Perfect Cake-Cutting Procedures with Money Steven J. Brams Department of Politics New York University New York, NY 10003 UNITED STATES
[email protected] Michael A. Jones Department of Mathematics Montclair State University Upper Montclair, NJ 07043 UNITED STATES
[email protected] Christian Klamler Institute of Public Economics University of Graz A-8010 Graz AUSTRIA
[email protected] December 2003 2 Abstract A cake-cutting procedure is perfect if it satisfies the properties of efficiency, envy- freeness, and equitability. For two players there is no perfect procedure, including cut- and-choose and Austin’s procedure. But the addition of money makes perfection possible, as we demonstrate by describing a 2-player, 1-cut perfect procedure that induces each player to be truthful in order to maximize the minimum portion of cake it can guarantee for itself. For n > 2 players, an envy-free procedure can be rendered equitable with the addition of money, but not necessarily efficient if it uses more than n - 1 cuts (the minimal number). Although there is a minimal 3-player, 2-cut envy-free procedure, no 4-player, 3-cut envy-free procedure is known to exist. 3 Perfect Cake-Cutting Procedures with Money 1. Introduction By perfect we mean a cake-cutting procedure that satisfies three desirable properties—efficiency, envy-freeness, and equitability—which we will describe and illustrate shortly. This ideal, as Jones (2002) showed, can in principle be achieved for n = 2 people with one cut. However, no procedure, or set of rules, is known that gives two people an incentive to make the perfect cut.