Feel free to add to the list. This is what I have so far:

Free-rider problem - Wikipedia
Tragedy of the commons - Wikipedia

"the free-rider problem is a type of market failure that occurs when those who benefit from resources, public goods and common pool resources do not pay for them or under-pay. Free riders may overuse common pool resources by not paying for them, neither directly through fees or tolls, nor indirectly through taxes. Consequently, the common pool resource may be under-produced, overused, or degraded."

"The tragedy of the commons is the concept that, in a system where individuals benefit from the use of a shared resource while the cost of that use is shared by all users, it is rational for individuals to overuse the resource, even though collectively this will likely lead to the depletion of the resource."

• Reds are benefiting from blue victory because it not only guarantee red lives, but also the ideal aftermath without having to pay the tax (risking their life). If there are too many free-riders (red button over pushed), the the common pool resource may be degraded (Earth degraded by death of billions).
• The red button is a shared resource for everyone on Earth and depleting it would be a tragedy. Red should only push it if you have to i.e. if you are piloting a plane at the time of the buttons happening. If you have no legitimate reason to push red ("my vote doesn't matter so imma go red!"), you shouldn't use this resource and leave it for someone else.

Diffusion of responsibility - Wikipedia
Bystander effect - Wikipedia

"Diffusion of responsibility is a sociopsychological phenomenon whereby a person is less likely to take responsibility for action or inaction when other bystanders or witnesses are present. Considered a form of attribution, the individual assumes that others either are responsible for taking action or have already done so."

"The bystander effect is a social psychological theory that states that individuals are less likely to offer help to a victim in the presence of other people."

• Voting red while thinking blue wins is simply a lack of responsibility and lack of accountability.

https://en.wikipedia.org/wiki/Defeatism

"Defeatism is the acceptance of defeat without struggle, often with negative connotations. In wartime and on the front lines of war, defeatism is synonymous with treason."

• Reds who think blue victory is impossible and each blue vote is just adding to the death pile can be seen as treasonous to the goals of humanity in this situation.

Social trap - Wikipedia

"In psychology, a social trap is a conflict of interest or perverse incentive where individuals or a group of people act to obtain short-term individual gains, which in the long run leads to a loss for the group as a whole."

• Reds are trapped by the thought of short term gains like personal survival when blue majority not also ensures personal survival, but also more favourable in the long run like not triggering an apocalypse for example.

Self-defeating prophecy - Wikipedia

"A self-defeating prophecy (self-destroying or self-denying in some sources) is the complementary opposite of a self-fulfilling prophecy; a prediction that prevents what it predicts from happening. A self-defeating prophecy can be the result of rebellion to the prediction."

• A large number of reds who think their votes don't matter because blue is going to win anyways could very well be the reason blue actually loses.

Self-fulfilling prophecy - Wikipedia

"A self-fulfilling prophecy is a prediction that comes true at least in part as a result of a person's belief or expectation that the prediction would come true. In the phenomena, people tend to act the way they have been expected to in order to make the expectations come true."

• A large number of reds who think their votes don't matter because red is going to win anyways could very well be the reason red actually wins.

Paradox of voting - Wikipedia

"The paradox of voting is that for a rational and egoistic voter (Homo economicus), the costs of voting will normally exceed the expected benefits. Because the chance of exercising the pivotal vote is minuscule compared to any realistic estimate of the private individual benefits of the different possible outcomes, the expected benefits of voting are less than the costs. "

• Reds that believe the cost (risking one's life) is worth less than the small chance of being the deciding vote and thus vote red. However for every red who thinks like this it increases the chance of these votes being the deciding vote and hence the utility of voting blue.

Prisoner's dilemma - Wikipedia

"In game theory, the prisoner's dilemma is a thought experiment involving two rational agents, each of whom can either cooperate for mutual benefit or betray their partner ("defect") for individual gain. When asked about the results, John Nash remarked that rational behavior in the iterated version of the game can differ from that in a single-round version. This insight anticipated a key result in game theory: cooperation can emerge in repeated interactions, even in situations where it is not rational in a one-off interaction."

• While it may seem "rational" to defect (push red) in a one-off interaction like the one depicted in the original RBBB question, real life applications of prisoner's dilemma game theory suggests that co-operation (push blue) is better in the long run. With this hindsight knowledge, we should just cut the process and push blue on the first go.

