Better Ways to Cut a Cake

Abstract

In this paper we show how mathematics can illuminate the study of cakecutting in ways that have practical implications. Specifically, we analyze cakecutting algorithms that use a minimal number of cuts (n−1if there aren people), where a cake is a metaphor for a heterogeneous, divisible good, whose parts may be valued differently by different people. These algorithms not only establish the existence of fair divisions—defined by the properties described below—but also specify a procedure for carrying them out. In addition, they give us insight into the difficulties underlying the simultaneous satisfaction of certain properties of fair division, including strategyproofness, or the incentive for a person to be truthful about his or her valuation of a cake. As is usual in the cakecutting literature, we postulate that the goal of each person is to maximize the value of the minimumsize piece (maximin piece) that he or she can guarantee, regardless of what the other persons do. Thus, we assume that each person is riskaverse: He or she will never choose a strategy that may yield a more valuable piece of cake if it entails the possibility of getting less than a maximin piece. First we analyze the wellknown 2person, 1cut cakecutting procedure, “I cut, you choose, ” or cutandchoose.Itgoesbackatleasttothe HebrewBible (Brams and Taylor, 1999, p. 53) and satisfies two desirable properties: (1) Envyfreeness: Each person thinks that he or she receives at least a tiedforlargest piece and so does not envy the other person.

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