The chicken dilemma: a deep dive under several voting methods

19 min readSep 26, 2021

In May of 2010, Hawaii held a special election for its 1st congressional district. It was not preceded by a primary election, and there were three significant candidates:

Colleen Hanabusa, Democrat
Ed Case, Democrat
Charles Djou, Republican

The results? Djou, the Republican, won with 39.4% of the vote; Hanabusa and Case got 30.8% and 27.6%, respectively. Either Hanabusa or Case would have probably defeated Sjouo head-to-head, and Hanabusa went on to defeat him head-to-head a few months later, 53.2% to 46.8%.

This election offers a clear case of Plurality Voting failing catastrophically. The race was not particularly close, and the candidate who least reflected the preferences of the voters still got elected. However, what makes this race especially interesting is that it would pose a challenge for other voting methods as well.

The chicken dilemma in game theory

First, a digression. Suppose you are one of two extremely foolish drivers playing the following game: Both you and the other driver are driving cars that start far away but are facing directly at one another. You each accelerate to your cars’ maximum speeds. If you swerve away, but the other driver does not, they get to call you a chicken. (That’s bad.) If they swerve away, and you keep going straight, you get to call them a chicken. (That’s good!) If you both swerve away, neither of you gets to insult the other without hypocrisy. (That’s neutral.) And if you both keep going straight at each other, each of you will die in the ensuing high-speed car crash. (Death is worse than being called a chicken.) This kind of situation is called the chicken dilemma.

Your incentives here are horrible. You don’t want to swerve before the last possible minute; if you do you’ll definitely be a chicken. Unless you value staying alive infinitely more than you value calling the other driver a chicken (which isn’t the case since you’re playing this game to begin with), it’s worth incurring some non-zero chance of death to improve your odds of winning. Whichever driver is willing to incur a greater chance of mutual death will have a major advantage, and if you know the other driver is a daredevil who will never swerve you will have to swerve yourself. In the chicken dilemma, insanity can be an asset.

The chicken dilemma in Approval Voting

Let’s imagine that the Hawaii election was being conducted under Approval Voting instead of Plurality. The good news is that a Democrat would most likely win. The bad news is that the Democrats would be in the chicken dilemma.

Suppose each Republican votes only for Djou, and each Democratic votes for both of the Democratic candidates (Hanabusa and Case). This would result in the Democrats tying for first, with Djou finishing a very distant third. Now suppose a few Democrats “bullet vote”, i.e. only vote for their favorite candidate. In this case, whichever of the Democratic candidates has more of their supposers bullet vote will win. Unless, that is, over two-thirds of the supporters of each Democrat bullet vote — in which case the Republican will get the most votes. (Two-thirds is approximate and depends on the exact support of each candidate.)

This scenario is the chicken dilemma all over again (it’s also called the Burr dilemma). Voting for both Democrats corresponds to swerving, bullet voting corresponds to driving straight, and letting the Republican win corresponds to the high-speed car crash.

One point to bear in mind: The Republican will usually lose. If the Republican was favored, the Democratic voters would be heavily incentivized to vote for both Democrats, leading to one of them getting elected. The expectation of a Republican victory would be a self-defeating prophecy. In the strategic equilibrium, the Democrats are heavily favored — but the Republican still wins occasionally.

That’s the simplified version. Let’s look at the assumptions that went into this and see what happens when we lift them.

