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Auctions and mass transportation

Alexander V. Kolesnikov

TL;DR

The paper surveys the deep link between auction design and optimal transportation, starting from the landmark Daskalakis–Deckelbaum–Tzamos duality for the 1-bidder case and extending to multi-bidder settings via Beckmann-type formulations. It develops a unified duality framework that connects the monopolist problem, convex/$b$-convex representations, and transport theory (Kantorovich, Beckmann, and weak transport), providing both structural results and explicit 1-item examples. Key contributions include the 1-bidder duality with transform measures, the Beckenmann-type duality for many bidders, closed-form 1-item solutions via ironing of virtual valuations, and a mechanism characterization under symmetry. The work highlights open problems (duality interpretation, regularity, finite-variation dual measures) and perspectives toward transshipment formulations, offering a rigorous pathway for analyzing and computing optimal mechanisms in multi-item/multi-bidder auctions.

Abstract

In this survey paper we present classical and recent results relating the auction design and the optimal transportation theory. In particular, we discuss in details the seminal result of Daskalakis, Deckelbaum and Tzamos \cite{DDT} about duality between auction design with $1$ bidder and the weak transportation problem. Later investigations revealed the connection of multi-bidder case to the Beckmann's transportation problem. In this paper we overview a number of works on related subjects (monopolist's problem, regularity issues, weak transportation, measure ordering etc.). In addition, we prove some new results on duality for unreduced mechanisms.

Auctions and mass transportation

TL;DR

The paper surveys the deep link between auction design and optimal transportation, starting from the landmark Daskalakis–Deckelbaum–Tzamos duality for the 1-bidder case and extending to multi-bidder settings via Beckmann-type formulations. It develops a unified duality framework that connects the monopolist problem, convex/-convex representations, and transport theory (Kantorovich, Beckmann, and weak transport), providing both structural results and explicit 1-item examples. Key contributions include the 1-bidder duality with transform measures, the Beckenmann-type duality for many bidders, closed-form 1-item solutions via ironing of virtual valuations, and a mechanism characterization under symmetry. The work highlights open problems (duality interpretation, regularity, finite-variation dual measures) and perspectives toward transshipment formulations, offering a rigorous pathway for analyzing and computing optimal mechanisms in multi-item/multi-bidder auctions.

Abstract

In this survey paper we present classical and recent results relating the auction design and the optimal transportation theory. In particular, we discuss in details the seminal result of Daskalakis, Deckelbaum and Tzamos \cite{DDT} about duality between auction design with bidder and the weak transportation problem. Later investigations revealed the connection of multi-bidder case to the Beckmann's transportation problem. In this paper we overview a number of works on related subjects (monopolist's problem, regularity issues, weak transportation, measure ordering etc.). In addition, we prove some new results on duality for unreduced mechanisms.
Paper Structure (25 sections, 18 theorems, 183 equations, 1 figure)

This paper contains 25 sections, 18 theorems, 183 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Auctions
  3. The model: $1$ bidder
  4. The model: many bidders
  5. Some closed-form solutions
  6. The monopolist's problem and representation of J.-P. Rochet
  7. Model example and regularity issues
  8. General monopolist's problem
  9. Stochastic domination and reduction to a maximization problem for functions
  10. Mass transportation problem
  11. General optimal transportation problem
  12. $L^1$-transportation problem
  13. Beckmann's problem
  14. Weak transport
  15. Mass transportation in $1$-bidder case
  16. ...and 10 more sections

Key Result

Theorem 1

(Rochet) For $m=1$ bidder the auctioneer's problem is equivalent to the problem of maximization of over all convex nonnegative functions $u$ on $[0,1]^n$ satisfying $0 \le u_{x_i} \le 1$ for all $i$. Equivalently maximization can be taken over functions belonging to $\mathcal{U}$.

Figures (1)

  • Figure 1: Partition of the square for $n=2, m=1$ and the Lebesgue measure

Theorems & Definitions (36)

  • Remark 1
  • Example 1
  • Example 2
  • Example 3
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Remark 2
  • Remark 3
  • ...and 26 more