On monotonicity in economies with indivisible goods

We consider the problem of fair allocation in economies with indivisible objects that may or may not be desirable (for instance, activities that may or may not be pleasurable but have to be carried out unless there are not enough agents for that). We search for efficient solutions satisfying two additional properties. First, each agent should find his bundle at least as desirable as the bundle that would be assigned to him in the hypothetical economy in which all agents have preferences identical to his, under equal treatment of equals and efficiency. In a preliminary step, we show that there is no logical relation between this requirement and no-envy, and between it and egalitarian-equivalence. We also establish the existence of efficient allocations satisfying it. The second property, object monotonicity, says that the availability of additional objects either has a negative impact on everyone's welfare, or it has a positive impact on everyone's welfare. We show that there is no object-monotonic selection from the correspondence that associates with each economy its set of efficient allocations meeting an even weaker version of the bound.I am grateful to Atila Abdulkadiroglu, Koichi Tadenuma, and a referee for their very helpful comments.