BetterWays to Cut a Cake

I n this paper we show how mathematics can illuminate the study of cake-cutting in ways that have practical implications. Specifically, we analyze cake-cutting algorithms that use a minimal number of cuts (n − 1 if there are n 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 strategy-proofness, or the incentive for a person to be truthful about his or her valuation of a cake. As is usual in the cake-cutting literature,we postulate that the goal of each person is to maximize the value of the minimum-size piece (maximin piece) that he or she can guarantee, regardless of what the other persons do. Thus, we assume that each person is risk-averse: 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 amaximinpiece. First we analyze the well-known 2-person, 1-cut cake-cutting procedure, “I cut, you choose,” or cutand-choose. It goesbackat least to theHebrewBible (Brams and Taylor, 1999, p. 53) and satisfies two desirableproperties: