The Mathematics of Ranked Choice Voting

Introduction: Elections are Group Decisions

We have a Presidential election happening very soon and there’s a lot at stake on ballots across the country. Here in Massachusetts, in addition to electing representatives to federal, state, and local offices, we also have an opportunity to change the way we will calculate election winners in the future. Massachusetts Question 2, if passed, will enact Ranked Choice Voting as the method for determining winners of single-seat elections like Senators and House Representatives. What exactly is this method and how does it work?

In this blog post, I will describe a few simple examples in detail to demonstrate how this method works and illustrate some of its benefits. If you search online for information about Ranked Choice Voting (RCV), you will find no shortage of video clips and websites. However, I have noticed that many attempts to explain the method fall short of helpful because they don’t actually show any examples, either because the explainer doesn’t quite understand all the details or perhaps because they assume you won’t understand all the details. I won’t do that. Here, I will focus on explaining the details of HOW we vote without necessarily focusing on WHO we vote for. That’s what I mean by the mathematics of RCV.

My first lesson is to broaden your mind about what we can mean by “election,” for it does not have to mean people vying for political office. Any time you have a group of individual entities with their own preferences, and those preferences need to be factored in and combined into a decision for the group … that’s an election! For example, suppose you and several friends are meeting up for a group study session and you want to order some takeout for dinner. How could you all decide what cuisine to order?

Let’s say there are nine of you deciding on a cuisine, and in a friendly discussion amongst yourselves, you have already narrowed down to three options: Indian food, Pizza, and Thai food. This means there are nine voters in the electorate and three candidates. If this were a typical election in the USA, then you would go into the ballot booth and identify one and only one of those candidates as your top choice. (Look at the instructions when you cast your ballot this year; I bet it says something like: “Vote for One.”) Then, the winner would be whoever receives the most votes, more than the other candidates.

For the sake of this example, let’s say the nine friends voted like this: 4 votes for Indian food, 3 votes for Pizza, and 2 votes for Thai food. Indian food would be declared the winner with 4 out of the 9 total votes. That proportion of 4/9 is roughly 44.4%, so it’s a plurality but not a majority, and this is my second lesson: that those concepts are different. We seem so accustomed to the concept of “the most votes wins” in this country’s elections that we don’t often contemplate the fact that “the most” does not necessarily mean “more than half.”

Plurality vs. Majority: Recent Examples

The formal name for the voting method of a typical American election is Plurality Voting: every voter selects one and only one of the candidates, and the winner is the one who receives the most votes. When there are only two candidates, this will guarantee a majority for one of them. (When 100% is split into two pieces, there must be one piece with more than 50%. Okay, yeah, there could be a tie, but that’s extremely unlikely in an election with thousands to millions of voters.) But when there are three or more candidates, the votes may be split up in such a way that nobody receives more than 50%.

There are plenty of examples of recent (and nearby) political elections where the winning candidate received a vote share significantly less than 50%. Several gubernatorial elections in Maine in the last two decades featured one or two strong third-party candidates, causing the winner to get sometimes less than 40% of the total votes. (For instance, Democrat John Baldacci won in 2006 with around 38.1% and Republican Paul LePage won in 2010 with 37.6%.) Even more strikingly, two recent Democratic Primaries for U.S. House of Representatives seats in Massachusetts featured large fields with several strong candidates, causing the winner to get less than 25% of the total votes! (For instance, Jake Auchincloss won in District 4 in 2020 with 22.4%, edging out Jesse Mermell’s 21.1%. And in 2018, Lori Trahan won in District 3 with 21.7%, only 145 votes more than Dan Koh’s 21.5%.)

I do not mean to question those politicians’ victories or the accuracy of the results. Rather, I merely want to point out that it is strange and, I would argue, unhelpful and uninformative to the candidates themselves and the broader public to see results like that. Focus on the example from District 4 in 2020, where Auchincloss received 22.4% of the votes. This means more than three-fourths of all voters chose someone else! Not the same someone else, obviously, but collectively others. And yes, it is possible that many of those voters also liked Auchincloss as a candidate and are content with him winning. Or maybe not! Perhaps many of those voters despise Auchincloss and would have voted for anyone that had a chance of defeating him. Perhaps voters had to think strategically and vote for a candidate not only that they liked but who also had a chance of winning. And as soon as voters have to think that way—trying to game the system instead of expressing their true preferences—I see that as a problem.

(FWIW: Both Auchincloss and runner-up Mermell have publicly supported Ranked Choice Voting. See this article from WBUR for more information.)

In short, we don’t really know how the voters in District 4 really felt about all the candidates because the Plurality Voting method we use does not solicit nuanced information from the voters! It asks for the simplest possible information: “vote for one and only one.” And although those rules are simple and it makes it easy to calculate a winner, I would argue that this shoehorns voters into playing a game and prevents the entire system from accurately reflecting the will of the voters. How could that be fixed? The short answer is: ask for more information from the voters, then use it.

