  We consider a game-theoretic model of a market where investors compete for
payoffs yielded by several assets. The main result consists in a proof of the
existence and uniqueness of a strategy, called relative growth optimal, such
that the logarithm of the share of its wealth in the total wealth of the market
is a submartingale for any strategies of the other investors. It is also shown
that this strategy is asymptotically optimal in the sense that it achieves the
maximal capital growth rate when compared to competing strategies. Based on the
results obtained, we study the asymptotic structure of the market when all the
investors use the relative growth optimal strategy.
