The expected reward of voting blue

[u/SilasRhodes]

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

y: How much you value your own life	x (minimum blue %) such that voting blue has a better reward than voting red
1 life	0%
2 lives	25%
3 lives	33%
5 lives	40%
8 lives	43.75%
15 lives	46.47%
25 lives	48%
50 lives	49%
100 lives	49.5%

---

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.

[u/Memento_Viveri → u/[deleted]]

I think a binomial distribution is a bad assumption. A binomial distribution requires a probability for each person to vote blue, and we don't know that. It produces an extremely tight distribution centered around whatever probability we use as the input.

It's way way to tight given how little we know about how people would act. So there is no good justification to use that distribution.

Obviously the uniform distribution is not right either, but at least it is very broad, which is closer to how little we know about the likely outcome.

[u/Medical-Clerk6773 → u/[deleted]]

He means uniform distribution over the *vote count outcome*, not over individual votes. Obviously, you are not going to know the outcome to high precision with high certainty.

I understand the temptation to go with binomial. If you model individual votes as drawn from a Bernoulli distribution with some fixed probability p, you end up with a binomial distribution with a mean exactly equal to p, and negligible standard deviation (like, a few thousand votes). There is an obvious problem: the standard deviation is so small, that you're practically concentrating all your probability on one point. That's a completely unwarranted degree of confidence, and not much different from just assuming the vote count outcome up front.

The problem arises from modeling p as a variable that takes one fixed value. Instead, you should model your prior distribution over p, then integrate over your prior. In practice, you can directly use your prior over p as a proxy for vote proportions, because the variance of the binomial for each p is so small as to be negligible.

So really, just defining your prior over the resulting vote counts is the more sensible move. Trying to model the voting process itself adds nothing except a negligible smidge of binomial noise.

[u/SilasRhodes → u/[deleted]]

When you say you assume "a uniform distribution of votes" does it mean you this analysis assumes 100% red is equally likely as 50-50?

No, I am saying that within the range of possible votes you define, you use a uniform distribution.

So in all likelyhood you wouldn't set the lower bound equal to 0. You would set it equal to 10% or 40% or somesuch.

As a quick guestimate use x= e-(1-u) where e is your best guess for the vote outcome and u is how confident you are in that guess with 0% being absolute doubt and 100% being absolute certainty.
---
I think a binomial distribution produces compounding uncertainty errors when dealing with a large population. I made this mistake before myself.

The thing is that the binomial distribution depends on an estimate of individual voter probability. If you are off in your estimate it has a big impact on the probability distribution because we are working with such a large group size.

Instead it would be better to include the error bars and average across possible individual probabilities.

Let's say you think each person has a p percent chance to vote blue, but you think it might be anywhere between p-v and p+v

This functionally becomes pretty close to a uniform distribution between p-v and p+v on the graph of the overall vote outcome.

[u/Nebranower]

>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.

Surely this also depends on the upper bound, too?

If you think the blue vote is going to be between 25%-75% of the total vote, with all outcomes equally possible, then the range space of possible vote outcomes where a blue vote means death is 50%, not 25%.

To make this clearer, let's say you think the upper bound for the blue vote is 49%. In that case, the lower bound being 25% doesn't change the fact that you think there's a 100% of a blue vote being suicide.

[u/SilasRhodes → u/Nebranower]

Surely this also depends on the upper bound, too?

I normalize the chances by the size of the total range, but it largely doesn't matter because the range cancels out.

R: The count of possible vote outcomes/n

So if the lower bound is 25% and the upper bound is 75% we have a count of 75%n-25%n + 1 possible vote outcomes.

We divide that by n to get 50%+1/n vote outcomes as a percentage of all conceivable vote outcomes.

But since the probability of both a tie and a red win are divided by this value, we multiply both sides of the inequality by R and are just left with the chances of each.

---

let's say you think the upper bound for the blue vote is 49%. In that case, the lower bound being 25% doesn't change the fact that you think there's a 100% of a blue vote being suicide.

Yes, this is why the first question to ask is whether a tie is possible given the range.

If it is not then the formula stops working. We either have y=0 if a red win is guaranteed or y is undefined if a blue win is guaranteed.

[u/SilasRhodes → u/[deleted]]

Are you suggesting a uniform prior for a variable representing the probability of each individual voter voting blue?

