  In this paper, we study nonzero-sum separable games, which are continuous
games whose payoffs take a sum-of-products form. Included in this subclass are
all finite games and polynomial games. We investigate the structure of
equilibria in separable games. We show that these games admit finitely
supported Nash equilibria. Motivated by the bounds on the supports of mixed
equilibria in two-player finite games in terms of the ranks of the payoff
matrices, we define the notion of the rank of an n-player continuous game and
use this to provide bounds on the cardinality of the support of equilibrium
strategies. We present a general characterization theorem that states that a
continuous game has finite rank if and only if it is separable. Using our rank
results, we present an efficient algorithm for computing approximate equilibria
of two-player separable games with fixed strategy spaces in time polynomial in
the rank of the game.
