Ranked Choice voting is having a bit of a moment in the public consciousness — at least on the very handicapped scale of voting systems. It promises to avoid some of the unsavory outcomes that occur in the “First Past the Post” Plurality system used in the vast majority of U.S. elections by allowing voters to rank all of the candidates in a race rather that vote for just a single candidate. Organizations like FairVote have even seen some success in promoting Ranked Choice voting (and other voting reforms), as Maine has adopted it for several federal and state elections, it has been partially adopted in some Democratic Presidential primaries, and some cities have begun using it for local elections. But figuring out how good this change is — and whether there are much better changes we could make — is a surprisingly challenging task (hence why this post is humbly subtitled “Part 1”).
Plurality systems, which dominate American elections, give each voter one vote, and whichever candidates gets the most votes, wins. The straightforward, intuitive nature of this system might make the existence of alternatives hard to imagine, but there are alternatives, and many of them have compelling advantages over the plurality system. In particular, we’ll look at three competitors to Plurality voting systems to try to get a handle on their potential strengths, weaknesses, and oddities.
A few alternatives to First Past the Post
One alternative to Plurality is Plurality Runoff. The “Jungle Primaries” seen in some U.S. elections are a particular kind of Plurality Runoff that replaces party primaries. In a Plurality Runoff election, you can run a Plurality election, eliminate all but the top two contenders, and then run a second election between just the two remaining candidates. Plurality Runoff elections in the U.S. are usually done this way instead of having an election with party primaries. This means that the final election could be between two candidates of the same party, as I occasionally see on my California ballots. This also makes an apples-to-apples comparison of these voting methods a bit tricky in the U.S. context, as Plurality can be used for primaries and then general elections individually, whereas Plurality Runoff can act as a system that incorporates both the primary and general election stages, or it could be used to replace either of the stages individually.
Ranked Choice operates somewhat similarly to Plurality Runoff, but instead of eliminating all but the top two finishers, we eliminate the last place candidate, over and over, until one candidate gets a majority. Instead of running elections over and over, we save taxpayer money by having voters rank the candidates on their ballots. (Plurality Runoff can be done in the same way.) Whenever a candidate is eliminated, all the voters whose votes were counted for that candidate have their votes moved to the next active (not yet eliminated) candidate in their ranking. We repeat this process until one candidate gets a majority of the votes and is declared the winner.
To illustrate Ranked Choice, let’s use everyone’s favorite Presidential election, the 1912 contest (and ignore the Electoral College). This race pitted Republican incumbent William Taft against Democrat Woodrow Wilson, Progressive Teddy Roosevelt, and Socialist Eugene Debs. (Tragically) Wilson won with about 42% of the vote, followed by Roosevelt with ~28%, Taft with ~23%, and Debs with ~6%. In a Ranked Choice system, rather than declaring Wilson the winner, we would first eliminate Debs, and move Debs’ voters to their second choices. My rough guess from my limited knowledge of politics at the time is that more of them would have gone to Teddy than the other candidates (though historical accuracy isn’t important for this explanation). That might leave us with something like:
Now we eliminate Taft. Taft had been in Roosevelt’s previous administration, and even with the hard feelings that led Roosevelt to run against Taft, I would imagine that Taft voters would typically rank Roosevelt over Wilson on their ballots. We then end up with a final result of:
Teddy wins and retroactively becomes the first Roosevelt to get a third term as President, saving minorities, political dissenters, and journalists from Woodrow Wilson. This kind of history-altering result is why people advocate for Ranked Choice.
Borda Count can use the same ballots as Ranked Choice, but instead of eliminating candidates until we have a winner, candidates are given points based on where they are ranked on each ballot. The candidate with the most total points from all the ballots wins. For instance, if someone fills out a ballot as
Then Roosevelt would get the most points from their ballot and Debs would get the least. The simplest way to allocate points is to give the last place candidate on a ballot zero points, the second to last place candidate one point, the third to last candidate two points, and so on. This is the only point assignment method we will use when investigating Borda Count, at least for now.
A comparison of how these rules would work in the real world requires a massive amount of information about how actual voters and candidates make decisions. This means that we have to make complicated models of how voters and candidates act and the strategies they implement in elections governed by various voting rules. Voters and candidates are clever enough to make strategic decisions about how to vote or position themselves in response to the rules of the election and the nature of the field running for office.
