# The center squeeze: a deep dive under several voting methods

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In 2009, Burlington, Vermont used Instant Runoff Voting (single-winner Ranked Choice Voting) to elect its mayor. (Readers who are already very familiar with this election may want to scroll down to the Approval section and start reading from there.) Here is the support for each of the candidates in the first round:

• Kurt Wright (Republican): 32.9%
• Andy Montroll (Democrat): 23.0%
• Bob Kiss (Progressive and current incumbent): 28.8%
• Dan Smith (Independent): 14.5%
• James Simpson (Green): 0.4%
• Write-ins: 0.4%

Since it was mathematically impossible for Smith or Simpson to win regardless of voters’ second choices, both of them were eliminated (along with the write-ins) in the first round. Second round:

• Kurt Wright (Republican): 36.7%
• Andy Montroll (Democrat): 28.4%
• Bob Kiss (Progressive): 33.2%
• Exhausted ballots: 1.7%

Montroll was eliminated due to having the fewest votes, leaving Kiss to win the final round:

• Kurt Wright (Republican): 45.2%
• Bob Kiss (Progressive): 48.0%
• Exhausted ballots: 6.7%

However, Montroll would have defeated either Kiss or Wright head to head; given the ballots that were cast, Montroll would have defeated Kiss 45.3% to 38.7% and would have defeated Wright 51.2% to 40.8%. Much of Montroll’s support in these matchups comes from being the second choice of voters who ranked Kiss or Wright first — and this second-choice support was irrelevant in the round where Montroll was eliminated.

## The Center Squeeze

The Burlington election is the only unambiguous example we have of the center squeeze: an election in which, of the top three candidates, the Condorcet winner (the candidate who would defeat anyone else head-to-head) is the first choice of the fewest voters. We could also define the center squeeze as occurring in any election in which there is a Condorcet winner who is not the candidate elected under IRV.

The most natural case in which to imagine a center squeeze happening is when the candidates are arranged on a left-to-right spectrum, where the far-left and far-right candidates have the most first-choice support but there’s a centrist who is the second choice of both the left-wing and right-wing voters. In Burlington, the Progressive (Bob Kiss) played the role of the left-wing candidate, the Republican (Kurt Wright) the role of the right-wing candidate, and the Democrat (Andy Montroll) the role of the centrist.

Intuitively, it seems like center squeezes should be extremely common; in our two-party system, the Democrat tends to be to the left of the center of public opinion and the Republican to the right. Also, Democrats and Republicans tend to hate one another enough that both parties should strongly prefer some unaligned centrist to the opposing party’s candidate. But we need to be careful here. The 2016 presidential election seems a lot like a center squeeze — Gary Johnson seems far more acceptable to Democrats than Trump and far more acceptable to Republicans than Clinton. Actual research shows that there might have been a center squeeze, but the “centrist” who got squeezed out was Bernie Sanders. Head-to-head, Johnson would have lost in a landslide to either Trump or Clinton. The one-dimensional model shown above gives an unrealistic picture of the electorate, and we should be careful about assuming the existence of a center squeeze without good data.

## Strategic Voting in IRV’s center squeeze

Looking back on the Burlington results, the Republicans who voted Wright>Montroll>Kiss would have been better off had they dishonestly voted Montroll>Wright>Kiss since this would have caused Montroll to win instead of Kiss — a more acceptable outcome to these voters, and presumably a better outcome from the perspective of the electorate as a whole. But hindsight is 20/20. When these voters were actually voting they didn’t know how everyone else would vote — they had to make their decision under uncertainty.

To the best of my knowledge, nobody ever conducted a poll prior to the Burlington election. More knowledgeable voters could still have made educated guesses as to how various head-to-head matchups might go, but there would have been a lot of uncertainty. Given this uncertainty, it seems highly unlikely that it would have been strategically sound for many Wright>Montroll>Kiss to dishonestly vote Montroll>Wright>Kiss. The Burlington election was close, and Kiss won the final round with less than a 3-point margin. Perhaps they could have guessed that Montroll was more likely to beat Kiss in the final round, but they couldn’t have guessed this with enough confidence to justify voting Montroll>Wright>Kiss unless they cared about stopping Kiss a lot more than they cared about electing Wright.

