An impossibility result for strongly group-strategyproof multi-winner approval-based voting
Ioannis Caragiannis, Rob LeGrand, Evangelos Markakis, Emmanouil Pountourakis
TL;DR
This work proves a strong impossibility for unanimous multi-winner approval voting: for $m≥3$ alternatives and any $k$ with $k ∈ {1,2,...,m-2}$, no voting rule can be both unanimous and strongly group-strategyproof. The authors build a reduction from ranking-based single-winner manipulation (Gibbard–Satterthwaite) to approval-based multi-winner settings, showing that a Pareto-efficient, strongly GSP rule would imply a SP, onto single-winner rule, contradicting GS. The result implies that strongly GSP mechanisms cannot achieve finite approximations for minimax approval voting, participatory budgeting with approvals, binary classification with shared inputs, or constrained facility location, signaling fundamental limits in strategyproof mechanism design without money. The paper also identifies specific regime boundaries (e.g., small numbers of agents) where positive results might still be possible and highlights the broader theoretical impact on approximation guarantees in strategic settings.
Abstract
Multi-winner approval-based voting has received considerable attention recently. A voting rule in this setting takes as input ballots in which each agent approves a subset of the available alternatives and outputs a committee of alternatives of given size $k$. We consider the scenario when a coalition of agents can act strategically and alter their ballots so that the new outcome is strictly better for a coalition member and at least as good for anyone else in the coalition. Voting rules that are robust against this strategic behaviour are called strongly group-strategyproof. We prove that, for $k\in \{1,2, ..., m-2\}$, strongly group-strategyproof multi-winner approval-based voting rules which furthermore satisfy the minimum efficiency requirement of unanimity do not exist, where $m$ is the number of available alternatives. Our proof builds a connection to single-winner voting with ranking-based ballots and exploits the infamous Gibbard-Satterthwaite theorem to reach the desired impossibility result. Our result has implications for paradigmatic problems from the area of approximate mechanism design without money and indicates that strongly group-strategyproof mechanisms for minimax approval voting, variants of facility location, and classification can only have an unbounded approximation ratio.