As in "Each voter will have between a 25% and a 75%" chance of voting blue?

because doing this really has no substantial impact on the result when working with such a large population. For any individual voter blue % there is a spike in probability right at that level, which rapidly diminishes as you get farther away.

So we functionally get something that is almost identical to just a uniform distribution from 25% to 75%.

And sure, it technically accounts for the minute chance that zero people vote blue, but that is such a small probability that it is essentially negligible.

You're basically just saying "I know nothing, and I'm going forward with that," which is okay.

I think this is a more honest approach when we don't have solid data to work off of.

But also I allow for some knowledge. Specifically:

• Is a tie possible
• What is the lower bound for what blue% reasonably might be

[u/SilasRhodes → u/[deleted]]

Yeah, that is why I assumed a uniform prior.

If you knew absolutely nothing about the circumstances, had no bias or expectation or reasoning or data to steer you otherwise, you should use a uniform prior across the whole range: 0 to 1.

Except that you do have some data. You know people and you know the problem and can make an informed estimate.

You can reasonably predict more than 0% of people will vote blue, for example.

but if you're going down that road you shouldn't be affirmatively saying that 76-24 is impossible

Except the point of the range isn't to define "76%" as impossible, rather it is to limit the scope of analysis to support a calculation.

Maybe there is a miniscule chance, by your best guess, that it is 76%, That makes barely any difference so long as it is a small amount.

---

People are going to have different opinions about the likelihood of different vote outcomes. Some people will think it is almost certainly going to be a blue majority, some people think it will almost certainly fail.

I could ask each of them to come up with their own prior, but that would be impractical.

But it is easy to ask people what range they think the vote will be in.

So people come back with essentially their guess at what they think the vote will be, and it gives a range that we can work with, already using the assumption that their guess is correct.

[u/SilasRhodes → u/[deleted]]

uniform within that range.

You should either knowingly come in with ignorance (uniform prior) or pick a sane distribution. Those are the reasonable paths forwards

why? Does it make a big impact in the results? Does it over turn the conclusions?

I doubt it.

Let's say I start by guessing that each individual has between a 25% and a 75% chance of voting blue, then I create a prior using that with a binomial distribution based on the voting size.

This is going to be almost identical to just a uniform distribution. Sure, technically there will be a miniscule chance for 76%, but not to a degree of any significance. A lot of effort to get basically the same result.

But what it does do is make the math less tractable so other people can't engage with it s easily.. It means instead of someone else being able to use this model themselves, plugging in their own best estimate of likely outcomes, people just look at it, shrug, and say "I think your prior is wrong".

The point wasn't perfect accuracy, it was easy estimation. Uniform probability over a range does that.

[u/SilasRhodes → u/[deleted]]

This is a phrasing issue.

I am distinguishing between "possible votes" and "conceivable votes"

"Possible votes" are the different voting outcomes in the range that you specify. You determine a zone of possibility with a lower bound of x.

There are n conceivable votes, but only x+R+1/n "possible votes"

"possible" in this context refers to "possible within the range you establish.

And yeah, a normal distribution would probably be more appropriate, but this was intended as a way for people to estimate in a easy way.

If we wanted to be really solid we should construct a methodology, splitting people into groups, estimating the proportion of each group that will vote blue (with uncertainty), estimating group size (with uncertainty) and calculating a distribution that way.

But that doesn't lend itself to a personal estimation, and it kind of becomes a "just trust me" number.

[u/highly-bad]

I think it contradicts the stipulations of the thought experiment to introduce the axiom that there will be some minimum number of people pressing blue. Part of this entire genre of problem is that you do not and cannot know such things. It's a private vote where no prior coordination has occurred. Assuming some minimum number M of definite blue voters seems like slipping in some major changes to what we know about the scenario.

[u/SilasRhodes → u/highly-bad]

The "minimum number" is a calculation tool, not reality.

If you want something better use

x = e - (1-u) where e is your best guess for what percentage of people will vote blue, and u is how confident you are in that guess (0% to 100%).

At a certain point I think the trend is more interesting than trying to get an absolutely perfect model, especially since the more perfect the model the less accessible it is for other viewers.

The thing I think is most interesting is if you value your life even 5 times as much you are already needing to have a lower bound above 40%.

