[Paper Review] A note on the undercut procedure

Haris Aziz|arXiv (Cornell University)|May 5, 2014

Game Theory and Voting Systems11 references7 citations

TL;DR

This paper simplifies and generalizes the undercut procedure for envy-free allocation of indivisible objects among agents with preferences. It extends the method to handle indifference between objects and non-responsive preferences, and proves its validity even when agents have unequal claims, broadening its applicability beyond the original strict, responsive preference assumptions.

ABSTRACT

The undercut procedure was presented by Brams, Kilgour and Klamler (2012) 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. 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.

Motivation & Objective

• To identify limitations in the original undercut procedure proposed by Brams, Kilgour, and Klamler (2012).
• To simplify the undercut procedure for broader applicability in fair division problems.
• To extend the procedure to cases where agents may express indifference between objects.
• To demonstrate that the procedure remains valid even when agents have unequal claims.

Proposed method

• The paper redefines the undercut procedure to operate under generalized preference structures that allow for indifference between objects.
• It replaces the responsive set extension with a more flexible preference extension that accommodates non-responsive preferences.
• The procedure is reformulated to maintain envy-freeness by checking undercut conditions on bundles under generalized preferences.
• It introduces a modified algorithmic structure that dynamically handles ties and non-responsive preferences during allocation.
• The method proves that the undercut condition still guarantees envy-freeness under the generalized conditions.
• It validates the procedure's correctness using logical consistency and fairness criteria under unequal claims.

Experimental results

Research questions

• RQ1 Does the undercut procedure remain valid when agents can express indifference between objects?
• RQ2 Can the undercut procedure be simplified while preserving envy-freeness under generalized preference models?
• RQ3 Is the undercut procedure still effective when agents have unequal claims over the objects?
• RQ4 Can the responsive set extension be relaxed without compromising the fairness guarantees of the procedure?
• RQ5 What conditions ensure that the undercut procedure produces an envy-free allocation beyond strict, responsive preferences?

Key findings

• The undercut procedure can be simplified without loss of fairness, making it more accessible and easier to implement.
• The procedure remains envy-free even when agents have indifferences between objects, relaxing the need for strict preferences.
• The method works under non-responsive preference extensions, broadening its applicability beyond the original assumptions.
• The procedure maintains fairness guarantees when agents have unequal claims, extending its use in real-world allocation settings.
• The generalized undercut procedure ensures an envy-free allocation under a wider range of preference structures than previously established.