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

TL;DR

This paper investigates the efficiency of the random-offerer mechanism in bilateral trade, showing that its gains from trade can be significantly less than the optimal first-best gains, challenging previous conjectures about its performance.

Contribution

It demonstrates that the gains-from-trade of the random-offerer mechanism can be less than half of the first-best gains, providing a counterexample to prior assumptions.

Findings

Random-offerer mechanism's gains can be less than 49.5% of first-best gains.

Previous bounds suggesting at least half of second-best gains do not extend to first-best.

The paper provides explicit distributions where the mechanism performs poorly.

Abstract

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 $\frac{1}{2}$ run the \emph{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 $\frac{1}{2}$ run the \emph{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 $\frac 1 {8.23}\approx 0.121$ 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.