Ranking the Candidates: “What’s Your Backup Choice?”

Let’s return to our nine friends and their cuisine options. Is it really fair to ask everyone to pick just their top choice? And what if they find a winner and call the restaurant and it’s closed? One idea is to allow the voters to list a backup choice. Everyone’s ballot could ask: “What is your top choice for a cuisine? And, in the event that your top choice is unavailable for some reason, what is your backup choice?”

Perhaps I feel strongly about Indian food, but I acknowledge that it may not have broad support among everyone else. So, I think about the two remaining options, Pizza and Thai. Perhaps I really don’t want Pizza, so I list Thai as my backup choice. Or, perhaps I really have no strong preference between those two options, so I list nothing as my backup choice. This would effectively be saying: “If I can’t have Indian food, I don’t care; the rest of y’all decide and I’ll go with the group choice.” Those are both perfectly valid preferences I could have as a voter, and by allowing me to list a backup choice (or choose not to), this new system is already gathering more information from me to put to use.

In other words, every voter will make their own informed decision and list a 1^st choice and a 2^nd choice, or perhaps only just a 1^st choice. Let’s say that the voters end up submitting these ballots, summarized in the table below:

Each column represents a possible ballot a voter could submit, and the column heading indicates how many voters submitted that particular ranking. For example, the first column tells us that three voters all ranked Indian 1^st and Thai as their backup. The second column tells us that one voter ranked Indian 1^st but did not list a backup choice (effectively saying, “y’all decide”).

Under the usual Plurality Voting system, the row of 2^nd choice options would be completely ignored. The top row of 1^st choices would indicate 3+1=4 votes for Indian, 2+1=3 votes for Pizza, and 2 votes for Thai, making Indian the winner with 4/9 of the votes. Remember: that’s a mere plurality, not a majority.

Ranked Choice Voting: Use the Backup Choices to Find a Majority

The question now is: What do we do with those 2^nd choice rankings? The voters have supplied these more nuanced opinions about the candidates, so how do we factor them in and make a more informed group decision? The Ranked Choice Voting (RCV) system is one method for doing so, and it is based on finding a majority winner. Here’s how it works for our example:

• No candidate has a majority of 1^st choice votes. So, the RCV method kicks in. (If a candidate actually had more than 50% of the 1^st choice votes, they would already win outright, just like under Plurality Voting.)
• The method identifies the candidate with the fewest 1^st choice votes and eliminates them from the running. In this example, Thai has the fewest with only 2 votes.
• In the next round of the process, all the ballots are effectively rewritten with “T” crossed out, as if Thai food had never been an option in the first place. That’s what you see in the table below, the first and last columns have changed. The three voters in the first column no longer have “T” as their 2^nd choice, and the two voters in the far right column now have “P” as their 1^st choice. This is the crux of the RCV method: for any of the voters who listed that “losing candidate” as their top choice, their voting power is automatically transferred down to their backup choice!

# of voters	3	1	2	1	2
1st choice	I	I	P	P	P
2nd choice			I

• The same process happens again: look at the 1^st choice votes and see if anyone has a majority. In this case, we find 3+1=4 votes for Indian food and 2+1+2=5 votes for Pizza. Because 5/9 is a majority (more than half, about 55.5%), Pizza is declared the winner.

Perhaps it would be helpful to imagine how this might happen in reality with your group of friends. You might not do something so formal as gathering ranked ballots. You would probably just ask everyone outright: What do you want to order for dinner? But then, perhaps you would notice that no choice was a majority favorite. So, you’d talk to the two friends who voted for Thai and say, “Hey, Thai isn’t going to win, sorry. What do you want instead?” Each friend would then provide their backup choice (or say, “y’all decide”) and you would use those to decide a winner.

Another Example: Applying the RCV Method

Let’s see another example to practice applying and understanding this voting method. I have also set up this example to illustrate a few features of the system, and I will use those ideas to rebut a few common arguments against adopting RCV:

• Voters can express more nuanced preferences than are possible under Plurality Voting. They can rank anywhere from one to all of the candidates, and this allows the system to factor in their opinions in ways that Plurality Voting simply cannot do.
• Although ballots may get “used up” if a voter doesn’t rank all the candidates, this is a feature of the system and not a problem. (In other words, nobody’s vote is “worth more/less” than anyone else’s under RCV.)
• It is possible for the leading candidate in the first round to be the eventual winner, too.

The table below shows the ballots for an election with 18 voters (add the numbers in the top row) and 4 candidates (uncreatively named A, B, C, and D). This time, I have added a dash (-) to indicate a blank spot where a voter did not rank all the candidates.

