In Ranked Choice Voting, How Likely Is It that the Winner Did Not Get a Majority at Any Rank?

Factual Questions
1

Thinking about the four-candidate ranked choice election for Alaska senator, my first thought was that the winner is close to being guaranteed a majority of ballots at some level. But this is not true in scenarios where voters mostly just voted for one candidate. So, I checked some AI prompts, and they all seem not to know the answer to this question:

“In ranked choice voting, how common is it that the number of valid ballots listing the winner at any rank is less than or equal to 50 percent of the number of valid ballots listing at least one candidate.”

I’m thinking this must be rare worldwide. Can dopers outthink AI and tell me?

2

By my understanding, a ranked choice vote only ends when one person has a majority. So, impossible, unless there’s another method to when a winner is declared using RCV that I don’t know.

3

It’s obviously possible. What do you mean by common? Are you asking how often it has happened in actual elections? Or do you mean statistically given X voters and Y candidates, how many voting permutations have the winning voter with less than 50%?

ETA: I can’t answer it either way, but I’m curious what you’re looking for.

4

When one person has the majority of the remaining ballots. Some ballots are eliminated entirely each round if none of their ranked candidates are in the top X.

5

Yes.

6

Yep, I missed that part. I didn’t account for voters failing to rank every available candidate, which is doubly bad given the first place I looked was the Fairvote organization that pointed out only 2/3 of voters ranked more than a single candidate in the 2022 House race.

7

I’ve seen local elections for something like school board or city council where there will be several open seats, and the top two, three or whatever vote-getters will be elected. In those cases there might be five candidates and no one gets more than something like 35% percent of the vote. Is that the kind of example the OP is looking for?

8

Re last post, that’s not ranked choice voting.

Also, I am thinking of elections, such as for members of the U.S. congress, where there is only one winner.

Ranked choice voting explanation:

9

This is called preferential voting in Australia. There are three variants used, depending on the jurisdiction:

a. All candidates numbered (lets say 1-6 of 6 candidates numbered according to your preference)

b. Set minimum number, at least (e.g.) 3 candidates in preference order. If the election is still going after your third preference allocated, your vote is considered to be exhausted.

c. Optional preferential - you can vote for your top preference [1], and as many others as you want after that. Your ballots are exhausted after your last preference is allocated.

Ballot results are called when either someone gets an absolute majority on first results, after each round of preferences is allocated or all votes are exhausted. Because in Australia voting is compulsory there should be a clear and stable understanding before the election of exactly what 50% of the vote would be. With your random turn-up-if-you-feel-like-it elections with an unknown number of valid ballots cast, you probably need to get to the end of the counting to be sure. It also helps that our elections are administered by an independent, non-political election authority who engage a very high level of institutional trust.

For the OP’s question, parties in Australia have a very predictable preference flow. Impressionistically, about a third will gain a 50%+ vote on the first count, another third will lead and win on preferences, and the remainder will win on what can be a very protracted allocation of preferences from behind.

10

It is possible in instant runoff voting (IRV) to elect a non-condorcet winner but it does have the feature that it can never elect a Condorcet loser, a candidate who loses every head-to-head matchup (a Condorcet winner is a candidate who would beat every other candidate head-to-head). That’s because the winner always has a majority in the final round.

I think the chances of it working like this are small but they are still there (I could not find stats on the chances…guessing it really depends on the particulars of each election which can’t really be compared from one to another).

ETA: I asked my AI for stats and it gave an answer. As with all AI be wary but I have read about this before and this seems reasonable:

U.S. elections: The main empirical study is by mathematicians Adam Graham-Squire and David McCune. They examined a database of 182 American ranked-choice elections for political office from 2004 to 2022 and concluded that RCV’s weaknesses rarely show up in real-world elections, with one exception: ballot exhaustion (voters not ranking enough candidates) frequently leads to winners without a majority of all ballots cast. Earlier, smaller studies of overlapping datasets found a Condorcet failure only in the 2009 Burlington election. The 2022 Alaska special election was added to that list later. In that race the Condorcet winner differed from the IRV winner, and the election also showed a monotonicity paradox and a no-show paradox. So the two cases I mentioned earlier are essentially the whole documented U.S. list among a couple hundred elections, roughly a 1% rate. arxiv.org +2

Elsewhere: Law professor Nicholas Stephanopoulos surveyed the existing research on American IRV elections and analyzed a new dataset of almost 200 foreign IRV races. Both approaches showed that IRV almost always elects the Condorcet winner in real elections, and that a Condorcet winner almost always exists. harvard

Simulations show higher rates. One computer study of artificial elections found that even under the conditions where mismatches were rarest, the IRV and Condorcet winners would be expected to differ about once in every 47 elections. Real elections seem to come out better than simulations. One likely reason is that many races are lopsided or are decided by a first-round majority, so there’s no squeeze to happen. arxiv

Why it’s rare: A Condorcet winner needs a real base of support to win all those head-to-head matchups, and it only has to survive to the IRV final round, which it will always win. Failures need a fairly specific setup: three competitive candidates, with the broadly acceptable one finishing third in first choices. researchsquare

Caveat: Critics argue that a low rate isn’t the whole story, because the failures that do happen tend to come in close, high-stakes, polarized races. Burlington and Alaska both fit that pattern. Supporters counter that plurality voting has no way to detect such failures at all, and probably produces them more often.

11

It should be noted that ranked choice voting isn’t a voting method; it’s a whole category of voting methods. It just means that voters fill out their ballot by putting the candidates in order of preference. But there are multiple different ways that you can use those ballots to determine a winner.

The most-commonly used ranked choice system is Instant Runoff Voting. But you could also use a variety of of Condorcet methods, or assign each candidate a point value based on their ratings, or many other methods.

The “Condorcet condition” says that, if there’s some candidate who would win a head-to-head against each other individual candidate, then that candidate should be the overall winner. This does not always happen in Instant Run-Off. Condorcet methods are engineered such that, if there is a Condorcet winner, that person always wins. It’s not guaranteed that there will be a Condorcet winner in every race; the various Condorcet methods are differentiated by what they do in such a case.

12

It’s easy to construct a scenario where the top vote getter in the first round fails to get a majority. Say, Albert is esposing doctrine A, and Bob, Charlie, and Dan are espousing doctrine B. Or, Albert has been in power long enough to antagonize 60% of the electorate. Then a vote for Bob,Charlie, or Dan ranks the other two as second and third choices.

Something similar happens in Canadian party leader races, although historically they have been sequential rounds of voting, so the choice is not fixed between rounds, people could change their minds.

In 1976 Joe Clark was third on the first two rounds, second of two on the 3rd round and won on the 4th. Presumably, the back-room-drafted lead candidate Claude Wagner was not acceptable to many rank and file.

Then in 1983 the opposite - Clarke was first on the first on the first three ballots and lost on the fourth, as “Anybody but Clarke” sentiment coalesced around Brian Mulroney.

The opposite type of election is what you saw in the first Egyptian election after the Arab Spring, where it was a run-off between the top 2 if nobody got 50%. There were a plethora of candidates of various political stripes. There were plenty of moderates, but none got enough votes, so the run-off was between the traditional (corrupt) Mubarak old guard and the Muslim Brotherhood strong Islam group effectively forcing moderate voters to choose between “the lesser of two evils”.