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tldr: Always flip the switch
- Let N be the size of the population that the villain abducts from
- Let X be the event that you are abducted
- Let R be the outcome of the villain’s roll
- If 1-5 is rolled, then the probability that you are abducted is P(X|R\in{1,2,3,4,5}) = 1/N
- If 6 is rolled, then P(X|R=6) = (N-1 choose 9)/(N choose 10)
- The probability of getting abducted at all is P(X) = P(X|R\in{1,2,3,4,5})P(R\in{1,2,3,4,5}) + P(X|R=6)P(R=6) = (1/N)*(5/6) + (N-1 choose 9)/(N choose 10)*(1/6)
- The probability that a six was rolled given that you were abducted: P(R=6|X) = P(X|R=6)P(R=6)/P(X) = ((N-1) choose 9)/(N choose 10)*(1/6)/((1/N)*(5/6) + ((N-1) choose 9)/(N choose 10)*(1/6)) = 2/3
So as it turns out, the total population is irrelevant. If you get abducted, the probability that the villain rolled a 6 is 2/3, and the probability of rolling anything else is its complement, so 1/3.
- If you flip the switch, you have a 1/3 chance of dying.
- If you don’t flip it, you have a 2/3 chance of dying.
If you want to maximize your own probability of survival, you flip the switch.
As for expected number of deaths:
- If you flip the switch, the expected number of deaths is (1/3)*1+(2/3)*0 = 1/3.
- If you don’t flip it, the expected number of deaths is (1/3)*0+(2/3)*10=6.67.
So to minimize the expected number of casualties, you still want to flip the switch.
No matter what your goal is, given the information you have, flipping the switch is always the better choice.
Excellent excellent!
If 6 is rolled, then P(X|R=6) = (N-1 choose 9)/(N choose 10)
Might as well reduce that to 10/N to make the rest of the lines easier to read.
If you don’t flip it, you have a 2/3 chance of dying.
There is also a chance that your switch is not connected and someone else has control of the real one. So there is an implicit assumption that everyone else is equally logical as you and equally selfish/altruistic as you, such that whatever logic you use to arrive at a decision, they must have arrived at the same decision.
No matter what your goal is, given the information you have, flipping the switch is always the better choice.
That is my conclusion too! I was surprised to learn though in the comment thread with @pancake that the decision may be different depending on the percentage of altruism in the population. E.g. if you are the only selfish one in an altruistic society, you’d benefit from deliberately not flipping the switch. Being a selfish one in a selfish society reduces to the prisoner’s dilemma.
There is also a chance that your switch is not connected and someone else has control of the real one. So there is an implicit assumption that everyone else is equally logical as you and equally selfish/altruistic as you, such that whatever logic you use to arrive at a decision, they must have arrived at the same decision.
Ah, yes. I forgot to account for that in my calculations. I’ll maybe rework it when I find time tomorrow.
Noise cancelling earphones sucks at blocking voices. Just yell and ask if there are others.
Or spit, or blow air at your potential neighbor, or fart in their general direction!
If I’m tied to a train track any potential fart risks coming with a little extra mustard on it.
How are they going to explain it to me when I am blindfolded with noise canceling headphones?
And dumb.
The time I understand the rules, the trolley is not there anymore.
The trolley is like really slow.
Through the headphones :P
Thank you! <3 My guess is that no to the first, since I have a 1/3 chance of being in the forked path, vs 1/15 of being in the straight path and my lever being connected. However, in the second situation I would flip it, since I’d only kill 1/3 of all people (myself every time), versus 2/3 (myself included) if I don’t flip it.
My guess is that no to the first, since I have a 1/3 chance of being in the forked path, vs 1/15 of being in the straight path and my lever being connected.
Suppose you live in a kingdom where everyone is as selfish as you, and you’ve seen on TV many situations exactly like this one where people were tied to the tracks - usually one at a time and occasionally 10 at a time. (The villain has been prolific.) You’ve seen them all follow this logic and choose not to flip their switch, yet out of ~1500 people you have seen in peril this way, ~1000 of them have died. If only their logic had convinced them (and you) otherwise, 1000 of them could have selfishly survived! Doesn’t seem very logical to follow a course of action that kills you more often than its opposite.
(If you don’t want to imagine a kingdom where everyone is selfish, you can imagine one where x% are selfish and (100-x)% are altruistic, or some other mixture maybe with y% of people who flip the lever randomly back and forth and z% who cannot even understand the question. The point is that the paradox still exists.)
