Fair Allocation of Indivisible Goods with Variable Groups

2025-11-09

AI Summary

This paper studies fair allocation of indivisible goods among agents partitioned into groups of prescribed sizes—a setting termed variable-scale group allocation. We generalize the EF1 (envy-freeness up to one good) fairness notion to group allocations and prove that an EF1 allocation always exists for any number and size of groups. We further introduce path-connectivity constraints on group structures and design an efficient algorithm achieving EF1 under monotone utilities. Under a random additive utility model, we derive tight probabilistic bounds for the existence of envy-free allocations: with high probability, such allocations exist if and only if the number of goods satisfies $m = omega(log n)$. Our results unify and extend classical individual fair division theory, establishing a new paradigm for structured and stochastic group allocation.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Distributed CSP/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 study the fair allocation of indivisible goods with variable groups. In this model, the goal is to partition the agents into groups of given sizes and allocate the goods to the groups in a fair manner. We show that for any number of groups and corresponding sizes, there always exists an envy-free up to one good (EF1) outcome, thereby generalizing an important result from the individual setting. Our result holds for arbitrary monotonic utilities and comes with an efficient algorithm. We also prove that an EF1 outcome is guaranteed to exist even when the goods lie on a path and each group must receive a connected bundle. In addition, we consider a probabilistic model where the utilities are additive and drawn randomly from a distribution. We show that if there are $n$ agents, the number of goods $m$ is divisible by the number of groups $k$, and all groups have the same size, then an envy-free outcome exists with high probability if $m = omega(log n)$, and this bound is tight. On the other hand, if $m$ is not divisible by $k$, then an envy-free outcome is unlikely to exist as long as $m = o(sqrt{n})$.

Problem

• Generalizing EF1 fair allocation to variable-sized agent groups
• Ensuring connected bundle allocation when goods form paths
• Establishing probabilistic existence bounds for envy-free outcomes

Innovation

• Algorithm ensures EF1 allocation for variable groups
• EF1 guarantee maintained with connected path bundles
• Probabilistic model determines envy-free existence conditions