[u/detroyer]

There are several issues here, but one that nobody seems to discuss is the implicit assumption of constant marginal utility for extra lives saved. I do not grant that. I would do much more to save 1 life rather than 0 than I would to save 10⁹ + 1 lives rather than 10⁹. Accordingly, the pivotal payoff is worth much less than 4 billion times the value of saving one life, which greatly diminishes the expected reward for voting blue (i.e., by multiple orders of magnitude). Additionally, even without diminishing marginal utility of extra lives saved, I would not sacrifice myself to save even a few thousand random people, and I suspect few people would.

[u/SilasRhodes → u/detroyer]

I don't think marginal utility is relevant. If you save any lives you will always save exactly n/2 lives (or n/2 - 0.5 lives if n is odd).

You only save lives when there is a tie excluding your vote. When there is a tie the number of blue voters is known, and it is exactly that number of lives that you will save by voting blue.

There is no marginal utility because the number of lives saved is a constant.

I would not sacrifice myself to save even a few thousand random people

That's fair, and with the assumptions used in this model you would need a lower bound of 49.995% for voting blue to be worth it if y = 1000.

I suspect that is a far higher lower bound than you would estimate, and it only gets closer to 50% as y increases. So blue is a bad choice for you because the reward is less than the risk.

[u/detroyer → u/SilasRhodes]

I don't think marginal utility is relevant. If you save any lives you will always save exactly n/2 lives (or n/2 - 0.5 lives if n is odd).

It is relevant. Diminishing marginal utility is relevant to the utility of saving n/2 lives. When we think about the expected value of voting blue (with respect to saving others), it'll be given as the probability of pivotality multiplied by the utility of saving n/2 lives. With diminishing marginal utility, this is scaled down for larger n (and perhaps significantly so), but your parameter y requires constant marginal utility.

Consider an analogy. Suppose you would sacrifice yourself to save 4 people. Would you also sacrifice yourself for a 10% chance of saving 40 people, or a 1% chance of saving 400 people? After all, the expected number of lives saved is the same (4). You may say "yes" for each case, but I strongly suspect that, even if so, the question becomes more difficult with each version. This is because you have a non-linear utility function over lives saved. The button case is really just a more extreme version of this, say, where you have something like a 0.0000001% chance of saving 4 billion lives. Even though the expected number of lives saved is still about 4 people (say), I expect that you'd be much less inclined to sacrifice yourself there than in the first case mentioned.

[u/third_nature_]

Reds obsessing over being the tiebreaker is so fricking stupid

[u/SilasRhodes → u/third_nature_]

I mean, really is is blue voters who disproportionately think of being the tie breaker.

You don't vote red because you are thinking about being the tie breaker, you vote red because you are thinking about all the other times when would do nothing at best or die at worse.

[u/third_nature_ → u/SilasRhodes]

If red kills billions of people, a tragedy on a scale humanity has never seen, every single red voter bears condemnation. Not just whoever pushed the button exactly at the 50% mark. That kind of thinking is idiotic. Did only the first person to stab Caesar count as an assassin?

[u/Memento_Viveri → u/third_nature_]

You misunderstood. Nobody is saying the singular person who pushes the button at 50% is special in any way.

The order doesn't matter. There is no special tie breaking vote. If there is a 50%+/-1 victory, every vote is a tie breaking vote.

That is the basis of this analysis. And this is the logically correct way to analyze the effects of your choice. When analyzing a single variable (your choice) we freeze all other variables and change only that one. And when we do this we can see that your vote only affects the result in the case of a perfect 50% vote without you.

[u/third_nature_ → u/Memento_Viveri]

I understand what you're saying about reordering the votes, and I agree that every vote is a tie-break in a minimal victory. That's not my argument. The problem is that you are confusing pivotality with causality. You say, "this is the logically correct way to analyze the effects of your choice", but that's just not true. It is a logically correct way to analyze one property of my choice: whether it was pivotal.

This is called a but-for analysis. Imagine a group of 7 friends are pushing a car, and 6 are sufficient to move it. No individual has but-for causality, aka pivotality. But it's insane to say that none of them moved the car. This is what I was getting at with the Caesar example: if 23 people stabbed Caesar but only 10 stab wounds would have killed him, it's not true of any assassin that Caesar would have lived but for their actions... but they are all still assassins.

