Evan Rosenman writes:
The implosion of Graham Platner’s Senate campaign in Maine has upended a marquee Senate race, leaving the state Democratic party just a few weeks to choose a substitute nominee. A planned nominating convention on July 25th has drawn considerable candidate interest. But the mathematical properties of ranked choice voting add a strange wrinkle to these deliberations.
The Maine Democratic Gubernatorial Primary
Three of the top contenders to replace Platner are former gubernatorial candidates: Nirav Shah, former director of the Maine Center for Disease Control and Prevention; Troy Jackson, former Maine State Senate president; and Shenna Bellows, Maine’s secretary of state. All three ran for the Democratic nomination for Governor, losing the primary to Hannah Pingree, former speaker of the Maine State House.
June’s primary results are given below. (Data from Wikipedia.) Maine uses ranked choice voting (RCV) in primaries and federal elections, so voters could rank up to six choices for Governor. Using the instant runoff algorithm, candidates were sequentially dropped based on who had the fewest first-choice votes, and ballots were reallocated to each voter’s next-ranked choice. Jackson, Shah, Bellows, and Pingree were highly competitive, each receiving between 20% and 27% of first-choice votes. Of the four, Bellows was eliminated first, then Jackson. Shah fell to Pingree in the final tabulation round.
Candidate Round 1 Round 2 Round 3 Round 4 Pingree 50,552 (23%) 55,360 (26%) 75,671 (36%) 111,750 (56%) Shah 58,606 (27%) 62,860 (30%) 72,681 (35%) 86,950 (44%) Jackson 45,959 (21%) 47,597 (22%) 60,010 (29%) Eliminated Bellows 44,770 (21%) 47,049 (22%) Eliminated — King III 17,860 (8%) Eliminated — — Exhausted ballots — 4,881 (2%) 9,385 (4%) 19,047 (9%) Continuing ballots 217,747 212,866 208,362 198,700 These results have taken on extra significance as the state party seeks democratic buy-in for the selection of a substitute Senate nominee. Media outlets, for example, have routinely referred to Shah as the “runner-up” in the Governor primary. But analyses of the individual ballots cast in the primary reveal a surprising mathematical fact: though she was eliminated before them, Bellows would have defeated either Shah or Jackson in one-on-one elections.
Mathematical Details
This unintuitive fact is a generalization of a well-known feature of ranked choice voting elections: it does not satisfy the Condorcet winner criterion.
First, some definitions. Suppose we have an election with a set of candidates C:
- A “Condorcet winner” is a candidate in C who would defeat all the other candidates in a head-to-head election. A Condorcet winner need not exist for any given C; think of rock-paper-scissors, where each option wins against one alternative and loses against the other. But Condorcet winners exist in many standard election settings.
- The Condorcet criterion is a feature of electoral methods: a method satisfies the criterion if it always selects a Condorcet winner when one exists.
Standard plurality elections – in which voters make one selection, and whomever gets the most votes wins – do not obey the Condorcet criterion. This is well-understood due to the “spoiler effect.” For example, a Libertarian candidate may attract voters who would otherwise prefer a Republican to a Democrat, siphoning enough voters such that a Democrat obtains the most votes.
Because voters express richer preferences in RCV elections, the method is considered better at identifying Condorcet winners. But it can easily be shown that RCV also does not satisfy the Condorcet criterion. This is not purely hypothetical. In a 2022 U.S. House special election in Alaska, Democrat Mary Peltola was elected against two Republican opponents: Sarah Palin and Nick Begich III. An analysis of the underlying ballot data revealed that Begich was a Condorcet winner. But he was eliminated in the first round because he received slightly fewer first-choice votes than Palin, allowing Palin to advance and lose to Peltola.
As RCV does not obey the Condorcet criterion, it stands to reason that the order of elimination need not correspond to who would win head-to-head elections. This is indeed true. A candidate eliminated in an earlier round may well have defeated one eliminated in a later round in a head-to-head election.
Results in Maine
We can understand the electorate’s preferences in Maine because the state releases its cast vote record: the anonymized set of rankings for every ballot cast. These data are available online and have also been analyzed extensively by the election advocacy group FairVote.
