Price of Fairness in Budget Division and Probabilistic Social Choice

A group of agents needs to divide a divisible common resource (such as a monetary budget) among several uses or projects. We assume that agents have approval preferences over projects, and their utility is the fraction of the budget spent on approved projects. If we maximize utilitarian social welfare, the entire budget will be spent on a single popular project, even if a substantial fraction of the agents disapprove it. This violates the individual fair share axiom (IFS) which requires that for each agent, at least 1/n of the budget is spent on approved projects. We study the price of imposing such fairness axioms on utilitarian social welfare. We show that no division rule satisfying IFS can guarantee to achieve more than an O(1/ √ m) fraction of maximum utilitarian welfare, in the worst case. However, imposing stronger group fairness conditions (such as the core) does not come with an increased price, since both the conditional utilitarian rule and the Nash rule match this bound and guarantee an Ω(1/ √ m) fraction. The same guarantee is attained by the rule under which the spending on a project is proportional to its approval score. We also study a family of rules interpolating between the utilitarian and the Nash rule, quantifying a trade-off between welfare and group fairness. An experimental analysis by sampling using several probabilistic models shows that the conditional utilitarian rule achieves very high welfare on average.