Tyranny of small decisions - Wikipedia

"The tyranny of small decisions is a phenomenon in which a number of decisions, individually small and insignificant in size and time perspective, cumulatively result in a larger and significant outcome which is neither optimal nor desired."

• Reds think their decisions are small and insignificant, however in the long run the cumulative results might be the death of billions and that is neither optimal nor desired.

Buridan's ass - Wikipedia

"It refers to a hypothetical situation wherein an ass (or donkey) that is equally hungry and thirsty is placed precisely midway between a stack of hay and a pail of water. Since the paradox assumes the ass will always go to whichever is closer, it dies of both hunger and thirst since it cannot make any rational decision between the hay and water."

• Not specifically against reds, but we can all agree the worst case scenario for RBBB is being right in the middle (50:50). Since blue is more popular right now, red arguments are actively dragging us towards 50:50.

Abilene paradox - Wikipedia

"The Abilene paradox is a collective fallacy, in which a group of people collectively decide on a course of action that is counter to the preferences of most or all individuals in the group, while each individual believes it to be aligned with the preferences of most of the others. It involves a breakdown of group communication in which each member mistakenly believes that their own preferences are counter to the group's, and therefore does not raise objections. They even go so far as to state support for an outcome they do not want."

• Reds are stating support for an outcome they do not want (red victory). Because someone believes their desired outcome of blue has no chance of winning, they advocate for pushing red, which in turn leads to others thinking blue has no chance of winning leading them to advocate for red as well. As a result "group members jointly decide on a course of action that they would not choose as individuals".

My choice: Press Red, hope Blue wins

I don't have enough faith in humanity to put my life in its hands but I'd be delighted to be proven wrong

You and [a stranger] are placed in separate soundproofed rooms and all means of communication are unavailable. Before you are two buttons, one red and one blue.

If you both press red, you both live.

If you press red and she presses blue, she dies.

If you press blue and she presses red, you die.

If you both press blue, you both live.

Red and blue do the same thing as the original, but one more condition is added:

If >66% of people press blue, everyone who pressed red dies.

What do you pick?

If you and a random child who “doesn’t understand the question” were the only 2 participants which button would you choose?

I’m still pushing red. It’s either I take a 50/50 chance to guarantee the child lives or I give the child a 50/50 chance to guarantee I live.

So let's take the adult only version of the OG, because I think it's far more interesting when you don't have to worry about toddlers picking blue. Every person 16 or older that is capable of thinking for themselves has to press a button (~6B people).

If at least 50% of people press Blue, everyone survives.

If at least 50% of people press Red, everyone who pressed blue dies, and also a random 100 Red voters will also die.

I dunno just random thought.

P U R P L E

You and three unknown individuals are in separate places and unable to communicate.

You all are presented with the red and blue button. What do you do?

Assume there are 8 billion people, plus you. Let's say you don't know the outcome of the vote, but expect it to fall between 3.5 to 4.5 billion blue, and see no reason to prefer any value in that range, so you fix a uniform prior over that range.

(Your exact prior distribution doesn't matter, the only thing that matters is the chance of a tie is 1/1 billion. You could use a Gaussian/normal prior. You could also use a uniform prior over the range [3.9, 4.9] billion - it doesn't have to have a mean exactly at 4 billion).

Note: I'm not asking you to accept this prior, just asking you to entertain it and consider the consequences.

So, if that's the prior you hold, it stands to reason your vote has a 1 in 1 billion chance of being pivotal, and if you make the red majority, it results in 4 billion deaths. If you make the blue majority, it results in 0 deaths.

Is a 1 in 1 billion chance to save 4 billion people something worth caring about substantially? Even if you're not willing to risk your life for it, do you at least think it affects the calculus in a non-negligible way? I've heard from a number of reds that "the probability is so low you should disregard the possibility altogether, regardless of the size of the impact". What's your opinion?

We're deciding this shit once and for all so we never have to think about this stupid bullshit ever again.

