St. Petersburg Paradox

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Anonymous

St. Petersburg Paradox 2016-01-10 01:57:13 Post No. 7772653

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St. Petersburg Paradox 2016-01-10 01:57:13 Post No. 7772653

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https://en.wikipedia.org/wiki/St._Petersburg_paradox

If I understand this paradox, the game has an infinite expected value, implying that you should pay any amount to play it. Given enough time, you'll always eventually win.

The proposed solutions to the paradox all seem to be based on the idea that the player wouldn't have infinite money to start with, therefore, he couldn't play long enough to actually get ahead.

I think there's something missing from these solutions, though. Even if the player did have infinite money to start, there is still some average payout over time. There's no way that average is "infinity." I ran some simulations and after many millions of games, the average payout is about $13. I'm now trying to figure out why that is. Any ideas?

Intuitively, that number seems about right. I wouldn't pay $20 to play the game, regardless of the size of my bankroll.

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hump

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Not a mathmatician, but this actually really interests me. Can any smart people give me a 5-year-old's explanation of some of the proposed solutions?

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>>7772653

I think it's just not a well-behaved problem, so not even millions of games are enough to be statistically significant. The expected value is infinity, but so is the standard deviation.

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>>7772712

So how do you answer the paradox's original question, which is "how much should you pay to play the game?"

Do you just say that the question has no answer?

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>>7772653

>something missing

They just tell you half the story. They tell you that the expected value tends to infinity but they don't tell you that as [math]k[/math] approaches infinity, your chances of winning again approach 0.

Thus, the only way of actually winning 'infinite' money is to pay them an infinitesimal amount of money. Give them a cent, perhaps. Anything beyond that will never approach infinity, anything more than 2 dollars would be meh, and anything beyond 8 dollars is irresponsible because the chances of getting your money back by then would be 1/4 and to get a profit 1/8.

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The problem is solved by introducing utility theory. The main idea is that money had decreasing marginal returns on utility. If M is the dollar amount of the payout, then it gives us u=log (M) utils of happiness. Economic research that human utility functions usually have a positive first derivative (nonsatiety) and negative second derivative (concavity), so u=log(M) is a good choice for a utility function.

Under this paradigm the amount to pay for the game is the amount [math]E[\math] that gives an equal utility as the payoff ie

[math] \logE= \sum \limits_{n=1}^{\infty} \frac{\log{2^n}}{2^n} [\math]

Thus the amount to pay for the game is E=$4

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>>7772717

No, the paradox is, what is the best course of action if you are offered the wager?

common sense says to pay approx $10, but mathmatically it appears that you could pay any huge some of money and still get an infinite profit. how do you resolve the difference?

at least that is my understanding of it.

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>>7772732

[math] \logE= \sum \limits_{n=1}^{\infty} \frac{\log{2^n}}{2^n} [/math]

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>>7772720

Like any other risk-reward problem, it's resolved by expected utility theory. Real humans don't value money linearly. "Infinite" or tremendously large amounts of money do not have infinite utility. When risk is taken into account, this means there is a finite limit to the amount of money they will pay for the game.

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>>7772720

Yeah, I think that's a good way of putting it. The expected value of infinity comes with the massive asterisk that it's infinitely unreliable.

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>>7772730

Problem with that explanation is that you can play as many times as you like. If you're playing for $2, you'll come out ahead after about 8 games, on average. If you're playing for $4, you'll come out ahead after about 128 games. If you're playing for $8, it will take a very long time, but you will eventually come out ahead (not sure the number of games at the moment, but I think it's 2^(2k-1) games).

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>>7772734

yes, and there is no problem there.

common sense tells me to follow math, and math tells me to wager whatever the cost.

I really see no paradox here. It's like saying "but if we build this with so much glass, won't it break down?" while having civil engineers tell you it won't.

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>>7772839

>common sense tells me to follow math

Would you bet anything if offered a chance to play? That's not what I would call common sense.

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>>7773648

Not him but the problem is your own bankroll IMO. Not sure why all the solutions on wiki talk about the casino's bankroll. If you can play the game for 500 bucks you should have a shitload of cash (and time) available to get to the point where you can amortize the very rare massive winnings with your losses.

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The paradox is based on the assumption of the classical school of economics that we humans are rational beings and decide according to these rational laws that is expected value. Hence, if you are offered a gamble of 50% chance of winning 100 and 0% chance of winning nothing, your expected value is 50. Hence, thats the maximum you should pay.

Obviously, this is bullshit, so people came up with scenarios to disprove the assumption that we are rational. The mentioned paradox is one of the many examples, if we are offered this gamble we should, according to the rational theory, pay an infinite amount. However, reality shows that we do not. Hence, this cant be true...Fast forward several years, behavioral finance is born with Kahneman as its leader to explain why we dont behave rational.

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>>7773648

I would.

I mean I wouldn't because because of religious reasons, but I would be tempted.

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