Adjusted Winner: from Splitting to Selling
Robert Bredereck, Bin Sun, Eyal Briman, Nimrod Talmon
TL;DR
This work extends the Adjusted Winner protocol to accommodate indivisible resources by permitting selling under a global budget, replacing fractional splits with revenue redistribution. The authors formalize the Dispute Settlement with Indivisible Resources and Sale (DSIRS) framework and define the Adjusted Winner without Splitting (AWNS) family of problems, analyze fairness trade-offs, and prove broad computational hardness results with an exception: an FPTAS exists for AWNS-ρ. They provide a detailed dynamic-programming-based approximation for AWNS-ρ and validate the approach via simulations on real-world Spliddit data, showing that enabling selling under a budget markedly improves equity, particularly under favorable price-cost modes. The results yield practical insight into fair division under indivisibility and feasible selling, with implications for real-world negotiations where splits are undesirable or impractical.
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.