Incompatibility of Efficiency and Strategyproofness in the Random Assignment Setting with Indifferences

TL;DR

This paper proves that in the random assignment problem with indifferences, no mechanism can be both ex post efficient and weak strategyproof, highlighting an inherent trade-off in designing such mechanisms.

Contribution

It demonstrates the fundamental incompatibility of efficiency and strategyproofness for the probabilistic serial mechanism when preferences include indifferences.

Findings

No extension of PS is ex post efficient and weak strategyproof with indifferences.

The incompatibility holds without additional symmetry or fairness assumptions.

The result clarifies the limitations of existing mechanisms in more general preference domains.

Abstract

A fundamental resource allocation setting is the random assignment problem in which agents express preferences over objects that are then randomly allocated to the agents. In 2001, Bogomolnaia and Moulin presented the probabilistic serial (PS) mechanism that is an anonymous, neutral, Pareto optimal, and weak strategyproof mechanism when the preferences are considered with respect to stochastic dominance. The result holds when agents have strict preferences over individual objects. It has been an open problem whether there exists a mechanism that satisfies the same properties when agents may have indifference among the objects. We show that for this more general domain, there exists no extension of PS that is ex post efficient and weak strategyproof. The result is surprising because it does not even require additional symmetry or fairness conditions such as anonymity, neutrality, or equal treatment of equals. Our result further demonstrates that the lack of weak SD-strategyproofness of the extended PS mechanism of Katta and Sethuraman (2006) is not a design flaw of extended PS but is due to an inherent incompatibility of efficiency and strategyproofness of PS in the full preference domain.