In Gale (1986) I developed a model of decentralized exchange and price formation. Agents meet at random and bargain over the terms of trade. In the absence of transaction costs every perfect equilibrium of this game implements a Walrasian equilibrium of the underlying exchange economy. In this paper I show, conversely, that every Walrasian equilibrium of the underlying exchange economy can be implemented as a perfect equilibrium of this game. In particular this shows the set of perfect equilibria is nonempty.