The paper examines a game-theoretic evolutionary model of an asset market with endogenous equilibrium asset prices. Assets pay dividends that are partially consumed and partially reinvested. The investors use general, adaptive strategies (portfolio rules), distributing their wealth between assets, depending on the exogenous states of the world and the observed history of the game. The main goal is to identify strategies, allowing an investor to "survive," i.e. to possess a positive, bounded away from zero, share of market wealth over the whole infinite time horizon. This work brings together recent studies on evolutionary finance with the classical topic of non-cooperative market games.