[Paper Review] Repeated Fair Allocation of Indivisible Items

Ayumi Igarashi, Martin Lackner+2 more|arXiv (Cornell University)|Apr 4, 2023

Game Theory and Voting SystemsEconomics, Econometrics and Finance3 citations

TL;DR

This paper introduces a novel model for the repeated fair allocation of indivisible goods and chores, showing that when the number of repetitions is a multiple of the number of agents, proportional and Pareto-optimal allocations always exist. For two agents with an even number of rounds, envy-free and Pareto-optimal sequences can be achieved, with strong per-round fairness guarantees such as EF1 or weak envy-freeness up to one item.

ABSTRACT

The problem of fairly allocating a set of indivisible items is a well-known challenge in the field of (computational) social choice. In this scenario, there is a fundamental incompatibility between notions of fairness (such as envy-freeness and proportionality) and economic efficiency (such as Pareto-optimality). However, in the real world, items are not always allocated once and for all, but often repeatedly. For example, the items may be recurring chores to distribute in a household. Motivated by this, we initiate the study of the repeated fair division of indivisible goods and chores, and propose a formal model for this scenario. In this paper, we show that, if the number of repetitions is a multiple of the number of agents, there always exists a sequence of allocations that is proportional and Pareto-optimal. On the other hand, irrespective of the number of repetitions, an envy-free and Pareto-optimal sequence of allocations may not exist. For the case of two agents, we show that if the number of repetitions is even, it is always possible to find a sequence of allocations that is overall envy-free and Pareto-optimal. We then prove even stronger fairness guarantees, showing that every allocation in such a sequence satisfies some relaxation of envy-freeness. Finally, in case that the number of repetitions can be chosen freely, we show that envy-free and Pareto-optimal allocations are achievable for any number of agents.

Motivation & Objective

• To formalize a new model for repeated fair allocation of indivisible items, distinguishing between per-round and overall fairness and efficiency criteria.
• To investigate whether fairness (envy-freeness, proportionality) and efficiency (Pareto-optimality) can be simultaneously achieved over a sequence of allocations.
• To explore the trade-offs between fairness and efficiency in repeated settings, especially when the number of repetitions is fixed or variable.
• To determine the conditions under which envy-free and Pareto-optimal sequences exist, particularly for two agents and for any number of agents with variable repetition counts.
• To bridge the gap between indivisible and divisible fair division by leveraging fractional and randomized allocations as theoretical tools.

Proposed method

• Proposes a formal model for repeated allocation where sequences of allocations are evaluated for fairness and efficiency both overall and per-round.
• Introduces the concept of 'repeated translation' to convert fractional or randomized allocations into sequences of discrete allocations.
• Uses fractional allocations as a theoretical foundation, proving that Pareto-optimality and envy-freeness in the fractional setting imply corresponding properties in the repeated discrete setting.
• Applies Lemma 21 to show that any envy-free and Pareto-optimal fractional allocation can be implemented via a randomized allocation with per-round PROP[1,1] guarantees.
• Employs a translation mechanism to convert a randomized allocation with rational probabilities into a sequence of discrete allocations over a finite number of rounds.
• Leverages connections to divisible fair division by showing that the existence of a PO and EF fractional allocation implies the existence of a PO and EF repeated allocation sequence when the number of rounds is variable.

Experimental results

Research questions

• RQ1 Under what conditions does a sequence of allocations exist that is both envy-free overall and Pareto-optimal?
• RQ2 Can strong per-round fairness guarantees (e.g., EF1 or weak envy-freeness up to one item) be achieved alongside overall Pareto-optimality in repeated allocation?
• RQ3 Is it possible to achieve envy-freeness and Pareto-optimality in repeated allocation when the number of repetitions is fixed?
• RQ4 How does the ability to choose the number of repetitions affect the feasibility of achieving strong fairness and efficiency guarantees?
• RQ5 Can the connection between divisible and indivisible fair division be exploited to construct repeated allocations with desired properties?

Key findings

• When the number of repetitions $k$ is a multiple of the number of agents $n$, there always exists a sequence of allocations that is proportional and Pareto-optimal.
• For $n > 2$ agents and a fixed $k$ not divisible by $n$, a proportional overall allocation may not exist, showing the necessity of the multiple condition.
• For $n > 2$ agents and any fixed $k$, there exist chore allocation instances where no envy-free and Pareto-optimal sequence exists, indicating a fundamental limitation.
• For two agents and an even number of rounds, there always exists a sequence that is both envy-free and Pareto-optimal overall, and per-round weak envy-free up to one item.
• In the case of two rounds, the per-round fairness guarantee can be strengthened to envy-freeness up to one item (EF1), and such allocations can be computed in polynomial time.
• When the number of repetitions $k$ can be chosen freely, an envy-free, Pareto-optimal, and per-round PROP[1,1] allocation always exists for any number of agents, achieved via a randomized implementation of a fractional allocation.