Ballot Length in Instant Runoff Voting

TL;DR

This paper investigates how restricting ballot length in Instant Runoff Voting can significantly influence election outcomes, providing theoretical bounds, constructions, and empirical analysis of real-world elections.

Contribution

It offers the first comprehensive analysis of the impact of ballot length restrictions on IRV election results, including bounds, winner sequences, and real-world implications.

Findings

Shorter ballots can change election outcomes in about 25% of real-world cases.

Preference restrictions like single-peakedness limit the number of possible winners.

Explicit constructions demonstrate the theoretical bounds on voter profiles and winner sequences.

Abstract

Instant runoff voting (IRV) is an increasingly-popular alternative to traditional plurality voting in which voters submit rankings over the candidates rather than single votes. In practice, elections using IRV often restrict the ballot length, the number of candidates a voter is allowed to rank on their ballot. We theoretically and empirically analyze how ballot length can influence the outcome of an election, given fixed voter preferences. We show that there exist preference profiles over $k$ candidates such that up to $k-1$ different candidates win at different ballot lengths. We derive exact lower bounds on the number of voters required for such profiles and provide a construction matching the lower bound for unrestricted voter preferences. Additionally, we characterize which sequences of winners are possible over ballot lengths and provide explicit profile constructions achieving any feasible winner sequence. We also examine how classic preference restrictions influence our results--for instance, single-peakedness makes $k-1$ different winners impossible but still allows at least $\Omega(\sqrt k)$. Finally, we analyze a collection of 168 real-world elections, where we truncate rankings to simulate shorter ballots. We find that shorter ballots could have changed the outcome in one quarter of these elections. Our results highlight ballot length as a consequential degree of freedom in the design of IRV elections.