Random Assignment of Indivisible Goods under Constraints

TL;DR

This paper studies the complex problem of assigning indivisible goods fairly and efficiently under constraints, revealing when such assignments exist and proposing mechanisms for cases where they do not.

Contribution

It characterizes conditions for the existence of efficient, envy-free assignments under constraints and develops mechanisms for cases where standard methods fail.

Findings

Efficient, envy-free assignments may not exist under certain constraints.

Special cases guarantee existence of such assignments.

New mechanisms are proposed for constrained assignment problems.

Abstract

We investigate the problem of random assignment of indivisible goods, in which each agent has an ordinal preference and a constraint. Our goal is to characterize the conditions under which there always exists a random assignment that simultaneously satisfies efficiency and envy-freeness. The probabilistic serial mechanism ensures the existence of such an assignment for the unconstrained setting. In this paper, we consider a more general setting in which each agent can consume a set of items only if the set satisfies her feasibility constraint. Such constraints must be taken into account in student course placements, employee shift assignments, and so on. We demonstrate that an efficient and envy-free assignment may not exist even for the simple case of partition matroid constraints, where the items are categorized, and each agent demands one item from each category. We then identify special cases in which an efficient and envy-free assignment always exists. For these cases, the probabilistic serial cannot be naturally extended; therefore, we provide mechanisms to find the desired assignment using various approaches.