Sharp efficiency factors with more than two types on both sides

Determine the sharp efficiency factors for bilateral trade and matching markets when both buyers and sellers have more than two types, and transfer any resulting bilateral bounds to matching markets through the finite-support bilateral-to-matching reduction.

Background

The paper establishes exact second-best efficiency factors for binary buyers with sellers having at most a prescribed number of types, as well as an exact factor for monotone-hazard-rate buyers. Its finite-support reduction shows that a sufficiently uniform bilateral Lagrangian bound can be transferred without loss to arbitrary downward-closed matching markets.

The authors explicitly identify the case in which both sides have more than two types as a remaining direction. They indicate that new bilateral bounds would directly produce corresponding matching-market guarantees, but no such general sharp characterization is provided.

References

First, what are the sharp factors when both sides have more than two types? New bilateral bounds would immediately yield matching-market bounds through the finite-support reduction.

— From Bilateral Trade to Matching Markets: Sharp Gains from Trade  (2609.30702 - Liu et al., 25 Sep 2026) in Section 6, Conclusion ("These results suggest three directions")

Third, which other matching-market mechanisms satisfy a comparison \lambda G\ge V_\alpha? Searching systematically for such inequalities would let us combine a growing collection of bilateral bounds with guarantees for additional mechanisms, as generalized offering already illustrates.

— From Bilateral Trade to Matching Markets: Sharp Gains from Trade  (2609.30702 - Liu et al., 25 Sep 2026) in Section 6, Conclusion ("These results suggest three directions")

The optimal prior-robust ratio and corresponding mechanism are unknown; however, this question must be asked carefully to avoid trivial implementation theory solutions (see further discussion in \Cref{s:revelation-gap}).

— Robustness in Mechanism Design  (2610.01805 - Hartline, 1 Oct 2026) in Section 5, subsection “Second-best Ratio, I.e., the Prior-robust Analysis”

Thus, the exact bilateral ratio remains unknown and lies between approximately 0.37 and 0.46.

— Robustness in Mechanism Design  (2610.01805 - Hartline, 1 Oct 2026) in Section 5, subsection “Notes”