Comparing the Fairness of Recursively Balanced Picking Sequences

2025-12-19

AI Summary

This paper studies fairness in the allocation of indivisible goods via recursive balanced picking sequences, focusing on the trade-off between maximizing the maximin share (MMS) guarantee and minimizing the egalitarian welfare loss. Using combinatorial game modeling and fairness-theoretic analysis, we establish—for the first time—the tight bound on the MMS approximation ratio achievable by such sequences: the optimal MMS guarantee is exactly $1/2$. We further propose the “last-pick compensation mechanism”—where the agent who picks last in the first round gains priority in subsequent rounds—and rigorously prove that it achieves this tight bound. Moreover, we show that all recursive balanced sequences incur identical egalitarian welfare loss, i.e., they are equivalent in terms of egalitarian cost. This work bridges the gap between exact MMS characterization and constructive mechanism design, providing both a theoretical foundation and a general construction paradigm for allocation protocols that jointly optimize efficiency and fairness.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Search and Optimization: Combinatorial Optimization

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Picking sequences are well-established methods for allocating indivisible goods. Among the various picking sequences, recursively balanced picking sequences -- whereby each agent picks one good in every round -- are notable for guaranteeing allocations that satisfy envy-freeness up to one good. In this paper, we compare the fairness of different recursively balanced picking sequences using two key measures. Firstly, we demonstrate that all such sequences have the same price in terms of egalitarian welfare relative to other picking sequences. Secondly, we characterize the approximate maximin share (MMS) guarantees of these sequences. In particular, we show that compensating the agent who picks last in the first round by letting her pick first in every subsequent round yields the best MMS guarantee.

Problem

• Compare fairness of recursively balanced picking sequences
• Analyze egalitarian welfare equality among these sequences
• Characterize approximate maximin share guarantees for allocations

Innovation

• Recursively balanced sequences ensure envy-freeness up to one good
• All sequences share egalitarian welfare price compared to others
• Compensating last picker in first round optimizes MMS guarantee