A Note on the Assignment Problem with Uniform Preferences

TL;DR

This paper investigates the assignment problem with uniform preferences, showing that extending the probabilistic serial mechanism to weak preferences fails strategyproofness and that envy-free, efficient mechanisms cannot be strategyproof.

Contribution

It demonstrates the limitations of mechanism design in uniform preference domains, extending known results to weak preferences and highlighting the trade-offs involved.

Findings

Extension of probabilistic serial fails strategyproofness

All ordinally efficient, equal-treatment mechanisms fail strategyproofness

Envy-free, ex post efficient mechanisms cannot be strategyproof

Abstract

Motivated by a problem of scheduling unit-length jobs with weak preferences over time-slots, the random assignment problem (also called the house allocation problem) is considered on a uniform preference domain. For the subdomain in which preferences are strict except possibly for the class of unacceptable objects, Bogomolnaia and Moulin characterized the probabilistic serial mechanism as the only mechanism satisfying equal treatment of equals, strategyproofness, and ordinal efficiency. The main result in this paper is that the natural extension of the probabilistic serial mechanism to the domain of weak, but uniform, preferences fails strategyproofness, but so does every other mechanism that is ordinally efficient and treats equals equally. If envy-free assignments are required, then any (probabilistic or deterministic) mechanism that guarantees an ex post efficient outcome must fail even a weak form of strategyproofness.