Every Democratic voter prefers one Democratic candidate to the other. In truth, there will be some Democrats who are indifferent between their two candidates. Maybe they’ll have decided that they’re equally good, or maybe they only know that they prefer the Democratic party but haven’t bothered researching the candidates. Either way, they’re obviously best off voting for both Hanabusa and Case.
Every voter either prefers both Democratic candidates to the Republican or the Republican to both Democrats. Some voters will have the preference Hanabusa > Djou > Case or Case > Djou > Hanabusa. They may vote for Djou as well as their preferred Democrat, but they’ll never vote for both Democrats. They’re not really part of the chicken dilemma, but they have an influence. If it was only Hanabusa running against the Republican, all of the Case > Djou > Hanabusa voters would simply vote for Djou; with Case also in the race, they’re guaranteed to vote for one of the Democrats and might also forgo voting for the Republican. The existence of these voters means that the Democratic party has advantages as well as disadvantages in fielding a second candidate.
Every Republican will bullet vote. The strategic incentives in the chicken dilemma mean that Djou has a chance of winning, but he’s still an underdog. Therefore, a Republican with the preferences Djou > Case > Hanabusa may want to vote for Case as well as Djou. The lower Djou’s odds of winning, the more reasonable it becomes for Republicans to vote for the more tolerable Democrat in addition to him. In the extreme case in which Djou had no chance of winning, voting only for him would be throwing your vote away — so it would always make sense for Republicans with a preference between the Democrats to vote for one of them as well.
Having some Republicans also vote for a Democrat makes it substantially more likely that a Democrat will win. It also means the Democratic candidates will be incentivized to reach out to Republicans; even if they have no real hope of becoming a Republican voter’s first choice, there’s a great deal of value in being the lesser of two evils.
Democrats won’t coordinate. Suppose you’re a Democrat who prefers Case and your spouse is a Democrat who prefers Hanabusa. If you each bullet vote, Hanabusa gets one vote, Case gets one vote, and Djou gets zero votes from the two of you. If each of you vote for both Democrats, Hanabusa and Case will each get two votes and the two of you will double your collective impact. No matter how strong your preferences are and how heavily favored the Democrats are, agreeing that both of you will vote for both Democrats will be advantageous. If the chicken dilemma is a major concern then the Democratic candidates would be heavily incentivized to encourage their supporters to make such agreements. I would be very surprised if this became popular enough to help the Democrats by more than 10%, but it should still help.
Republicans won’t coordinate. Analogously, if one Republican votes for Djou and Hanabusa and another votes for Djou and Case their combined impact will be to give Djou one more vote than each Democrat. If they agree to both bullet vote, the two of them will give Djou two more votes than each Democrat. Such agreements would have the opposite effect of agreements between Democrats, but their net impact will probably be much smaller. This is partly because there are more Democrats than Republicans (and thus more agreements), and partly because I expect a baseline of more Democrats voting for exactly one Democrat than Republicans voting for exactly one Democrat.

When these complications are factored in, the chicken dilemma looks a lot more manageable. It’s still a concern, but probably not a massive one.

Why is the chicken dilemma bad, exactly?

While it’s intuitively obvious that the chicken dilemma is an unhealthy dynamic, it’s worth reviewing exactly how it’s harmful.

First, it occasionally results in the Republican winning. I’m not claiming this is bad because Republicans are bad; it’s bad because it fails to reflect the wishes of the electorate. Hawaii residents tend to be quite liberal, and electing a Republican “by accident” goes against their preferences. The scenario could just as easily be reversed, with two equally popular Republicans competing against a single Democrat in a red district. It’s also worth noting that electing the Republican in Hawaii is presumably a bigger miss than electing a suboptimal Democrat would be.

Second, it arguably leads to ugly incentives for the Democratic candidates. It would be very valuable for Hanabusa to convince half her supporters that Case is almost as bad as Djou; this will make them far more inclined to bullet vote. In the chicken dilemma (and also, to some extent, with approval voting more generally) it’s valuable to get your supporters to increase their dislike of the other candidates; being a voter’s first choice is good, but being their first choice by a large margin is even better.

I don’t really buy the second point, however. Whenever there are exactly two frontrunners (in this case, the Democrats), they will be heavily incentivized to attack one another under any voting method. Under virtually every other voting method, the voters that the Democratic candidates most want to appeal to will be the swing voters who think the two Democrats are about equally good. Suppose you’re one of the Democratic candidates. If you get a voter to go from infinitesimally preferring the other Democrat to infinitesimally preferring you in a plurality or IRV election, you’ve just gained a full vote. Under Approval, the infinitesimal differences won’t matter in either case since someone with such a weak preference between the two of you will be better off voting for both of you anyway. Instead, the swing voters under Approval Voting will be:

Democratic voters who are ambivalent between voting only for you and voting for both you and the other Democrat
Democratic voters who are ambivalent between voting only for the other Democrat and voting for both of you
Republican voters who are ambivalent between voting only for the Republican candidate (Djou) and voting for both Djou and the Democrat they deem more tolerable.

