Market welfare loss due to private values quantified sharply
Sharp Second-Best Welfare in Bilateral and Matching Markets
Computer Science and Game Theory
Summary
Markets often lose some social welfare because buyers and sellers keep their values private. This paper finds the exact worst-case ratio of achievable welfare when these private values are considered, compared to a scenario where all values are public. It shows that under common assumptions, the welfare can still be at least about 89% or 91% of the best possible, depending on conditions. The results apply to simple two-player trades as well as complex matching markets, providing precise mathematical characterizations of these limits.
What this means in practice
- •For marketplace platform designers: Design and analyze pricing and matching rules ensuring near-optimal welfare despite participants’ private values.
- •For auction software engineers: Create auction algorithms that guarantee strong welfare outcomes while respecting incentive and budget constraints.
A theory result. No direct application yet.
Authors
Zhengyang Liu, Ying Qin, Zihe Wang
Abstract
How much social welfare must a market lose because values are private? We determine the sharp ratio of second-best to first-best welfare under independent nonnegative types, Bayesian incentive compatibility, interim individual rationality, and no expected budget deficit. The ratio is approximately $0.8882516903$ for arbitrary priors and $0.9113893681$ when buyers have monotone hazard rates and sellers are unrestricted. The arbitrary-prior guarantee holds for every downward-closed family of feasible matchings; the MHR guarantee holds when every matching of a compatibility graph is feasible. Both constants are \revise{attained}{sharp} already in bilateral trade. The proof keeps the sellers' initial endowment inside the budget Lagrangian. For arbitrary priors, a common threshold decomposition reduces the problem to three-parameter power-law distributions. For monotone hazards, common transformations of buyer and seller scores reduce it to shifted capped exponential buyers; a typewise allocation certificate and an explicit worst-case seller satisfy the same boundary equation. The bilateral-to-matching framework, with a welfare endowment charge and an MHR-preserving packing argument, transfers these affine inequalities without loss. We give exact variational characterizations of both constants and reproducible interval certificates for their numerical evaluation. Unrestricted signed transfers also permit pointwise strong budget balance.