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Temporal Fair Division

Benjamin Cookson, Soroush Ebadian, Nisarg Shah

Abstract

We study temporal fair division, whereby a set of agents are allocated a (possibly different) set of goods on each day for a period of days. We study this setting, as well as a number of its special cases formed by the restrictions to two agents, same goods on each day, identical preferences, or combinations thereof, and chart out the landscape of achieving two types of fairness guarantees simultaneously: fairness on each day (per day) and fairness over time (up to each day, or the weaker version, overall). In the most general setting, we prove that there always exists an allocation that is stochastically-dominant envy-free up to one good (SD-EF1) per day and proportional up to one good (PROP1) overall, and when all the agents have identical preferences, we show that SD-EF1 per day and SD-EF1 overall can be guaranteed. For the case of two agents, we prove that SD-EF1 per day and EF1 up to each day can be guaranteed using an envy balancing technique. We provide counterexamples for other combinations that establish our results as among the best guarantees possible, but also leaving open some tantalizing questions.

Temporal Fair Division

Abstract

We study temporal fair division, whereby a set of agents are allocated a (possibly different) set of goods on each day for a period of days. We study this setting, as well as a number of its special cases formed by the restrictions to two agents, same goods on each day, identical preferences, or combinations thereof, and chart out the landscape of achieving two types of fairness guarantees simultaneously: fairness on each day (per day) and fairness over time (up to each day, or the weaker version, overall). In the most general setting, we prove that there always exists an allocation that is stochastically-dominant envy-free up to one good (SD-EF1) per day and proportional up to one good (PROP1) overall, and when all the agents have identical preferences, we show that SD-EF1 per day and SD-EF1 overall can be guaranteed. For the case of two agents, we prove that SD-EF1 per day and EF1 up to each day can be guaranteed using an envy balancing technique. We provide counterexamples for other combinations that establish our results as among the best guarantees possible, but also leaving open some tantalizing questions.

Paper Structure

This paper contains 38 sections, 24 theorems, 9 equations, 3 figures, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Our Results & Techniques
  3. Related Work
  4. Repeated fair division.
  5. Online fair division.
  6. Repeated (or many-to-many) matching.
  7. Constrained fair division.
  8. Preliminaries
  9. Model
  10. Agents, goods, and valuations.
  11. Preferences.
  12. Allocations.
  13. Restrictions.
  14. Fairness Desiderata
  15. Temporal Fairness
  16. ...and 23 more sections

Key Result

Proposition 1

Let $A$ be an allocation of a set of goods $S$.

Figures (3)

  • Figure 1: Hierarchy of temporal fairness notions.
  • Figure 2: Restrictions placed on the allocation by SD-EF1 up to any day
  • Figure 3: Representation of the "up to each day" and "per day" constraints for three days as a laminar set family. An allocation that is EF1 up to each day and EF1 per day would be an allocation that is EF1 with respect to every set in this family. To just represent the "up to each day" constraints, only the sets $M_1$, $M_1 \cup M_2$ and $M_1 \cup M_2 \cup M_3$ would be required.

Theorems & Definitions (59)

  • Definition 1: Envy-Freeness (EF)
  • Definition 2: Envy-Freeness Up to One Good (EF1)
  • Definition 3: Proportionality (PROP)
  • Definition 4: Proportionality Up to One Good (PROP1)
  • Definition 5: SD-EF
  • Definition 6: SD-EF1
  • Definition 7: SD-PROP1
  • Proposition 1
  • Definition 8: Per Day Fairness
  • Definition 9: Overall Fairness
  • ...and 49 more