Some Theoretical Results on Monotonicity-Related Properties of Voting Rules

This chapter gives an overview of some of the most important theoretical results related to monotonicity properties of choice rules in fixed and variable electorates. The earliest one is Smith’s theorem from early 1970s. It deals with runoff systems based on scoring functions. Fishburn’s theorem, on the other hand, links non-monotonicity to positional dominance and pairwise majority rule by showing that a reasonable combination of these is incompatible with monotonicity. Another theorem by Fishburn is related to runoff systems where a winner may be found on a single round if its support is high enough. The theorem shows that also these systems are non-monotonic. Of results pertaining to variable electorates Moulin’s theorem is well-known. It shows that with more than three alternatives and at least 25 voters, all Condorcet extension rules are vulnerable to the no-show paradox. Pérez strengthens this result by establishing the incompatibility of Condorcet consistency and invulnerability to the strong version of the no-show paradox for nearly all known Condorcet extensions. Saari’s theorem establishes the extreme variation of electoral outcomes in subsets of alternatives in some profiles. He also shows that in those profiles a particularly dramatic version P-BOT paradox occurs. The Muller-Satterthwaite theorem shows that all positively non-imposed Maskin monotonic choice rules are dictatorial. We present these theorems and sketch the proofs of some of them. Some comments of their relevance in choice rule evaluation are also presented.