Price of Fairness for allocating a bounded resource

We study the problem faced by a decision maker who wants to allocate a scarce resource among several agents in order to maximize the total utility. An optimal solution may present a very unbalanced allocation of the resource to the agents and hence be perceived as unfair. On the other hand balanced allocations may be far from the optimum. In this paper we are interested in assessing the quality of fair solutions, i.e., in measuring the system efficiency loss under a fair allocation compared to the one that maximizes the total utility. This indicator is called the Price of Fairness and we study it under three different definitions of fairness, namely maximin, Kalai-Smorodinski and proportional fairness. Our results are twofold. We first formalize a number of properties holding for any general multi agent problem without any special assumptions on the agent utilities. Then we introduce an allocation problem, in which each agent can consume the available bounded resource in given discrete quantities (items). The utility of each agent is given by the sum of these quantities (weights of allocated items). We distinguish two cases depending on whether disjoint sets or a shared set of items is available to the agents. Clearly, the maximization of the total utility is given by a Subset Sum Problem. For the resulting Fair Subset Sum Problem, in the case of two agents, we provide upper and lower bounds on the Price of Fairness as functions of an upper bound on the items size. (C) 2016 Elsevier B.V. All rights reserved.