On the other hand, the voters with the preferences Kiss>Montroll>Wright were also facing uncertainty and were in an analogous position. Had they guessed that Montroll stood a better chance of beating Wright in the final round (as was indeed the case) then they might have been justified in strategically voting Montroll>Kiss>Wright even though this would not have helped them in hindsight.

My best guess is that the Wright>Montroll>Kiss and Kiss>Montroll>Wright voters who considered Montroll to be almost as good as their first choice should have dishonestly ranked Montroll first, but everyone else was best off voting honestly. Increased strategic voting would have helped Montroll some, but I highly doubt that it could have been enough to put in him the final round.

## Plurality

After the 2009 election, Burlington repealed IRV and switched to using Plurality. Was this an improvement? In general, it’s a mistake to judge a voting method by looking at a single election, but let’s see how Plurality would have fared in the 2009 election anyway.

If everyone voted as for their first choice (as expressed in the IRV election), Wright would have won:

• Kurt Wright (Republican): 32.9%
• Andy Montroll (Democrat): 23.0%
• Bob Kiss (Progressive): 28.8%
• Dan Smith (Independent): 14.5%
• James Simpson (Green): 0.4%
• Write-ins: 0.4%

Given that Wright would have lost head-to-head to either Kiss or Montroll, this is a very bad outcome. Presumably Kiss (and also Wright) would have received a few more votes than the IRV results would suggest on account of strategic voting, but Wright still looks like the heavy favorite.

• An even worse candidate would have been elected
• Many votes would have been completely wasted on non-viable candidates like Dan Smith
• Dan Smith would have seen less support due to strategic voters deserting him

The Burlington election was the greatest failure of IRV we know of. Plurality still would have done worse.

## Plurality + Runoff

Adding a runoff would prevent Wright from being elected and mitigate the loss of influence for voters who wasted their votes on Smith or one of the other underdogs. However, all of the problems that occur under IRV would remain.

## Approval

There’s a big limitation of the IRV election results that makes it hard to assess the Burlington election under Approval Voting: it doesn’t tell us the relative strengths of voters’ preferences. That is to say, a voter with the utilities {Wright: 10, Montroll: 9, Kiss: 0} would vote exactly the same as a voter with the utilities {Wright: 10, Montroll: 1, Kiss: 0} under IRV — but, under Approval Voting, the first voter would probably vote for both Wright and Montroll whereas the second voter would only vote for Wright. To get around this lack of data we have to make assumptions.

Here’s a very simple model: Every voter votes for their first choice out of the top three (Montroll, Kiss, and Wright). In addition, some percentage of voters votes who expressed a second choice in the IRV election will for their second choice out of the top three, and this percentage is constant across all preference profiles; a Wright>Montroll>Kiss voter is just as likely to vote for Montroll as a Kiss>Wright>Montroll voter is to vote for Wright. To help with computations, here’s the total first choice and second choice support for each of the three main candidates (where I’m counting a ballot like Smith>Kiss>Simpson>Montroll>Wright as a first choice vote for Kiss and a second choice vote for Montroll):

• Wright: 36.7% #1, 12.7% #2
• Kiss: 33.2% #1, 20.3% #2
• Montroll: 27.4% #1, 39.6% #2

Here’s who wins based on what fraction of the voters who indicated a second choice vote for that second choice:

• < 24.8%: Wright wins, Kiss finishes second
• 24.8%-30.7%: Wright wins, Montroll finishes second
• 30.7%-45.7%: Montroll wins, Wright finishes second
• > 45.7%: Montroll wins, Kiss finishes second

Factoring in the 27% of voters who only marked one of the top three in the IRV election, Montroll wins if at least 22.3% of voters vote for multiple candidates (among the top three). For comparison, in St. Louis’ four-candidate Approval Voting mayoral election, the average ballot had a vote for 1.563 candidates, and this poll found that 52% of voters voted for two candidates and 7% voted for three. Based on this it seems probable, but by no means inevitable, that Montroll would have been elected had Burlington used Approval Voting.

Would things be substantially different if voters behave strategically? Probably not. Strategic Approval Voting 101 says that if there are two candidates who are substantially more viable than all the rest put together then you should vote for exactly one of them (and also for all the less-viable candidates you like more). But who’s more viable here: Kiss or Wright? Given that Kiss finishes second either if the vast majority of voters bullet vote (i.e. they vote for only one candidate) or only a minority of voters bullet vote, but not if an intermediate number bullet votes, it’s very hard to say. Montroll looks like the frontrunner, but Kiss and Wright seem roughly neck-and-neck.

