Fair Welfare Maximization

We discuss the general problem of finding fair constrained resource allocations. As criterion for fairness we propose an inequality index, termed fairness ratio, the maximization of which produces Lorenz-undominated Pareto-efficient allocations. Since the fairness ratio does not depend on the choice of any particular social welfare function, it can be used for an a priori evaluation of any given feasible resource allocation. The ratio 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. Our method yields a unique and balanced fair optimum for an important class of problems with linear budget constraints. We provide a fundamental lower bound for the optimal fairness ratio and show that as the population size increases, the optimal fairness ratio decreases at most logarithmically in what we call the inhomogeneity of the problem