A law of large numbers for weighted plurality

TL;DR

This paper proves that in elections with many voters and a preferred candidate, a weighted plurality voting rule ensures the preferred candidate's victory with high probability, unlike other voting rules, extending previous results for two candidates.

Contribution

It extends a known law of large numbers for the preferred candidate winning under weighted plurality to multiple candidates, highlighting its unique robustness.

Findings

Weighted plurality ensures the preferred candidate wins with high probability.

Other reasonable voting rules do not guarantee the preferred candidate's victory.

The result generalizes previous two-candidate cases to multiple candidates.

Abstract

Consider an election between k candidates in which each voter votes randomly (but not necessarily independently) and suppose that there is a single candidate that every voter prefers (in the sense that each voter is more likely to vote for this special candidate than any other candidate). Suppose we have a voting rule that takes all of the votes and produces a single outcome and suppose that each individual voter has little effect on the outcome of the voting rule. If the voting rule is a weighted plurality, then we show that with high probability, the preferred candidate will win the election. Conversely, we show that this statement fails for all other reasonable voting rules. This result is an extension of H\"aggstr\"om, Kalai and Mossel, who proved the above in the case k=2.