Allocating multiple estates among agents with single-peaked preferences

We consider the problem of allocating multiple social endowments (estates) of a perfectly divisible commodity among a group of agents with single-peaked preferences when each agent’s share can come from at most one estate. We assume that an agent only cares about the size of her share and not the identity of the estate from which it comes. We inquire if well-known single-estate rules, such as the Uniform rule, the Proportional rule or the fixed-path rules can be coupled with a matching rule so as to achieve eciency in the multi-estate level. We first observe that on the class of problems where all agents have symmetric preferences, any ecient single-estate rule can be extended to an ecient multi-estate rule. We however obtain negative results on the full preference domain.