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Thanks a lot for participating in these discussions! Looking back, I realized that my view of this paradox actually changed quite a lot. Mainly, my past self thought the paradox was supposed to be about something which it is not. So I resolve as YES, again thanks for the discussions!

Thank you all for the discussion! I will now go through it once more time and then resolve.

I think paradoxes in general are quite interesting, but I don't know what it would take for you to be "impressed" by a paradox, so I won't trade on this market (Ok maybe 1 mana :P)

I agree that the snake-eyes paradox is only relevant at exactly n=infinity, so I won't comment on that. But the St. Petersburg "paradox" has interesting implications even for finite values of n.

Let's take your statement: "if you say you cannot win more than C dollars for some C, there does not seem to be any paradox" where C=2^(10^9).

Would you (or anybody) be willing to play this game once for $999,999,999? (The EV is 1 billion.) The answer is likely not. Most people would not be willing to bankrupt themselves to play this game once. Perhaps if you had infinite funds and you could play infinite times, then it would be profitable.

The main issue is that in real life, people's utility functions are not u(m)=km (where m is money that the person has, and k is some constant) Imagine a small child getting their allowance of $5. They're very happy in that moment, and the delta u is a lot higher than the delta u of a grown man with thousands of dollars getting $5.

Since people maximize for expected utility and not for expected money, getting 2^(n+1) money is not twice as good as getting 2^n money, it's only probably marginally better.

Basically, for a real person, having a 1/x chance to get x dollars is worth less and less the larger x is.

That's how I would explain the St. Petersburg "paradox." I wouldn't call it "impressive" though. In my opinion, most paradoxes are just a "spot the mistake in reasoning" problem.

@CharlesLien All beautifully said, and I think diminishing marginal utility is a great way out of the corner St Petersburg backs us into. But what if we have an unlimited hoard of magic utility pellets and can offer someone a game where their actual utility doubles on each successful roll? Do we bite the bullet? Maybe as long as we bound the number of utility pellets around the number of atoms in the universe, standard expected utility maximization is still perfectly reasonable?

Between diminishing marginal utility and limitations of a physical universe it seems like we have to go to literally fantastical lengths to give us bullet-biters indigestion.

Thanks, Charles, Daniel. I would actually bite the bullet in St. Petersburg with utility and play it as long as possible, which would reastically be the point when I feel my brain is too dumb to begin to understand what my utility function on that scale is. :)

@VaclavRozhon I like this comment in the snake eyes market:

https://manifold.markets/dreev/is-the-probability-of-dying-in-the#UbCUkBlaXL7KYSQRpLvu

It does all the math and then characterizes succinctly where the confusion and controversy comes from.

Is the probability of dying in the Snake Eyes Paradox 1/36?

90% chance. You’re offered a gamble where a pair of six-sided dice are rolled and unless they come up snake eyes you get a bajillion dollars. If they do come up snake eyes, you’re devoured by snakes. So far it sounds like you have a 1/36 chance of dying, right? Now the twist. First, I gather up an…

@dreev Thanks for linking it, I also like it!

Btw, @VaclavRozhon, I'm curious how you're betting or what you think the correct answer is to the snake eyes question as posed in the original market.

@dreev Baring the use of alt accounts, @VaclavRozhon has never traded in the Snake Eyes Market and has only made the one comment to advertise this meta market.

@ShitakiIntaki Yeah, I did not trade there. When it comes to the finite variant of the question, I think there is no paradox there. Where it comes to the infinite variant of the question, I simply don't know how the relevant math works there. I would first ask you to rephrase the question extremely formally, telling me very precisely what the relevant probability space is, and as Daniel pointed out, already there seems to be an issue with how to define it.

If I compare snake eyes paradox with sleeping beauty, I see how sleeping beauty really forces me to think hard about what probability is, while snake eyes seem "only" to force me to think hard about what can break if I try to generalize finite questions into infinite ones. Maybe my problem is that I read about the paradox in Quantum Computing since Democritus where it was presented in a section that was supposed to make me "feel uneasy about Bayesianism" or something like that I just don't see how I can take away something from this paradox in this spirit.

@VaclavRozhon

When it comes to the finite variant of the question, I think there is no paradox there.

I agree. But there is still something very puzzling about even the finite version. For example, is it a good idea or not a good idea to play the finite version with population 2^1000 ? I'm honestly not sure.

Personally, I feel the reverse about the two paradoxes. I don't see much paradox in Sleeping Beauty. The answer to me is very clearly 1/3. The Halfer argument leads to all kinds of nonsensical consequences it seems. I need to study it more. I'm sure there's more to it. A colleague explained the Halfer argument to me and at least I sort of see it.

