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Pricing Valid Cuts for Price-Match Equilibria

Robert Day, Benjamin Lubin

Abstract

We use valid inequalities (cuts) of the binary integer program for winner determination in a combinatorial auction (CA) as "artificial items" that can be interpreted intuitively and priced to generate Artificial Walrasian Equilibria. We thus provide a method for converting a CA problem that admits only non-anonymous, nonlinear bundle prices into one that admits anonymous linear prices over the augmented item space, forestalling ex-post bidder complaints about opaque and strongly discriminatory pricing. To this end, we introduce a refinement of the Walrasian equilibrium which we call a "price-match equilibrium" (PME) in which all prices are justified by providing an iso-revenue reallocation for the hypothetical removal of any single bidder. We prove the existence of PME for any CA and characterize their economic properties and computation. We implement minimally artificial PME rules and compare them with other prominent CA payment rules in the literature.

Pricing Valid Cuts for Price-Match Equilibria

Abstract

We use valid inequalities (cuts) of the binary integer program for winner determination in a combinatorial auction (CA) as "artificial items" that can be interpreted intuitively and priced to generate Artificial Walrasian Equilibria. We thus provide a method for converting a CA problem that admits only non-anonymous, nonlinear bundle prices into one that admits anonymous linear prices over the augmented item space, forestalling ex-post bidder complaints about opaque and strongly discriminatory pricing. To this end, we introduce a refinement of the Walrasian equilibrium which we call a "price-match equilibrium" (PME) in which all prices are justified by providing an iso-revenue reallocation for the hypothetical removal of any single bidder. We prove the existence of PME for any CA and characterize their economic properties and computation. We implement minimally artificial PME rules and compare them with other prominent CA payment rules in the literature.

Paper Structure

This paper contains 21 sections, 16 theorems, 34 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background and Related Literature
  3. Model Preliminaries
  4. Winner Determination
  5. Payments, Prices, and Equilibrium
  6. Computing Dual Prices
  7. Pricing Valid Cuts as Artificial Items
  8. Motivation and examples
  9. Price-Match Equilibria for CAs
  10. Comparison to Min-Revenue Core-Selecting and VCG Mechanisms
  11. Generating Cuts for WE and PME
  12. Cuts for WE
  13. Cuts for any Core Point and PME
  14. Computing MAP prices
  15. Conclusions and Future Directions
  16. ...and 6 more sections

Key Result

Proposition 1

Minimal Walrasian Equilibrium prices may be strictly greater than winning-level payments.

Figures (3)

  • Figure 1: A generic diagram of the core in payment space with five alternative CA pricing rules, including our newly introduced MAP prices and the PME region of prices. The central dark gray region consists of non-PME points that can be supported by AWE prices.
  • Figure 2: The core of Example \ref{['ex:Four']} in the $\bm{\rho_1}$,$\bm{\rho_2}$ plane. The lighter region consists of NWE payment outcomes while the AWE $\bm{=}$ PME outcomes form the gray shaded triangle. The thicker line segment intersection of these two regions is the image of MAP prices on the core. WDP-quad-dual selects MAP$^*$ (red point) from among the PME points. The pay-as-bid and MRC $\bm{=}$ VCG points are also labeled.
  • Figure 3: The core (in payment space) for Example \ref{['ex:Seven']}. The pay-as-bid, NWE, and VCG $=$ MRC points are labeled. PME points form the shaded triangle on the near face, with the unique MAP point $\bm{(12,4,4)}$ in red.

Theorems & Definitions (35)

  • Definition 1
  • Example 1
  • Example 2
  • Example 3
  • Proposition 1
  • Definition 2: Price Match Equilibrium
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Definition 3
  • ...and 25 more