Fair welfare maximization

We consider the general problem of finding fair constrained resource allocations. As a criterion for fairness we propose an inequality index, termed “fairness ratio,” the maximization of which produces Lorenz-undominated, Pareto-optimal allocations. The fairness ratio does not depend on the choice of any particular social welfare function, and hence it can be used for an a priori evaluation of any given feasible resource allocation. The fairness ratio for an allocation provides a bound on the discrepancy between this allocation and any other feasible allocation with respect to a large class of social welfare functions. We provide a simple representation of the fairness ratio as well as a general method that can be used to directly determine optimal fair allocations. For general convex environments, we provide a fundamental lower bound for the optimal fairness ratio and show that as the population size increases, the We would like to thank Kenneth Arrow, James Foster, Krishna Pendakur, Clemens Puppe, George Shanthikumar, participants of the 2005 Econometric Society World Congress at University College London, and seminar participants at UC Berkeley for helpful comments. Goel acknowledges NSF CAREER Award 0133968, NSF ITR Grant 0428868, a Terman Engineering Fellowship, a 3-COM Faculty Scholar Award, an Alfred P. Sloan Faculty Fellowship, and gifts from Google, Cisco, and Microsoft. Meyerson acknowledges NSF grant CCR-0122581 and ARO grant DAAG55-98-1-0170. Weber acknowledges a David T. Morgenthaler II Faculty Scholar Award and grant 1094280-1-WAKAB by the Woods Institute for the Environment. A. Goel · T. A. Weber (B) Department of Management Science and Engineering, Stanford University, Stanford, CA 94305, USA e-mail: [email protected] A. Goel e-mail: [email protected] A. Meyerson Computer Science Department, UCLA, Los Angeles, CA 90095, USA e-mail: [email protected]