The Impossibility of Strategyproof Rank Aggregation
Manuel Eberl, Patrick Lederer
TL;DR
The paper analyzes strategyproofness in rank-aggregation SWFs under Kendall/Kemeny distance, proving a sweeping impossibility: no anonymous SWF can be both unanimous and strategyproof when the number of alternatives satisfies $m\geq 4$ (and similarly under certain $n$ for larger $m$). The authors combine SAT-based computer-aided proofs with an Isabelle/HOL verification to establish base-case nonexistence and lift it via inductive arguments, also showing that majority-consistency conflicts with strategyproofness. They further quantify manipulability via incentive ratios, showing that natural SWFs (Kemeny, distance scoring, positional scoring) are highly manipulable, often with large ratios or unbounded incentives. The results suggest that circumventing the impossibility requires relaxing assumptions (randomization, set-valued outcomes, or restricted domains) and point to rich avenues for future research in robust, manipulation-resistant rank aggregation. Overall, the work provides a rigorous, computer-verified analogue of Gibbard–Satterthwaite for social welfare functions in rank aggregation and maps out the practical limits of incentive-robust design.
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.