Adjusted Winner: from Splitting to Selling

The Adjusted Winner (AW) method is a fundamental procedure for the fair division of indivisible resources between two agents. However, its reliance on splitting resources can lead to practical complications. To address this limitation, we propose an extension of AW that allows the sale of selected resources under a budget constraint, with the proceeds subsequently redistributed, thereby aiming for allocations that remain as equitable as possible. Alongside developing this extended framework, we provide an axiomatic analysis that examines how equitability and envy-freeness are modified in our setting. We then formally define the resulting combinatorial problems, establish their computational complexity, and design a fully polynomial-time approximation scheme (FPTAS) to mitigate their inherent intractability. Finally, we complement our theoretical results with computer-based simulations.

💡 Research Summary

The paper revisits the classic Adjusted Winner (AW) procedure, a two‑person fair‑division protocol that guarantees equitability, envy‑freeness, and Pareto optimality by allowing a single item to be split fractionally. Recognizing that many real‑world assets (real estate, heirlooms, vehicles) cannot be physically divided and that splitting may be socially or legally unacceptable, the authors propose an extension that replaces fractional splits with the option to sell selected items.

They introduce the Dispute Settlement with Indivisible Resources and Sale (DSIRS) framework, formally defined as an instance ⟨R, u₁, u₂, p, c, B⟩ where R is a finite set of indivisible items, u₁ and u₂ are additive utility functions with equal total sum, p(r) is the revenue obtainable by selling item r (bounded above by the higher of the two utilities), c(r) is the selling cost, and B is a global budget on total selling costs. A solution (plan) partitions R into three disjoint subsets: S₀ (items sold), S₁ (allocated to agent 1), and S₂ (allocated to agent 2). The total sale revenue p(S₀) is divided between the agents by a parameter q∈