One not-very-intuitive fact about strategic approval voting is that, in determining whether to vote for a particular candidate, it doesn’t matter how viable that candidate is — it only matters how viable all the other candidates are relative to one another. The effects of Montroll’s frontrunner status are to make his first-choice supporters more inclined to bullet vote and to make the (relatively few) voters who have him as their last choice less inclined to bullet vote, but, from the perspective of Kiss>Montroll>Wright and Wright>Montroll>Kiss voters, his lead is strategically irrelevant. Overall, strategic voting should slightly improve Montroll’s already-good chances.

## Approval + Runoff

Adding a top-two runoff to the Approval contest yields some obvious benefits:

• Montroll has an easier time winning: Since he’d win the runoff, it’s only necessary for 24.8%, instead of 30.7%, of voters with a second choice to vote for that second choice in order for Montroll to win under the simple model above.
• Wright never wins. Even if Montroll misses the runoff, Kiss will still beat Wright so the worst-case outcome will be avoided.
• Everyone would get an equal vote in the runoff. Without a runoff, the handful of voters who vote in a strategically foolish manner (e.g. only voting for Dan Smith) are throwing their votes away, and the runoff mitigates this issue.

Adding a runoff also introduces a non-obvious downside: it may be reasonable for some voters to vote dishonestly. Suppose your utilities are {Kiss: 10, Montroll: 1, Wright: 0}. Without a runoff, it’s pretty clear that you should just bullet vote for Kiss. With a runoff, there’s a decent argument for voting for both Kiss and Wright in the first round. If you believe that (a) Montroll is significantly favored against either Kiss or Wright, and (b) Kiss is slightly favored against Wright, then the best possible runoff is between Kiss and Wright. While bullet voting for Kiss would maximize the probability that he’d advance to the runoff, voting for both Kiss and Wright would maximize the probability that Kiss would win it.

There are big downsides to this dishonest strategy: Wright might win the runoff (he did, after all, fall a mere 3% short under IRV). Wright might also edge Kiss out of the runoff even if Kiss would beat Montroll. But supposing you have a very confident read on the runoff, the dishonest strategy is all upside. If Montroll is assured of winning any runoff and Kiss is assured of winning against Wright, the only thing to aim for in the first round is having the runoff be Kiss vs. Wright. Realistically though, there’s going to be a ton of uncertainty in a municipal election like Burlington’s. The dishonest strategy helps Wright more than it helps Kiss, so it wouldn’t make sense if your utilities were something like {Kiss: 10, Montroll: 5, Wright: 0}; if Montroll is more than a little bit better than Wright in your judgment you’re best off just voting honestly. And even if you think Montroll and Wright are equally bad, you need to be quite confident that Montroll would beat Kiss in a runoff to justify voting for Wright as well.

(There’s a symmetric case for a voter with utilities {Wright: 10, Montroll: 9, Kiss: 0}. This voter could be best off bullet voting for Montroll.)

On balance, I think the advantages of having a runoff outweigh the strategic downside. The dishonest strategies require an excellent understanding of the electorate’s preferences, can easily backfire, and are not particularly obvious, so I would be surprised if many voters made use of them. (A particularly astute reader may disagree with this on the grounds that that St. Louis poll I mentioned found that 10% of people who voted for two candidates did so because “Voting for more than one candidate would HELP favorite candidate chances”. My belief is that respondents interpreted this in the context of having two favorite candidates rather than only one; the exact wording in the poll did not make the number of favorite candidates explicit.)

## STAR

Of the voting methods discussed so far, STAR behaves most similarly to Approval + Runoff. But there are some important differences.

STAR lets voters rate candidates on a gradated scale instead of just a thumbs-up or thumbs-down. Suppose your Utilities are {Kiss: 10, Montroll: 3, Wright: 0} and you think all three candidates are about equally likely to win. Under Approval Voting, you’re best off bullet voting. Under STAR, you’re best off giving Montroll a 1; STAR is better than Approval at factoring in such weak support. (The flip side is that STAR factors in strong support less Approval does; if your utilities were {Kiss: 10, Montroll: 7, Wright: 0} you’d vote for both Kiss and Montroll under Approval but only give Montroll a 4 under STAR. Factoring in weak support is more important for this election, however; if Montroll has a lot of strong second-choice support he’ll easily win anyway.)

