Counterexamples in the theory of fair division

TL;DR

This paper reviews the history of fair division theory, identifies common errors and misconceptions, and provides concrete counterexamples to clarify the correct understanding of hypotheses and assumptions in the field.

Contribution

It offers a systematic correction of misconceptions in fair division theory through concrete counterexamples, clarifying the necessity and sufficiency of key hypotheses.

Findings

Identified common errors in fair division literature

Provided concrete counterexamples to misconceptions

Clarified the role of hypotheses in fair division

Abstract

The formal mathematical theory of fair division has a rich history dating back at least to Steinhaus in the 1940's. In recent work in this area, several general classes of errors have appeared along with confusion about the necessity and sufficiency of certain hypotheses. It is the purpose of this article to correct the scientific record and to point out with concrete examples some of the pitfalls that have led to these mistakes. These examples may serve as guideposts for future work.