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Winner-Pays-Bid Auctions Minimize Variance

Preston McAfee, Renato Paes Leme, Balasubramanian Sivan, Sergei Vassilvitskii

TL;DR

This work tackles risk management in mechanism design by showing that among all payment rules implementing a fixed allocation, the winner-pays-bid rule minimizes convex risk measures of revenue, extending the analysis from ex-post IR to interim-IR and beyond standard auctions. In symmetric (IID) settings, WPB also minimizes the variance of total revenue, with a clean decomposition that leverages the revenue-equivalence principle. However, in asymmetric (non-IID) environments WPB need not be optimal; the authors provide an optimality condition and a constructive counterexample demonstrating how alternative payment rules can reduce risk while preserving the same allocation. The results offer a principled risk-averse justification for choosing WPB in many practical auctions and contribute to the broader understanding of variance-minimization in auction design under different IR notions.

Abstract

Any social choice function (e.g., the efficient allocation) can be implemented using different payment rules: first-price, second-price, all-pay, etc. All of these payment rules are guaranteed to have the same expected revenue by the revenue equivalence theorem, but have different distributions of revenue, leading to a question of which one is best. We prove that among all possible payment rules, winner-pays-bid minimizes the variance in revenue and, in fact, minimizes any convex risk measure.

Winner-Pays-Bid Auctions Minimize Variance

TL;DR

This work tackles risk management in mechanism design by showing that among all payment rules implementing a fixed allocation, the winner-pays-bid rule minimizes convex risk measures of revenue, extending the analysis from ex-post IR to interim-IR and beyond standard auctions. In symmetric (IID) settings, WPB also minimizes the variance of total revenue, with a clean decomposition that leverages the revenue-equivalence principle. However, in asymmetric (non-IID) environments WPB need not be optimal; the authors provide an optimality condition and a constructive counterexample demonstrating how alternative payment rules can reduce risk while preserving the same allocation. The results offer a principled risk-averse justification for choosing WPB in many practical auctions and contribute to the broader understanding of variance-minimization in auction design under different IR notions.

Abstract

Any social choice function (e.g., the efficient allocation) can be implemented using different payment rules: first-price, second-price, all-pay, etc. All of these payment rules are guaranteed to have the same expected revenue by the revenue equivalence theorem, but have different distributions of revenue, leading to a question of which one is best. We prove that among all possible payment rules, winner-pays-bid minimizes the variance in revenue and, in fact, minimizes any convex risk measure.
Paper Structure (16 sections, 10 theorems, 39 equations, 2 figures)

This paper contains 16 sections, 10 theorems, 39 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Related Work
  4. Setting
  5. Auction Setting.
  6. Revenue Equivalence Theorem.
  7. Variance and Risk Minimization.
  8. Direct Revelation Mechanisms.
  9. Notation.
  10. Winner-Pays-Bid Payment Rule.
  11. Jensen's inequality
  12. Warm-Up: Risk-Minimizing Ex-post IR mechanisms
  13. Risk-Minimization Beyond Ex-post IR
  14. Symmetric Settings: IID Valuations
  15. Asymmetric Settings: Non-IID Valuations
  16. ...and 1 more sections

Key Result

Lemma 2.1

For any implementable allocation function $\bm{{A}}(\bm{v})$ and corresponding interim allocation $x_i(\bm{v})$ and interim payments $z_i(\bm{v})$, the function $b_i^{\textsf{WPB}}(v_i) := z_i(v_i) / x_i(v_i)$ is monotone non-decreasing.

Figures (2)

  • Figure 1: The revenue distribution of the second-price, first-price, and all-pay auctions in case of two bidders with i.i.d. Uniform [0,1] value distributions. Observe that while all three formats give the same expected revenue of $1/3$, the underlying distribution of revenue varies greatly across them.
  • Figure 2: In the efficient mechanism that minimizes revenue variance for $2$ asymmetric bidders with pdf $F_1(v_1) = v_1$ and $F_2(v_2) = v_2^2$, we allocate to bidder $1$ in regions A and to bidder $2$ in regions B+C, but agent $1$ pays in regions A+B while bidder $2$ pays in regions B+C. Hence, in the shaded region, bidder $2$ is allocated but bidder $1$ pays.

Theorems & Definitions (20)

  • Lemma 2.1
  • proof
  • Theorem 3.1: Generalization of Waehrer et al.waehrer1998auction
  • proof
  • Corollary 3.2
  • proof
  • Corollary 3.3
  • proof
  • Lemma 4.1
  • proof
  • ...and 10 more