TY - JOUR

T1 - Relaxed notions of Condorcet-consistency and efficiency for strategyproof social decision schemes

AU - Brandt, Felix

AU - Lederer, Patrick

AU - Romen, René

N1 - Publisher Copyright:
© The Author(s) 2024.

PY - 2024

Y1 - 2024

N2 - Social decision schemes (SDSs) map the preferences of a group of voters over some set of m alternatives to a probability distribution over the alternatives. A seminal characterization of strategyproof SDSs by Gibbard (Econometrica 45(3):665–681, 1977) implies that there are no strategyproof Condorcet extensions and that only random dictatorships satisfy ex post efficiency and strategyproofness. The latter is known as the random dictatorship theorem. We relax Condorcet-consistency and ex post efficiency by introducing a lower bound on the probability of Condorcet winners and an upper bound on the probability of Pareto-dominated alternatives, respectively. We then show that the randomized Copeland rule is the only anonymous, neutral, and strategyproof SDS that guarantees the Condorcet winner a probability of at least 2/m. Secondly, we prove a continuous strengthening of Gibbard’s random dictatorship theorem: the less probability we put on Pareto-dominated alternatives, the closer to a random dictatorship is the resulting SDS. Finally, we show that the only anonymous, neutral, and strategyproof SDSs that maximize the probability of Condorcet winners while minimizing the probability of Pareto-dominated alternatives are mixtures of the uniform random dictatorship and the randomized Copeland rule.

AB - Social decision schemes (SDSs) map the preferences of a group of voters over some set of m alternatives to a probability distribution over the alternatives. A seminal characterization of strategyproof SDSs by Gibbard (Econometrica 45(3):665–681, 1977) implies that there are no strategyproof Condorcet extensions and that only random dictatorships satisfy ex post efficiency and strategyproofness. The latter is known as the random dictatorship theorem. We relax Condorcet-consistency and ex post efficiency by introducing a lower bound on the probability of Condorcet winners and an upper bound on the probability of Pareto-dominated alternatives, respectively. We then show that the randomized Copeland rule is the only anonymous, neutral, and strategyproof SDS that guarantees the Condorcet winner a probability of at least 2/m. Secondly, we prove a continuous strengthening of Gibbard’s random dictatorship theorem: the less probability we put on Pareto-dominated alternatives, the closer to a random dictatorship is the resulting SDS. Finally, we show that the only anonymous, neutral, and strategyproof SDSs that maximize the probability of Condorcet winners while minimizing the probability of Pareto-dominated alternatives are mixtures of the uniform random dictatorship and the randomized Copeland rule.

UR - http://www.scopus.com/inward/record.url?scp=85189068064&partnerID=8YFLogxK

U2 - 10.1007/s00355-024-01519-0

DO - 10.1007/s00355-024-01519-0

M3 - Article

AN - SCOPUS:85189068064

SN - 0176-1714

JO - Social Choice and Welfare

JF - Social Choice and Welfare

ER -