The price of imposing vertical equity through asymmetric outcome constraints

Vertical equity or fairness is an important aspect in many settings, yet has received relatively little attention in the literature. Recent developments underline the practical relevance (e.g., COVID-19 vaccination policies). It plays an important role in the performance evaluation of many (nongovernmental) organizations. For example, donors might require a family-planning organization to allocate a minimum fraction of the total utility (client volume) to a particular player (the "high-impact"subgroup of the population, e.g., young and poor clients). However, the price (decrease in client volume) of such requirements is not well-understood. Consequently, this price is not accounted for in decisionmaking. We provide an analytical upper bound on the price (i.e., loss of overall utility) of vertical equity considerations in resource allocation. We assume that these concerns are expressed via outcome constraints, specifying a minimum percentage of the total utility for each player. Our set-up considers a decision maker maximizing total utility over a general convex set, subject to outcome constraints. The set-up is general and applicable to many practical problems. Our results depend only on high-level parameters and are therefore well-suited for strategic decision making. We conclude with two applications. First, we apply our results to practical instances in health delivery.We confirm that outcome constraints can entail a substantial price and analyze the factors driving this price close to the worst-case bound. Second, we analyze how our results can help bound the impact of prioritization in vaccine allocation.