Pareto-Efficient Multi-Buyer Mechanisms: Characterization, Fairness and Welfare

TL;DR

This work characterizes the full Pareto frontier of ex-ante seller revenue and buyers' surplus in Bayesian single-item, multi-buyer auctions with i.i.d. values, under regular, MHR, and anti-MHR distributions. It plugs a constrained-revenue optimization into a Lagrangian dual framework, showing that frontier extreme points are implementable by indirect Vickrey auctions or second-price auctions with reserves, with frontier structure strongly dictated by the distribution class. The authors then analyze fairness-driven solution concepts—Kalai-Smorodinsky and Nash (and Cross-Side Nash)—revealing robust welfare guarantees for KS (at least 50% in general, asymptotically optimal under regular ∩ anti-MHR) and more fragile performance for Nash under adversarial distributions, while Cross-Side Nash mirrors KS in guarantees. They also extend the discussion to multi-unit scenarios, providing supply-to-demand-based welfare bounds, and prove auxiliary results on budget balance and price-posting revenue approximations, including tight bounds for quasi-regular distributions. Overall, the paper offers a comprehensive framework linking frontier structure, bargaining fairness, and welfare in two-sided Bayesian mechanism design, with practical implications for fairness-aware marketplaces.

Abstract

A truthful mechanism for a Bayesian single-item auction results with some ex-ante revenue for the seller, and some ex-ante total surplus for the buyers. We study the Pareto frontier of the set of seller-buyers ex-ante utilities, generated by all truthful mechanisms when buyers values are sampled independently and identically (i.i.d.). We first provide a complete structural characterization of the Pareto frontier under natural distributional assumptions. For example, when valuations are drawn i.i.d. from a distribution that is both regular and anti-MHR, every Pareto-optimal mechanism is a second-price auction with a reserve no larger than the monopoly reserve. Building on this, we interpret the problem of picking a mechanism as a two-sided bargaining game, and analyze two canonical Pareto-optimal solutions from cooperative bargaining theory: the Kalai-Smorodinsky (KS) solution, and the Nash solution. We prove that when values are drawn i.i.d. from a distribution that is both regular and anti-MHR, in large markets both solutions yield near-optimal welfare. In contrast, under worst-case MHR distributions, their performance diverges sharply: the KS solution guarantees one-half of the optimal welfare, while the Nash solution might only achieve an arbitrarily small fraction of it. These results highlight the sensitivity of fairness-efficiency tradeoffs to distributional structure, and affirm the KS solution as the more robust notion of fairness for asymmetric two-sided markets.

Figures (2)

• Figure 1: Distribution hierarchy. MHR distributions and anti-MHR distributions intersect at exponential distributions.
• Figure 2: The welfare approximation guarantee of the KS Solution in \ref{['thm:KS solution welfare approximation multi-unit']}. The gray dashed (solid) curve is $\frac{2 - \ln(\tau)}{\,2 - 2\ln(\tau)\,}$ ($\frac{e-1}{e}+\frac{1}{e+e^2\cdot \tau}$) and the black solid curve is $\Psi(\tau)$.

Theorems & Definitions (81)

• Definition 2.1: Pareto domination, Pareto optimality, and Pareto frontier
• Definition 2.2: Regularity, MHR, and anti-MHR
• Definition 2.3: Price-posting revenue curve
• Proposition 2.1: BR-89har-16
• Definition 2.4: Indirect Vickrey auctions
• Proposition 3.0
• Proposition 3.1
• Theorem 3.2
• Theorem 3.3
• Corollary 3.4