Non-Borda elections under relaxed IIA conditions

TL;DR

This paper explores the relaxation of Arrow's IIA condition to MIIA, constructing social welfare functions that satisfy this weaker criterion and differ from Borda, revealing new possibilities and limitations in election rules.

Contribution

It introduces a modified IIA condition (MIIA), constructs SWFs satisfying MIIA that are distinct from Borda, and identifies cases where only Borda-like rules are possible.

Findings

SWFs satisfying MIIA can differ significantly from Borda.

Certain election scenarios enforce Borda as the only SWF.

Relaxing IIA broadens the class of acceptable election rules.

Abstract

Arrow's celebrated Impossibility Theorem asserts that an election rule, or Social Welfare Function (SWF), between three or more candidates meeting a set of strict criteria cannot exist. Maskin suggests that Arrow's conditions for SWFs are too strict. In particular he weakens the "Independence of Irrelevant Alternatives" condition (IIA), which states that if in two elections, each voter's binary preference between candidates $c_i$ and $c_j$ is the same, then the two results must agree on their preference between $c_i$ and $c_j$. Instead, he proposes a modified IIA condition (MIIA). Under this condition, the result between $c_i$ and $c_j$ can be affected not just by the order of $c_i$ and $c_j$ in each voter's ranking, but also the number of candidates between them. More candidates between $c_i$ and $c_j$ communicates some information about the strength of a voter's preference between the two candidates, and Maskin argues that it should be admissible evidence in deciding on a final ranking. We construct SWFs for three-party elections which meet the MIIA criterion along with other sensibility criteria, but are far from being Borda elections (where each voter assigns a score to each candidate linearly according to their ranking). On the other hand, we give cases in which any SWF must be the Borda rule.