On fair and efficient solutions for budget apportionment

This article deals with an apportionment problem involving n agents and a common budget B. Each agent submits some demands which are indivisible portions of the budget, and a central authority has to decide which demands to accept. The utility of an agent corresponds to the total amount of her accepted demands. In this context, it is desirable to be fair among the agents and efficient by not wasting the budget. An ideal solution would be to spend exactly B/n for every agent but this is rarely possible because of the indivisibility of the demands. Since combining fairness with efficiency is highly desirable but often impossible, we explore relaxed notions of fairness and efficiency, in order to determine if they go together. Our approach is also constructive because polynomial algorithms that build fair and efficient solutions are also given. The fairness criteria under consideration are the maximization of the minimum agent utility (max–min), proportionality, a customized notion of envy-freeness called jealousy-freeness, and the relaxations up to one or any demand of the previous two concepts. Efficiency in this work is either the maximization of the utilitarian social welfare or Pareto optimality. First we consider fairness and efficiency separately. The existence and computation of solutions that are either fair or efficient are studied. A complete picture of the relations that connect the fairness and efficiency concepts is provided. Second, we determine when fairness and efficiency can be combined for every possible instance. We prove that Pareto optimality is compatible with two notions of fairness, namely max–min and proportionality up to any demand. In contrast, none of the fairness concepts under consideration can be paired with the maximization of utilitarian social welfare. Therefore, we finally conduct a thorough analysis of the price of fairness which bounds the loss of efficiency caused by imposing fairness or one of its relaxations.