This article studies a market game under uncertainty in which agents may submit multiple limit and market orders. When agents know their preferences at all states, the competitive equilibrium can be supported as a Nash equilibrium of the market game, that is, agents behave as if they were price takers. Therefore, if the associated competitive economy has a fully revealing rational expectations equilibrium, then so does the market game. This resolves the puzzle that agents behave as if prices were given, even though prices aggregate private information, at least for this “private values” case. Necessary conditions for Nash equilibrium show that the resulting allocation cannot deviate too far from a competitive equilibrium. When agents do not know their preferences at some states, though, a characterization result shows that the Nash equilibria of the market game tend to be far from competitive.