[Paper Review] Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

Aurélie Beynier, Sylvain Bouveret+3 more|arXiv (Cornell University)|Jul 28, 2018

Economic theories and models7 citations

TL;DR

This paper establishes a formal hierarchy of efficiency concepts in fair division of indivisible goods by linking picking sequences (sequenceability), cycle-deals, and Pareto-optimality. It shows that sequenceability is equivalent to optimality for 1-cycle deals, that every Pareto-optimal allocation is sequenceable but not vice versa, and that competitive equilibrium with equal incomes (CEEI) implies sequenceability, revealing deep connections between efficiency, fairness, and market-based mechanisms under additive preferences.

ABSTRACT

In fair division of indivisible goods, using sequences of sincere choices (or picking sequences) is a natural way to allocate the objects. The idea is as follows: at each stage, a designated agent picks one object among those that remain. Another intuitive way to obtain an allocation is to give objects to agents in the first place, and to let agents exchange them as long as such "deals" are beneficial. This paper investigates these notions, when agents have additive preferences over objects, and unveils surprising connections between them, and with other efficiency and fairness notions. In particular, we show that an allocation is sequenceable iff it is optimal for a certain type of deals, namely cycle deals involving a single object. Furthermore, any Pareto-optimal allocation is sequenceable, but not the converse. Regarding fairness, we show that an allocation can be envy-free and non-sequenceable, but that every competitive equilibrium with equal incomes is sequenceable. To complete the picture, we show how some domain restrictions may affect the relations between these notions. Finally, we experimentally explore the links between the scales of efficiency and fairness.

Motivation & Objective

• To formalize the relationship between sequenceable allocations and picking sequences in fair division.
• To investigate the connections between sequenceability and various types of deals, especially cycle-deals involving a single object.
• To clarify the relationship between sequenceability, Pareto-optimality, and fairness concepts such as envy-freeness and competitive equilibrium with equal incomes (CEEI).
• To experimentally analyze the distribution of allocations across efficiency and fairness levels under different preference domains.
• To provide a computational framework for testing efficiency and fairness criteria, including NP-hard problems like MMS and CEEI.

Proposed method

• Formalizes sequenceability as the property that an allocation can be generated by a picking sequence, with a characterization based on the existence of a valid order of agent choices.
• Introduces the concept of cycle-deal-optimality, particularly focusing on 1-cycle (swap) deals, and proves that sequenceability is equivalent to optimality under such deals.
• Uses linear programming and ILP solvers to test Pareto-optimality, MMS, mMS, and CEEI fairness criteria on generated instances.
• Employs a systematic experimental evaluation on 50 MARA instances with 3 agents and 8 objects, using both uniform and single-peaked preference models.
• Applies a 6-level fairness scale (CEEI to non-fair) and a 4-level efficiency scale (PO → Seq → Swap → non-sequenceable) to map allocation distributions.
• Develops and releases a free, documented Python library for fair division evaluation, supporting efficient testing of fairness and efficiency properties.

Experimental results

Research questions

• RQ1 Is every Pareto-optimal allocation sequenceable, and is every sequenceable allocation Pareto-optimal?
• RQ2 What is the precise relationship between sequenceability and optimality under cycle-deals, particularly 1-cycle (swap) deals?
• RQ3 How do domain restrictions such as single-peaked preferences affect the hierarchy of efficiency concepts?
• RQ4 To what extent does competitive equilibrium with equal incomes (CEEI) imply sequenceability?
• RQ5 What is the empirical distribution of allocations across efficiency and fairness levels in random and structured preference domains?

Key findings

• Sequenceability is equivalent to optimality under 1-cycle (swap) deals: an allocation is sequenceable if and only if it is optimal for some swap deal involving a single object.
• Not all Pareto-optimal allocations are sequenceable, showing that Pareto-optimality and sequenceability are not equivalent in the general additive preference model.
• Every competitive equilibrium with equal incomes (CEEI) allocation is sequenceable, establishing a strong link between market-based mechanisms and picking sequences.
• In single-peaked preference domains, all swap-optimal allocations are also sequenceable, confirming that the hierarchy collapses to PO → Seq → Swap → non-sequenceable in this domain.
• Empirically, a majority of allocations lack both fairness and efficiency properties, but envy-free and CEEI allocations show higher proportions of sequenceable and Pareto-optimal allocations.
• The implementation of fairness and efficiency checks, including NP-hard problems like CEEI and MMS, is made practical via an open-source Python library for reproducible evaluation.