@DavidPennock Ok, now I thought about sleeping beauty one more time and I am also a happy thirder (though I don't understand very well all posible positions and arguments for them). But whenever I think of the problem, I get confused about what probability actually is. Ultimately, I always justify probability by thought experiments in which I am betting and want to maximize my winnings, and this works great until the events on which I am betting start correlating with how many bets I make. This is the case in sleeping beauty or the doomsday problem and the confused feeling I always have when thinking about these is the reason why I think they are great paradoxes.

@VaclavRozhon I also like to equate subjective probability with betting behavior. FWIW, after reading a bit more and thinking and talking more, my position as a Thirder is still strong but now I can see a bit more the Halfer's argument and the subtlety and "beauty" of the SB paradox.

Repeating my comment from the original market:

Sounds like you have your head fully around the situation to me! It's only a veridical paradox so it goes away when you find the right way to think about it. For many of us, that's been quite a struggle! A gratifying one though. I'm learning a lot from this.

So, yes, it technically is nonsensical to talk about starting with a literally infinite number of people (there's no such thing as sampling uniformly from an infinite set, for starters) and so the only interpretation we can make is as a limit as the set grows arbitrarily large. And that turns out to be relatively straightforward. The trickiest part for me was, where the heck does the argument based on the fraction of people who end up dead go wrong?

With St Petersburg, one way to resolve the paradox is to have diminishing marginal utility for money. But we can unresolve it by saying your actual utility doubles on each successful dice roll. In that case the expected value of the game really does go to infinity in the limit. But, as you say, if you just factor in a cap -- the maximum amount the casino can give you -- then the expected value of the game is eminently reasonable. And I think the seeming paradox is similar to the snake eyes paradox: you'd think you could ignore the vanishingly unlikely event that the casino runs out of money, but you can't.

I think there's definitely more to it than that, but also that it's ultimately still objectively answerable. If I were to convince you of that, would that make you substantially more impressed with the paradox?

Specifically, would it be enough just to show that there's interesting mathematical stuff going on in a way that's only possible due to the infinite nature?

@Julian Thanks! Maybe see my replies to other comments. I guess my problem is that after reading about this in this book I mentioned in the other comment, I expected the paradox to go deeper than it actually goes.

Shooting Room/Snake Eyes is an annoying paradox since it has several tricks, so it doesn't clarify any individual principle very well. It's also easy to ask subtly different questions and not realize they are different, as Daniel did.

Sleeping Beauty Problem is a better demo of anthropics. Which I do think is interesting.

/MartinRandall/is-the-answer-to-the-sleeping-beaut

Part of the paradox is the definition of probability, ie frequentists might give a different answer. That's maybe interesting but I don't think anyone would pick their definition of probability based on the paradox.

There's lots of math about infinite series and infinite sets, but it's easier to learn about those things via specific problems.

Well, what's interesting to me is that the probability is affected in an unusual way.

Thanks for this market, @VaclavRozhon ! Let me try to convince you that there is something mind-bending about the Snake Eyes paradox.

Let's remove the life or death aspect and suppose that the snake deity pays each person $100 if he does not roll snake eyes, and takes from each person $200 if he does roll snake eyes. This is clearly a favorable bet. Each person wins $100 with 35/36 probability and loses $200 with only 1/36 probability. This is a great bet to take for almost any level of risk aversion. The paradox is exactly the same without the life or death aspect.

To me, one of the craziest aspects is the following. If the only thing you know is that you have been chosen for the game, then you should play. It's a very favorable bet. Your chance of losing is only probability 1/36. However, if you know one more piece of information -- that the game ended in snake eyes -- now suddenly you do NOT want to play. Your chance of losing is 1/2.

Think about an extremely large finite game with population 2^1000. If you're told you are chosen, you should play. But if you're told that some time in the 1000 rounds snake eyes is rolled -- something you are already extremely confident about -- you will want to change your mind. You won't want to play.

Now consider the infinite game. If you chosen to play, you should. BUT if some omniscient being peeks into the future and tells you that snake eyes was rolled within the first 1 billion rounds -- something that you for all practical purposes already know -- you should change your mind and not play. Here is a game that (1) you should play, and (2) you are guaranteed to regret playing.

