Update
This post got a shout-out on an episode of the Modern Wisdom podcast, where Chris Williamson interviews Alex O’Connor (the ‘Cosmic Skeptic’). Very exciting! Here’s a link to where it gets brought up. I should also attribute the source of this puzzle: I first heard of it here, in Scott Aaronson’s online version of his quantum computing course (which I highly highly recommend, by the way). He attributes the puzzle to the philosopher John Leslie, though I haven’t been able to track down that source. (The addition of the 50%-escape-route is due to me.)
At the time I wrote this (about 3 years ago), I probably did think that the ‘anthropic’ calculation was the right one. Coming back to the puzzle after ~3 years, I am less sure. Like Alex, what I find most interesting about the puzzle is really that there are these two arguments that both feel fairly independently plausible, but end up at opposite conclusions. On the one hand: I’m in the room waiting to see how the roll of the dice comes out, and it seems crazy (if not blatantly contradictory) to say both that the dice are fair and that nonetheless my credence that they will land snake-eyes is > 90%. This is a very strong argument!
On the other hand: I am one of the captives, and I know that >90% of the captives end up dead (no matter how many rounds occur). So if I stay in the room I have a >90% chance of ending up dead. Stated with a more decision-theoretic flavor: Suppose the captives accept the first line of reasoning. Then they will all forsake the 50%-chance-of-survival route and stay in the room. But then >90% of them will die! On the other hand, if the captives follow the anthropic line of reasoning, then they all take the 50% route, and only 50% of them end up dying. Is it really rational for me to follow a strategy that I know does significantly worse in expectation?
Original post starts now
Today we discuss anthropic reasoning.
The Problem
Imagine the following scenario:
A mad killer has locked you in a room. You are trapped and alone, with only your knowledge of your situation to help you out.
One piece of information that you have is that you are aware of the maniacal schemes of your captor. His plans began by capturing one random person. He then rolled a pair of dice to determine their fate. If the dice landed snake eyes (both 1), then the captive would be killed. If not, then they would be let free.
But if they are let free, the killer will search for new victims, and this time bring back ten new people and lock them alone in rooms. He will then determine their fate just as before, with a pair of dice. Snake eyes means they die, otherwise they will be let free and he will search for new victims (ten times as many as he just let free).
His murder spree will continue until the first time he rolls snake eyes. Then he will kill the group that he currently has imprisoned and retire from the serial-killer life.
Now. You become aware of a risky way out of the room you are locked in and to freedom. The chances of surviving this escape route are only 50%. Your choices are thus either (1) to traverse the escape route with a 50% chance of survival or (2) to just wait for the killer to roll his dice, and hope that it doesn’t land snake eyes.
What should you do?
(Think about it before reading on)
A plausible-sounding answer
Your chance of dying if you stay and wait is just the chance that the dice lands snake eyes. The probability of snake eyes is just 1/36 (1/6 for each dice landing 1).
So your chance of death is only 1/36 (≈ 3%) if you wait, and it’s 50% if you try to run for it. Clearly, you are better off waiting!
But…
You guessed it, things aren’t that easy. You have extra information about your situation besides just how the dice works, and you should use it. In particular, the killing pattern of your captor turns out to be very useful information.
Ask the following question: Out of all of the people that have been captured or will be captured at some point by this madman, how many of them will end up dying? This is just the very last group, which, incidentally, is the largest group.
Consider: if the dice land snake eyes the first time they are rolled, then only one person is ever captured, and this person dies. So the fraction of those captured that die is 100%.
If they lands snake eyes the second time they are rolled, then 11 people total are captured, 10 of whom die. So the fraction of those captured that die is 10/11, or ≈ 91%.
If it’s the third time, then 111 people total are captured, 100 of whom die. Now the fraction is just over 90%.
In general, no matter how many times the dice rolls before landing snake eyes, it always ends up that over 90% of those captured end up being in the last round, and thus end up dying.
So! This looks like bad news for you… you’ve been captured, and over 90% of those that are captured always die. Thus, your chance of death is guaranteed to be greater than 90%.
The escape route with a 50% survival chance is looking nicer now, right?
Wtf is this kind of reasoning??
What we just did is called anthropic reasoning. Anthropic reasoning really just means updating on all of the information available to you, including indexical information (information about your existence, age, location, and so on). In this case, the initial argument neglected the very crucial information that you are one of the people that were captured by the killer. When updating on this information, we get an answer that is very very different from what we started with. And in this life-or-death scenario, this is an important difference!
You might still feel hesitant about the answer we got. After all, if you expect a 90% chance of death, this means that you expect a 90% chance for the dice to land snake eyes. But it’s not that you think the dice are biased or anything… Isn’t this just blatantly contradictory?
This is a convincing-sounding rebuttal, but it’s subtly wrong. The key point is that even though the dice are fair, there is a selection bias in the results you are seeing. This selection bias amounts to the fact that when the dice inevitably lands snake-eyes, there are more people around to see it. The fact that you are more likely than 1/36 to see snake-eyes is kind of like the fact that if you are given the ticket of a random concert-goer, you have a higher chance of ending seeing a really popular band than if you just looked at the current proportion of shows performed by really popular bands.
It’s kind of like the fact that in your life you will spend more time waiting in long lines than short lines, and that on average your friends have more friends than you. This all seems counterintuitive and wrong until you think closely about the selection biases involved.
Anyway, I want to impress upon you that 90% really is the right answer, so I’ll throw some math at you. Let’s calculate in full detail what fraction of the group ends up surviving on average.
By the way, the discrepancy between the baseline chance of death (1/36) and the anthropic chance of death (90%) can be made as large as you like by manipulating the starting problem. Suppose that instead of 1/36, the chance of the group dying was 1/100, and instead of the group multiplying by 10 in size each round, it grew by a factor of 100. Then the baseline chance of death would be 1%, and the anthropic probability would be 99%.
We can find the general formula for any such scenario:
IF ANYBODY CAN SOLVE THIS, PLEASE TELL ME! I’ve been trying for too long now and would really like an analytic general solution. 🙂
There is a lot more to be said about this thought experiment, but I’ll leave it there for now. In the next post, I’ll present a slight variant on this thought experiment that appears to give us a way to get direct Bayesian evidence for different theories of consciousness! Stay tuned.