To assess how two candidates A and B would fare in a head-to-head election, we look at the set of ballots that rank at least one of them. Any ballot in which A appears before B, or A is ranked and B is not, represents a voter who prefers A to B; any ballot in which B appears before A, or B is ranked and A is not, represents a voter who prefers B to A.
In the table below, we summarize all the head-to-head matchups among the top four candidates. Note that if the final column is positive, then A defeats B; if it is negative, B defeats A.
Candidate A Candidate B % of Ballots
Listing Neither% Who Prefer A % Who Prefer B A vs. B Margin Bellows Pingree 14% 41% 44% –3% Bellows Jackson 19% 48% 33% 15% Bellows Shah 12% 45% 43% 3% Shah Pingree 10% 39% 50% –11% Shah Jackson 13% 50% 37% 12% Jackson Pingree 14% 33% 52% –19% Pingree wins all three of her matchups, indicating she was indeed the Condorcet winner. But notably, Bellows wins every matchup except the one against Pingree. She was preferred to Jackson on 48% of ballots while he was preferred on 33%, with the remaining ballots listing neither candidate. Bellows had a narrower margin against Shah, but she was preferred on 45% of ballots to his 43%.
These results reflect the strengths and pitfalls of RCV. Because voters’ ranked choices are recorded, we can better assess the electorate’s head-to-head preferences among many candidates. But elimination orders under instant runoff needn’t reflect these preferences. In closely contested elections like the Maine Democratic gubernatorial primary, this can yield unintuitive results – with big implications for the next big question: whom to choose as a substitute Senate nominee.
Following up on Rosenman’s analysis, I have a few points to raise:
- Why should I care who would win in a head-to-head race? I’m not trying to ask this in an aggressive way; it’s just not clear to me why this should be the question to ask, or why we should care about a Condorcet winner. Another way to say this is that intensity of preference could matter too.
- A related issue is that there are lots of people who could potentially be qualified to be the senator from Maine–after all, a senator doesn’t really have to do much, their staff does all the work, right? Just ask Senator Grassley from Iowa! My point here is not to trivialize the election–people live or die based on who is elected to Congress–just that the steps of choosing a candidate involve a winnowing from many many possible choices. The Condorcet winner criterion and other similar rules apply only after drastically limiting the number of options. From that perspective, I’d be more inclined to rate candidates based on a summing of pluses and minuses for various attributes, rather than head-to-head comparisons. I get that the general election is a head-to-head race so you need to think about such things, but from a political theory perspective, or from a which candidate-to-choose perspective, I see this Condorcet thing as a blind alley.
- Who you’d want to run for governor isn’t necessarily the same as who you’d want to run for senator. I say this for two reasons. First, they’re different jobs: what it takes to run the executive branch of a state is different than what it takes to be a member of the national legislature. Second, the main goal of a political party is to win the election, and it could take different things to win in the two races in Maine this year. I don’t know how important this is, as I have no sense of politics in that state. I’m just raising the issue.
I don’t know if I have ever been to Maine, but is that “whomever” a hyper-correction for “whoever”?
“whomever gets the most votes wins.”
Paul:
Hey, give Rosenman a break! He was just writing a blog post, and everyone makes mistakes. OK, maybe not David Brooks–I can’t recall him ever acknowledging error–but everyone else.
P.S. I take it back. Paul pointed to a couple places where Brooks did admit mistakes.
Good for you, David! Just do it more often.
Your point about the Condorcet condition is well taken. A major point of ranked vote schemes since the days of Thomas Hare is to equalize voting power, the extent to which each individual’s vote affects the final outcome. As you say, that property will often lead to non-Condorcet outcomes, and so be it.
>”This unintuitive fact is a generalization of a well-known feature of ranked choice voting elections: it does not satisfy the Condorcet winner criterion.”
Huge terminological pet peeve of mine here: that is a feature only of *instant runoff*, not all ranked choice voting. There are Condorcet approaches to deciding the election (with embellishments to break rock-paper-scissor cycles if the aggregate preferences happen to be intransitive).
Connecting this failure mode in the public consciousness to RCV per se undermines one of the best hopes for breaking loose of FPTP.
