A Note on the Gains from Trade of the Random-Offerer Mechanism

We study the classic bilateral trade setting. Myerson and Satterthwaite show that there is no Bayesian incentive compatible and budget-balanced mechanism that obtains the gains from trade of the first-best mechanism. Consider the random-offerer mechanism: with probability frac12 run the seller-offering mechanism, in which the seller offers the buyer a take-it-or-leave-it price that maximizes the expected profit of the seller, and with probability frac12 run the buyer-offering mechanism. Very recently, Deng, Mao, Sivan, and Wang showed that the gains from trade of the random-offerer mechanism is at least a constant factor of frac 1 8.23≈ 0.121 of the gains from trade of the first best mechanism. Perhaps a natural conjecture is that the gains-from-trade of the random-offerer mechanism, which is known to be at least half of the gains-from-trade of the second-best mechanism, is also at least half of the gains-from-trade of the first-best mechanism. However, in this note we exhibit distributions such as the gains-from trade of the random-offerer mechanism is smaller than a 0.495-fraction of the gains-from-trade of the first-best mechanism.