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%

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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.

I put together a graph that shows the expected reward (in lives) of pressing blue.

https://www.desmos.com/calculator/teogjmy8yo

• N is the number of people in the experiment. I set it to 8 billion for approximately the world population.
• x is the probability of a person pressing blue.
• v is how much you value your own life compared to the lives of other people. Essentially how many people would be enough for you to be willing to die to save them.

Note that the graph is pretty zoomed in on the X axis, and pretty zoomed out on the y axis.

What this graph gives you is a threshold for when you should press blue based on how likely you think the average other person in the experiment is to press blue.

My take on this is that essentially whether or not you should press red or blue has very little to do with "selfishness". If v = 1, you value your own life exactly equal to anyone else's life, then you should press blue if you think the probability of people pressing blue is 49.9974%. If v = 1 million then the cutoff is 50.0062%.

When v ≈ 70,000 the cutoff is right around 50%

But we are talking about such a small difference that, honestly, I don't think people are going to be able to estimate x with that level of precision.

What matters is your estimation x. If you estimate x to be anything less than 50%, you basically shouldn't press the blue. If you estimate x to be 50% or higher you should press blue.

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If you would press blue and you think essentially x<50% then you probably shouldn't press blue. You are more likely to kill someone than save someone. Even if we account for fallout after the vote killing more people you are still more likely to kill someone (yourself) than save anyone.

If you would press red and you think x=50% or higher, then if you would be willing to personally die to save millions of people, you should press blue. It will be more in line with your actual values.