1. Introduction
1.1 The scenario
Recently, a dilemma has been shared on X and other places. Here’s the question:1
For simplicity, we may suppose that people cannot coordinate in advance, offer incentives/disincentives to each other, or reveal individual votes afterward. Indeed, we may suppose that you are right now tasked with pressing one of the buttons. What’s the rational choice?
I will also assume that the votes are statistically independent. Perhaps this isn’t exactly correct, but I will assume that any statistical dependence is sufficiently weak such that the independence assumption is admissible in this case. I will revisit this issue in §4.1.
I approach this scenario like any other decision problem. I want to determine what the expected outcome is if I vote blue, what the expected outcome is if I vote red, and then choose the option with the better expected outcome. Notably, the value of an outcome isn’t merely a function of how likely I am to survive. It may be worthwhile to take on some risk to myself to increase the likelihood that others will survive. Accordingly, it would be too quick to say that voting red is rational merely because there’s only risk to me if I vote blue.
1.2 Overview
I wanted to give this problem a careful analysis, since I’ve found that most of the discourse concerning it has been very poor. Most of the arguments I’ve seen, both for red and for blue, are not good arguments. As such, I think that at least some people might benefit from a well-thought-out argument, or at least find it interesting.
I will argue that voting red is the correct choice. The basic reason is that voting blue takes on a substantial risk to yourself and others who would survive you if you die, while creating a negligible difference in the likelihood that blue will win and everyone will survive. While there are other factors worth considering, the given consideration in favor of red is clearly decisive.
In §2, I will consider the risks to life that you create by voting red and voting blue. I will estimate the risks to yourself by voting blue in §2.1, and the risks to others by voting red in §2.2. In §3.1, I will explain why those expectations favor voting red. In §3.2-3.6, I will consider motivations aside from those risks for voting blue. In §4, I will discuss some of the modeling assumptions made in my analysis. I will conclude in §5.
2. Risks to life
2.1 Risk to myself
If I vote red, there is no risk to myself. If I vote blue, the risk to myself is that less than 50% of people will vote blue, and I will die. Accordingly, the chance that I will die if I vote blue is simply the probability that red wins if I vote blue.
Our expectation of this risk, therefore, will depend on what evidence we have about how people will likely vote. Polls on X and other websites indicate a moderate victory for blue. I suspect, however, that more people would vote red if faced with this problem, and for a few reasons. I will revisit this issue in §4.2. However, my analysis here does not require the expectation that red will win. I will assume, as I think is reasonable, that the chance that red will win if I vote blue are about 50%, and so the chance that I will die if I vote blue is about 50%. I will revisit this assumption in §4.2.
2.2 The risk to everyone else
Here, I want to consider how much worse off I expect everyone else to be if I vote red than if I vote blue. After all, if I vote red, there’s a greater chance that many people will die than if I vote blue. Sure, it’s only very slightly greater, but contributing even slightly to the chance of a very terrible outcome might be too much risk. Accordingly, we have to figure out how serious the risk is in order to determine whether voting red is sensible. I will argue that the risk is very tiny, but it will take some math to give a plausible estimate.
We will let p̄ stand for the average bias in the vote toward blue. That is, if voters are on average disposed with 55% likelihood to vote blue, then p̄ = 0.55. We will also let X = the number of blue votes out of the total population.2 By the linearity of expectation, E[X] = np̄, where n is the population size. Thus, E[X]/n gives us the expected fraction of blue votes, which equals p̄. Importantly, however, we don’t know ahead of time what the value of p̄ is. We can thus consider an expectation for p̄, which I’ll write as E[p̄].3 We then have E[X]/n = E[p̄].
We may also want to estimate the variance of X, of the total number of blue votes. While the mean total votes depends only on the average voter bias, the variance does not; it also depends on individual voter biases. However, for somewhat technical reasons that I won’t fully explain here, the variance will not ultimately matter for our calculations. Nevertheless, when p̄ is very close to 0.5, Var[X] ≈ 0.25n, scaled down a bit if we expect greater voter heterogeneity.
Because X is a sum of many independent random variables (n is quite large), the distribution for X can be modeled, for our purposes, as a Gaussian with mean E[X] and variance 0.25n.4 Before we proceed, we need one more thing: a specification of the prior distribution for p̄. The exact shape won’t matter very much; all that really matters is how much of the probability density is focused in a very tiny window around 0.5. I will let f(p̄) be our probability density function given our prior beliefs, and I will suppose that it’s a uniform distribution between 0.25 and 0.75. We may think that another distribution is more plausible. If it’s narrower or more Gaussian, more of the density will be focused around 0.5. However, if it’s skewed or bimodal, less will be.
