Scarce Societal Resource Allocation and the Price of (Local) Justice

We consider the allocation of scarce societal resources, where a central authority decides which individuals receive which resources under capacity or budget constraints. Several algorithmic fairness criteria have been proposed to guide these procedures, each quantifying a notion of local justice to ensure the allocation is aligned with the principles of the local institution making the allocation. For example, the efficient allocation maximizes overall social welfare, whereas the leximin assignment seeks to help the "neediest first." Although the "price of fairness" (PoF) of leximin has been studied in prior work, we expand on these results by exploiting the structure inherent in real-world scenarios to provide tighter bounds. We further propose a novel criterion - which we term LoINC (leximin over individually normalized costs) - that maximizes a different but commonly used notion of local justice: prioritizing those benefiting the most from receiving the resources. We derive analogous PoF bounds for LoINC, showing that the price of LoINC is typically much lower than that of leximin. We provide extensive experimental results using both synthetic data and in a real-world setting considering the efficacy of different homelessness interventions. These results show that the empirical PoF tends to be substantially lower than worst-case bounds would imply and allow us to characterize situations where the price of LoINC fairness can be high.