In this paper the process of exchange is formulated as a noncooperative game. The game is analogous to the familiar institution of competitive bidding in a sealed-tender auction in which one agent (the auctioneer) chooses among trades offered by the other agents (the bidders). Using various regularity assumptions it is shown that this noncooperative game has a Nash equilibrium which yields an allocation in the core of the corresponding cooperative game of exchange. Also, as the bidders are replicated by division this allocation (aggregated by types) converges to a Walrasian allocation in which each type's budget constraint (using the efficiency prices) is satisfied. Thus it appears that bidding is a competitive process for achieving a cooperative outcome, and in the limit, a market outcome.