Simple Approximate Equilibria in Games with Many Players

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