Optimal Allocations under Strongly Pigou-Dalton Criteria: Hidden Layer Structure & Efficient Combinatorial Approach

2025-11-02

AI Summary

This paper studies the problem of allocating $m$ items to $n$ agents with binary additive or submodular valuations under the strong Pigou–Dalton fairness criterion, aiming to maximize social welfare. The key insight is that optimal allocations are exactly the stable allocations, all of which share an inherent hierarchical structure. Leveraging this, we design the first combinatorial exact algorithm for the divisible-item setting and extend it to the indivisible case via inter-layer reduction techniques. We propose two efficient algorithms with time complexities $O(m^2 n)$ for additive valuations and $O(m^2 n^5)$ for submodular valuations. Furthermore, we prove that for divisible items, the Chebyshev distance between any two optimal allocations is zero; for indivisible items with submodular valuations, this distance is at most one—demonstrating remarkable consistency among optimal solutions.

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
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We investigate optimal social welfare allocations of $m$ items to $n$ agents with binary additive or submodular valuations. For binary additive valuations, we prove that the set of optimal allocations coincides with the set of so-called emph{stable allocations}, as long as the employed criterion for evaluating social welfare is strongly Pigou-Dalton (SPD) and symmetric. Many common criteria are SPD and symmetric, such as Nash social welfare, leximax, leximin, Gini index, entropy, and envy sum. We also design efficient algorithms for finding a stable allocation, including an $O(m^2n)$ time algorithm for the case of indivisible items, and an $O(m^2n^5)$ time one for the case of divisible items. The first is faster than the existing algorithms or has a simpler analysis. The latter is the first combinatorial algorithm for that problem. It utilizes a hidden layer partition of items and agents admitted by all stable allocations, and cleverly reduces the case of divisible items to the case of indivisible items. In addition, we show that the profiles of different optimal allocations have a small Chebyshev distance, which is 0 for the case of divisible items under binary additive valuations, and is at most 1 for the case of indivisible items under binary submodular valuations.

Problem

• Finding optimal allocations under SPD welfare criteria
• Designing efficient algorithms for stable allocation computation
• Analyzing distance between optimal allocation profiles

Innovation

• Hidden layer partition of items and agents
• Reduction from divisible to indivisible items
• Combinatorial algorithm for stable allocations