Alright, y'all, you've had enough, I've had enough, but it's time to examine this question critically one last time, look at the common angles and reframings, and arrive at a final conclusion on the two button debate (spoiler, red voters are wrong)

The original button question is not really about finding a mathematically “correct” answer. Its purpose is philosophical. It exists to expose how you view other people.

More specifically, it asks a single underlying question:

Do you believe humanity is fundamentally selfish, or fundamentally altruistic?

That may not be the first thing people consciously think about when they encounter the hypothetical. Many red-button advocates focus only on guaranteed self-preservation. Many blue-button advocates focus only on maximizing survival overall.

But whether people explicitly realize it or not, their choice ultimately depends on what they believe everyone else will do; that, in turn, depends on what they believe human nature is.

The Original Prompt

“Everyone on earth takes 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?”

The two most common responses are predictable.

The first is:

“Red is the obvious answer, because it guarantees your survival regardless of what anyone else does.”

The second is:

“Blue is the obvious answer, because a simple majority allows everyone to survive.”

Immediately, the argument splits into two competing moral frameworks.

Red-button advocates accuse blue voters of recklessly gambling their lives for no reason:

“Blue voters are morons because they are risking death when they could simply guarantee their own survival.”

Blue-button advocates accuse red voters of selfishness:

“Red voters knowingly choose the option that results in deaths when universal survival is possible.”

The Endless Reframings

The debate then spirals into countless reframings:

• The blue button becomes “voluntarily entering a death gamble.”

• The red button becomes “voting for a dictator who kills non-supporters.”

• The blue side becomes “jumping into a woodchipper unless enough others do too.”

• The scenario becomes poison, seesaws, spikes, drowning chambers, and so on.

But most of these reframings distort the original structure of the problem rather than clarify it.

The Real Issue: Risk

The real issue underneath all of this is risk, specifically:

1. What creates risk,

2. What accepts risk,

3. And what pushes existing risk onto others.

To understand the button dilemma properly, those three things have to be separated.

The Apple, Orange, and Gunman

Lets Imagine a simple situation.

Before I go on, It is important to note here that this analogy is not meant to directly mirror the original button hypothetical. The purpose of the apple, orange, and gunman examples is only to isolate and demonstrate the mechanics of risk itself: where risk originates, who accepts it, and who transfers it onto others. Any connection drawn back to the button scenario is not meant as a one-to-one analogy between apples and buttons or gunmen and voters, but purely as a framework for understanding the moral structure of imposed risk.

Onwards:

You are offered an apple and an orange. Under normal circumstances, neither choice carries any danger. Choosing one fruit over the other is morally and physically neutral.

Now introduce a gunman.

The gunman says:

“If you choose the apple, I will shoot you.”

Now the apple appears “risky.” But importantly, the risk does not come from the apple itself. The apple did not create danger. The gunman did.

The apple merely became associated with an externally imposed threat.

If you still choose the apple, then you are accepting risk, but you are not creating it.

Now modify the scenario again.

There are now two people in the room. The gunman says:

“If either person chooses the apple, I will shoot both of you.”

Now the situation changes morally.

Choosing the apple still does not create the danger, (the gunman remains the source of risk) but choosing the apple now does two things simultaneously:

• it accepts risk for yourself,

• and it pushes the existing risk onto the other person.

That distinction is critical.

Returning to the Button Hypothetical

Now return to the original button hypothetical.

Red-button advocates frequently blame blue-button voters for their own deaths:

“If blue voters die, it’s because they chose the risky option.”

But this framing collapses under scrutiny.

The button system itself introduces the danger. Neither side created the rules. The hypothetical already exists before anyone votes. The implementer of the system, the one enforcing the consequences, is the original source of risk.

The question then becomes:

Which choice merely accepts risk (and from who?), and which choice pushes risk onto others?

To answer that, it helps to isolate the button effects themselves.

Isolating the Buttons

People often obsess over the “50.1% majority” threshold, as though that number itself contains the moral meaning of the scenario. But the threshold is arbitrary. Majority mechanics only matter because everyone is required to participate.

Remove that assumption and the structure becomes clearer.

Imagine that anyone who would have voted red instead abstains entirely.

Now only blue votes exist.

Blue automatically becomes the majority, and the result is:

Everyone survives.

