A Solution to the Envy-Free Selection Problem in Economies with Indivisible Goods

Enriqueta Aragones

No 984, Discussion Papers from Northwestern University, Center for Mathematical Studies in Economics and Management Science

Abstract: We consider the problem of selecting envy-free allocations in economies with indivisible objects and qusi-linear utility functions. We study the set of envy-free allocations for these economies and characterize the minimal amount of money necessary for its nonemptiness when negative distributions of money are not allowed. We also find that, when this is precisely the available amount of money, there is a unique way to combine objects and money such that these bundles may form an envy-free allocation. Based on this property, we propose a solution that selects a unique utility profile for any economy. When there is more money than is needed to solve the envy-free problem, our solution allocates it equally and we retain the uniqueness of the utility profile. Among the properties satisfied by this solution we find that it is invariant with respect to shifts in the utility scale of any agent, not obviously manipulable and can be computed by a polynomincal-bounded algorithm. We show that when some agents leave the economy, the sent of envy-free allocations for the new economy may offer new possible combinations of objects and money. Based on this, we argue that one should not expect a solution to this selection problem to satisfy any property related to consistency, as has been suggested in the literature.

Date: 1992-04
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