Now, suppose that my initial expectation that blue will win is 0.5 and n = 8⋅109.5 My expectation that blue will win should be slightly higher if I vote blue and slightly lower if I vote red. We are now equipped to estimate how much higher and how much lower that is. There are more complicated ways to calculate this, but the probability that blue will lose if I vote blue equals my prior probability that blue will lose, minus the probability that my vote will be pivotal.6 The probability that my vote will be pivotal is essentially the probability that the rest of the votes will be an exact tie. The corresponding distribution will be basically the same as for X, and so it suffices to determine P(X = n/2). This is very well approximated as f(0.5)/n, although I will forgo the tedious explanation for why this is the case. We then have:
Unsurprisingly, we find that the chance that blue loses if you vote blue is very slightly lower than the chance that blue loses if you vote red.7 We are then equipped to determine the expected number of deaths in the rest of the population conditional on you voting blue vs. voting red. Roughly speaking, the expected number of deaths to others if blue loses is approximately n⋅E[p̄|p̄ < 0.5], which is 8⋅109⋅0.375 = 3⋅109. However, this value is not quite the same if you vote blue vs. if you vote red. In fact, it turns out that:
I will forgo a derivation of these values.8 We are finally equipped to estimate the expected number of deaths to others, conditional on my voting blue and conditional on my voting red.
This result makes sense. Regardless of how we vote, we expect close to 20% of the rest of the population to die on average. If we vote red, the increased chance that blue loses translates to one extra expected death to others compared to voting blue. Of course, the estimation here required several assumptions. First, it assumes that individual votes are statistically independent from each other. Second, it assumes that the expected likelihood that blue wins is 0.5. Third, it assumes that our prior expectation on p̄ is a uniform distribution across [0.25,0.75] and 0 outside that range. These assumptions will be discussed further in §4. In the next section, I will compare the expected outcomes and determine the right course of action.
3. Values of outcomes
3.1 Risks of death
I will first weigh the risks to lives (of myself and others) as estimated in §2. To be clear, this isn’t the only factor relevant to the decision, although it is arguably the most significant. I will consider other factors in §3.2, §3.3, §3.4, §3.5, and §3.6.
Regardless, here’s what I consider the headline observation. If you vote blue, you take on a 50% chance that you’ll die to lower the expected number of deaths to others from 1,500,000,000 to 1,499,999,999. The exact expectations may differ somewhat, but this is representative of a reasonable expectation. The point is, you should think that your vote is almost certainly not going to make a difference to anyone else, but is likely to make a difference to your life. For the vast majority of people, the negligible increase in the likelihood that everyone survives from you voting blue is not worth the significant likelihood that you will die.
It’s even worse if you have family, friends, and people who care for you who might survive your death. The risk you take to your own life is thereby also a risk to them. How would you prefer them to vote? How would they prefer that you vote? I would hope that the people I care about voted red, since I don’t want them to die, and I hope that they’d feel the same for me. The negligible benefit of their voting blue is not even remotely worth the significant chance of death. Imagine you have a young child, and you could tell them what to do. You know that if they vote blue, there’s about a 50% chance that they’ll die, and about a 0% chance that they’ll save anyone else. If you told them to vote blue, I would think that you either didn’t fully understand the situation or that you’re some sort of moral monster. These considerations will be revisited in §3.5.
3.2 Easier to save everyone with blue
Many people commenting on this problem have observed that it’s much easier to save everyone if more people vote blue. After all, the only scenarios where everyone survives are if blue wins with >50% or if everyone votes red. However, it’s simply not going to happen that everyone votes red. Indeed, the number of people who vote blue will at least be in the hundreds of millions, if not billions. Given this, so the argument goes, we should try to save these people by helping blue to win.
It is true that among the plausible outcomes, the best are those in which blue wins. However, this fact was already assumed by the analysis in §3.1. Blue winning is a good outcome, and the fact that voting blue makes it more likely that blue wins is some reason to vote blue. My point is that the expected contribution to that outcome by voting blue is so negligible that it’s not worth taking a substantial risk to your life.
3.3 Contributing to death
Another relatively common remark is that, if you vote red and red wins, you’re contributing to the death of many people. Even if your contribution was negligible and they all would have died regardless of what you do, it’s still bad to have contributed to that outcome. Furthermore, even if blue wins and everyone survives, it’s bad that you were part of a collection of voters who, had more people voted along with you, would have led to the death of many people.
