Exact Maximin Share Fairness via Adjusted Supply

2025-02-06

AI Summary

This paper investigates the achievability of exact Maximin Share (MMS) fairness in the allocation of indivisible items, focusing on overcoming the fundamental impossibility of exact MMS fairness via bounded resource augmentation—specifically, item replication for goods or chore deletion for chores. For monotone utility and cost functions, we establish the first tight bounds: for goods, replicating at most (m) items suffices, with each item copied at most (3log m) times (or at most twice, totaling (leq 2m) copies, under additive valuations); for chores, deleting at most (m/e) chores ensures exact MMS fairness, improvable to (widetilde{O}(m/n^{1/4})) under ordered additive costs. Our approach integrates combinatorial analysis, extremal arguments, monotonicity characterization, probabilistic methods, and constructive algorithms. We derive multiple tight bounds and demonstrate that supply-side adjustment—rather than approximation—is a viable and essential pathway to circumvent the inherent inapproximability of MMS under monotonicity constraints.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

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

Abstract

This work addresses fair allocation of indivisible items in settings wherein it is feasible to create copies of resources or dispose of tasks. We establish that exact maximin share (MMS) fairness can be achieved via limited duplication of goods even under monotone valuations. We also show that, when allocating chores under monotone costs, MMS fairness is always feasible with limited disposal of chores. Since monotone valuations do not admit any nontrivial approximation guarantees for MMS, our results highlight that such barriers can be circumvented by post facto adjustments in the supply of the items. We prove that, for division of $m$ goods among $n$ agents with monotone valuations, there always exists an assignment of subsets of goods to the agents such that they receive at least their maximin shares and no single good is allocated to more than $3 log m$ agents. Also, the sum of the sizes of the assigned subsets (i.e., the total number of goods assigned, with copies) does not exceed $m$. For additive valuations, we prove that there always exists an MMS assignment in which no good is allocated to more than $2$ agents and the total number of goods assigned, with copies, is at most $2m$. For additive ordered valuations, we obtain a bound of $O(sqrt{log m})$ on the maximum assignment multiplicity and an $m + widetilde{O}left(frac{m}{sqrt{n}} ight)$ bound for the total number of goods assigned. For chore division, we upper bound the number of chores that need to be discarded to ensure MMS fairness. We prove that, under monotone costs, there always exists an MMS assignment in which at most $frac{m}{e}$ remain unassigned. For additive ordered costs, we establish that MMS fairness can be achieved while keeping at most $widetilde{O} left(frac{m}{n^{1/4}} ight)$ chores unassigned. We also prove that the obtained bounds for monotone valuations and costs are essentially tight.

Problem

• Achieve exact maximin share fairness via resource duplication.
• Ensure MMS fairness with limited disposal of chores.
• Establish tight bounds for MMS assignments under monotone valuations.

Innovation

• Limited duplication achieves MMS fairness.
• Limited disposal ensures MMS chore allocation.
• Bounds on goods and chores assignment.