff function is δ‑Lipschitz in the opponents’ joint strategy. The authors denote by L(n,m,δ) the class of games with n players, at most m strategies per player, and Lipschitz constant at most δ.

Basic Existence Result
Proposition 2.3 shows that if δ ≤ ε/(2n) then every game in L(n,m,δ) possesses a pure ε‑equilibrium. The proof is elementary: start from any profile a, replace each player’s action by a best response to the opponents’ actions, and use the Lipschitz bound to show that no unilateral deviation can improve a player’s payoff by more than ε. This result mirrors the intuition that when a player’s payoff is essentially independent of the opponents’ actions, a pure approximate equilibrium is trivial.

Main Existence Theorem
The central contribution is Theorem 2.5, which dramatically relaxes the required bound on δ. It proves that if

  δ ≤ ε / (8 n log(2 m n))

then a pure ε‑equilibrium is guaranteed. The proof adapts Kalai’s “self‑purification” argument: take a mixed‑strategy Nash equilibrium μ = (μ_1,…,μ_n) of the game, and consider for each player i and each pure strategy h the event

 E_{i,h} = { a ∈ A : | f_i(h, a_{−i}) – E_{a_{−i}∼μ_{−i}}