Divide-and-Conquer: A Proportional, Minimal-Envy Cake-Cutting Algorithm

We analyze a class of proportional cake-cutting algorithms that use a minimal number of cuts (n − 1 if there are n players) to divide a cake that the players value along one dimension. While these algorithms may not produce an envy-free or efficient allocation—as these terms are used in the fair-division literature—one, divide-and-conquer (D&C), minimizes the maximum number of players that any single player can envy. It works by asking n ≥ 2 players successively to place marks on a cake—valued along a line—that divide it into equal halves (when n is even) or nearly equal halves (when n is odd), then halves of these halves, and so on. Among other properties, D&C ensures players of at least 1/n shares, as they each value the cake, if and only if they are truthful. However, D&C may not allow players to obtain proportional, connected pieces if they have unequal entitlements. Possible applications of D&C to land division are briefly discussed. 1. Introduction. A cake is a metaphor for a heterogeneous good, whose parts each of n players may value differently. A proportional division of a cake is one that gives each player, as it values the cake, at least a 1/n portion, which we call a proportional share. We represent a cake by the interval [0, 1], over which each player's preference is given by a probability density function with a continuous cumulative distribution function. There exist several algorithms for cutting this cake into pieces such that each player receives a proportional share, but we know of only one algorithm, due to Dubins and Spanier (1961), that does so using only n − 1 cuts (the minimal number), which are assumed to cut the interval at points in (0, 1). However, this algorithm, which we will describe later, requires a knife to move continuously across a cake and players to make cuts by calling " stop. " By contrast, a discrete algorithm specifies when and what kinds of cuts will be made that do not depend on the continuous movement of knives.