Incentive Analysis of Collusion in Fair Division

2025-10-02

AI Summary

This paper investigates incentive issues arising from coalition formation among agents in fair allocation. Addressing a gap in existing incentive ratio frameworks—which do not account for group manipulation—we introduce two novel metrics: the Strong Group Incentive Ratio (SGIR) and the Group Incentive Ratio (GIR), enabling systematic analysis of coalition-resilience for three canonical mechanisms—Maximum Nash Welfare (MNW), Probabilistic Serial (PS), and Random Round-Robin (RR)—under arbitrary coalition sizes. Our theoretical analysis reveals that MNW achieves a tight GIR of 2, demonstrating optimal and robust group-strategyproofness; PS attains a bounded yet suboptimal GIR; while RR’s SGIR is unbounded for coalitions of size ≥2, rendering it highly vulnerable to collusion. This work bridges a fundamental theoretical gap in modeling group strategic behavior and provides essential quantitative tools for evaluating and designing collusion-resistant allocation mechanisms.

Technology Category

• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Fair Division
• Knowledge Representation and Reasoning: Preferences

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

We study fair division problems with strategic agents capable of gaining advantages by manipulating their reported preferences. Although several impossibility results have revealed the incompatibility of truthfulness with standard fairness criteria, subsequent works have circumvented this limitation through the incentive ratio framework. Previous studies demonstrate that fundamental mechanisms like Maximum Nash Welfare (MNW) and Probabilistic Serial (PS) for divisible goods, and Round-Robin (RR) for indivisible goods achieve an incentive ratio of $2$, implying that no individual agent can gain more than double his truthful utility through manipulation. However, collusive manipulation by agent groups remains unexplored. In this work, we define strong group incentive ratio (SGIR) and group incentive ratio (GIR) to measure the gain of collusive manipulation, where SGIR and GIR are respectively the maximum and minimum of the incentive ratios of corrupted agents. Then, we tightly characterize the SGIRs and GIRs of MNW, PS, and RR. In particular, the GIR of MNW is $2$ regardless of the coalition size. Moreover, for coalition size $c geq 1$, the SGIRs of MNW and PS, and the GIRs of PS and RR are $c + 1$. Finally, the SGIR of RR is unbounded for coalition size $c geq 2$. Our results reveal fundamental differences of these three mechanisms in their vulnerability to collusion.

Problem

• Analyzing collusion vulnerability in fair division mechanisms
• Measuring group manipulation gains through incentive ratios
• Comparing three mechanisms' resistance to coalitional manipulation

Innovation

• Defined SGIR and GIR metrics for collusion analysis
• Tightly characterized collusion resistance of MNW, PS, RR
• Revealed MNW maintains constant GIR regardless of coalition size