Steven J. Brams (NYU), D. Marc Kilgour (Wilfrid Laurier University), Christian Klamler (University of Graz) & Fan Wei (Princeton)
Journal of Philosophy, 2023
The early literature on fair division focused on a divisible good like cake, over which two people may have different preferences. We focus on indivisible items, like the marital property in a divorce, which cannot so easily be split. This switches the problem from making cuts to allocating whole items to each person. We show that in the two-person case, there are several desiderata one might aspire to, but a central one, championed by Rawls, was the prevention of envy by helping the least well off in a society. This was not a concern of Bentham. We show consequences of the different principles that each figure espoused.
The paper’s abstract:
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 there may be no EF division, in which case some division must satisfy a modification of Bentham’s (1789/2017) “greatest satisfaction of the greatest number” property, called maximum Nash welfare (MNW), that satisfies PO. However, an MNF division may be neither MM nor EFX, which is a weaker form of EF. We conjecture that there is always an EFX allocation that satisfies MM, ensuring that an allocation is maximin, precisely the property that Rawls (1971/1999) championed. We discuss four broader philosophical implications of our more technical analysis.