N-person Envy-free Chore Division

In this paper we consider the problem of chore division, which is closely related to a classical question, due to Steinhaus [7], of how to cut a cake fairly. We focus on constructive solutions, i.e., those obtained via a well-defined procedure or algorithm. Among the many notions of fairness is envy-freeness: an envy-free cake division is a set of cuts and an allocation of the pieces that gives each person what she feels is the largest piece. Much progress has been made on finding constructive algorithms for achieving envy-free cake divisions; a landmark result was that of Brams and Taylor [1], who gave the first general n-person procedure. In contrast to cakes, which are desirable, the dual problem of chore division is concerned with dividing an object deemed undesirable. Here, each player would like to receive what he considers to be the smallest piece, of say, a set of chores. This problem appears to have been first introduced by Martin Gardner in [4]. Oskui (see [6]) referred to it as the dirty work problem and gave the first discrete and moving-knife solutions for exact envyfree chore division among 3 people. Peterson and Su [5] gave the first explicit 4-person moving-knife procedure for chore division, adapting ideas of Brams, Taylor, and Zwicker [3] for cake-cutting. The purpose of this article is to give a general n-person solution to the chore division problem. Su [9] gives an n-person chore division algorithm but it only yields an ǫ-approximate solution after a finite number of steps. Brams and Taylor suggest in [2] how cake-cutting methods could be adapted to chore division without working out the details, and our algorithm owes a great debt to their ideas. But we also show where some new ideas are needed, and why the chore division problem is not exactly a dual or straightforward extension of the cake-cutting problem.