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Equitable Auctions

Simon Finster, Patrick Loiseau, Simon Mauras, Mathieu Molina, Bary Pradelski

Abstract

We initiate the study of how auction design affects the division of surplus among buyers. We propose a parsimonious measure for equity and apply it to the family of standard auctions for homogeneous goods. Our surplus-equitable mechanism is efficient, Bayesian-Nash incentive compatible, and achieves surplus parity among winners ex-post. The uniform-price auction is equity-optimal if and only if buyers have a pure common value. Against intuition, the pay-as-bid auction is not always preferred in terms of equity if buyers have pure private values. In auctions with price mixing between pay-as-bid and uniform prices, we provide prior-free bounds on the equity-preferred pricing rule under a common regularity condition on signals.

Equitable Auctions

Abstract

We initiate the study of how auction design affects the division of surplus among buyers. We propose a parsimonious measure for equity and apply it to the family of standard auctions for homogeneous goods. Our surplus-equitable mechanism is efficient, Bayesian-Nash incentive compatible, and achieves surplus parity among winners ex-post. The uniform-price auction is equity-optimal if and only if buyers have a pure common value. Against intuition, the pay-as-bid auction is not always preferred in terms of equity if buyers have pure private values. In auctions with price mixing between pay-as-bid and uniform prices, we provide prior-free bounds on the equity-preferred pricing rule under a common regularity condition on signals.
Paper Structure (34 sections, 32 theorems, 94 equations, 10 figures)

This paper contains 34 sections, 32 theorems, 94 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related literature
  4. Outline
  5. Setup
  6. Model
  7. Auction mechanisms.
  8. Interim values, payments, and utilities
  9. Equilibrium bidding
  10. Surplus equity
  11. The surplus-equitable mechanism
  12. Uniform, pay-as-bid, and mixed auctions
  13. Equity comparisons in uniform, pay-as-bid, and mixed auctions
  14. Challenging the intuition: private values and uniform pricing
  15. Equity-optimal pricing for log-concave signal distributions
  16. ...and 19 more sections

Key Result

Proposition 1

The unique equilibrium bidding strategy in the uniform-price auction, i.e., the case $\delta=0$, is given by $\beta^{U}(s) := \widetilde{V}(s,s) = \mathbb{E}[v(s_i,{\bm{s}}_{-i}) \mid s_i = s, Y_{k}({\bm{s}}_{-i}) = s]$.

Figures (10)

  • Figure 1: Equilibrium bid functions, $\beta^{\delta}$, for uniform signal distributions as a function of the signal, $s$, for common value parameters $c\in\{0,0.5,0.8,1\}$.
  • Figure 2: WEV as a function of $\delta$ for uniform signals and various common value proportions $c$
  • Figure 3: Surplus-equitable payments, $\widetilde{p}$, for uniform signal distributions as a function of the signal $s$ and the first rejected signal, $y$, for common value parameters $c\in\{0,0.5,0.8,1\}$.
  • Figure 4: Equilibrium bid as a function of quantiles for $n = 5$ and $\eta = 0.01$
  • Figure 5: Bounds on equity-optimal combinations of $c$ and $\delta$
  • ...and 5 more figures

Theorems & Definitions (49)

  • Definition 1: Mixed auctions
  • Proposition 1: e.g., Krishna-2009
  • Proposition 2
  • Example 1: label=uniform-example
  • Definition 2: Dominance in pairwise differences
  • Definition 3: Dominance in pairwise differences among winners
  • Definition 4: Winners' empirical variance
  • Proposition 3
  • Lemma 1
  • Example 2: continues=uniform-example
  • ...and 39 more