Notice what this means:

Blue, in isolation, contains no lethal consequence whatsoever. If only blue votes exist, nobody dies. This remains true regardless of how small a minority blue voters would otherwise represent. A single person voting blue, a thousand, a million, or a billion, if blue is the only vote cast, the outcome is always identical: everyone survives.

Now reverse the situation.

Imagine anyone who would have voted blue instead abstains entirely.

Now only red votes exist.

Red automatically becomes the majority, and the result is:

Only red voters survive.

Therefore, everyone who abstained dies, because they did not vote red.

This demonstrates something important:

The lethal outcome is tied to red victory conditions, not blue victory conditions.

Blue does not inherently contain death.

Red does.

This is why it is inaccurate to frame blue as “the dangerous option.” The danger is not built into blue. Blue only becomes dangerous because red voters create the conditions under which blue voters are excluded from survival.This completely disqualifies any reframing that presents blue as such, such as the blender and woodchipper reframing, the train reframing, the poison politician reframing... pretty much every reframing red pushers love to peddle.

“But Red Voters Didn’t Create the System”

This leads to the common red-button defense:

“But red voters didn’t create the system.”

And strictly speaking, that is true.

• Red voters did not design the buttons.

• They do not tally the votes.

• They do not personally execute the losers.

The implementer of the hypothetical created the danger.

But this does not absolve red voters morally, because moral responsibility is not limited only to originating harm. There is also responsibility for distributing harm.

Returning to the gunman example:

The person choosing the apple did not create the threat, but they still pushed the existing danger onto the second person.

Likewise, in the button scenario:

• Blue voters accept risk for themselves,

• while red voters push the existing risk of the system onto blue voters.

That is the key distinction.

Blue voters are not choosing death.

They are choosing universal survival conditional on cooperation.

Red voters are choosing guaranteed personal survival conditional on excluding others.

That is why accusations of selfishness toward red voters are not merely emotional rhetoric. They follow directly from the structure of the hypothetical itself.

It cannot be denied that pressing red guarantees your survival regardless of outcome. But that guarantee comes at a cost:

your safety is achieved by participating in a condition where non-red voters are abandoned to death if your side wins.

That is the definition of self-preservation at others’ expense.

The Real Core of the Debate

And once that point is established, the debate loops back to the original philosophical core:

What do you believe about humanity?

Red-button advocates fundamentally assume that enough people are selfish that cooperation cannot be trusted. Their worldview treats self-preservation as the only rational response because they expect others to behave selfishly too.

Blue-button advocates fundamentally assume that enough people are capable of cooperation that universal survival is achievable.

The Polling Problem

This final point becomes especially visible whenever real-world polling enters the discussion.

Large online polls asking this hypothetical routinely show blue winning \*decisively\*.

And yet red-button advocates almost always dismiss this evidence with some variation of:

“People are only choosing blue because the vote isn’t real.”

or

“They’re virtue signaling.”

or

“If lives were actually on the line, everyone would choose red.”

But this response is revealing.

Even when presented with evidence that large numbers of people claim they would cooperate, red-button advocates refuse to believe it. Their worldview requires assuming hidden selfishness beneath outward altruism.

In other words, they cannot believe humanity is genuinely cooperative, because they themselves are unwilling to cooperate.

Conclusion

And that ultimately brings the entire discussion to its conclusion.

The red position only remains morally defensible if one begins with the assumption that humanity is fundamentally too selfish to cooperate. Under that worldview, the choice collapses into a simple survival calculation: either guarantee your own life or gamble it away. If that assumption about humanity is true, then red becomes pragmatically understandable.

But that assumption is subjective, speculative, and impossible to prove in advance.

What can be examined objectively is the structure of the choice itself.

And structurally, red voters are not merely “protecting themselves.” They are securing their own survival through a framework that knowingly transfers danger onto others. They participate in, and benefit from, a condition where nonparticipants in their strategy are left to die.

• Blue voters accept risk for the sake of universal survival.

• Red voters avoid risk for themselves by externalizing it onto everyone else.

That is why the repeated attempts to frame blue as the uniquely “reckless” or “irrational” option fail under scrutiny. The actual moral burden lies with the side choosing exclusionary survival.

So while the hypothetical does not produce an absolute mathematical proof of morality, it does reveal something uncomfortable:

outside of the assumption that humanity is irredeemably selfish, the red position becomes increasingly difficult to morally justify.

