Simple approximate equilibria in games with many players

We consider $\epsilon$-equilibria notions for constant value of $\epsilon$ in $n$-player $m$-actions games where $m$ is a constant. We focus on the following question: What is the largest grid size over the mixed strategies such that $\epsilon$-equilibrium is guaranteed to exist over this grid. For Nash equilibrium, we prove that constant grid size (that depends on $\epsilon$ and $m$, but not on $n$) is sufficient to guarantee existence of weak approximate equilibrium. This result implies a polynomial (in the input) algorithm for weak approximate equilibrium. For approximate Nash equilibrium we introduce a closely related question and prove its \emph{equivalence} to the well-known Beck-Fiala conjecture from discrepancy theory. To the best of our knowledge this is the first result introduces a connection between game theory and discrepancy theory. For correlated equilibrium, we prove a $O(\frac{1}{\log n})$ lower-bound on the grid size, which matches the known upper bound of $\Omega(\frac{1}{\log n})$. Our result implies an $\Omega(\log n)$ lower bound on the rate of convergence of dynamics (any dynamic) to approximate correlated (and coarse correlated) equilibrium. Again, this lower bound matches the $O(\log n)$ upper bound that is achieved by regret minimizing algorithms.

💡 Research Summary

The paper investigates the relationship between the granularity of a strategy grid and the existence of various ε‑approximate equilibrium concepts in normal‑form games with a constant number of actions (m) per player but an arbitrary number of players (n). The authors focus on k‑uniform distributions, where each mixed strategy assigns probabilities that are integer multiples of 1/k; larger k corresponds to a denser grid. The central question is: for a given ε‑equilibrium notion, how large must k be (as a function of ε, n, and m) to guarantee that every n‑player, m‑action game possesses a k‑uniform ε‑equilibrium?

1. Weak Approximate Nash Equilibrium
The first major result shows that for any constant ε, δ > 0 and fixed m, a constant k suffices to guarantee the existence of a (ε, δ)‑weak Nash equilibrium. Specifically, any k ≥ 32·(ln 8 + ln m − ln ε − ln δ)/ε² works, and crucially this bound does not depend on n. The proof samples k independent draws from an exact Nash equilibrium and uses a Hoeffding‑type concentration inequality (Theorem 2) to argue that with probability at least 1 − δ each player’s payoff deviation from the exact equilibrium is at most ε/2 for all actions. Consequently, at least a (1 − δ) fraction of players are ε‑best‑responding, establishing a (ε, δ)‑weak equilibrium. Because the number of k‑uniform profiles per player is bounded by km, exhaustive search over (km)ⁿ profiles runs in polynomial time in the input size N = n·mⁿ, yielding a polynomial‑time algorithm for weak approximate equilibrium (Corollary 1). This improves on the previously known O(log n) bound for exact Nash equilibrium existence on grids and shows that the naive exhaustive‑search algorithm is optimal even when n is large.

2. Standard ε‑Nash Equilibrium and the Beck‑Fiala Conjecture
The authors then ask whether a constant‑size grid also guarantees a standard ε‑Nash equilibrium. They introduce a related question: given an exact Nash equilibrium x, does at least one vertex of the surrounding 1/k‑cube Cₖ(x) constitute an ε‑Nash equilibrium? They construct binary‑action games where, for k < √(log n)/8, no point in Cₖ(x) is a 0.1‑Nash equilibrium (Proposition 1). Moreover, any ε‑Nash equilibrium on the grid can be far from the exact equilibrium, with some players playing the opposite pure strategy.

To understand the fundamental barrier, they reduce the grid‑existence problem to a classic discrepancy‑theory question: given a 0‑1 matrix M with at most t ones per column, can one assign signs ±1 to rows so that each column sum is O(√t)? This is precisely the Beck‑Fiala conjecture (1981). Theorem 3 proves that for a natural class of games, the existence of a constant‑size grid guaranteeing an ε‑Nash equilibrium is equivalent to the Beck‑Fiala conjecture holding for the associated incidence matrix. Since Beck‑Fiala remains open, the paper establishes a deep and previously unknown connection between game‑theoretic equilibrium computation and discrepancy theory. Consequently, unless Beck‑Fiala is resolved, we cannot claim a constant‑grid guarantee for standard ε‑Nash equilibria.

3. Approximate Correlated Equilibrium
For correlated equilibrium (CE), prior work showed that a grid of size k = O(log n) suffices for the existence of an ε‑CE, and regret‑minimizing dynamics converge to an ε‑CE in O(log n) rounds. The authors complement this with a matching lower bound: any k‑uniform ε‑CE must have k = Ω(log n). Theorem 4 constructs games where, if the grid is too coarse (k < c·log n), every k‑uniform distribution violates the CE incentive constraints for at least one player. This implies that no learning dynamic—regardless of its design—can converge to an ε‑CE faster than Ω(log n) steps. Hence the O(log n) convergence rate of regret‑minimization algorithms is optimal.

4. Algorithmic and Learning Implications

• Weak Approximate Nash: Constant‑size grids give a polynomial‑time algorithm via exhaustive search, confirming that the naive approach is optimal for this equilibrium notion.
• Standard ε‑Nash: The grid‑size question is tied to the Beck‑Fiala conjecture; a positive resolution would yield constant‑grid guarantees and polynomial‑time algorithms, while a negative resolution would imply inherent super‑constant dependence on n.
• Correlated Equilibrium: Θ(log n) grid size is both necessary and sufficient, establishing tight bounds on the convergence speed of any learning dynamic.

5. Overall Contribution
The paper systematically characterizes the minimal grid granularity required for three central equilibrium concepts: weak approximate Nash, standard ε‑Nash, and approximate correlated equilibrium. It shows that weak approximate Nash equilibria are extremely robust (constant grid), standard ε‑Nash equilibria sit at the frontier of a major open problem in discrepancy theory, and approximate correlated equilibria require logarithmic granularity, matching known algorithmic upper bounds. These results have immediate consequences for the design of equilibrium‑finding algorithms and for understanding the fundamental limits of learning dynamics in multi‑agent systems.