Two-Person Fair Division of Indivisible Items: Compatible and Incompatible Properties

Suppose two players wish to divide a finite set of indivisible items, over which each distributes a specified number of points. Assuming the utility of a player’s bundle is the sum of the points it assigns to the items it contains, we analyze what divisions are fair. We show that if there is an envy-free (EF) allocation of the items, two other desirable properties—Pareto-optimality (PO) and maximinality (MM)—can also be satisfied, rendering these three properties compatible, but other properties—balance (BL), maximum Nash welfare (MNW), maximum total welfare (MTW), and lexicographic optimality (LO)—may fail. If there is no EF division, as is likely, it is always possible to satisfy EFx, a weaker form of EF, but an EFx allocation may not be PO, BL, MNW, MTW, or LO. Moreover, if one player considers an item worthless (i.e., assigns zero points to it), an EFx division may be Pareto dominated by a nonEFx allocation that is MNW. Although these incompatibilities suggest that there is no “perfect” 2-person fair division of indivisible items, EFx and MNW divisions—if they give different allocations when there is no EF-PO-MM division—seem the most compelling alternatives, with EFx, we conjecture, satisfying the Rawlsian objective of helping the worse-off player and MNW, a modification of MTW, suggesting a more Benthamite view.