I don’t think that ranked choice is really designed for anything but getting the first place winner. It’s interesting as a question, maybe, but in winner take all elctions it doesn’t matter what place you come in unless it is first.
A few replies to follow-up points:
– The substitute for Platner may well be someone other than Jackson, Shah, or Bellows. But there are several reasons they’ve emerged as top contenders:
– A lot of top-tier Maine political talent opted to run for Governor — rather than Senator — because Collins is viewed as a very skilled campaigner. As a result, Platner was able to consolidate support due to the lack of a big-name opponent, and the late entry of Governor Janet Mills was unable to change that dynamic. With Platner exiting the stage, it’s now the case that a lot of ambitious Maine Dems have pre-existing campaign operations at the same time that a huge statewide race needs a candiate.
– The Maine Dems want to lend a sense of democratic legitimacy to a Platner replacement so as not to anger his insurgent base (the ghosts of the 2024 presidential candidate swap are omnipresent). Because the gubernatorial primary was concurrent with the Senate primary, there is a sense that voters recently weighed in on these candidates, so they could potentially better serve as consensus picks.
– My goal with the post was to highlight the unintuitive aspects of ranked choice voting, since they are relevant to the media coverage of the Maine convention. But, to be clear: Maine Dems certainly shouldn’t only be interested in the results of the gubernatorial primary. Their goal is to select a nominee who can win a general election with a significantly larger electorate.
– To that end, I am *definitely* not arguing that Bellows should be the nominee because she was preferred to Shah and Jackson by the primary electorate. Putting on my pundit hat, I think the cast vote records reflect two slightly unexpected results. The first is that Bellows is more popular with primary voters than you might expect given that she was eliminated early. The second is that Troy Jackson’s support is narrower than you might expect; he got a decent chunk of first-place votes, but primary voters had a pretty clear preference for both Shah and Bellows over him.
– Your point is well-taken regarding Governors and Senators. Interestingly, four incumbent Senators ran for Governor this cycle (Michael Bennet just lost the primary in Colorado; Tuberville already won in Alabama, and Klobuchar and Blackburn will likely win their primaries and generals in Minnesota and Tennessee, respectively). The more typical pathway is the other direction (12 current Senators are former Governors). Geoffrey Skelley has a nice piece on this: https://decisiondeskhq.substack.com/p/greener-pastures-senators-running
My comment on the prior ranked choice post was kind of hinting that Condorcet is better. You address one potential issue, which is that Condorcet methods are tricky when you have too many candidates. The obvious solution is a two-stage election. Stage one limits the number of candidates down to 3-5. This could be done by many means. Stage two then uses a Condorcet method to pick the winner.
But I think your ultimate question is, why should I care about pairwise votes.
First of all, we only really care about Condorcet when there are more than two viable candidates. If the top two candidates will get 90%+ of the vote, then it doesn’t matter that much. The remaining candidates are de minimis and you’re more likely in a situation when one candidate receives a majority. Of course, the structure of elections may have an impact that kind of outcome. A plurality vote rule has the tendency center on two main candidates. But I also see that as the main deficiency of the plurality voting system. It silences potential alternative voices.
So to me, the question is more about instant run-off vs Condorcet methods. And the differences between these are most obvious in a situation with more than 2 main candidates. If you’re in a situation where one candidate has more than a majority of the first place vote, then the outcome is the same as plurality. They are the winner under all three systems. The main difference between Condorcet method and instant run-off in this situation is that Condorcet methods comes closer to matching the preferences of the median of the population (under some assumptions, this is Black’s theorem). This means that the winners are more likely to be more moderate (and perhaps people are less enthusiastic about them). By contrast, if a candidate is widely favored, but fewer people are enthusiastic about them (as an example, in a 3 candidate election, almost everyone ranks a candidate second but only a few first rank them first), then they can get knocked out early. So to me, the question of “why should we care about pairwise elections” is more about do we want to adopt voting methods that knock out candidates early when they have broad support even if they aren’t the top choice.
Personally, I like the idea of voting methods that meet the Condorcet winner criterion, but I think its a personal preference and not something objective. What’s objective is that the some voting methods meet some criterion or another. But whether you care about that is a personal preference in terms of what it implies for the characteristics of the subsequent winners of the elections.