Generalizing Instant Runoff Voting to Allow Indifferences

TL;DR

The paper generalizes IRV to weak orders by introducing Approval-IRV, and extends the idea to STV via Approval-STV. It provides two foundational axioms—independence of clones and respect for cohesive majorities—and a monotonicity notion for indifferences, showing Approval-IRV uniquely satisfies these within elimination scoring rules, while Split-IRV does not. The multi-winner analogue Approval-STV is shown to preserve generalized PSC, addressing proportional representation in the weak-order setting. Empirical results on synthetic and real data indicate Approval-IRV tends to select higher-quality winners (in Borda and distortion metrics) and achieves competitive alignment with Condorcet outcomes, supporting its practical appeal as a richer, less burdensome voting input. Overall, the work offers a theoretically grounded, practically attractive alternative to traditional IRV when voters may legitimately express indifference among candidates.

Abstract

Instant Runoff Voting (IRV) is used in elections for many political offices around the world. It allows voters to specify their preferences among candidates as a ranking. We identify a generalization of the rule, called Approval-IRV, that allows voters more freedom by allowing them to give equal preference to several candidates. Such weak orders are a more expressive input format than linear orders, and they help reduce the cognitive effort of voting. Just like standard IRV, Approval-IRV proceeds in rounds by successively eliminating candidates. It interprets each vote as an approval vote for its most-preferred candidates among those that have not been eliminated. At each step, it eliminates the candidate who is approved by the fewest voters. Among the large class of scoring elimination rules, we prove that Approval-IRV is the unique way of extending IRV to weak orders that preserves its characteristic axiomatic properties, in particular independence of clones and respecting a majority's top choices. We also show that Approval-IRV is the unique extension of IRV among rules in this class that satisfies a natural monotonicity property defined for weak orders. Prior work has proposed a different generalization of IRV, which we call Split-IRV, where instead of approving, each vote is interpreted as splitting 1 point equally among its top choices (for example, 0.25 points each if a vote has 4 top choices), and then eliminating the candidate with the lowest score. Split-IRV fails independence of clones, may not respect majority wishes, and fails our monotonicity condition. The multi-winner version of IRV is known as Single Transferable Vote (STV). We prove that Approval-STV continues to satisfy the strong proportional representation properties of STV, underlining that the approval way is the right way of extending the IRV/STV idea to weak orders.

Figures (24)

• Figure 1: Ballot from 2016 Australian federal election
• Figure 2: The fraction of voters in the 2022 Australian federal election that needed to rank each number of candidates between 5 and 12.
• Figure 3: An example of Approval-IRV with voters $v_1, \dots, v_5$. The first eliminated alternative is $c$, which is ranked on top only once. Then $d$ is eliminated, and finally $a$ wins the majority vote against $b$. Thus, $a$ is the winner.
• Figure 4: Ballot data from the 2019 mayoral election in San Francisco.
• Figure 5: Examples of ballots that can be interpreted as weak orders (2019 mayoral election in San Francisco).

Theorems & Definitions (19)

• theorem \ref{thm:clonemajority}
• theorem \ref{thm:monocharacterization}
• theorem \ref{thm:approval-stv-satisfies-gen-psc}
• definition 1
• theorem \ref{thm:approval-stv-satisfies-gen-psc}
• definition 2
• proposition 1
• theorem \ref{thm:approval-stv-satisfies-gen-psc}
• theorem \ref{thm:approval-stv-satisfies-gen-psc}
• definition 3