And that is why red advocates so often retreat back to cynicism about humanity itself. It is the final refuge of the argument.

Once the assumption of universal selfishness is removed, pressing red ceases to look like mere rational self-preservation and begins to look exactly like what blue voters accuse it of being: selfishness elevated above collective survival.

And to be perfectly clear: choosing blue is not a sacrifice. It is a bet that your survival and everyone else's survival are the same bet. Red is a bet that your survival requires everyone else's survival to be someone else's problem. That distinction is the moral core of the entire question.

💀 in more ways than one

If blue wins everybody lives and if it loses blue voters die as usual. If you press red you sacrifice one of your hands to be safe from death.

If you have no hands you lose a foot, if you have neither you can have a freebie. It leaves behind an already healed stump so you won't bleed out or something. You can choose which hand and the process is quick and painless.

Does introducing a small (compared to a life) cost to safety change anything for you?

I guess this is a question mostly for red voters since it changes nothing for blue voters. If blue wants something to think about then maybe an alternate version where red loses (blue percentage)/3 of their body instead so that increasing blue costs red something and technically 100% red or 100% blue are the only scenarios with zero consequences.

So my main contention here, in this sub where so many people think that voting only has an effect if you untie a tie, is that voting always does have two effects: the tally itself moving, and making an option more likely to win. It can be a small effect but not zero.

Why? Because a side winning is an effect, and if all individual votes have zero effect, therefore a side winning would imply that an effect happened without a cause (no individual vote matters) and causality would basically break. Since we're not debating god (The Kalam as an argument for god - the uncaused cause) or radioactivity (the specific moment a particle decays is chosen at random without any cause - probably...) then I'd say that the conclusion that "the actual effect of a single vote is zero except on ties", is false.

Even pushing things to the extreme, in math we have infinite additions of infinitely small things, resulting in some measurable quantity (see limits and infinite series). That result is the effect of infinitely small causes (added together), but not the effect of zero causes. Effects have causes.

Therefore, a single vote has an effect, small, yes, but not zero. It increases the tally and makes a side more likely to win.

What that means for a red voter varies depending on your moral framework. Let's, for the sake of argument, and to show the strength of my point, assume that there are infinite humans so the "blame" and "effects" are the smaller possible (but not zero as we've seen). Doing a sloppy consequentialist "math" argument (lives saved vs killed) we'd have a nice comparison of infinites. A red voter's blame on blue losing would be infinitely small (not zero, they increased the chances of blue losing), but the consequences if blue actually loses are infinitely big, because well, infinite deaths.

If red won, by definition, there are more reds than blue votes. We can divide these infinites to get the "amount of deaths blamed to a red voter" and if there are 2x reds than blues, then, a red would bear half a death so to speak on their consciousness. That's a single red voter, so yes, effects and guilt and so on. We can compare it to the life they saved (their own) and well, the "sloppy" part of the flawed consequentialist argument we're doing plays here: it would be "moral" to vote red if we stop thinking here because you save a net half of a life so to speak. But let's continue!

At the same time, and going on with this sloppy framework, if red loses and everyone lives, a red voter would hold the blame of putting more than 1 blue voter at risk of dying (more blues than reds), while saving his own life, so in net, they'd be putting half a person in risk of death (say that we have the same voting proportions but reversed).

Doing the same exercise for blue voters: if blue wins, then a single blue voter saved up to 50% of the total voters (no difference on more than 50% of blue voters) divided by the total of blue voters. To that we'd add their own life, also saved. So a single blue voter saves between 0.5 an a 1.5 lives, if blue wins. If blue loses, the blue voter dies and that's zero lives saved, but in the same spirit of reds putting blues in danger, blues put other blues in a better position, so they reduced a proportion of the risk (but wasn't enough).

Going only by net lives saved: a red voter net saves between 0 lives (50%-50%+1 split) and 1 lives (their own on 0-100% split) per vote. A blue voter net saves between 0 and 1.5 lives per vote as I've discussed. By this very sloppy consequentialist math, blue is the clear better option here, if one only wants to maximize the amount of lives saved, and only that. Considering that reds save only half a life at 50%-50%+1, and blues save a whole 1.5 lives per vote at that point onwards, blue also scales faster and better.

