Whoever Said Money Won't Solve All Your Problems? Weighted Envy-free Allocation with Subsidy
Noga Klein Elmalem, Haris Aziz, Rica Gonen, Xin Huang, Kei Kimura, Indrajit Saha, Erel Segal-Halevi, Zhaohong Sun, Mashbat Suzuki, Makoto Yokoo
TL;DR
This work addresses the problem of fairly allocating indivisible items to agents with heterogeneous entitlements by introducing weighted envy-freeness (WEF) and the notion of WEF-ability with subsidies. It develops a general, polynomial-time framework and bounds that guarantee WEF with bounded subsidies across broad valuation classes, including monotone, additive, identical, binary, and matroid valuations, using tools such as weighted envy-graphs and network-flow based one-to-many matchings. Key contributions include tight subsidy bounds for general monotone valuations, additive valuations with gcd-aware bounds, and specialized algorithms for identical and binary valuations, as well as a two-agent procedure achieving WEF(1,1). The paper also introduces MWEF for budget-constrained subsidies and reports preliminary experiments showing practical subsidies often well below worst-case guarantees, highlighting both theoretical and practical impact for fair division in weighted contexts.
Abstract
We explore solutions for fairly allocating indivisible items among agents assigned weights representing their entitlements. Our fairness goal is weighted-envy-freeness (WEF), where each agent deems their allocated portion relative to their entitlement at least as favorable as any others relative to their own. Often, achieving WEF necessitates monetary transfers, which can be modeled as third-party subsidies. The goal is to attain WEF with bounded subsidies. Previous work relied on characterizations of unweighted envy-freeness (EF), that fail in the weighted setting. This makes our new setting challenging. We present polynomial-time algorithms that compute WEF allocations with a guaranteed upper bound on total subsidy for monotone valuations and various subclasses thereof. We also present an efficient algorithm to compute a fair allocation of items and money, when the budget is not enough to make the allocation WEF. This algorithm is new even for the unweighted setting.