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Fair Division with Market Values

Siddharth Barman, Soroush Ebadian, Mohamad Latifian, Nisarg Shah

Abstract

We introduce a model of fair division with market values, where indivisible goods must be partitioned among agents with (additive) subjective valuations, and each good additionally has a market value. The market valuation can be viewed as a separate additive valuation that holds identically across all the agents. We seek allocations that are simultaneously fair with respect to the subjective valuations and with respect to the market valuation. We show that an allocation that satisfies stochastically-dominant envy-freeness up to one good (SD-EF1) with respect to both the subjective valuations and the market valuation does not always exist, but the weaker guarantee of EF1 with respect to the subjective valuations along with SD-EF1 with respect to the market valuation can be guaranteed. We also study a number of other guarantees such as Pareto optimality, EFX, and MMS. In addition, we explore non-additive valuations and extend our model to cake-cutting. Along the way, we identify several tantalizing open questions.

Fair Division with Market Values

Abstract

We introduce a model of fair division with market values, where indivisible goods must be partitioned among agents with (additive) subjective valuations, and each good additionally has a market value. The market valuation can be viewed as a separate additive valuation that holds identically across all the agents. We seek allocations that are simultaneously fair with respect to the subjective valuations and with respect to the market valuation. We show that an allocation that satisfies stochastically-dominant envy-freeness up to one good (SD-EF1) with respect to both the subjective valuations and the market valuation does not always exist, but the weaker guarantee of EF1 with respect to the subjective valuations along with SD-EF1 with respect to the market valuation can be guaranteed. We also study a number of other guarantees such as Pareto optimality, EFX, and MMS. In addition, we explore non-additive valuations and extend our model to cake-cutting. Along the way, we identify several tantalizing open questions.

Paper Structure

This paper contains 25 sections, 22 theorems, 4 equations, 1 figure, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Related Work
  4. Preliminaries
  5. Simultaneous Envy-Freeness
  6. Extensions to Other Criteria
  7. Pareto Optimality
  8. MMS and EFX
  9. Monotone Valuations
  10. Cake Division
  11. Envy-Free Cake Division
  12. Number of Cuts
  13. Number of Queries
  14. Pareto Optimality
  15. Conclusion and Future Work
  16. ...and 10 more sections

Key Result

Lemma 1

Let $\pi : \mathcal{M} \leftrightarrow [m]$ be a strict ranking over $\mathcal{M}$. If $u_i$ induces $\pi$ for every $i \in \mathcal{N}$, then $A$ is SD-EF1 with respect to $\left( u_i \right)_{i \in \mathcal{N}}$ iff, for all agents $i \in \mathcal{N}$ and each index $\ell \in \{1, 2, \ldots, \lcei

Figures (1)

  • Figure 1: An instance in which no balanced EF+PO allocation exists.

Theorems & Definitions (51)

  • Definition 1: Envy-free up to one good
  • Definition 2: SD-preference bogomolnaia2001new
  • Definition 3: SD-EF1 aziz2023best
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 41 more