Minimizing and Balancing Envy Among Agents Using Ordered Weighted Average

In fair resource allocation, envy freeness (EF) is one of the most interesting fairness criteria as it ensures no agent prefers the bundle of another agent. However, when considering indivisible goods, an EF allocation may not exist. In this paper, we investigate a new relaxation of EF consisting in minimizing the Ordered Weighted Average (OWA) of the envy vector. The idea is to choose the allocation that is fair in the sense of the distribution of the envy among agents. The OWA aggregator is a well-known tool to express fairness in multiagent optimization. In this paper, we focus on fair OWA operators where the weights of the OWA are decreasing. When an EF allocation exists, minimizing OWA envy will return this allocation. However, when no EF allocation exists, one may wonder how fair min OWA envy allocations are. After defining the model, we show how to formulate the computation of such a min OWA envy allocation as a Mixed Integer Program. Then, we investigate the link between the min OWA allocation and other well-known fairness measures such as max min share and EF up to one good or to any good. Finally, we run some experiments comparing the performances of our approach with MNW (Max Nash Welfare) on several criteria such as the percentage of EF up to one good and any good.