Mathematical Flaws in Ranked Choice Voting Are Rare but Real

A political movement in the U.S. is encouraging municipalities and states to adopt ranked choice voting as a supposedly more representative voting method. In new research, David McCune and Adam Graham-Squire analyze the theoretical and historically observed flaws of ranked choice voting and argue that politicians and voters must weigh both its benefits and shortcomings when considering adoption.

Should more cities and states adopt ranked choice voting (RCV) for municipal and state elections? Is RCV a “good” way to select election winners? These questions are not easily resolved, as the answer depends on what one wants from a voting method and whether a method’s benefits outweigh its downsides (all voting methods seemingly have downsides). Furthermore, the study of how best to select election winners is decidedly interdisciplinary, involving tools from political science, economics, mathematics, and psychology, among others. As mathematicians, we approach these questions by examining how frequently RCV’s deficiencies manifest in actual elections, where we focus on deficiencies which have historically been of interest in the mathematically oriented social choice literature.

How does RCV work? Voters cast a preference ballot where they rank candidates from first to last. If a candidate receives a majority of first-place preferences, that candidate is declared the winner. Otherwise, the voting process eliminates the candidate with the fewest first-place votes. Ballots that ranked the eliminated candidate first are then reallocated to the candidates they ranked second (or, if the candidate ranked second has previously been eliminated, are reallocated to the third candidate, etc.). The process continues in this fashion until a candidate has earned a majority of the remaining votes. RCV is commonly referred to as “instant runoff voting” because it uses preference ballots to instantaneously mimic the outcomes of potential future runoff elections, thereby saving the jurisdiction the cost and time of holding such runoffs.

To illustrate this process, consider the table below, which contains the preference ballot information for the August 2022 Special Election for the U.S. House in Alaska. This election was the first ranked choice election for state or federal office in the state. Neglecting write-in candidates, this election contained three candidates: Republicans Nick Begich and Sarah Palin and Democrat Mary Peltola. 

The table lists every observed ballot permutation from the Special Election. So, the number 7,623 in the table denotes that 7,623 voters ranked Begich as their first choice, Palin as their second choice, and Peltola as their last choice. Remember that voters are not required to select a second or third choice. The first row of data conveys that 11,262 voters chose Begich as their first preference but did not select a second or third preference.

Counting the number of first-place votes for each candidate yields vote totals of 53,810, 58,974, and 75,799 for Begich, Palin, and Peltola, respectively. No candidate earns a majority and thus Begich is eliminated. As a result, 27,070 of his votes are transferred to Palin and 15,478 are transferred to Peltola, and Peltola wins the election with 91,277 first-place votes to Palin’s 86,044.

This election demonstrates several of RCV’s less desirable features, from a social choice or mathematical point of view. First, when using RCV, it is possible that a candidate can be hurt by receiving more support from voters. In this election, if 6,000 of the voters who ranked only Palin were to instead rank Peltola first and Palin second, Peltola would lose the resulting election. The reason is that even though Peltola receives more initial voter support with this hypothetical change to these 6,000 ballots, in the resulting election Palin would be eliminated first and then Begich would receive enough votes from her elimination to defeat Peltola. In other words, in this election, if Peltola had done a better job reaching out to Palin voters, it would have cost her the election. The reason RCV is susceptible to this kind of problem is that changing ballots can cause a change in the order in which candidates are eliminated, potentially changing the eventual winner.

Second, RCV is susceptible to the so-called “spoiler effect,” which is usually defined as an outcome in which the removal of a losing candidate from the election changes the winner. In this election, Palin is a spoiler candidate: if we remove her from the election then (assuming all voters still choose to vote, excepting the 21,237 who voted just for Palin) Begich would win the election with 87,888 votes to Peltola’s 79,458.

Third, when using RCV, it is possible to have a set of voters that cause their least favorite candidate to win by ranking their favorite candidate in first place. In this election, if 6,000 of the voters who ranked Palin first, Begich second, and Peltola third had instead ranked Begich in first, then Begich would have won the election and these voters would have had their second favorite as the winner, rather than their least favorite. This is undesirable because typically we would like voters to rank their favorite candidate in first place without worrying that by doing so they are creating a less desirable electoral outcome.

In our work, we examine the frequency with which these kinds of issues arise in RCV elections in the U.S. We wanted to investigate if this Alaska election is a “typical” RCV election or if it is an outlier. We collected 182 RCV elections for political office in the U.S. where no candidate received an initial majority and we wrote code to check for the kinds of RCV deficiencies mentioned above. We found three elections in which the winner can be made into a loser by shifting them up the rankings on some ballots, three elections which demonstrate the spoiler effect, and seven elections in which some voters should have ranked a different candidate in first place to avoid having elected their least favorite candidate. Elections like what occurred in Alaska are outliers. 

However, even with a low failure rate, it is reasonable to reject RCV because of its susceptibility to these kinds of outcomes. While we can say that these kinds of issues occur rarely, they do occur sometimes, and such outcomes are perhaps not worth the benefits of RCV when weighed against the benefits of other methods. 

In defense of RCV, at least in relation to the election in Alaska, we note two pertinent facts: One, the RCV data indicates that Peltola still would have won if the election were run using the election rules prior to implementation of RCV, in which there was an initial round of plurality voting followed by a runoff election. Thus, switching to RCV seemingly did not alter the electoral outcome. Two, a reasonable reaction to the RCV issues in the Alaska election is that these anomalies are present because RCV chose the wrong winner. If Begich had been elected instead we would not observe the spoiler effect, for example. The voters in Alaska seem to disagree with this reaction—the House election with Peltola, Palin and Begich was repeated three months later in November 2022, and voters again elected Peltola with RCV, by a larger margin, in an election with none of the issues outlined above. 

We note that RCV has one issue which occurs frequently in our elections database. RCV proponents often claim that one of its benefits is that the eventual winner earns majority support from the electorate. This is not true—the eventual winner is only guaranteed to win a majority of the remaining votes, after eliminating other candidates.  In the Alaska election, a true majority does not occur in the final round of vote counting, as Peltola earns a victory with 91,277 votes, far short of a majority of the approximately 189,000 voters who cast a ballot in the election. The reason she does not obtain a true majority is that when Begich is eliminated, the 11,262 ballots which only rank him are removed from the election and are not transferred to Palin or Peltola. This kind of “majoritarian failure” occurs in 95 of the 182 elections in our database. In practice, if no candidate receives an initial majority then it is very likely that no candidate ever will. It is possible, however, that this will improve if voters decide to rank more candidates in RCV elections.

Our research does not settle any debates about RCV as an election method, but it gives  important contextual information about the practical use of RCV: majoritarian failures are common, other issues rarely occur, but when other issues occur they are nontrivial. When deciding if to implement RCV in a given state or municipality, those facts should be weighed against the benefits of RCV, as well as the benefits and drawbacks of other voting methods.

Articles represent the opinions of their writers, not necessarily those of the University of Chicago, the Booth School of Business, or its faculty.