[Paper Review] Fair and Efficient Allocations under Lexicographic Preferences

Hadi Hosseini, Sujoy Sikdar+2 more|arXiv (Cornell University)|Dec 14, 2020

Game Theory and Voting Systems41 references4 citations

TL;DR

This paper establishes the first algorithmic characterization of envy-freeness up to any good (EFX) and Pareto optimality (PO) under lexicographic preferences, proving that EFX+PO allocations always exist and can be computed in polynomial time. It further characterizes strategyproof, non-bossy, and neutral mechanisms satisfying EFX+PO, while showing that stronger efficiency notions like rank-maximality lead to incompatibility and NP-hardness, unless fairness is relaxed to maximin share (MMS).

ABSTRACT

Envy-freeness up to any good (EFX) provides a strong and intuitive guarantee of fairness in the allocation of indivisible goods. But whether such allocations always exist or whether they can be efficiently computed remains an important open question. We study the existence and computation of EFX in conjunction with various other economic properties under lexicographic preferences--a well-studied preference model in artificial intelligence and economics. In sharp contrast to the known results for additive valuations, we not only prove the existence of EFX and Pareto optimal allocations, but in fact provide an algorithmic characterization of these two properties. We also characterize the mechanisms that are, in addition, strategyproof, non-bossy, and neutral. When the efficiency notion is strengthened to rank-maximality, we obtain non-existence and computational hardness results, and show that tractability can be restored when EFX is relaxed to another well-studied fairness notion called maximin share guarantee (MMS).

Motivation & Objective

• To investigate the existence and computation of EFX and Pareto optimal (PO) allocations under lexicographic preferences, a domain where EFX and PO are incompatible under additive valuations.
• To characterize mechanisms that satisfy EFX, PO, strategyproofness, non-bossiness, and neutrality under lexicographic preferences.
• To examine the impact of strengthening efficiency to rank-maximality, revealing incompatibilities and computational hardness.
• To explore whether relaxing EFX to MMS restores tractability under rank-maximality, and to evaluate the empirical likelihood of such allocations.

Proposed method

• Proposes a family of polynomial-time algorithms for computing EFX+PO allocations under lexicographic preferences, with a characterization that any such allocation can be generated by one of these algorithms.
• Introduces a class of quota-based serial dictatorship mechanisms to characterize EFX+PO+strategyproof mechanisms under neutrality and non-bossiness.
• Proves NP-completeness for checking existence of EFX+rank-maximal allocations, even when EFX is relaxed to EFk (envy-freeness up to k goods).
• Demonstrates that EFX+rank-maximal allocations are incompatible with strategyproofness, and that the problem remains NP-hard under EFk.
• Shows that efficient computation of EFX+rank-maximal allocations becomes feasible when EFX is relaxed to MMS (maximin share guarantee), providing a tractable alternative.
• Empirically evaluates the frequency of {EF, EFX, EF1, MMS}+rank-maximal allocations using the Mallows model, varying the number of goods and dispersion parameter.

Experimental results

Research questions

• RQ1 Do EFX and Pareto optimal allocations always exist under lexicographic preferences, and can they be computed efficiently?
• RQ2 Which mechanisms satisfy EFX, PO, strategyproofness, non-bossiness, and neutrality under lexicographic preferences?
• RQ3 Is it possible to achieve EFX and rank-maximality simultaneously, and what is the computational complexity of checking such allocations?
• RQ4 Does relaxing EFX to MMS restore tractability for rank-maximal allocations?
• RQ5 How does the probability of existence of EFX+rank-maximal allocations vary with the number of goods and preference dispersion?

Key findings

• A family of polynomial-time algorithms exists for computing EFX+PO allocations under lexicographic preferences, and any such allocation can be generated by one of these algorithms, providing a complete algorithmic characterization.
• Under neutrality and non-bossiness, mechanisms satisfying EFX, PO, and strategyproofness are fully characterized by a special class of quota-based serial dictatorship mechanisms.
• EFX and rank-maximality are incompatible, and checking the existence of EFX+rank-maximal allocations is NP-complete.
• Even when EFX is relaxed to EFk (envy-freeness up to k goods), the existence of EFX+rank-maximal allocations remains NP-complete.
• Relaxing EFX to MMS restores tractability: EFX+rank-maximal allocations can be computed efficiently under MMS relaxation.
• Empirical results show that EFX+rank-maximal allocations are more frequent than EF+rank-maximal allocations for small numbers of goods, but the gap narrows as the number of goods increases, with both types becoming increasingly likely asymptotically.