The paper is concerned with a continuum model of the limit order book, viewed as a noncooperative game for n players. An external buyer asks for a random amount X > 0 of a given asset. This amount will be bought at the lowest available price, as long as the price does not exceed a given upper bound (P) over bar. One or more sellers offer various quantities of the asset at different prices, competing to fulfill the incoming order, whose size is not known a priori. The first part of the paper deals with solutions to the measure-valued optimal pricing problem for a single player, proving an existence result and deriving necessary and sufficient conditions for optimality. The second part is devoted to Nash equilibria. For a general class of random variables X and an arbitrary number of players, the existence and uniqueness of the corresponding Nash equilibrium is proved, explicitly determining the pricing strategy of each player. For a different class of random variables, it is shown that no Nash equilibrium can exist. The paper also describes the asymptotic limit as the total number of players approaches infinity, and provides formulas for the price impact produced by an incoming order.