The fact that each decision is taken by a single person make it even more mind bending. I can see how humanity overall should reject this game -- for the same reason that no one should pay too much money to play the St. Petersburg lottery, and the same reason a casino should not accept arbitrarily large bets from a Martingale bettor with infinite wealth. However, why shouldn't a single person play? The single person is only risking $200 at most. The single person has a 35/36 chance to win $100 and only 1/36 chance to lose. This is an extraordinarily good deal. The single person is not facing a series of exponentially growing bets. The single person is not being asked to pay to play a St. Petersburg lottery. The single person is just facing a single bet and knows nothing and cares nothing about the doubling series he is part of. It seems that this single person should play. But we know that the group of all people collectively should NOT play. So if I am a random one of this group of people who should not play, then it seems like I should NOT play. Hence the paradox. Honestly, I am still not sure what each single person should do. Are you convinced there's an obvious answer?

@DavidPennock If you object to the infinite version, think about the finite version with population 2^1000. Should each single person play? With probability 1 - 5.8*10^-13 -- effectively with probability 1 -- humanity overall will lose, and each person who plays will expect to lose. But in the round that the single person was chosen, that person has quite a high positive expected profit. Assume that you only care about your own profit or loss, not humanity's overall profit/loss. Would you play? Why or why not?

@DavidPennock Thanks! I really like the analogy with martingale betting strategy. Basically, my view is that the unintuitive part of martingale betting is 1) I can trade very large probability of very small gain with very small probability of very large loss 2) If we generalize to the infinite setup, somehow the small probability becomes zero probability and we need to be super-careful with definitions.

In the same sense, even in the finite version of snake-eyes paradox, the very nonsuspicious conditioning on snake eyes happening at some point which happens with probability almost 1 changes everything. I did not think about it this way and really like it!

Perhaps me being not impressed about snake-eyes stems from reading about this paradox from Quantum computing since Democritus (Chapter 18 in that book). The author is talking about anthropic principle, doomsday argument etc. and then throws in the snake-eyes paradox in the middle of the chapter. But I don't get how it's relevant to these discussions. Paradoxes like the doomsday or sleeping beauty seem to me to go deep into the question of how should we think about probability, whereas martingale betting/snake-eyes paradox seem to me to be "just" about how we can have small probability events substantially affecting expectation. Am I missing something?

I'm not sure what you're expecting to get out of this. The two paradoxes are only superficially related anyway. Like, St Petersburg at least has some history behind it. You can read Bernoulli's original paper, and I think Samuelson had some interesting things to say about it. It's not especially deep though. If you're just looking for interesting paradoxes then you could try Bertrand's paradox or the two envelopes.

The whole snake eyes thing sadly just boils down to a lack of rigor. That other market's a debacle, the blind leading the blind.

@TrevorG Actually the connection between Snake Eyes and St. Petersburg is very direct. The snake deity who is running the game is the St. Petersburg lottery. He is offering a doubling prize to humanity at very favorable odds. The question to humanity is, what price should humanity pay for this doubling prize? The Snake Eyes paradox phrases the price it in term of lives lost, but if you replace 1 life with 2 bajillion dollars, it becomes a very concrete price, and the paradox is exactly the same. And now you can see that Snake Eyes is exactly the Martingale betting paradox, which is itself as variant of the St. Petersburg paradox. The snake deity is the Martingale bettor and humanity is the casino.

There is a twist in the Snake Eyes paradox that I have not seen before in descriptions of St. Petersburg or Martingale. In the Snake Eyes paradox, it's not humanity, with infinite population, making the decision collectively. It's each individual person deciding what to do. And each individual person only has finite wealth -- 1 life or 2 bajillion dollars. So Snake Eyes is more like a casino with infinitely many finite investors trying to decide whether to accept arbitrarily large bets from a Martingale bettor. Each finite investor only risks their money in a single round of the Martingale series. Each round of the series has very favorable odds -- so it seems like each finite investor should be perfectly happy to play. However, the Martingale bettor is guaranteed to walk away a winner at the end, and the group of finite investors are guaranteed to come out as losers collectively. So it seems like each finite investor should NOT play. Hence the paradox.

@DavidPennock I think you could also look at the Snake Eyes deity as selling a Ponzi Scheme investment and each person is an individual investor in a doubling funding/investment round with a 1/36 chance that their investment round is the last round and will never see a return on their investment because there will not be a subsequent funding round to pay them.

@ShitakiIntaki Interesting! And that makes a good case for why it's not a good idea to play, even for the individual. Thanks. I will think about that analogy

@DavidPennock @Wamba Thanks, I like these analogies!

@ShitakiIntaki to be fair the analogy doesn't work 100%, because In a finite sense running out of players results in a forced loss for the last round of "investors"

I guess in the finite case maybe the Snake Eyes diety pays the last round of investors 35/36 of the time to avoid being accused of running a ponzi scheme.