Edit: I can see now how in a 100% altruistic kingdom, where you are the only selfish one and you know for sure that everyone else will logically altruistically pull the lever, it makes sense for you to not pull the lever. Presumably there is some population x% split (44% selfish/56% altruistic?) where your selfish decision will have to reverse. Weird to think that your estimate of the selfishness of the rest of the population has a relevance on your decision!
It’s a really cool puzzle, nice job! The solution being a huge prisoner’s dilemma makes it all the more interesting and deep. I guess an iterated version resulting in collaboration would be difficult in this particular case, though ;)
From a purely utilitarianism perspective, assuming all utility is linear and unscaled:
5/6 chance I’m on the side track * 1 person saved = 5/6
1/6 chance on the main track * 1/10 chance my switch is real * 10 people saved = 1/6
Seems pretty clear that you should not flip the switch. However, if I am on the main track, this thinking will lead to no-one flipping the switch and no lives saved whereas everyone thinking it will lead to a guaranteed save -> utility of 10/6.
If I can assume more than half the people can be rational and will think like me then I should flip the switch.
If I cannot, I should not flip the switch.
Except that if people are chosen randomly there is 2/3 chance that you are on the main track according to Bayes. Let’s assume there are 10 people.
The probability to be chosen is 1/6 (all are chosen if 6 is rolled) + (5/6) × (1/10) (only one is chosen to go to the side track if 1-5 is rolled) = 15/60 = 1/4.
The probability that you are on the side track knowing that you have been chosen is the probability that you have been chosen knowing that the side track is selected (1/10) × the probability that the side track is selected (5/6) divided by the probability for you to be selected at all (1/4), so (1/10)×(5/6)/(1/4) = 20/60 = 1/3. So there is a 2/3 chance that you are on the main track.
If you do not flip the switch, (2/3)×10 = 20/3 people die.
If you flip the switch, 1/3 (you if on side track) + 10 × 2/3 × 9 / 10 (switch misfires 9 out of 10 times if on the main track) = 190/30 die. This is slightly better than not flipping the switch.
I wouldn’t assume anything about the number of people and how anyone was chosen. All I know is I’m on the track.
But if you say you are on each track with p=1/2 then you also assume (different) details about how you were chosen
The task is unclear here, you have to make an assumption. I don’t know which is intended.
Most people would fail to understand the question. So many will flip the switch a bunch of times randomly. In other words: This would be super frustrating for the villain.
I love the idea of the villain explaining the whole thing to the captive and at the end being like, “okay, I’m about to put the blindfold and noise canceling headphones on, so this is the last chance for any other questions about how this works”
There surely must’ve been a more comprehending way to phrase the dilemma.
Yeah, if I woke up tied to train tracks and had someone explain that to me, I’d zone out and then panic because I had no idea what the fuck was going on
Half the fun of trolley problems is adapting them to puzzles for which they are utterly unsuitable:
I know it’s not the point of the question, but remember whatever happens is 100% the villain’s fault, not yours.
Spit to the left and right. If it spits back then there is someone else and I’m on the main track.
I am confused as to why anyone would not flip the switch? Flipping the switch seems to have somewhere between a 10% and 100% chance of saving your life, and not flipping the switch seems to guarantee death?
Is there some kind of penalty to flipping the switch that I am missing?
Or is the drawing misleading, and in Scenario B, there is also supposed to be a person drawn on the other track?
I think in Scenario A you die if you flip the switch.
Info: How many people live in the kingdom?
There’s a five in six chance you are picked 1 out of X, and a one in six chance you are 10 out of X.
If you’ve been picked, there are three possible outcomes.
Flipping the lever kills you. 5/6 x 1/X
Flilling the lever saves you and 9 other people. 1/6 x 1/X
Flipping the lever does nothing at all. 1/6 x 9/X
From a purely statistical standpoint, you’re five times more likely to die flipping the lever, but the expected value, measured in lives saved, for flipping the lever is twice as high as not.
From a purely altruistic measure, you should always flip the lever, because at worst you kill yourself, at best you save 10 people, and you can do it with significant confidence that it doesn’t actually matter.
But back to my original question, 5/6X vs 1/6X vs 9/6X where as X approaches infinity, the difference becomes negligible.