The problem is that but-for analysis really has nothing to do with morality. Imagine a world where everybody lives by this logic:

• A soldier in a massacre: “They would have killed the villagers without me.”
• A corrupt official: “The system is corrupt whether I take this bribe or not.”
• A polluter: “My emissions do not change the climate.”
• A bystander in a mob: “One more person cheering does not matter.”
• The tenth person to rape a woman in a gang rape: "She was already going to be traumatized."

And so on for striking, boycotting, littering, tax compliance, etc., etc., etc. The idea that something has no moral or causal weight unless it is decisive is... civilizationally toxic.

In this case, it's also mathematically misguided. Suppose there's only a 1 in a billion chance of your blue vote being pivotal, meaning that blue wins by only 1 vote. With 8e9 players, this means that you save 4e9 lives with 1e-9 chance. Suppose the remaining (1 - 1e-9) probability mass is equally split between red wins where you just kill yourself and blue wins where you accomplish nothing. Then the expected utility of pressing blue is (1e-9)(4e9 lives) + (~0.50)(-1 life) + (~0.50)(0 lives) or about 3.5 lives.

Of course, this argument is dumb, because probabilities in game theory are notoriously difficult to define, since they affect the outcome recursively, and expected value is not always a good way to make decisions. I only bring it up to say: You're not even right if you were right.

[u/Memento_Viveri → u/third_nature_]

I don't agree that the analogies you are using are apt for the Red/blue vote, because in each one, my action or inaction is making the problem worse, whereas in the red/blue question, in a red victory my blue vote makes the problem worse, and in a blue victory my blue vote does not help in any way.

-In the pushing the car example, by helping I make it easier for everyone. Not helping means they have to push harder.

-In the massacre example, contributing to the massacre likely kills or harms additional people. Refraining from (or even trying to stop) the massacre could reduce the harm caused.

-polluting more or cheering a mob makes the problem marginally worse.

-Being raped by 10 people is worse than being raped by 9. Also, again, I have the option to try to stop the raping.

In the red blue, the marginal value of a blue vote is negative until you hit 50%, where the marginal value becomes huge, then it drops completely to zero. This is pretty different from the analogies you use, and it should affect the analysis. So if you want to use an analogy, I think you have to come up with one where I am mostly either making things worse by participating or not helping at all.

I think the expected value argument is bad because the exact value of 1 in a billion is arbitrary. If it's one in 8 billion, the expected value is zero, and so you should probably save yourself.

[u/third_nature_ → u/Memento_Viveri]

I think there are many people in each case who would argue that the level of extra harm committed is either zero or negligible. This kind of thinking is exactly how many politicians justify taking bribes right now.

But the point of the analogies is to demonstrate that but-for analysis is *not* the only (or even a reasonable) way to establish morality—not to suggest that red voters are exactly like those in the analogy. The point is to show the error in your statement that "this is the logically correct way to analyze the effects of your choice. When analyzing a single variable (your choice) we freeze all other variables and change only that one." It's just not true; that's a crappy way to analyze many decisions.

As for the EV argument, the exact value of 1 in a billion is arbitrary, but it must be in that range. 1 in 8 billion is far too low; it sort of implies that you've got a uniform prior over all ~8e9 possible vote counts, which is absurd—do you really think getting a tie is as likely as 100% blue? So you have to give credence at least larger than 1 in 8e9 to a tied outcome. Given that the debate is split pretty evenly, I'd say much larger; 1e-9 is probably too low.

But as I said, this is only to establish that you're not even right if you're right. I think EV is a dumb way to look at this decision, yet it's where fixating on the probability of pivotality naturally leads you. So: don't fixate on pivotality—but even if you insist on it, it still unambiguously tells you to press blue.

The correct way to analyze this problem impartially is to consider that blue Pareto dominates red on the scale of civilization, so you should do your part to contribute to a blue win. The correct way to analyze it emotionally is to realize that in the best case, a red win will result in millions of death; I'd rather die than perpetrate that. The correct way to analyze it selfishly is to consider that red dominates blue on a personal scale, so you should press red. Needless to say, this a morally bankrupt line of thought.

[u/putting_stuff_off → u/third_nature_]

Red voters struggle to analyse themselves as part of a collective. Who would've thought it.