Optimal Trading Mechanisms with Ex Ante Unidentified Traders

We analyze optimal trading mechanisms in an exchange economy where each trader owns some units of a good to be traded and may be either a seller or a buyer, depending on the realization of the privately observed valuations. The concept of virtual valuation is extended to ex ante unidentified traders; contrary to the case where each trader is assigned a role as either a buyer or a seller, the traders' virtual valuations now depend on the choice of the trading mechanism and are generally non-monotonic even if the distribution of valuations is regular. We show that the trading mechanisms that maximize a broker's expected profit or expected total gains from trade are generalized double auctions which maximize the gains from trade measured in some modified monotonic virtual valuations for the traders. The bunching phenomena, which are here specific to ex ante unidentified traders, will be a general feature in these mechanisms. Furthermore, the randomization rule by which ties are broken is now part of the design of the optimal mechanisms. Finally, we show that the optimal mechanism converges toward a simple bid-ask mechanism as the number of participants in the market increases.