Another assessment of the red/blue button case (or: why I vote red and so should you)

This is my analysis of the problem, which I initially posted to Substack [here](https://open.substack.com/pub/fric/p/red-or-blue-button).

# 1. Introduction

# 1.1 The scenario

Recently, a dilemma has been shared on [X](https://x.com/waitbutwhy/status/2047710215265730755) and other places. Here's the question:¹

>Everyone in the world has to take a private vote by pressing a red or blue button. If more than 50% of people press the blue button, everyone survives. If less than 50% of people press the blue button, only people who pressed the red button survive. Which button would you press? > >— Tim Urban (@waitbutwhy), April 24, 2026

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:

P(blue loss | blue) = 0.5 − f(0.5)/n = 0.5 − 2/(8·10⁹) = 0.5 − 2.5·10⁻¹⁰ P(blue loss | red) = 0.5

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:

E[deaths to others | blue loses ∧ blue] ≈ 3·10⁹ − 0.5 E[deaths to others | blue loses ∧ red] ≈ 3·10⁹

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.

E[deaths to others | blue] = P(blue loss | blue) · E[deaths to others | blue loses ∧ blue] = (0.5 − 2.5·10⁻¹⁰)(3·10⁹ − 0.5) ≈ 1,499,999,999 E[deaths to others | red] = P(blue loss | red) · E[deaths to others | blue loses ∧ red] = (0.5)(3·10⁹) = 1,500,000,000

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.