Problem
Research questions and friction points this paper is trying to address.
bilateral markets
matching markets
social welfare
private values
second-best
Sharp Second-Best Welfare in Bilateral and Matching Markets
🤖 AI Summary
This study addresses the welfare loss induced by quantized private information, aiming to determine the exact lower bound on the ratio between second-best and first-best social welfare. To this end, it proposes a novel proof framework that preserves initial endowments within the budget Lagrangian under Bayesian incentive compatibility constraints. This approach integrates threshold decomposition, power-law distribution reduction, and variational characterization techniques, complemented by interval arithmetic for rigorous numerical verification. The primary contribution lies in establishing, for the first time, the exact approximation ratio constants in this domain and achieving a lossless generalization to matching markets. Specifically, the welfare ratio is proven to be approximately 0.8883 under arbitrary priors and 0.9114 under monotone hazard rate conditions. These tight bounds are shown to be attainable even in bilateral trade, ensuring theoretical completeness.
📝 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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Second-Best Welfare
Bilateral Trade
Matching Markets
Monotone Hazard Rate
Mechanism Design
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