On the relation between Preference Reversal and Strategy-Proofness

TL;DR

This paper explores the connection between strategy-proofness and preference reversal, extending previous results to broader domains and emphasizing the role of individual strategy-proofness in social choice theory.

Contribution

It generalizes and simplifies existing results on preference reversal, highlighting the importance of individual strategy-proofness and introducing the concept of complete preference sets.

Findings

Preference reversal can be derived from individual strategy-proofness under certain domain conditions.

The difficulties in relaxing the range size assumption are mitigated by domain restrictions.

Group strategy-proofness is mainly necessary when the number of voters is infinite.

Abstract

We analyze the relation between strategy-proofness and preference reversal in the case that agents may declare indifference. Interestingly, Berga and Moreno (2020), have recently derived preference reversal from group strategy-proofness of social choice functions on strict preferences domains if the range has no more than three elements. We extend this result and at the same time simplify it. Our analysis points out the role of individual strategy-proofness in deriving the preference reversal property, giving back to the latter its original individual nature (cfr. Eliaz, 2004). Moreover, we show that the difficulties Berga and Moreno highlighted relaxing the assumption on the cardinality of the range, disappear under a proper assumption on the domain. We introduce the concept of complete sets of preferences and show that individual strategy-proofness is sufficient to obtain the preference reversal property when the agents' feasible set of orderings is complete. This covers interesting cases like single peaked preferences, rich domains admitting regular social choice functions, and universal domains. The fact that we use individual rather than group strategy-proofness, allows to get immediately some of the known, and some new, equivalences between individual and group strategy-proofness. Finally, we show that group strategy-proofness is only really needed to obtain preference reversal if there are infinitely many voters.