It is special that, under Approval Voting, candidates are incentivized to appeal more towards for who are already their first-choice supporters (and get their opponents to appeal less to them). This is counterbalanced by a reduced desire to appeal to voters who are currently indifferent between the two Democrats (and to make the other Democrat less appealing). Overall, I don’t think Approval Voting creates a new incentive for negative campaigning; it merely redistributes the incentives that exist under any voting method.

Another point: Even if this concern over increased negativity is valid, do we really mind all that much if Democrats become more inclined to attack their Democratic opponents and Republicans become more inclined to attack other Republicans? America is a divided country, and these divisions pose a real threat to our continued democracy. However, it’s the division between the two major parties which leads to constitutional hardball and the possibility of democratic backsliding. Since Approval Voting incentivizes Democratic candidates to appeal to Republican voters (or, at the very least, to come across as less intolerable), even if it increases negativity within each party it should still reduce the most dangerous form of animosity.

So why again is the chicken dilemma in Approval Voting bad? If we reject the argument of ugly incentives and negative campaigning we’re still left with the fact that it should occasionally result in the election of the candidate who least reflects the will of the electorate. This is indisputably bad, and there are other voting methods that do a better job here.

Approval + Runoff

With the addition of a runoff, the Republican will never win. This means that Democratic voters should always bullet vote (unless they’re flat-out indifferent between Hanabusa and Case) but Republicans still have an incentive to vote for a Democrat as well as Djou. Adding a runoff election essentially eliminates the problems of the chicken dilemma, albeit at the cost of conducting an additional election.

Instant Runoff Voting

How would the Hawaii election have gone if Instant Runoff Voting (IRV, aka single-winner Ranked Choice Voting) was used instead? As a reminder, here are the Plurality results:

Colleen Hanabusa (D): 30.8%
Ed Case (D): 27.6%
Charles Djou (R): 39.4%
Other candidates: 2.2%

After the other candidates were eliminated, Case would have the fewest votes and would then be eliminated. A substantial majority of Case’s votes would be transferred to Hanabusa, giving her more votes than Djou and resulting in her victory. That’s the advantage of IRV over Approval Voting in the chicken dilemma: a candidate from the larger party always wins. (I’ll keep referring to this as the chicken dilemma, even though, under most voting methods, the scenario lacks the toxic dynamics from the game of chicken that are present in approval voting.)

The downside of IRV is that the votes of Republicans won’t matter in determining which of the Democrats gets elected. Djou’s votes never get redistributed to Hanabusa or Case, so his votes are completely wasted (unless Djou has enough support to win, in which case we’re not in the chicken dilemma at all). This leads to some largely unpleasant consequences:

The Democratic candidates have hardly any incentive to appeal to Republicans. There’s no benefit to being the second choice of a voter whose first-choice candidate will make it to the final round.
Republicans are incentivized to dishonestly rank the more tolerable Democrat ahead of Djou. This is actually a case of strategic voting being socially beneficial, and this behavior leads to a more representative outcome for the electorate as a whole. But, insofar as we dislike dishonesty, it’s still suboptimal. It also makes Djou appear less popular than he actually is, and, if Djou’s actual popularity is far greater than expected, such strategic voting could backfire and cause the election of a Democrat when Djou ought to win.

With only three candidates, Plurality + Runoff functions basically the same as IRV. The presence of the minor candidates means that IRV will perform better, however.

Score Voting

Under Score Voting (assuming allowed scores go from 0 to 5), Democratic voters can give their favorite Democrat a 5, the less-preferred Democrat something like a 3, and the Republican a 0. If enough of them do this, a Democrat should win. But voting like this isn’t strategically optimal. There are two possibilities:

It’s better for the less-preferred Democrat to have a higher score than you expect them to get (due to the risk of the Republican winning). In this case, it’s best to give them a 5.
It’s better for the less-preferred Democrat to have a lower score than you expect them to get (since they’re mainly competing with the more-preferred Democrat). In this case, it’s best to give them a 0.

