Relaxed Notions of Condorcet-Consistency and Efficiency for Strategyproof Social Decision Schemes

TL;DR

This paper introduces relaxed notions of Condorcet-consistency and efficiency in strategyproof social decision schemes, characterizing optimal schemes that balance these properties and extending Gibbard's theorem.

Contribution

It provides a novel characterization of strategyproof SDSs with relaxed conditions, identifying the Copeland-based scheme as optimal under certain bounds and strengthening Gibbard's theorem.

Findings

Copeland-based SDS guarantees a Condorcet winner probability of at least 2/m.

No strategyproof SDS can surpass this Condorcet probability bound.

Mixtures of random dictatorship and randomized Copeland are optimal for balancing Condorcet and Pareto-dominated probabilities.

Abstract

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 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 SDS that assigns probabilities proportional to Copeland scores is the only anonymous, neutral, and strategyproof SDS that can guarantee the Condorcet winner a probability of at least 2/m. Moreover, no strategyproof SDS can exceed this bound, even when dropping anonymity and neutrality. 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.