A Game of Fair Division

The " divide-and-choose " method has played an important role in the literature on fair division.' This technique for allocating bundles of goods seems impartial, requires little cooperation from agents, and is nearly free of administrative costs. It is therefore somewhat puzzling that it has found so few applications in the real world, where sometimes even prolonged and costly negotiations produce only imperfect agreements. Either the method has drawbacks not yet well understood, or it is underutilized. This paper examines the game that arises when two agents agree to use the divide-and-choose method. The analysis leads to a resolution of the puzzle mentioned above and identifies a class of situations where replacing conventional arbitration procedures with the divide-and-choose method can be strongly recommended. In the sequel, an agent who would prefer another agent's bundle of goods to his own will be said to envy the other agent. An allocation at which no agent envies another will be called a fair allocation.2 A well-known property of the two-person version of the divideand-choose game3 is that each player can insure that he does not envy the other. The divider (D) can accomplish this by dividing so that lie is indifferent to his opponent's choice; the chooser (C) need only choose his most preferred bundle after D divides. It is interesting that the players can insure that the outcome of the game is fair, but more information about the allocations actually generated by the game is needed to judge its usefulness as a fair division device. Conceivably, with players motivated by self-interest, the game could generate an unfair allocation in spite of the above result. To learn more about the divide-and-choose method, I assume that players seek to obtain the most desirable bundle possible. They are also assumed to behave noncooperatively, since negotiating a mutually acceptable settlement would be relatively easy if they were willing to cooperate, and the method would then be superfluous. In Section 2 of this paper D's problem is formulated and his optimal non-cooperative strategy is characterized. As is suggested by Kolm [6, p. 61], Luce and Raiffa [8, p. 365] and Singer [12], the common belief that D should divide the bundle so that D is indifferent about C's choice is false. If D knows C's preferences with certainty, under very general conditionsroughly, that players' behaviour can be described by the maximization of continuous and strongly monotonic utility functions and that goods are homogeneous and perfectly divisible-his optimal strategy involves dividing the bundle so that it is C, rather than D, who is indifferent about his choice. Once D's optimal strategy has been characterized, several interesting conclusions follow. As Kolm [6, p. 31] points out, Luce and Raiffa's belief [8, pp. 364-365] that the