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Polyhedral Clinching Auctions for Indivisible Goods

Hiroshi Hirai, Ryosuke Sato

Abstract

In this study, we propose the polyhedral clinching auction for indivisible goods, which has so far been studied for divisible goods. As in the divisible setting by Goel et al. (2015), our mechanism enjoys incentive compatibility, individual rationality, and Pareto optimality, and works with polymatroidal environments. A notable feature for the indivisible setting is that the whole procedure can be conducted in time polynomial of the number of buyers and goods. Moreover, we show additional efficiency guarantees, recently established by Sato for the divisible setting: The liquid welfare (LW) of our mechanism achieves more than 1/2 of the optimal LW, and that the social welfare is more than the optimal LW.

Polyhedral Clinching Auctions for Indivisible Goods

Abstract

In this study, we propose the polyhedral clinching auction for indivisible goods, which has so far been studied for divisible goods. As in the divisible setting by Goel et al. (2015), our mechanism enjoys incentive compatibility, individual rationality, and Pareto optimality, and works with polymatroidal environments. A notable feature for the indivisible setting is that the whole procedure can be conducted in time polynomial of the number of buyers and goods. Moreover, we show additional efficiency guarantees, recently established by Sato for the divisible setting: The liquid welfare (LW) of our mechanism achieves more than 1/2 of the optimal LW, and that the social welfare is more than the optimal LW.
Paper Structure (18 sections, 27 theorems, 59 equations, 2 figures, 2 algorithms)

This paper contains 18 sections, 27 theorems, 59 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our Model
  3. Polyhedral Clinching Auctions for Indivisible Goods
  4. Tight Sets Lemma
  5. Unsaturation Relation
  6. Proof of Proposition \ref{['dropping_of_buyers']}
  7. Efficiency
  8. Pareto Optimality
  9. Liquid Welfare and Social Welfare
  10. Proof Outline
  11. Proof of Proposition \ref{['optimal']}
  12. Proof of Lemma \ref{['prepare']}
  13. Proof of Lemma \ref{['payment']}
  14. Concluding Remarks
  15. Relation to No Trading Path Property
  16. ...and 3 more sections

Key Result

Lemma 3.1

In Algorithm 2, it holds $\delta_i=f_{x,d}(N)-f_{x,d}(N\setminus i)$ for each $i\in N$.

Figures (2)

  • Figure 1: Illustration of the dropping of buyers in Algorithm 1. The white circles represent buyers $i_1, i_2,\ldots, i_t$ and the gray shaded circles represent other buyers. For each $k\in [t]$, the dropping price $c^{\rm f}_i$ of buyer $i\in X_k\setminus X_{k-1}$ is equal to $c^{\rm f}_{i_k}$, as shown in Theorem \ref{['tightsets']} (ii).
  • Figure 2: Diagram of dependencies between theorems, propositions, and lemmas in Section 5.2.

Theorems & Definitions (62)

  • Lemma 3.1
  • Lemma 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Lemma 3.5
  • proof
  • proof : Proof of Theorem \ref{['BF']}
  • Theorem 3.6
  • proof
  • Theorem 3.7
  • ...and 52 more