The Adjusted Winner Procedure: Characterizations and Equilibria

TL;DR

The paper analyzes the Adjusted Winner fair division procedure, revealing its characterizations, strategic properties, and equilibrium existence, including conditions for pure and approximate Nash equilibria, with implications for fair division outcomes.

Contribution

It provides new characterizations of the Adjusted Winner procedure and analyzes its strategic behavior, including equilibrium existence and properties under various tie-breaking rules.

Findings

Pure Nash equilibria may not always exist in the procedure.

The procedure guarantees $psilon$-Nash equilibria for all psilon > 0.

Exact pure Nash equilibria exist under informed tie-breaking and are Pareto optimal.

Abstract

The Adjusted Winner procedure is an important fair division mechanism proposed by Brams and Taylor for allocating goods between two parties. It has been used in practice for divorce settlements and analyzing political disputes. Assuming truthful declaration of the valuations, it computes an allocation that is envy-free, equitable and Pareto optimal. We show that Adjusted Winner admits several elegant characterizations, which further shed light on the outcomes reached with strategic agents. We find that the procedure may not admit pure Nash equilibria in either the discrete or continuous variants, but is guaranteed to have $\epsilon$-Nash equilibria for each $\epsilon$ > 0. Moreover, under informed tie-breaking, exact pure Nash equilibria always exist, are Pareto optimal, and their social welfare is at least 3/4 of the optimal.