In support of this reply, we might point out that there are other cases where our action makes no or a negligible difference to the outcome, but that it’s arguably still bad that we act in that way. For example, imagine an election where one of the candidates is a murderous maniac. You might think, perhaps correctly, that your vote is almost certainly not going to make a difference to whether they win. Nevertheless, there’s arguably something bad about voting for them, even if your vote is fully anonymous. Furthermore, it’s still bad even if the maniac loses the election. Similar points may be made about, for example, our negative contribution to the environment or our negative contribution to animal welfare, even if individual inefficacy concerns are legitimate.9
While I think the general point is fair, this is not a good argument for voting blue. The difference is that, in the red/blue button case, the cost of voting blue is much higher. Consider the maniac candidate. Part of the reason why it’s reasonable to vote against him (or at least abstain) is that there’s basically no cost in doing so. If, on the other hand, there were a 50% chance of you dying unless you voted for him, the general badness of contributing even negligibly to his election is a much less significant concern. Or suppose that every time you wanted to make an eco-friendly choice, you had to flip a coin to decide whether you would die. The point is that you’re in a regrettable situation where the stakes are so high that you basically have to do an action that, without those stakes, would be worse for you to do. Accordingly, red remains the better option.
3.4 Character traits
It might be argued that voting blue is simply a more virtuous choice. In particular, voting blue exemplifies kindness, concern for others, altruism, self-sacrifice, and so on. On the other hand, voting red exemplifies selfishness, lack of caring, cold indifference, etc. Since blue is the more virtuous choice, we should vote blue.
There are two main problems with this argument. First, this does not fairly describe the relative virtuousness of voting blue and voting red. After all, voting blue is rather foolish and shortsighted, whereas voting red is appropriately sensitive to the cost that your death would cause to yourself and those around you. This is a kindness and concern for others that only voting red exemplifies. Additionally, while self-sacrifice is often commendable, it’s not good when done for poor reasons. Your sacrifice here is almost certainly not going to make any positive difference to the welfare of others. On the contrary, as discussed, it’s relatively likely to harm yourself and others.
Second, the argument overstates the importance of exemplifying certain virtues on one particular occasion. Even if voting blue were more virtuous in certain ways, the costs to voting blue clearly outweigh the good of cultivating and exemplifying those virtues on this one occasion. Consider, again, the maniac candidate from §3.3. Are there some virtues that are better cultivated/exemplified by refusing to vote for him, even if you’re likely to die? Perhaps, but not so much that it’s worth the terrible cost in this scenario.
Ultimately, I think that voting red is the more virtuous choice. However, even if this is wrong, it’s still not the case that this consideration outweighs the negative stakes of voting blue. At best, then, the considerations adduced in §3.3 and §3.4 provide some reason to vote blue, but not nearly enough to make blue the better choice.
3.5 Concern for particular individuals/groups
Probably one of the most common remarks made in support of voting blue is that we know that there will be many people who end up voting blue. This will include children, the elderly, people who don’t fully understand the problem, or who are disadvantaged in some way or another. It will also include family, friends, and others for whom we care directly. With this in mind, so the thought goes, it is clear that we should try to save them, and that requires voting blue.
There are two problems with this idea. First, the fact that there will be many people who vote blue was already built into the analysis in §3.1. The fact that there will be many people who vote blue, and that our voting blue contributes to the likelihood that they will survive, is some reason to vote blue. Nevertheless, the significant risk to us and others around us by voting blue is not worth the negligible contribution our vote would make to the chance that everyone will survive.
Second, as suggested in §3.1, this consideration actually cuts the other way. By voting blue, you’re creating a substantial risk to others who might survive you. Suppose that blue loses, and you vote blue. In that scenario, any family, friends, loved ones, etc., who voted red will have to deal with losing you. Had you voted red, although things as a whole would still be bad (many other people would still have died), they wouldn’t have to deal with losing you. Think about children losing a parent, or parents losing a child, or someone losing their spouse, and so on. The person died from voting blue when their vote made no positive difference to anyone else, but rather just made things clearly worse.
This is the key observation: if you vote blue, there’s basically a 50% chance that you’ll die and things will be significantly worse for yourself and others. If you vote red, there’s basically a 0% chance that things will be worse for anyone. Again, the risk you take on to yourself and others by voting blue is not at all worth the negligible contribution to the likelihood that everyone will survive. With this in mind, it is simply unavoidable that red is the rational and moral choice.