The dishonest strategies under Approval + Runoff backfire worse under STAR Voting. Under Approval + Runoff, a voter with the preferences Kiss>Montroll>Wright who dishonestly votes for Kiss and Wright still gets to vote for Montroll in a runoff between him and Kiss. Under STAR, the corresponding strategy is to vote Kiss: 5, Montroll: 0, Wright: 4 — and this carries the added downside of counting as a vote for Wright in a runoff between him and Montroll. The already-dubious strategy looks even worse.

STAR looks like it would have done the best of the voting methods mentioned so far; Montroll is heavily favored and the dishonest strategies look pretty bad. But there’s an interesting argument that more honest forms of strategic voting could cost Montroll the election:

Montroll wins if all voters with a 2nd choice give their second choice the same rating. Montroll, who was 3rd place on 1st-place rankings, relies on stars from supporters of the other two candidates to get to the runoff. Can the supporters of those candidates get their candidates to win by lowering their ratings for Montroll? Yes.

Kiss wins if K>M>W supporters lower the stars given to Montroll to 1. Montroll then fails to get to the runoff and Kiss beats Wright 4314–4064.

It seems that 5–1–0 voting [giving your favorite 5 stars, your second choice 1 star, and your last 0 stars] is the most common strategy suggested for 3-candidate STAR elections. If that is how people voted with the preferences expressed, then Kiss would win in a runoff against Wright 4314–4064.

Even though Montroll fails to make the runoff based on 5–1–0 voting based on the historical vote totals, Montroll could make the runoff and win if M>K>W voters (initially 1332) switch to bullet voters. If 251 of M>K>W switch to M bullet voters, Montroll still fails to make the runoff but Wright beats Kiss in the runoff. However, if more than 411 of M>K>W switch to M bullet voters, than Kiss no longer makes the runoff and Montroll beats Wright in the runoff. If the bullet voting is not constrained to M>K voters and includes M>W voters, than Kiss supporters could get Kiss into the runoff ahead of Montroll by bullet voting. Of course, W>M voters could decide that they don’t want to help make Wright lose and stop giving stars to Montroll. If the bullet voting rate among all voters increases uniformly to around 55% or above, then the results become the same as plurality with Wright winning and Kiss losing in the runoff.

It is clear that STAR elections can have Later-Harm and can give incentives to bullet vote. Furthermore, STAR can result in the worst candidate winning the election.

This analysis is correct as a “What would happen if…” story, but it overlooks two crucial factors.

First, these voters didn’t know one another’s preferences. Heading into the election, it was entirely plausible that Wright was the Condorcet winner, for example. It’s true that there was a potential for one-sided strategy to change the outcome of the election given perfect knowledge of how all other blocs were voting, but the talk of bullet voting makes little sense in the face of uncertainty.

Second, this analysis overlooks an important symmetry and the role of Wright>Montroll>Kiss voters. Under a voting method like STAR that passes the opposite cancellation criterion, when two voters have completely opposite preferences, if it is strategically optimal for one of those voters to cast one ballot then it must be strategically optimal for the vote with the opposite preferences to cast the opposite ballot. For example, if it is optimal for a voter with the utilities {Kiss: 10, Montroll: 7, Wright: 0} to vote Kiss: 5, Montroll: 1, Wright: 0, it must also be optimal for a voter with the opposite utilities, namely {Kiss: 0, Montroll: 3, Wright: 10}, to vote Kiss: 0, Montroll: 4, Wright: 5. Because of this symmetry, it’s impossible for everyone to be strategically justified in voting 5–1–0 unless people care far more about getting their first choice elected than about preventing their second choice from being elected. Statements like “Kiss wins if K>M>W supporters lower the stars given to Montroll to 1” which imply that K>M>W supporters should give Montroll a 1 also imply that W>M>K supporters should give Montroll a 4 — and this fact is in no way factored into the analysis.

