We analyze optimal trading mechanisms in environments 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 privately observed valuations. First, the concept of virtual valuation is extended to ex ante unidentified traders; contrary to the case with identified traders, the traders' virtual valuations now depend on the trading mechanisms and are generally not 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 completely characterized by some modified monotonic virtual valuations. Here, the bunching phenomenon, which is specific to ex ante unidentified traders, will be a general feature in these mechanisms. We also show that the randomization rule by which ties are broken is now an important instrument in the design of the optimal mechanisms.

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