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Comment by hbrav | Hacker News Reader

The dice room puzzle described here is a little frustrating, because no-one seems to get it right. In the finite population case the anthropic argument for "I'm likely to be the in the last round" is wrong. In the countably infinite case it can't be applied (no discrete uniform distribution on a countably infinite set). I wrote some ramblings about it here, though I'm not sure it's super-clear: https://harrybraviner.github.io/posts/2024-01-28-anthropic_d...

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I thought the writeup was convincing and addresses the heart of the paradox. The only nitpick: you can have uniform probability distributions on infinite sets, like [0,1]: https://en.wikipedia.org/wiki/Continuous_uniform_distributio... There p(x) = 0 for any x, but for fixed e, p(x +/- e) is the same for all x.

But you can't have such a distribution on an unbounded set, which is where the paradox fails. If we had a uniform distribution on an unbounded set, p(x +/- e) has to be the same for all x and therefore nonzero, but

  p(1 +/- e) + p(2 +/-e) + ...

has to sum to <= 1. It is an infinite sum of nonzero terms so this is a contradiction. (The same argument works if you drop the epsilon for thinking of a distribution on the integers).

I think your writeup was basically clear on this in terms of the math, just some of the language was a bit confused.

Yeh, U[0, 1] is different because it assigns non-zero probabilities to intervals, not points. In this case we're assuming that we live in an uncountable population (each real in [0, 1] is a person), so you can't do things like assign a unique number to each person. There, even if the maniac goes on kidnapping forever, he will only kidnap a countable subset of the population. Thinking about this honestly makes my brain hurt a little.

Thanks for writing this.

Is this concept (dice room puzzle, doomsday argument) at all related to the st Petersburg paradox? https://en.m.wikipedia.org/wiki/St._Petersburg_paradox

I think it's a little different. It's like asking "what is the probability that a given coin is in the round that the player wins?" But the St Petersburg paradox isn't about that, it's purely about how many coins the player wins. I suppose it runs into similar problems when you ask about prizes that are so large that the bank runs out of coins, but I think it remains interesting even if you cap it at some finite number of coin flips. It still has the small-probability-of-huge-payout property.

If you've ever looked at the Kelly Criterion, that seems related (and in fact is one of the articles linked to from that Wikipedia page). There you maximise expected log return at each round, and I think that tames the infinity in this case (though I have _not_ checked that).

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