Strongly nonmanipulable multi-valued collective choice rules

Abstract

A collective choice rule is 'manipulated' whenever some individual misrepresents his preferences in order to secure an outcome preferred to the outcome when he his honest. According to the theorem of Gibbard [6] and Satterthwaite [10] (G-S in what follows), the only collective choice rule that cannot be manipulated is a dictatorship. While this result looks overwhelmingly final, and depressing, it does raise a few questions. In G-S, a collective choice rule is single-valued: for each specification of all individuals' preferences, or each 'preference profile', it produces a single winner, a unique alternative. What if, as it is often the case, the collective choice rule allows for ties? What happens to the nonmanipulability implies dictatorship theorem then? These are essentially the questions asked by Kelly [9] and Barbera [2]. If a collective choice rule is multi-valued, however, a new issue arises. Properly defined, alternatives are mutually exclusive, and ultimately only one can be chosen. A multi-valued collective choice rule is then a first step; if it produces ties, the ties must be broken. The obvious tie-breaker is a coin toss, that is, a random device. What happens to the nonmanipulability implies dictatorship theorem when the collective choice rule is random, or when it is a mixture of nonrandom and random parts? This is the question asked by Gibbard in [7]. The answers of Kelly [9] and Barbera [2] can be loosely characterized this way: Nonmanipilable multi-valued collective choice rules necessarily make some people 'weak dictators'. Now, a person is a 'weak dictator' if, whenever he prefers x to y and the social choice agenda is restricted to x versus y, x must at least tie y. However, it is arguable that the existence of weak dictatorship is per se objectionable; since the welfare economist's mainstay, the Pareto rule, makes everyone a weak dictator. In another