On the Resolution of Ties in Fair Convex Allocation Problems

We study the emergence of indistinguishable, but structurally distinct, outcomes in convex resource allocation models with fairness-oriented objectives. Such outcomes arise when differently prioritized users receive identical coverage despite differences in initial conditions. We formalize this phenomenon as indeterminate priority (IP) outcomes and provide a structural taxonomy, based on Karush-Kuhn-Tucker conditions, that characterizes when they can occur and how they depend on resource scarcity and overlap. We are motivated by applications where visibly distinct outcomes are desired for prioritized users. To this end, we introduce a preference for distinctness such that users are discouraged from ties in final coverage whenever they compete for shared resources. We do so via a smooth pairwise penalty that perturbs the objective by penalizing ties in final coverages, and show that for sufficiently small coefficients this perturbation preserves the underlying fairness structure. As a case study, we apply our framework to a previously developed vaccine distribution model from Texas. IP outcomes arise in this real-world instance under both equal and unequal weights, and persist even under increased resource availability. By appropriately choosing the perturbation parameter's scale, our approach yields distinct interpretable solutions without distorting the fairness objective. We further illustrate how the choice of penalty shape and perturbation magnitude influences the extent of separation among tied users.