  The Nash equilibrium is an important benchmark for behaviour in systems of
strategic autonomous agents. Polymatrix games are a succinct and expressive
representation of multiplayer games that model pairwise interactions between
players. The empirical performance of algorithms to solve these games has
received little attention, despite their wide-ranging applications. In this
paper we carry out a comprehensive empirical study of two prominent algorithms
for computing a sample equilibrium in these games, Lemke's algorithm that
computes an exact equilibrium, and a gradient descent method that computes an
approximate equilibrium. Our study covers games arising from a number of
interesting applications. We find that Lemke's algorithm can compute exact
equilibria in relatively large games in a reasonable amount of time. If we are
willing to accept (high-quality) approximate equilibria, then we can deal with
much larger games using the descent method. We also report on which games are
most challenging for each of the algorithms.
