Twofold optimality of the relative utilitarian bargaining solution

Given a bargaining problem, the relative utilitarian (RU) solution maximizes the sum total of the bargainer’s utilities, after having first renormalized each utility function to range from zero to one. We show that RU is ‘optimal’ in two very different senses. First, RU is the maximal element (over the set of all bargaining solutions) under any partial ordering which satisfies certain axioms of fairness and consistency; this result is closely analogous to the result of Segal (2000). Second, RU offers each person the maximum expected utility amongst all rescaling-invariant solutions, when it is applied to a random sequence of future bargaining problems which are generated using a certain class of distributions; this is somewhat reminiscent of the results of Harsanyi (1953) and Karni (1998). Let I be a finite group of individuals, and let A be a set of social outcomes (e.g. allocations of some finite stock of resources). If each i ∈ I has an ordinal preference relation overA and also over the set of all lotteries between elements in A, and if these lottery preferences satisfy the von Neumann-Morgenstern (vNM) axioms of minimal rationality, then we can define a cardinal utility function ui : A−→R 6− := [0,∞) such that i’s lottery preferences are consistent with maximization of the expected value of ui. Let u := (ui)i∈I : A−→R6− be the ‘joint’ utility function, and let B be the convex, comprehensive closure of the image set u(A) ⊂ R6−; then any element of B represents an assignment of a vNM utility level to each player, obtainable through some lottery between elements of A. Let ℘B be the Pareto frontier of B. We assume that the members of I can obtain any social outcome in ℘B, but only through unanimous consent. Let