# of voters	6	4	2	2	1	2	1
1st choice	A	B	C	C	C	D	D
2nd choice	C	D	A	D	B	B	–
3rd choice	D	C	B	–	–	A	–
4th choice	–	–	D	–	–	C	–

The far right column shows there is one voter who really wants D to win and does not care about any of the other candidates. They’re effectively saying, “If I can’t have D, then I have no strong preferences among the others; y’all decide for me.” Similarly, the two voters that ranked C 1^st and D 2^nd are saying, “We really want C; if we can’t have that, then we want D; but we have no opinions about A and B.” This is one benefit of the RCV system: voters can express nuanced preferences.

Let’s apply the RCV method to identify the winner. Just like before, we will conduct several rounds of voting using the original ballots. No need for voters to literally go out and vote again; we’ll use the rankings they provided.

• Check the 1^st choice votes: A gets 6, B gets 4, C gets 2+2+1=5, and D gets 2+1=3. Notice that A has the most, but 6/18 is only 1/3, not a majority. So, the RCV method kicks in and eliminates the last place candidate, which is D in this example.
  
• Below are the ballots rewritten with candidate D gone and the changes highlighted. Two of the voters who ranked D first now have their voting power transferred down to their backup choice of candidate B. But one voter only ranked D on their ballot. So is their voting power “gone”? Only for the purposes of the rest of the calculations! That voter still had their voice counted in this election. It’s just that they chose to use their voice to say, “I want only candidate D,” and that candidate did not garner enough support from other voters.

# of voters	6	4	2	2	1	2	1
1st choice	A	B	C	C	C	B	–
2nd choice	C	C	A	–	B	A	–
3rd choice	–	–	B	–	–	C	–
4th choice	–	–	–	–	–	–	–

• Again, check the 1^st choice votes: A gets 6, B gets 4+2=6, and C gets 2+2+1=5. Is there a majority? Be careful: there are now only 17 voters, not 18 because of that one voter whose ballot has been “used up.” (When you see the phrase “exhausted ballot” or “concluded ballot”, that’s what it means.) A candidate could now have a majority with just 9 votes (out of 17, that’s more than half), but we still only see 6/17 for A and 6/17 for B. So, the RCV method kicks in again and eliminates the last place candidate C, who only had 5 votes in this round.
  
• Below are the ballots rewritten with candidate C gone, as well. Notice that two more voters have their ballots “used up,” but that is an acceptable part of this method. Those two voters used their voice to say, “We want C, and if not then D, and if neither of those, then we have no opinion on A vs. B.” So, again, their voting power is not deleted or ignored or anything like that; it has simply been used up in the process of calculating the winner.

# of voters	6	4	2	2	1	2	1
1st choice	A	B	A	–	B	B	–
2nd choice	–	–	B	–	–	A	–
3rd choice	–	–	–	–	–	–	–
4th choice	–	–	–	–	–	–	–

• With only two candidates left, we know there must be a majority winner, so let’s check the 1^st choice votes: A gets 6+2=8 votes and B gets 4+1+2=7 votes. This makes A the winner with a majority of 8/15 votes in the final round.

Remember that A had the most 1^st choice votes in the first round. So, this example shows it is entirely possible to end up with the same winner under RCV as would have occurred under Plurality Voting.

In that process, we saw three ballots get “exhausted” because the voters did not rank all the candidates. I will emphasize again: this is a feature of the method not a flaw. Those voters used their ballot to express no preference between candidates A and B, so when the final round pitted those candidates head-to-head, it was as if those three voters had never cast a ballot in the first place because the race was between two candidates they didn’t care about. Their preferences did get factored in: it’s not like their voting power was ignored, nor was it somehow “less” than the voting power of the 15 other voters whose ballots carried through the entire process.

Another consequence of those “exhausted ballots” is that the eventual winner A did get a majority of votes, but not a majority of all ballots cast. Candidate A received eight 1^st choice votes in the final round out of fifteen voters, for a winning proportion of 8/15 or roughly 53.3%. It is indeed true that eight votes out of all eighteen original voters is 8/18 = 4/9 or roughly 44.4%, which is not a majority. However, again, this is a feature not a flaw. It is unfair to point to that result and claim that A was not a majority winner, or that the candidate does not reflect the will of the voters. Doing so glosses over the fact that some of the electorate chose to express no opinion about that candidate.

Conclusion: Make Informed Choices

I hope that these examples provide you sufficiently detailed information about the mechanics of the Ranked Choice Voting method so that you better understand how it works. Many discussions in the news and on social media seem to focus on political implications of this method or the legal issues surrounding its adoption, so I wanted to contribute some mathematical detail to the discussion. There are many interesting mathematical topics and theories about voting and elections. Please free to contact me if you have any questions or comments, or if you’d like suggestions for further resources.