To be fair, there are some features of the Burlington election which mean that more voters should vote 5–1–0 than 5–4–0. Montroll was probably closer to Kiss than to Wright ideologically, and there were probably more {Kiss: 10, Montroll: 5, Wright: 0} voters than {Kiss: 0, Montroll: 5, Wright: 10} voters, and it’s the Kiss and Montroll voters who had the strongest strategic incentives to vote 5–1–0 since Kiss is favored over Wright in the runoff. Still, there’s essentially no way that it made sense for enough voters to vote 5–1–0 (let alone 5–0–0) for Montroll not to get elected unless a wide swath of the electorate was nearly indifferent between their second and last choices.

## Condorcet

Condorcet methods use the same ballots as IRV but use a tabulation algorithm which ensures that the Condorcet winner — Montroll — always wins.

Could Kiss or Wright’s supporters cause their preferred candidate to win by doing something like ranking Montroll last? Under my preferred Condorcet method — Minimax — the answer is no.

For simplicity, let’s consider this a three-candidate election. Wright’s supporters can’t make him win because he loses to both Montroll and Wright head-to-head and they can’t change this. Kiss’ supporters can’t make Kiss win either, though to see we have to look more closely at Minimax.

Minimax elects the candidate whose greatest defeat head-to-head is the smallest. Kiss loses to Montroll by a 6.6% margin, but Kiss only defeats Wright by a 2.8% margin. The voters who prefer Kiss to anyone else can’t improve upon either margin. What they can do is vote Kiss>Wright>Montroll such that Montroll loses to Wright by more than a 6.6% margin — but this simply hands the election Wright. It would be hypothetically possible for some Kiss>Montroll>Wright voters to team up with some Wright>Kiss>Montroll voters to increase Kiss’ margin over Montroll and get Kiss elected that way, and such an arrangement would be beneficial to all the voters involved. But it’s completely implausible for many reasons, not least of which is that the voters with the preferences Wright>Kiss>Montroll would have the opportunity to get Wright elected instead just by voting honestly instead of by playing along and voting Kiss>Wright>Montroll.

(It might be possible for strategic voters to cause a different outcome by incorporating Dan Smith into their strategy; I haven’t run the numbers, but even if it’s mathematically possible it would require an implausibly large number of strategists.)

## What if Montroll shouldn’t win?

Looking at the results of the Burlington election, it seems obvious that Montroll is the “best” winner. But what does being the “best” winner even mean? There are two definitions:

• The Condorcet definition: The best candidate is the one who would defeat anyone else head-to-head (when such a candidate exists).
• The utilitarian definition: The best candidate is the one whose utility, summed over all voters (presumably after being normalized somehow), is the highest.

The utilitarian definition gets rather thorny when you consider questions like how exactly different voters’ utilities should be normalized. (Luckily, these difficulties are a non-issue when using it in computer simulations.) Nonetheless, the utilitarian definition is the better one. If 51 voters think candidate A is infinitesimally better than candidate B and 49 voters think that candidate B is dramatically better than candidate A, B should win. (A short argument why: Apply the veil of ignorance by imagining this scenario is repeated 100 times. You’re in the group that slightly prefers A 51 of those times and in the group that greatly prefers B 49 times. On average, you’ll be a lot happier if B wins.)

Based on the election results, we cannot determine which candidate’s victory would have maximized group utility. There are some possibilities that are irrelevant for comparing voting methods — for example, maybe Dan Smith’s supporters cared far more about the outcome than anyone else so electing Smith would have maximized group utility. But it’s impossible to make a voting method that is sensitive to things like one group of voters just caring more (outside of something like a VCG mechanism, which doesn’t really qualify as a voting method). The relevant possibilities are the ones that some voting methods (most notably Approval and STAR) can distinguish between. For example:

• All voters who expressed a second choice thought their second choice (among the top three) was 60% as good as their first choice. (They have utilities like {Kiss: 10, Montroll: 6, Wright: 0}). In this case, Montroll should win.
• Voters are far more concerned about helping their first choice win than preventing their last choice from winning. They have utilities like {Kiss: 10, Montroll: 1, Wright: 0}. In this case, Wright should win.
• Kiss>Montroll>Wright voters and Montroll>Kiss>Wright voters have utilities like {Kiss: 100, Montroll: 90, Wright: 0} and {Kiss: 90, Montroll: 100, Wright: 0}, but Wright voters (who expressed a second choice) have utilities like {Kiss: 0, Montroll: 1, Wright: 100}. In this case, Kiss should win.