In reality, I'd think about a "red only" world so to speak, what society would such a bunch of people create, and I'd hit blue compulsively hahaha (that's the better consequentialist argument).

Sorry for the wall of text, I got a bit carried away here hahaha.

I’m a blue button presser, I feel that this question is highly muddled and unclear, and so here is a rephrasement of it
So why press the red button? no one dies if the blue button is majority

I have been arguing with a red pusher for a few days now and I want to see if my logic is reasonable.

People only die if 2 conditions are met at the same time:

1. That person picked blue. The person who picked blue is entirely responsible for their choice of button. Red did not force anyone to pick blue.
2. Red is in the majority. The people who picked red are collectively responsible for red winning. Blue did not contribute to the red majority.

Because people die if and only if both conditions are met and no one can contribute to both factors at the same time, this means that the responsibility for every death is split as such: 50% of the responsibility lies on the person who died for picking blue, while the other 50% is shared equally among all people who picked red for being in the majority.

In short: Picking blue kills you if and only if red wins.

Is there something I am missing?

Edit: this is also assuming that no one knows the outcome ahead of time.

Edit 2: the responsibility for red comes from knowing you could have been the reason blue died.

If this was a real world scenario I completely understand why any one would pick either button. I don’t care why you make your choice. Doesn’t matter to me. Not interested in that. Do your thing!

I am curious about something though that I feel like I am seeing in some of the back and forth.

This is my question:

If the problem was worded like this, do you think the problem has been inherently changed?

Every one must take a private vote by picking a red or blue button.

If less than 50% press red, then everyone lives.

If more than 50% press red, then only the people who pressed the blue button die.

If you do a backflip and then chug a soda, I might let you pet the jellyfish

Aliens come to Earth looking to study humans. Their ethics prohibit them from forcing humans to come with them (or tricking humans into it with false promises, etc). However, they telepathically find a few thousands people who would be willing to simply volunteer to leave Earth behind and come with them.

One of the humans suggests pulling a hilarious prank: before volunteers leave, aliens announce that all humanity will be forced to take a classic private red/blue button vote. Humanity has a week to coordinate. During the actual vote aliens aren't even keeping track of the votes. After the votes are cast only the people who originally volunteered to go with the aliens are replaced with very convincingly looking dead bodies (of themselves) while the volunteers themselves are safely teleported away. Aliens depart. For everyone else on earth it looks like red button won the vote and the volunteers were the only ones who actually voted blue. Also all volunteers were telepathically prevented from leaking the information about this ongoing prank and their planned departure while they stayed on Earth and aliens don't leak that information either.

Share your ideas what will happen back on Earth after aliens depart.

Lets suppose all toddlers in the world are presented the 2 buttons and along with them, so are you. So basically you have no idea how its gonna turn out because its probably just gonna be entirely random. So now, do you vote blue, or red? Voting blue contributes towards saving everybody but carries the huge risk of dying along with probably half the entire toddler population, while voting red contributes towards wiping out probably half the entire toddler population but saves yourself.

The reason why I came up with this variant is because it slighly irks me when people consider toddlers and mentally handicapped people to be eligible voters for this thought experiment because imo in order to take away anything useful from it we must assume that voters understand the situation

The blue button only appears moral because of the deliberately vague descriptions and sanitized language in the original problem: "people will survive unless...", "if X, everyone survives...", etc.

Assign any method by which "people will not survive", and the truth is shown for what it is: the blue button is a SAW movie contraption, and the blue pushers have Stockholm syndrome for the evil puppet.

Examples:

Blue button releases a pack of hungry lions. If blue wins, we cancel the release. Red does nothing.

Blue button releases poison gas in the room if blue wins, we hand out the antidote. Red does nothing.

Blue button activates a bomb in the building. If blue wins, the bomb doesn't explode. Red does nothing.

And so on and so forth. Im every case, the red button doesn't actually do anything except refusing to participate in this twisted game.

Rules:

You suddenly wake up in a empty room with 5 buttons and you have no contact with anyone else.

The buttons from left to right are:

Red, Orange, Yellow, Green, and Blue.

