Adjusted Winner: from Splitting to Selling

TL;DR

This paper extends the Adjusted Winner method to include resource selling under budget constraints, aiming for fairer allocations, and analyzes its theoretical properties, computational complexity, and practical performance.

Contribution

It introduces a novel extension of AW allowing resource sales, provides an axiomatic analysis, and develops an FPTAS for the associated complex problems.

Findings

Extended AW with resource selling improves fairness.

Proved computational complexity of the new problems.

Designed an FPTAS for practical approximation.

Abstract

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.