How to Re-Divide a Cake Fairly

We study the classic problem of fairly dividing a heterogeneous resource (”cake”) among several agents with different preferences. In our setting, the resource is already divided among several ”old” agents. When ”young” agents come, the resource should be re-divided in a way that is fair both for the young and for the old agents. Motivated by land redivision and other two-dimensional division problems, we assume that the cake is a rectilinear polygon and require that the allotted pieces be rectangles. Our re-division protocol has an implication on another problem: the price-of-fairness the loss of social welfare caused by fairness requirements. Our protocol allows us to prove upper bounds on the price-of-fairness in two-dimensional cake-cutting with rectangular pieces. Our protocol also handles the more general case in which the cake is convex and the pieces can be arbitrary convex figures.