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.