The Distortion of Approval Voting with Runoff

Recent work introduces approval with runoff voting, in which voters cast approval ballots, two finalists are selected, and a runoff election is conducted between them to choose the final winner by majority voting. While the more common plurality with runoff voting admits only one reasonable choice of the two finalists (the two candidates with the most plurality votes), the use of approval ballots in the first stage opens up the possibility of using many reasonable ways to choose the two finalists. What is the optimal way to choose the two finalists? In this work, we answer this question using the distortion framework, in which the performance of every voting system is quantitatively measured by its worst-case social welfare approximation ratio, also known as distortion. We prove that the best distortion achievable by approval voting with (majority) runoff is Θ(m2) with deterministic finalist selection and Θ(m) with randomized finalist selection, where m is the number of candidates. This is actually worse than what simple approval voting without any runoff achieves (Θ(m) and Θ(√m), respectively). We pinpoint the use of majority runoff in the second stage as the culprit, propose a candidate proportional runoff system that declares each finalist the winner with probability equal to the fraction of voters who prefer it, and analyze the extent to which it can help curb the distortion.