From Bilateral Trade to Matching Markets: Sharp Gains from Trade

We study gains from trade in matching markets with independent private values and costs, Bayesian incentive compatibility, interim individual rationality, and no expected budget deficit. A second-best guarantee for finite bilateral trade extends without loss to matching markets with independent Borel priors, arbitrary downward-closed feasibility, and finite expected first-best gains. For bounded buyers with monotone hazard rates and arbitrary bounded sellers, we determine the exact worst-case ratio of second-best to first-best gains, approximately $0.72490721$. For binary buyers and sellers with at most $m$ types, we determine the exact ratio for every $m$, including $8\mathrm{/}9$ when $m=2$ and a limit of $4\mathrm{/}5$ as $m$ grows. Both families of bounds are tight already in bilateral trade.