Score may elect Djou less frequently than Approval due to the behavior of non-strategic voters, but the dynamics of the chicken dilemma, along with the very real risk of a Republican triumph, are still there.

STAR

With STAR, it shouldn’t matter if Djou makes it to the automatic runoff; a Democrat will still defeat him. However, it’s worth looking into the possibilities in depth.

Case one: Both Democrats advance to the runoff

In this case, everything goes smoothly, and the winner will be whichever Democrat is preferred by the electorate as a whole — Republicans included. Both Democrats will be equally incentivized to appeal to both Democratic and Republican voters, and it will be a two-person race between the two best candidates without strategy playing any significant factor. All voters need to avoid is giving both Democrats the same score; beyond that, they can’t really go wrong.

Case two: Djou and a Democrat advance to the runoff

This can happen if a lot of Democratic voters vote a 5–1–0 ballot (that is, giving 5 stars to their first choice, one to their second choice, and none to their last choice (Djou)). A Democrat will still win in this case, but strategy matters a lot more. Here, the winner will be whichever Democrat gets a higher score — so it’s important to spread out the scores of the Democrats as much as possible by voting 5–1–0 instead of 5–4–0 (if you’re a Democrat). This will, of course, reinforce the pattern of Democrats getting lower scores than Djou. On the other hand, Republicans are incentivized to vote 5–4–0 (5 for Djou, 4 for the more tolerable Democrat, and 0 for the other one), which will elevate the scores of the Democrats.

Whether you vote 5–4–0 or 5–1–0 matters a ton. One option gives your preferred Democrat four more points than your less preferred Democrat; the other option grants one more point. Choosing correctly will quadruple the impact of your ballot.

(Since it doesn’t matter which of these options you go for in the event that both Democrats make advance to the final round, in the presence of uncertainty it’s best to vote as you would if only one Democrat would make it to the final round.)

Could it makes sense for a Democrat to vote 5–0–0 instead of 5–1–0? Hypothetically yes, but it’s a high risk for a very low reward. If the finalists are the less-preferred Democrat and Djou, voting 5–0–0 means abstaining in the runoff. If Djou is more popular than expected, such behavior could hand him the election. By contrast, the reward is a mere 20% additional support to the preferred Democrat. If Djou has enough support to make it to the runoff while the Democrats are neck and neck, he’s probably within a couple standard polling errors of actually winning. In practice, I don’t foresee this extreme strategy ever being worth the risk.

Regardless of whether Djou makes it to the runoff, the Democrat who is preferred either in terms of score or the head-to-head comparison will win in the chicken dilemma. And, unlike under IRV, it’s the Democrat who is more popular among all voters (not just Democrats) that will win. The biggest downside is that, if Djou makes it to the runoff, voters who cast strategetic suboptimal ballots will have disproportionately less influence over which Democrat gets elected. Still, this is unlikely to cause an especially bad outcome, in part because the voters who care the most strongly about which Democrat gets elected are the least likely to give them similar scores. Ultimately, I think STAR handles the chicken dilemma a lot better than Approval or IRV.

Condorcet methods

With a Condorcet method, assuming everyone votes honestly, a Democrat will always be elected, and the preferences of Republicans and Democrats as to which Democrat is elected will matter equally. It’s the best of all possible worlds — assuming everyone votes honestly.

What if people don’t vote honestly? Is there a way to gain an advantage?

(Short answer: None that’s remotely viable. Feel free to skip ahead.)

Two technical points: First, it is impossible to use strategic voting to cause there to be a Condorcet winner you like more than the actual Condorcet winner. (If C is the Condorcet winner and you prefer D, you’re already doing your utmost to make D the Condorcet winner instead just by honestly ranking D ahead of C.) Second, if everyone votes honestly there is guaranteed to be a Condorcet winner since Djou is the Condorcet loser (i.e. the candidate who loses to any other candidate head-to-head), so whichever Democrat is preferred to the other head-to-head will be the Condorcet winner.