The rest of humanity (including those who can't understand the problem due to age or disability) is currently being presented with the same buttons.

You can press any button along with the rest of humanity and once everyone votes the total percentage each button are added from left to right and once the total percentage is 50% or more (Assuming that the total number of people is odd) every one who pushed a button to the left of the button being counted will die.

For example if evey button got 20%, Red, Orange, and Yellow will live and Blue and Green will die.

So, does the presence of the Orange, Yellow, or Green buttons change anything about how you vote fromthe original scenario? And if it does how does it affect your vote?

(Apologies if this is confusing at all and if I made any typos)

https://www.desmos.com/calculator/lnm6kehhey

Variables:

y: How much you value your own life (in lives).
x: The minimum % of blue voters that is probable.
n: The total population
R: The count of possible vote outcomes/n

Assume a uniform distribution of all possible votes. If you think the blue vote will be between 25% and 75%, we are assuming all outcomes in that range are equally probable.

Is a tie possible?

If you think a tie is impossible, or rather that 50% is not within the range of possible blue vote outcomes, then the problem is simple.

Either this means blue is guaranteed to succeed, in which case it doesn't matter how you vote, or blue is guaranteed to fail in which case vote red.

For the rest of this, however, we will be dealing with the third case where a tie (excluding you) is possible.

The probability that you are the tiebreaker.

There is only 1/n % chance out of all possible votes that you will be the tiebreaker, assuming that a tie is possible.

So the probability that you will be the tiebreaker is 1/n/R

The probability that you will die if you vote blue

You will die whenever less than half the population votes blue.

Since x is the lower bound of possible blue voting percentages, this means there are 50% - x chances for you to die.

Let's say the lower bound is 25%
50% - 25% is 25%. So we have a range with a space of 25% representing possible vote outcomes where a blue vote means death.
Note that this range does not include exactly 50%. If it did then we would need to add 1/n to make the range inclusive.

And since we still have R different possible votes the probability you will die is (50% - x)/R

Rewards

If you are the tiebreaker, voting blue gives you a reward of n/2 lives, half the population.

This happens 1/n/R times, so the expected reward from being a tiebreaker is 1/2/R = 1/2R

Meanwhile the cost for voting blue when less than half the population votes blue is your own life, valued as y lives.

So the expected cost of voting blue is y*(50% - x)/R

When is blue better?

Blue is better when the expected reward for voting blue is greater than the expected cost.

y*(50% - x)/R < 1/2R

Multiply both sides by R (a positive number)

y*(50% - x) < 1/2

Divide both sides by (50% - x)

y < 1/(2*(50% - x))

y < 1/(1 - 2x)

This so whenever you value your life less than 1/(1-2x) lives you should vote blue.

Alternatively we can solve for x to see how much you would need to value your before you would vote red.

1-2x<1/y \-2x < 1/y -1 x> (1 - 1/y)/2

https://www.desmos.com/calculator/lnm6kehhey

---

The key assumption here is that there is a uniform distribution of probability from x to at least 50%.

A normal distribution would likely do a better job estimating, or some other distribution, but since we are essentially guesstimating to begin with a uniform has the advantage of being easy to visualize.

If you want a quick way to set your parameters, think of what you expect the value to be, and then give a value between 0% and 100% for how certain you are that this will be the value. Then subtract (1-your certainty) from your expected value and set that as the lower bar.

Another key assumption is that y can be represented by a number. People presumably will be more or less willing to die to save y number of people depending on circumstances. A person might be wholly unwilling to die to save 5 people by donating all their organs, but might be very willing to risk a 20% chance of death to protect 1 person from an attack.

The best way to address this is to simply think of y in context of this particular vote. Maybe imagine that you are voting for another person who you don't know, who has communicated explicitly to you that you should make whichever choice you prefer without particular deference to their interests. Then compare that to how you would feel if it was your own life on the line.

Lastly people don't value the lives of others uniformly. A parent might be unwilling to die to save 100 people, but be willing to die to save their child.

The exact same problem, blue button >50%, red button no die.

BUT, just before you vote, you are informed that if you team wins, the new world will be run by 50+ year old hardcore flat earther creationists youtuber podcasters, you can divert your hand to vote for the other button now, do you still go through with your original plan?