Re: S. Petersburg paradox

From: jem (xxx_at_xxx.xxx)
Date: 02/19/05


Date: Sat, 19 Feb 2005 07:09:35 -0500

beda pietanza wrote:

> "jem" <xxx@xxx.xxx> ha scritto nel messaggio
> news:zHlRd.82793$bu.13471@fed1read06...
>
>>beda pietanza wrote:
>>
>>>"Yan Zhang" <yanzhang@fas.harvard.edu> ha scritto nel messaggio
>>>news:cv2vd2$3lh$1@us23.unix.fas.harvard.edu...
>>>
>>>
>>>>beda pietanza wrote:
>>>>
>>>>
>>>>>Sunny wrote:
>>>>>
>>>>>
>>>>>
>>>>>>>>Since the Expectation value of the game is infinite, what is the
>>>>>>>>even bet a bettor should be willing to pay in order to play the
>>>>>>>>game?????
>>>>>>
>>>>>>Here's something off the top of my head:
>>>>>>The probability of winning 2^n units is (1/2)^n, so on average you
>>>>>>need to play 1/((1/2)^n) = 2^n times to win 2^n units. However to
>>>>>
>>>>>play
>>>>>
>>>>>
>>>>>
>>>>>>that many times, you pay x*2^n units in entrance fees, where x is the
>>>>>>entrance fee for one throw.. So for every amount you win from a draw,
>>>>>>you expect to have lost x*2^n units. So to come out on top, you'd pay
>>>>>>x<1 units to enter.
>>>>>>
>>>>>>There's probably some silly mistake there, but I'm sure someone will
>>>>>>point it out soon enough.
>>>>>>
>>>>>>-Sunny
>>>>>
>>>>>
>>>>>I am not sure I got your reasoning, I only point out to you that the
>>>>>SPP is meant to be played only once: the Expectation value of the game
>>>>>is said to be infinite and it is expected that a bettor should be
>>>>>willing to pay a infinite amount of money in order to play the game
>>>>>just once !!!
>>>>>
>>>>
>>>>I think the confusion here is using "expected value" too loosely as a
>>>>decision device in a situation where only one probabilistic check is
>>>>made. Consider the following simpler game:
>>>>
>>>>You flip a coin. On the 0.0001% chance it lands on its side, I give you
>>>>$100000000000000000. On the 99.9999% chance it lands heads or tails, you
>>>>give me $100. Now, you have no doubt that the expected value is positive
>>>>(huge in fact, but I don't feel like doing simple arithmetic since I'm a
>>>>lazy person), but are you really willing to play the game just once and
>>>>pay me, say, $1000000 to do it? I don't think so, though the expected
>>>>value is much higher than that!
>>>>
>>>>The St. Petersburg "Paradox" is not a paradox for the precise reason as
>>>>above. It has a high expectation value, infinite, actually. But you are
>>>>talking about a situation where we just play "once" and playing into
>>>>the psychological factors involved. There is no paradox though, for if
>>>>we keep playing, forever and ever, the player will indeed come out ahead
>>>>in the long run on average.
>>>>
>>>>-Yan Z.
>>>
>>>
>>>I don't think the SPP has really a infinite expectation value if you pay
>
> a
>
>>>infinite bet in order to play it once:
>>>the really payoff scheme for a infinite paid bet is :
>>>1 outcome payoff 1- infinite net win = -infinite
>>>2 outcome payoff 2-infinite net win = -infinite
>>>3 outcome payoff 4-infinite net win = -infinite
>>>and so on....
>>>if you pay a infinite amount of money you will loose it for sure
>>>and if you play N numbers of games you surely will loose N time
>>>the infinite bets you have paid.
>>
>>Nonsense. It's not possible to bet an infinite amount.
>
>
> You don't have to you only calculate it.
> In any case substituting infinite with a very large amount the destiny
> of the bettor doesn't changes.
>

The amount of any bet is necessarily finite, and for any and all bets
the gambler's Expected gain is infinite.