A simple model of elections
Instead of trying to dive in the deep end right way, we’re going to start with a simple, idealized model of how voters and candidates will behave. Our voters will be simple, honest folk who will vote exactly according to how they feel about the candidates. They will like candidates more based only on how similar those candidates views are to their own, and they will be perfectly aware of the ideological positions of all the candidates. These voters don’t care whether they could help a stronger candidate that they also like by voting differently. Both the candidates and voters will have political beliefs that are randomly distributed in an ideology space (think of something like the political compass of /r/politicalcompassmemes infamy), where the further people are from each other in the space the more they disagree on political issues. How much a voter likes a candidate, will be -d, where d is the distance between the voter’s and candidate’s ideological positions in the ideology space. This means that a voter will always prefer a candidate whose views are more similar to her own over a candidate whose views are further away from her views.
We’ll run elections using Plurality, Plurality Runoff, Ranked Choice, and Borda Count and then measure how well each rule does by capturing how much the voters, on average, liked the winner — the “Average Utility” of the voters. After running 1,000 elections with 1,000 voters for each number of candidates, here is what we find:
So what is happening here? Each of these voting rules is identical to Plurality when there are two candidates, but as the number of candidates grow, the voting rules begin to distinguish themselves. Plurality — the voting rule that dominates elections in the U.S. — quickly distinguishes itself as the worst performer, although it does at least do significantly better than just randomly selecting candidates. Ranked choice and Plurality Runoff linger in the middle and start to flatten out as the number of candidates gets large. Borda Count, despite being able to use the exact same ballots that Ranked Choice uses, performs better than its competitors and, uniquely, steadily improves as more candidates enter the race.
The comparison becomes more clear as the number of candidates gets really big. Borda Count continues to improve, Ranked Choice flattens out after about a half dozen candidates, and Plurality Runoff and Plurality get worse results, with Plurality in particular falling behind. (Similar outcomes occur if our ideology space has one dimension or more than two dimensions.)
This seems to suggest that Borda Count is the superior voting method and only gets better as the field of candidates grows, Plurality is as bad as advertised, and Ranked Choice and Plurality Runoff are in between, with all but Borda Count performing worse as the field of candidates gets very large.
This model is way, way too simple
A particularly clever reader might realize that this way of measuring how well voting rules perform is really just measuring how well they do at selecting the candidate who is in the ideological center of voters. This might sound like a reasonable metric, but whether this is actually what people want is dubious. For instance, people value the direction candidates will move their country/state/city/etc. in addition to how closely candidates agree with them on issues. For instance, a voter who is slightly to the left of the status quo might really like a candidate who is much further to the left of the status quo because the voter likes the direction that candidate would move things. We’ll look at how this affects how voting rules perform in the Part 2.
Also, we have assumed that both voters and candidates are honestly and unstrategically voting and positioning themselves. This world looks more like the world of that terrible “The Invention of Lying” movie than the real world. (Most) voters aren’t total nincompoops and aren’t going to vote for candidates if their vote for that candidate can’t help them get an outcome they like. And (most) candidates aren’t total nincompoops and will at least try to sell themselves as being someone agreeable to voters if not actually change their positions to be more in line with voters’ beliefs.
In the opposite direction, people are working with limited information. If you had to rank twenty candidates for an office, could you? I certainly couldn’t. Just as voters and candidates can be more clever than this first-pass model assumes, they can also be less clever. How well realistic voters can fill out Ranked Choice/Borda Count ballots according to their own preferences could dramatically affect how those elections play out.
So there is a lot more to do to get a realistic idea of how Ranked Choice would compare to these other systems. We have a lot of building on this current model to do in terms of measuring what voters want, how voters and candidates act strategically, and how voters and candidates act irrationally. Each of these impacts how well different voting rules perform, and perhaps we will find that, under more realistic conditions, we find something completely different than we found in Part 1.
5 thoughts on “Is Ranked Choice Voting the Hero We Need? (Part 1)”
[…] our first look into this topic, we used a simple model of voters and candidates to find some mixed results for […]
[…] previous posts (Part 1 and Part 2) on the topic of modelling elections, we’ve looked at how well a few different […]
[…] my past posts on voting (here, here, here, and here), Borda Count has come out looking quite strong compared to the other ordinal […]
I suspect your graphs would be smoother if you ran more simulations, I don’t think that jaggedness is real.
[…] look at elections where voters and candidates are uniformly distributed in a two dimensional ideological space (they are equally likely to have any ideological views). I […]