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 optimal fairness ratio decreases at most logarithmically in what we call the "inhomogeneity" of the problem. Our method yields a unique and "balanced" fair optimum for an important class of problems with linear budget constraints.