Equitable Auction Design: With and Without Distributions

TL;DR

Equitable Auction Design confronts the problem of allocating a single good to multiple bidders under an ex-post equity constraint that reserves a fraction $\frac{\gamma}{1+\gamma}$ of allocations for a minority group. It delivers a closed-form optimal mechanism for stochastic design under independence and regularity, and a closed-form regret-based mechanism with a provable constant-factor guarantee (the worst-case regret is within a factor $e\theta$, with the maximum factor about 1.31 across $\gamma$). Both mechanisms admit a set-aside interpretation, enabling practical implementation and policy alignment with minority prioritization. Through numerical experiments, the stochastic mechanism performs best when value distributions are well-estimated, while the regret-based approach offers robustness to distributional misspecification and estimation errors, making it valuable in data-poor settings.

Abstract

We study a mechanism design problem where a seller aims to allocate a good to multiple bidders, each with a private value. The seller supports or favors a specific group, referred to as the minority group. Specifically, the seller requires that allocations to the minority group are at least a predetermined fraction (equity level) of those made to the rest of the bidders. Such constraints arise in various settings, including government procurement and corporate supply chain policies that prioritize small businesses, environmentally responsible suppliers, or enterprises owned by historically disadvantaged individuals. We analyze two variants of this problem: stochastic mechanism design, which assumes bidders' values follow a known distribution and seeks to maximize expected revenue, and regret-based mechanism design, which makes no distributional assumptions and aims to minimize the worst-case regret. We characterize a closed-form optimal stochastic mechanism and propose a closed-form regret-based mechanism, and establish that the ex-post regret under the latter is at most a constant multiple (dependent on the equity level) of the optimal worst-case regret. We further quantify that this approximation constant is at most 1.31 across different equity levels. Both mechanisms can be interpreted as set-asides, a common policy tool that reserves a fraction of goods for minority groups. Furthermore, numerical results demonstrate that the stochastic mechanism performs well when the bidders' value distribution is accurately estimated, while the regret-based mechanism exhibits greater robustness under estimation errors.

Figures (4)

• Figure 1: The approximation factor of the mechanism $(\bm{\hat{q}}, \bm{\hat{m}})$ as a function of the equity level $\gamma$.
• Figure 2: Normalized expected revenues (left) and the upper quartile regrets (right) of the three mechanisms when $\rho = 0$: the stochastically optimal mechanism $(\bm{q}^\star,\bm{m}^\star)$ (blue), its variant $(\overline{\bm{q}}, \overline{\bm{m}})$ that enforces equity only in expectation (yellow), and the regret-based mechanism $(\hat{\bm{q}},\hat{\bm{m}})$ (red).
• Figure 3: Normalized expected revenues (left) and the upper quartile regrets (right) of the three mechanisms when $\rho = 0.5$: the stochastically optimal mechanism $(\bm{q}^\star,\bm{m}^\star)$ (blue), its variant $(\overline{\bm{q}}, \overline{\bm{m}})$ that enforces equity only in expectation (yellow), and the regret-based mechanism $(\hat{\bm{q}},\hat{\bm{m}})$ (red).
• Figure 4: Normalized expected revenues (left) and the upper quartile regrets (right) of the three mechanisms when $\rho = -0.5$: the stochastically optimal mechanism $(\bm{q}^\star,\bm{m}^\star)$ (blue), its variant $(\overline{\bm{q}}, \overline{\bm{m}})$ that enforces equity only in expectation (yellow), and the regret-based mechanism $(\hat{\bm{q}},\hat{\bm{m}})$ (red).

Theorems & Definitions (18)

• Remark 1
• Lemma 1: See, e.g., krishna2009auction
• Lemma 2: See, e.g., krishna2009auction
• Proposition 1
• Theorem 1
• proof
• Remark 2: Set-aside Interpretation
• Theorem 2: Lower Bound
• proof
• Proposition 2