Good election rules with more than three candidates are Borda

TL;DR

This paper proves that for elections with four or more candidates, the only acceptable voting rules satisfying the MIIA condition are the unweighted Borda rules, extending previous results that were limited to three candidates.

Contribution

It establishes that the uniqueness of Borda rules as the only good election rules under MIIA holds for four or more candidates, unlike the three-candidate case.

Findings

For three candidates, non-Borda MIIA rules exist.

For four or more candidates, only unweighted Borda rules satisfy MIIA.

The phenomenon of non-Borda MIIA rules is unique to three candidates.

Abstract

Arrow proved that for three or more candidates, the IIA condition is enough to forbid all non-dictatorial election rules (or Social Welfare Functions). Maskin introduced the weaker MIIA condition, which permits the ``Borda'' election rules where each voter assigns points linearly to each candidate according to their order of preference. However, in previous work we demonstrated that there exist Social Welfare Functions between three candidates and satisfying the MIIA condition which are far from being Borda rules. We demonstrate that this phenomenon is unique to the case of three candidates. As soon as a fourth candidate is introduced, and indeed for any larger number of candidates, the only good election rules are the unweighted Borda rules.