Preference relations and measures in the context of fair division

Abstract

One of the most well-known metaphors in the mathematical theory of fair division concerns the problem of dividing a cake among n people in such a way that each person is satisfied with the piece he or she receives, even though different people value different parts of the cake differently. Our concern here is with how an individual’s preferences are formalized. David Gale has pointed out that although most of the deeper results in the field assume that preferences are given by an additive measure, the fundamental algorithms in the field require only that preferences be given by a binary relation satisfying a few natural properties. We introduce here one additional condition—an Archimedean property that obviously holds if the relation is induced by a measure—and we show that a preference relation satisfying Gale’s conditions is induced by a finitely additive measure if and only if it satisfies this Archimedean property.