A note on the undercut procedure

TL;DR

This paper refines the undercut procedure for envy-free allocation, allowing for indifference, non-responsive preferences, and unequal claims, thus broadening its applicability in fair division problems.

Contribution

The authors simplify and extend the undercut procedure to handle more general preference structures and claim scenarios, improving its robustness and practicality.

Findings

The simplified procedure works with indifference among objects.

It applies to non-responsive preferences over sets.

It remains valid with unequal claims.

Abstract

The undercut procedure was presented by Brams et al. [2] as a procedure for identifying an envy-free allocation when agents have preferences over sets of objects. They assumed that agents have strict preferences over objects and their preferences are extended over to sets of objects via the responsive set extension. We point out some shortcomings of the undercut procedure. We then simplify the undercut procedure of Brams et al. [2] and show that it works under a more general condition where agents may express indifference between objects and they may not necessarily have responsive preferences over sets of objects. Finally, we show that the procedure works even if agents have unequal claims.