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Merging and splitting endowments in object assignment problems

We consider a group of agents, each endowed with a set of indivisible goods, called “objects”, and equipped with a preference relation over subsets of the union of their endowments. An “allocation rule” associates with each economy of this type a redistribution among them of their endowments. We are concerned about two kinds of opportunities to manipulate a rule to their advantage that agents may have. First, imagine an agent giving control of his endowment to some other agent, and withdrawing; the rule is applied; the agent with the augmented endowment may now be assigned a bundle that can be partitioned into two bundles, one for himself and one for the agent who withdrew, that make both of them at least as well off as they would have been otherwise and at least one of them better off. Alternatively, an agent may invite some outsider, someone who has no endowment, and give him control over part of his endowment. In the enlarged economy, the two of them may be assigned bundles whose union the agent who invited the outsider prefers to what he would have been assigned otherwise. We are interested in selections from the correspondence that selects for each economy its efficient allocations that each agent finds his component of it at least as desirable as his endowment. We inquire about the existence of