What can be done is to create a Condorcet cycle — a scenario where candidate A beats candidate B, B beats C, and C beats A. This would happen if, for example, Hanabusa is the sincere Condorcet winner, the Republican voters are equally split between Hanabusa and Case, and a large fraction of the Case voters dishonestly vote Case > Djou > Hanabusa. If this happens, Hanabusa would defeat Case by the same (small) margin of victory that exists when everyone votes honestly, Case defeats Djou by the same (substantial) margin of victory that exists when everyone votes honestly, and Djou beats Hanabusa by a margin that depends on the number of Case supporters who vote dishonestly.

How the Condorcet cycle is resolved depends on which Condorcet method is used. Let’s assume that it’s Minimax, which elects whichever candidate is defeated by the smallest margin. In this case, assuming enough voters use this strategy, that’s Case — so the strategy is successful. Is this strategy realistic? Let’s compare it to the strategy of bullet voting under Approval Voting. With bullet voting in Approval, even a small number of voters using it can sway the election (provided it’s close enough). By contrast, voting Case > Djou > Hanabusa in Minimax will never be effective unless a large number of voters do it. Specifically, the number of strategists must exceed Hanabusa’s margin over Djou (to go from Hanabusa defeating Djou head-to-head to Djou defeating Hanabusa head-to-head) plus Hanabusa’s margin over Case (so that Case has the smallest margin of defeat).

Hanabusa’s margin over Case could be arbitrarily small; there’s no real downside to Case’s supporters using this strategy if Case turned out to be the sincere Condorcet winner. However, this strategy is risky unless Hanabusa’s margin over Djou is enormous. If Case is trailing Hanabusa and Djou is only a little behind Hanabusa, it’s quite possible that Case would lose to Djou head-to-head. In this case, dishonestly voting Case > Djou > Hanabusa could make Djou the Condorcet winner.

Let’s look at some actual numbers in combination with some made-up assumptions. The Plurality results were Colleen Hanabusa (D): 30.8%, Ed Case (D): 27.6%, Charles Djou (R): 39.4%, and Others: 2.2%. Let’s ignore the 2.2% by assuming they‘re evenly split in all pairwise comparisons among the major candidates, and let’s assume Djou’s supporters are evenly split between preferring Case to Hanabusa and preferring Hanabusa to Case. Finally, let’s assume that 80% of Hanabusa’s supporters prefer Case to Djou (the others preferring Djou to Case) and 80% of Case’s supporters prefer Hanabusa to Djou. This leads to the following head-to-head comparisons:

Hanabusa over Case: 51.6% to 48.4%
Hanabusa over Djou: 54.0% to 46.0% (She actually beat him 53.2% to 46.8% in November.)
Case over Djou: 53.3% to 46.7%

Thus, over 11.2% of the electorate would have to dishonestly vote Case > Djou > Hanabusa (in addition to the ones who vote that way honestly) in order for Case to win. This is over half the voters with the preference Case > Hanabusa > Djou. Even if it was practical to recruit this many strategists, doing so would require a massive publicity effort which would have the side effect of making Case look like a horrible person bent on subverting democracy. And this is in spite of the fact that the election was reasonably close. Case’s margin over Djou is a mere 6.6 points, so, given a plausible polling error, this strategizing could have handed Djou the election.

Looking at the numbers, I don’t think this strategy is remotely viable. Condorcet methods will do exactly what we what them to do in the chicken dilemma when voters are honest, and voters have no practical means of gaining an advantage via dishonesty.

The chicken dilemma, more broadly

Looking beyond the Hawaiian election, under what circumstances does the chicken dilemma appear in Approval Voting, and does the same analysis hold for the other voting methods?

