Characterizing the Top Cycle via Strategyproofness

TL;DR

This paper characterizes the top cycle as the unique set-valued social choice correspondence that is strategyproof, non-imposing, and solely based on pairwise comparisons, extending classical impossibility results.

Contribution

It provides a complete characterization of the top cycle as the only strategyproof, non-imposing SCC depending only on pairwise comparisons, generalizing Gibbard-Satterthwaite.

Findings

Top cycle is uniquely strategyproof and non-imposing under mild conditions.

Characterization extends Gibbard-Satterthwaite to set-valued SCCs.

Outcome depends solely on pairwise comparisons.

Abstract

Gibbard and Satterthwaite have shown that the only single-valued social choice functions (SCFs) that satisfy non-imposition (i.e., the function's range coincides with its codomain) and strategyproofness (i.e., voters are never better off by misrepresenting their preferences) are dictatorships. In this paper, we consider set-valued social choice correspondences (SCCs) that are strategyproof according to Fishburn's preference extension and, in particular, the top cycle, an attractive SCC that returns the maximal elements of the transitive closure of the weak majority relation. Our main theorem implies that, under mild conditions, the top cycle is the only non-imposing strategyproof SCC whose outcome only depends on the quantified pairwise comparisons between alternatives. This result effectively turns the Gibbard-Satterthwaite impossibility into a complete characterization of the top cycle by moving from SCFs to SCCs. It is obtained as a corollary of a more general characterization of strategyproof SCCs.