Equal-quantile rules in resource allocation with uncertain needs

A group of agents have uncertain needs on a resource, and the resource has to be divided before uncertainty resolves. We propose a class of division rules we call equal-quantile rules, parameterized by λ ∈ (0, 1]. The parameter λ is a common maximal probability of satisfaction — the probability that an agent’s realized need is no more than his assignment — imposed on all agents. Thus, the maximal assignment of each agent is his λ-quantile assignment. If the endowment of the resource exceeds the sum of the agents’ λ-quantile assignments, each agent receives his λ-quantile assignment and the resource is not fully allocated to the agents. Otherwise, the resource is fully allocated and the rule equalizes the probability of satisfaction across agents.We provide justifications for the class of equal-quantile rules from two perspectives. First, each equal-quantile rule maximizes a particular utilitarian social welfare function that involves an outside agent, who provides an alternative use of the resource, and aggregates linear individual utilities. Equivalently, it minimizes a particular utilitarian social cost function that is the sum of the aggregate expected waste and the aggregate expected deficit, weighted, respectively, by a unit waste cost and a unit deficit cost. Second, four familiar axioms, consistency, continuity, strict ranking, and ordinality, when extended to the uncertain context, characterize the class of equal-quantile rules. Thus, requiring the four axioms is equivalent to imposing either of these utilitarian objective functions.