Infinities in the anthropic dice killer thought experiment

~ squarishbracket

It’s time for yet another post about the anthropic dice killer thought experiment. 😛

In this post, I’ll point out some features of the thought experiment that have gone unmentioned in this blog thus far. Perhaps it is in these features that we can figure out how to think about this the right way.

First of all, there are lots of hidden infinities in the thought experiment. And as we’ve seen before, where there are infinities, things start getting really wacky. Perhaps some of the strangeness of the puzzle can be chalked up to these infinities.

For instance, we stipulated that the population from which people are being kidnapped is infinite. This was to allow for the game to go on for arbitrarily many rounds, but it leads to some trickiness. As we saw in the last post, it becomes important to calculate the probability of a particular individual being kidnapped if randomly drawn from the population. But… what is this probability if the population is infinite? The probability of selecting a particular person from an infinite population is just like the probability of picking 5 if randomly selecting from all natural numbers: zero!

Things get a little conceptually tricky here. Imagine that you’re randomly selecting a real number between 0 and 1. The probability that you select any particular number is zero. But at the same time, you will end up selecting some number. Whichever number you end up selecting, is a number that you would have said had a 0% chance of being selected! For situations like these, the term “almost never” is used. Rather than saying that any particular number is impossible, you say that it will “almost never” be picked. While this linguistic trick might make you feel less uneasy about the situation, there still seems to be some remaining confusion to be resolved here.

So in the case of our thought experiment, no matter how many rounds the game ends up going on, you have a 0% chance of being kidnapped. At the same time, by stipulation you have been kidnapped. Making sense of this is only the first puzzle. The second one is to figure out if it makes sense to talk about some theories making it more likely than others that you’ll be kidnapped (is \frac{11}{\infty} smaller than \frac{111}{\infty} ?)

An even more trouble infinity is in the expected number of people that are kidnapped. No matter how many rounds end up being played, there are always only a finite number of people that are ever kidnapped. But let’s calculate the expected number of people that play the game.

Number \  by \ n^{th} \ round = \frac{10^n - 1}{9} \\~\\ \sum\limits_{n=1}^{\infty} { \frac{35^{n-1}}{36^n} \cdot \frac{10^n - 1}{9} }

But wait, this sum diverges! To see this, let’s just for a moment consider the expected number of people in the last round:

Number \  in \ n^{th} \ round = 10^{n-1} \\~\\ \sum\limits_{n=1}^{\infty} { \frac{35^{n-1}}{36^n} \cdot 10^{n-1} } = \sum\limits_{n=1}^{\infty} { (\frac{350}{36})^{n-1} }

Since \frac{350}{36} > 1 , this sum diverges. So on average there are an infinite number of people on the last round (even though the last round always contains a finite number of people). Correspondingly, the expected number of people kidnapped is infinite.

Why might these infinities matter? Well, one reason is that there is a well known problem with playing betting games against sources with infinite resources. Consider the Martingale betting system:

A gambler makes a bet of $1 at some odds. If they win, then good for them! Otherwise, if they lose, they bet $2 on the same odds. If they lose this time, they double down again, betting $4. And so on until eventually they win. The outcome of this is that by the time they win, they have lost \$(1 + 2 + 4 + ... + 2^n) and gained \$ 2^{n+1} . This is a net gain of $1. In other words, no matter what the odds they are betting on, this betting system guarantees a gain of $1 with probability 100%.

However, this guaranteed $1 only applies if the gambler can continue doubling down arbitrarily long. If they have a finite amount of money, then at some point they can no longer double down, and they suffer an enormous loss. For a gambler with finite resources, they stand a very good chance of gaining $1 and a very tiny chance of losing massively. If you calculate the expected gain, it turns out to be no better than what you expect from any ordinary betting system.

Summing up: With finite resources, continually doubling down gives no advantage on average. But with infinite resources, continually doubling down gives a guaranteed profit. Hopefully you see the similarity to the dice killer thought experiment. With an infinite population to draw from, the killer can keep “doubling down” (actually “decupling” down) until they finally get their “payout”: killing all of their current captives. On the other hand, with a finite population, the killer eventually loses the ability to get a new group of 10x the population of the previous one and lets everybody free. In this case, exactly like the Martingale system, the odds for a kidnappee end up coming out to the prior odds of 1/36.

What this indicates is that at least some of the weirdness of the dice killer scenario can be chalked up to the exploitability of infinities by systems like the Martingale system. If you have been kidnapped by the dice killer, you should think that your odds are 90% only if you know you are drawn from an infinite population. Otherwise, your odds should come out to 1/36.

But now consider the following: If you are a casino owner, should you allow into your casino a person with infinite money? Clearly not! It doesn’t matter how much of a bias the games in the casino give in favor of the house. An infinitely wealthy person can always exploit this infinity to give themselves an advantage.

But what about allowing a person with infinite money into your casino to place a single bet? In this case, I think that the answer is yes, you should allow them. After all, with only a finite number of bets, the odds still come out in favor of the house. This is actually analogous to the original dice killer puzzle! You are only selected in one round, and know that you will not be selected at any other time. So perhaps the infinity does not save us here.

One final point. It looks like a lot of the weirdness of this thought experiment is the same type of weirdness as you get from infinitely wealthy people using the Martingale betting system. But now we can ask: Is it possible to construct a variant of the dice killer thought experiment in which the anthropic calculation differs from the non-anthropic calculation, AND the expected number of people kidnapped is finite? It doesn’t seem obvious to me that this is impossible. Since the expected number of captives takes the form of an infinite sum with the number of people by the Nth round multiplied by roughly (\frac{35}{36})^N , all that is required is that the number of people by the Nth round be less than (\frac{36}{35})^N . Then the anthropic calculation should give a different answer from the non-anthropic calculation, and we can place the chance of escape in between these two. Now we have a finite expected number of captives, but a reversal in decision depending on whether you update on anthropic evidence or not. Perhaps I’ll explore this more in future posts.

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