  We consider existence and uniqueness of Nash equilibria in an $N$-player game
of utility maximization under relative performance criteria of multiplicative
form in complete semimartingale markets. For a large class of players' utility
functions, a general characterization of Nash equilibria for a given initial
wealth vector is provided in terms of invertibility of a map from
$\mathbb{R}^N$ to $\mathbb{R}^N$. As a consequence of the general theorem, we
derive existence and uniqueness of Nash equilibria for an arbitrary initial
wealth vector, as well as their convergence, if either (i) players' utility
functions are close to CRRA, or (ii) players' competition weights are small and
relative risk aversions are bounded away from infinity.
