Nonsymmetric equal sacrifice solutions for claim problem

Abstract

A claim (allocation) problem is the problem of distributing a given amount of a divisible resource among agents with unequal claims on the resource. The main result of the paper is the following representation of a strictly monotonic, consistent, path independent, and individually unbounded solution of the claim problem. There exists a family of strictly increasing continuous utility functions of agents such that all agents have equal differences between values of their utility functions at the solution point and at the claim point. These solutions of the claim problem are called equal sacrifice solutions with respect to the family of utility functions. Young [Journal of Economic Theory 44 (1988) 321] obtained a similar representation under an additional anonymity condition. Due to this result, we get the solution of goal programming problems with convex feasible sets satisfying natural axioms. The solution of the goal programming problem is the point minimizing the measure of proximity to the goal point. Here the measure of proximity is defined as the sum of antiderivatives of differences between utility functions. Under additional positive homogeneity condition, we get maximal weighted entropy solution.