3.6 The world we live in
A few people have remarked that they’d rather not live in a world where red wins, perhaps because it would be such a terrible outcome (likely billions of people died), and it would mean that a majority of people didn’t vote in a virtuous manner. They would rather die with blue voters than live in a world like that.
There are a few things to say about this. If someone genuinely doesn’t care about their own life (or the welfare of those who would survive them), then they would discount the costs to voting blue that I’ve discussed. If, on reflection, you really would rather die than live in a world where red wins, then it’s in your interest to vote blue. However, I find the values required for that to be rather strange. Suppose you were the last person to vote, and you know that red is winning by billions of votes. Suppose you know that many of your family and friends had voted red. Would you really prefer to kill yourself by voting blue?
If your answer is “yes”, then I don’t have a lot to add. I do find that perspective rather insane, and I suspect few people will share it. For the vast majority of people, the considerations that support voting red will easily dominate. The large risk to yourself and others you incur by voting blue is not at all worth the entirely negligible chance that you’ll help others.
4. Modeling assumptions
4.1 Statistical independence
The calculations in §2 assumed that individual votes are statistically independent. That is, if we knew how a particular person or group voted, that wouldn’t change our expectation about how another individual or disjoint group voted. In reality, this is certainly incorrect. There are all sorts of factors that would lead to correlations between votes. People with similar psychology, values, background, culture, education, and so on are at least somewhat more likely to vote in the same way.
It turns out that this introduces two effects on the expected outcome of our choice, but they pull in different directions. First, if there are relatively broad correlations, ties become less likely, and so the likelihood that my vote will be pivotal decreases. What we would have, then, is a distribution for E[p̄] that is more bimodal, and so the likelihood that p̄ falls in the tiny window around 0.5 is reduced. How much it is reduced will depend on how sharply bimodal it is. Realistically, though, it probably won’t scale it down by more than an order of magnitude.
Second, if how I vote is somewhat positively correlated with how a relevantly similar group of people votes, then I should expect the total number of blue votes conditional on my voting blue to be greater than the expected number of blue votes conditional on my voting red by more than one. And the chance that the corresponding voting bloc is pivotal may be orders of magnitude greater than the likelihood that my uncorrelated vote would have otherwise been.10
Combined, these two effects might raise the impact of my vote by a couple of orders of magnitude. While that’s a relatively substantial difference, the chance that my voting blue will make a positive difference is still negligible, something around 10-8 or 10-9. On the other hand, if I’m updating on expected correlations between myself and others, I should think that there are a lot more people who will reason similarly to me who will ultimately vote red than there are in the bloc that would correlate with me from before. If this is right, then updating on this broader correlative data will lead me to expect more total red voters conditional on my voting blue than I would have otherwise expected without that evidence as background.
Ultimately, I think that taking these correlations seriously will count slightly against my voting blue, not for it.11 Nevertheless, what exactly we say here isn’t so important. Maybe these considerations count slightly in favor of blue, maybe they count slightly against it. Either way, the contribution of my voting blue to the likelihood that blue wins remains negligible, and the motivations for voting red previously adduced remain decisive.
4.2 Likelihood of blue winning
For my analysis, I assumed a 0.5 probability that blue would win. However, this can be questioned. In particular, polls conducted on this scenario found that ~58% of people preferred blue, give or take. On the other hand, there are some reasons to think that these polls may not be highly representative of how people would behave if actually facing the scenario. In addition to the sample population likely not being very representative, I strongly suspect that a greater portion of people would vote red when actually put into a life-or-death scenario. If I had to guess what the final vote would be, I think it would be ~35-40% for blue.
In any case, my analysis will not change significantly. If we suppose that E[p̄] = 0.58 but otherwise keep f(p̄) as a uniform distribution over [0.33,0.83], we’ll find that our chance of dying by voting blue has gone down to 0.34, although the likelihood that our voting blue will save others is unchanged.12 However, while voting blue is less risky here, it’s still quite risky, and not worth the negligible contribution to the likelihood that everyone will survive. On the other hand, if E[p̄] = 0.42, the cost of voting blue is higher, since you should think that your chance of dying by voting blue is about 0.66, with the same negligible contribution to the likelihood that everyone will survive. All else equal, changing the expected average voter bias only affects your expected chance of death by voting blue. Either way, the cost of voting blue goes up or down a bit, but it remains a terrible choice. Now, this assumes that the distribution is uniform, which I covered briefly in §4.1.