There are several requirements for the chicken dilemma to appear in a three-candidate election:

The candidates must be split into two factions, with a larger faction fielding two candidates and a smaller faction fielding only one candidate. The factions don’t have to be political parties; it’s possible to have a Republican and a Libertarian be from one faction and a Democrat be from the other, or you could get a similar dynamic in a primary or a non-partisan municipal race.
The candidate from the smaller faction must be the plurality winner; otherwise, everyone who preferred the more popular candidate of the large faction could bullet vote with impunity.
Either of the large faction candidates must beat the small-faction candidate head-to-head. If only one of them does, the supporters of the less-popular large-faction candidate can’t benefit from bullet voting since that candidate will lose no matter what. Moreover, they need to defeat the small-faction candidate by a sizeable margin; otherwise, the large-faction voters are best of playing it safe and voting for both of their candidates without the dynamics of the chicken dilemma playing a major role.
Most voters must care more about their preferred faction winning than about having their preferred candidate within the larger faction winning. If this doesn’t hold, the smaller-faction candidate could best reflect the preferences of the electorate.

These conditions require the smaller-party candidate to have between 33% and 50% first-choice support. It also can’t be too close to those extremes, lest voters feel the need to play it particularly safe in the face of uncertainty. I don’t consider the chicken dilemma to be especially contrived, but it does require a fairly narrow set of circumstances.

When these conditions are met, the same analysis holds for all the voting methods I’ve considered here — not just Approval Voting.

What if we add more candidates?

If we add additional large-party candidates we reduce the incentive to bullet vote since many voters will have two or more large-party candidates they significantly prefer to the other large-party candidates. However, adding more candidates should have a relatively small effect on the number of votes each viable large-party candidate receives; if all the candidates are comparably electable then it’s typically optimal to vote for around half of them, regardless of how many are in the race (the presence of the smaller party pushes this fraction up). Additional candidates should also have only a minimal impact on other voting methods; the biggest exception is probably IRV, where, if there are more large-party candidates than voters have room to rank on their ballots, some amount of Plurality-style vote-splitting becomes a possibility.

If we add an additional small-party candidate we could hypothetically give the smaller party the ability to win in STAR, Approval + Runoff, and Plurality + Runoff. This would happen if both of the smaller-party candidates advanced to the runoff. It could definitely happen under Plurality + Runoff due to classic vote-splitting (imagine 10 Democrats running against 2 Republicans), but I’m skeptical of it happening under the other methods. There’s a strong incentive to vote for (in Approval + Runoff) or score highly (in STAR) plenty of large-faction candidates, and all the mitigating factors from the chicken dilemma in (no-runoff) Approval Voting are still there. I think the expanding chicken dilemma with 2+ smaller-party candidates and many major-party candidates is a valid concern in STAR and Approval + Runoff, but it’s a pretty minor concern. Also, when you have that many candidates the scenario starts sounding rather contrived.

Conclusion

At first glance, the scenario of the chicken dilemma looks exactly like the kind of situation that Approval Voting is meant for. (In fact, the Center for Election Science’s introductory video features a similar scenario.) A closer look reveals a toxic strategic dynamic and a real possibility of the wrong candidate getting elected as a result, but, when real-world complications are factored in, these problems seem much smaller. They are still present though, and there are other voting methods that handle the chicken dilemma better.

To recap, here’s how each voting method performs in terms of how often Djou wins (or, more generally, how often the smaller party candidate wins) and how much the Democratic candidates are incentivized to appeal to Republicans:

Plurality: Djou always wins.
Approval: Djou occasionally wins. The Democratic candidates have a significant incentive to appeal to Republican voters, but it’s probably less than their incentive to appeal to Democrats.
Instant Runoff Voting: Djou never wins. Democratic candidates have hardly any incentive to appeal to Republicans.
Approval + Runoff: Djou never wins. The Democratic candidates have the same incentives as in no-runoff Approval.
STAR: Djou never wins. The Democratic candidates are incentivized to appeal to Republicans about as strongly as they’re incentivized to appeal to Democrats.
Condorcet methods: Djou never wins. Democratic candidates are equally incentivized to appeal to Republicans as to Democrats.

Reasonable people can disagree on whether IRV or Approval Voting performs better in the chicken dilemma. However, it seems clear that STAR, Approval + Runoff, and Condorcet methods perform better than both IRV and no-runoff Approval, and one conclusion is indisputable: Plurality performs the worst by far.