Allocation Inequality in Cost Sharing Problem

Abstract

This paper considers the problem of cost sharing, in which a coalition of agents, each endowed with an input, shares the output cost incurred from the total inputs of the coalition. Two allocations---average cost pricing and the Shapley value---are arguably the two most widely studied solution concepts to this problem. It is well known in the literature that the two allocations can be respectively characterized by different sets of axioms and they share many properties that are deemed reasonable. We seek to bridge the two allocations from a different angle---allocation inequality. We use the partial order: Lorenz order (or majorization) to characterize allocation inequality and we derive simple conditions under which one allocation Lorenz dominates (or is majorized by) the other. Examples are given to show that the two allocations are not always comparable by Lorenz order. Our proof, built on solving minimization problems of certain Schur-convex or Schur-concave objective functions over input vectors, may be of independent interest.

Keywords: cost sharing problem, average cost pricing, Shapley value, majorization, cooperative game