Plurality, IRV, and Condorcet methods are almost completely insensitive between these possibilities; they’ll elect the same winner (Wright for Plurality, Kiss for IRV, and Montroll for Condorcet) in each. Approval, Approval + Runoff, and STAR are different. When voters think their second choice is almost as good as their first choice, each of these methods will elect Montroll. When voters think their second choice is almost as bad as their last choice, Approval will elect Wright, Approval + Runoff will elect Kiss, and STAR will elect either Kiss or Wright (it’s tough to say which).

What would be nice is if we could say that one of these voting methods would always elect the utility-maximizing candidate, at least given some basic constraints (e.g. assuming that every voter considers their favorite among the top three to be 10 utility better than their least favorite). Unfortunately, we can’t quite go this far. Here are some very contrived examples for each voting method:

• Approval voting: Suppose everyone only votes for their second choice if that second choice is at least half as good as their first choice (i.e. if they have utilities like {A: 10, B: X, C: 0}, where X ≥ 5); this is the correct strategy if they believe all three candidates are equally viable. If everyone thinks their second choice is only 40% as good as their first choice ({A: 10, B: 4, C: 0} utilities), Montroll would be the utility-maximizing candidate — but everyone would bullet vote so Wright would win.
• Approval + Runoff is incapable of ever electing Wright no matter how little voters care about their second choices compared to their first choices.
• STAR: If everyone thinks their second choice is 20% as good as their first choice, Wright is the utility-maximizing candidate but Kiss will win in the runoff if everyone casts an honest (and, in this case, strategically sound) 5–1–0 ballot.

In reality, some voters will think their second choice is almost as good as their first choice, some will think their second choice is almost as bad as their last choice, and others will think their second choice is right in the middle. Precisely describing a realistic scenario would take us too far afield, but it’s worth noting how these kinds of preferences are related to the simple model given for Approval Voting: if relatively few voters vote for multiple candidates, that’s evidence that they don’t think their second choices are much better than their last choices (and therefore evidence that Montroll shouldn’t win). Similarly with STAR: If most voters only give their second choice one star, they probably only care a great deal about electing their favorite.

For all of these methods, Montroll is far more likely to be elected when he is the utility-maximizing candidate than when he isn’t. Concocting a realistic scenario in which Montroll is the utility-maximizing candidate but loses under Approval, Approval + Runoff, and STAR is probably doable, but I expect it would still end up looking weird or feature a lot of voters behaving in a strategically foolish manner.

## Conclusion

The center squeeze presents challenges to many voting methods, not just IRV. It seems plain that Plurality fares the worst here, followed by Plurality + Runoff and then IRV. How the other voting methods compare is much more debatable since they each have their plusses and minuses.

Condorcet methods are the simplest to analyze. Their strength is that they ensure Montroll’s election when he’s the best candidate; their weakness is that they also ensure Montroll’s election when he’s the worst candidate. But it seems highly likely that Montroll really was the best candidate, so the former matters a lot more than the latter. Condorcet looks quite good in the center squeeze.

The main weakness of Approval Voting is that it’s the least likely to elect Montroll of these four options when he should win; a secondary weakness is that a handful of non-strategic voters may essentially throw their votes away by bullet voting for Dan Smith. Its strengths are a lack of dishonest strategies, being by far the best at electing Wright when he should win, and simplicity.

Approval + Runoff avoids the weakness of no-runoff Approval, but at the cost of gaining some new ones. While the dishonest strategies are probably more of a curiosity than a significant concern, the incentive is definitely there. The total inability to elect Wright, even if voters care drastically more about their favorites than their #2 choices, is another downside. While I prefer Approval + Runoff to Approval overall (at least if you ignore the hassle of holding a runoff), in the center squeeze it’s much closer than usual.

STAR seems strictly better than Approval + Runoff. Aside from Condorcet it’s probably the best at electing Montroll when he should win, and it retains some limited ability to elect Wright when he should win. The dishonest strategies of Approval + Runoff are less viable here, though still perhaps a relevant factor. On balance, I think STAR performs the best in the center squeeze, though there are also solid arguments to be made for Approval and Condorcet.