The Impossibility of Strategyproof Rank Aggregation

TL;DR

This paper proves that no social welfare function can be both strategyproof and satisfy certain fairness criteria in rank aggregation with four or more options, highlighting fundamental limitations in designing manipulation-resistant ranking systems.

Contribution

It establishes the impossibility of strategyproof rank aggregation functions satisfying unanimity and anonymity for four or more alternatives, using computer-aided proofs and manual reasoning.

Findings

No anonymous SWF satisfies unanimity and strategyproofness with ≥4 alternatives.

Strategyproofness conflicts with majority consistency in SWFs.

All SWFs in certain classes are highly manipulable with large incentive ratios.

Abstract

In rank aggregation, the goal is to combine multiple input rankings into a single output ranking. In this paper, we analyze rank aggregation methods, so-called social welfare functions (SWFs), with respect to strategyproofness, which requires that no agent can misreport his ranking to obtain an output ranking that is closer to his true ranking in terms of the Kemeny distance. As our main result, we show that no anonymous SWF satisfies unanimity and strategyproofness when there are at least four alternatives. This result is proven by SAT solving, a computer-aided theorem proving technique, and verified by Isabelle, a highly trustworthy interactive proof assistant. Further, we prove by hand that strategyproofness is incompatible with majority consistency, a variant of Condorcet-consistency for SWFs. Lastly, we show that all SWFs in two natural classes have a large incentive ratio and are thus highly manipulable.