  In an $\epsilon$-Nash equilibrium, a player can gain at most $\epsilon$ by
unilaterally changing his behaviour. For two-player (bimatrix) games with
payoffs in $[0,1]$, the best-known$\epsilon$ achievable in polynomial time is
0.3393. In general, for $n$-player games an $\epsilon$-Nash equilibrium can be
computed in polynomial time for an $\epsilon$ that is an increasing function of
$n$ but does not depend on the number of strategies of the players. For
three-player and four-player games the corresponding values of $\epsilon$ are
0.6022 and 0.7153, respectively. Polymatrix games are a restriction of general
$n$-player games where a player's payoff is the sum of payoffs from a number of
bimatrix games. There exists a very small but constant $\epsilon$ such that
computing an $\epsilon$-Nash equilibrium of a polymatrix game is \PPAD-hard.
Our main result is that a $(0.5+\delta)$-Nash equilibrium of an $n$-player
polymatrix game can be computed in time polynomial in the input size and
$\frac{1}{\delta}$. Inspired by the algorithm of Tsaknakis and Spirakis, our
algorithm uses gradient descent on the maximum regret of the players. We also
show that this algorithm can be applied to efficiently find a
$(0.5+\delta)$-Nash equilibrium in a two-player Bayesian game.