5. Conclusion
The correct answer is red. You should vote red if you faced a scenario like this. You should hope that your family, loved ones, and people you care about would too. After all, you don’t want them to die, and you don’t want whoever survives them to have to deal with losing them. That would be a terrible outcome, and it’s not worth a substantial risk to each of their lives just to make a negligible difference to the likelihood that everyone lives.
Nevertheless, I would hope that enough people vote blue, since that would ensure the best of the feasible outcomes. But hope is not a plan. If I could ensure that enough people would vote blue, perhaps by running a global campaign, then I would do so. In the actual stipulated scenario, I only have my own vote, and I have to weigh the expected outcomes conditional on my voting red and conditional on my voting blue. Again, there are some reasons to vote blue. It slightly increases the likelihood that everyone will survive. Other reasons include the factors discussed in §3.3 and 3.4. However, red is nevertheless the far better option. It’s not worth the large risk to yourself and others to make a negligible difference to the likelihood that everyone will survive. When facing the problem, you should think that if you vote blue, there’s about a 50% chance that you will die, leaving those who survive you to mourn your loss, and about a 0% chance that you’ll save anyone from death. That’s a horrifically bad trade for you to make, and I hope that you would recognize that were you to ever have to face such a terrible situation as the one under consideration.
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The problem, as stated, leaves unclarified what happens if exactly 50% of people press the blue button. We may assume that an exact tie is sufficient to count as a blue win, although this assumption will not relevantly affect the analysis.
Strictly speaking, X is a random variable counting the number of blue votes, not the actual number of blue votes. X could be many different values.
To be clear, p̄ denotes the actual population statistic, which is some unknown but fixed number. E[p̄] denotes my expectation of that number given my prior beliefs.
Modeling the distribution as a Gaussian requires a couple of other assumptions, but they are satisfied here. In any case, the exact distribution shape will not matter very much.
The current world population is a bit higher, but that’s not critical to the problem or my analysis of it.
By “pivotal”, I mean that my vote will swing the result one way or the other. In that scenario, the rest of the votes, apart from mine, are exactly evenly split. Of course, if the total number of other votes is odd, then the rest of the votes cannot form an exactly even split. However, the calculations here will not be relevantly different if we account for that possibility.
Here, I’m assuming that my prior expectation that the majority of other votes are blue is 0.5. Strictly speaking, it would be more accurate to include expectations of our own vote and calculate conditional expected deaths with that in mind. However, that would unnecessarily complicate the calculation without any meaningful increase in precision.
For an intuitive explanation, observe that in the cases where blue just barely loses, the number of blue votes is higher than the average number of blue votes when blue loses. In some of those cases where you vote blue, blue now wins, and so those cases no longer contribute to the average of blue voters when blue loses, thereby slightly lowering the average number of blue voters when blue loses. However, there is no such correction when voting red.
These cases are interesting and deserve further discussion, although it’s outside of the scope here.
I’m employing a sort of evidentialist reasoning that I accept, though it is contested. I will not discuss it further or argue for it here.
Those expectations may be somewhat different for someone else, depending on how they reason, how many people they think will reason and decide in the same sort of way that they do, and so on.
That probability is essentially entirely a function of how much probability density is very close to the tiny window around 0.5, which is the same here.
"First, if there are relatively broad correlations, ties become less likely, and so the likelihood that my vote will be pivotal decreases"
If I am considering a form of evidential decision theory where my own vote influences the probability I assign to many others voting similarly, wouldn't I no longer be interested in the question of whether my individual vote is "pivotal" (in the sense of being a tie-breaker) but just the total probability that blue vs. red wins given the shift in probability distribution by my voting either red or blue?
"Combined, these two effects might raise the impact of my vote by a couple of orders of magnitude. While that’s a relatively substantial difference, the chance that my voting blue will make a positive difference is still negligible, something around 10-8 or 10-9."
Is this number premised on the idea that my vote must be the tie-breaker to "make a positive difference"?
"On the other hand, if I’m updating on expected correlations between myself and others, I should think that there are a lot more people who will reason similarly to me who will ultimately vote red than there are in the bloc that would correlate with me from before. If this is right, then updating on this broader correlative data will lead me to expect more total red voters conditional on my voting blue than I would have otherwise expected without that evidence as background."
I don't understand this part, can you elaborate on why, if my vote is correlated with others, my voting blue could be associated with believing there are more total red voters?