The Bipartite Rationing Problem

In the bipartite rationing problem, a set of agents share a single resource available in different “types,” each agent has a claim over only a subset of the resource types, and these claims overlap in arbitrary fashion. The goal is to divide fairly the various types of resources between the claimants when resources are in short supply. With a single type of resource, this is the standard rationing problem [O'Neill B (1982) A problem of rights arbitration from the Talmud. Math. Soc. Sci. 2(4):345--371], of which the three benchmark solutions are the proportional, uniform gains, and uniform losses methods. We extend these methods to the bipartite context, imposing the familiar consistency requirement: the division is unchanged if we remove an agent (respectively, a resource), and take away at the same time his share of the various resources (respectively, reduce the claims of the relevant agents). The uniform gains and uniform losses methods have infinitely many consistent extensions, but the proportional method has only one. In contrast, we find that most parametric rationing methods [Young HP (1987a) On dividing an amount according to individual claims or liabilities. Math. Oper. Res. 12(3):397--414], [Thomson W (2003) Axiomatic and game-theoretic analysis of bankruptcy and taxation problems. Math. Soc. Sci. 45(3):249--297] cannot be consistently extended.