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

Abstract

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.

Keywords: mechanism design, fair division, divisible good, cake-cutting, divide-and-choose

JEL Classification: C7, D6, D7