Good election rules with more than three candidates are Borda
Gabriel Gendler
TL;DR
The paper shows that for more than three candidates ($k>3$), any Social Welfare Function satisfying Unrestricted domain (U), Modified IIA (MIIA), Anonymity (A), and Neutrality (N) collapses to a Borda rule (specifically one of the unweighted Borda variants) in both infinite and finite voter settings, provided measurability conditions are met in the infinite case. It develops a toolkit of transitive election tables and relative weight-based SWFs to demonstrate that the pairwise weight differences $d_1$ determine the full voting rule, thereby proving Borda-determinacy. The results reinforce Maskin’s and Arrow’s dichotomies, showing that with more than four candidates the Borda rule is uniquely compatible with the axioms, while relaxing measurability permits non-Borda constructions under AC in the infinite-voter case. The paper also clarifies how the finite-voter case aligns with these conclusions and sharpens the boundaries between Borda and non-Borda possibilities across domains.
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.