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Related: St. Petersburg Paradox

I was reading today the Wikipedia page on the St. Petersburg Paradox.

The posted expected value is: $ 1/2 * 1 + 1/4*2 + 1/8*4 ... $

This seems very wrong to me. Here is a game which would lead to the same expected value calculation:
Flip a coin. If it's heads, you win \$1.
Regardless of result of previous game: Flip 2 coins. If both are heads, you win \$2.
Regardless of result of previous game: Flip 3 coins. If all three are heads, you win \$4.
...
Regardless of result of previous game: Flip N coins. If all are heads, you win $\$2^{N-1}$.

And you take this as N goes to infinity

Now this game obviously has an infinite expected value, and I would pay any amount of money to play it. It's hard to believe that the original game has the same expected value as this one.

Is there a flaw in my reasoning?

Community
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Cruncher
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  • Could you explain the purpose of the -1? I made an observation, and decided to ask about it. – Cruncher May 26 '14 at 13:33
  • When does your game terminate? – Peter Franek May 26 '14 at 13:35
  • @PeterFranek It doesn't. Which is why you get the infinite expected value. If you could theoretically play the whole game at once, you would have won an infinite number of the sub games. But I fail to see how this game wouldn't have the expected value that I posted. – Cruncher May 26 '14 at 13:36
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    In other words, it's a game in which you get $\infty$ with probability $1$. What is interesting at the St. Petersburg paradox is, that you always get a finite payoff and despite that, it doesn't have a finite mean. – Peter Franek May 26 '14 at 13:38

1 Answers1

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I don't think your constructed game and St. Petersburg are equivalent even though they have the same expected value. Consider this: in SP, your payout is always a power of 2; in your game you can have any whole number payout. That is, if you look at the probability of winning \$3 in SP, it's obviously $0$, whereas in your game, it's some finite non-zero value.

And we can't even claim that the expected payout of both the games is same. Sure both tend to infinity, but you can't say that their expected payout is the same on that basis. Consider $E_1$ to be the payout of SP and $E_2$ the payout of your game. $$E_1= 1 P_1(1)+2 P_1(2) + 3P_1(3) + \cdots $$ $$E_2= 1 P_2(1)+2 P_2(2) + 3P_2(3) + \cdots$$

Clearly, $P_1(3)=0$ but $P_2(3) \neq0$ and is some finite value. Same goes for the other terms. So you can see that $E_2>E_1$, hence the expected value of your game is more than SP.

sayantankhan
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  • Right. I never claimed that this game is equivalent. It just seems rather odd to me that this game could have the same expected value. – Cruncher May 26 '14 at 14:32
  • Actually. Winning exactly $3 in my game seems to have probability 0 to me as well. Sure winning the $1 and $2 is easy. It's just 1/8. But the chance of losing ALL other games seems like probability 0 to me. – Cruncher May 26 '14 at 14:35
  • In Crunchers game, the payoff can really be any natural number, but with probability $0$. With probability $1$, it is infinite. – Peter Franek May 26 '14 at 14:55
  • Not really. To win 3 dollars, the probability goes something like this $$\frac{1}{8} \times \frac{7}{8} \times \frac{15}{16} \times \cdots$$ That converges to somewhere around 0.096 – sayantankhan May 26 '14 at 14:56
  • I don't understand the proof of saying that the expected value of my game is more than SP. But if that were true I think that would make my point. That is what I said to begin with. This would mean that the SP should have lower expected value than is traditionally posted? Unless somehow my game has more expected value than that, but that seems difficult to believe. How could it be any different than what I posted? – Cruncher May 26 '14 at 15:16
  • I probably still don't understand Crunchers game and its payoff. Is it an infinite sequence of coin tossing and finite payoff iff only a finite number of them were heads? Or something else? – Peter Franek May 26 '14 at 15:35
  • @PeterFranek Read the question again and consider it for a finite N. Then see how that expands as $N\rightarrow\infty$ – Cruncher May 26 '14 at 15:46