Re: S. Petersburg Paradox
From: David C. Ullrich (ullrich_at_math.okstate.edu)
Date: 02/14/05
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Date: Mon, 14 Feb 2005 06:22:58 -0600
On Sun, 13 Feb 2005 23:14:12 GMT, "beda pietanza"
<beda-pietanza@libero.it> wrote:
>I would like to discuss about the S. Petersburg Paradox.
>
>This is an attempt to crossposting,(I have never done it).
>
>Do you agree on the folloing presentation of it ??
>
>A coin is tossed untill head comes (N=number of tosses)the
>player is paid 1 if head turns out at 1° toss; 2 if it turns out at
>2 toss, 4 if it turns out at 3° toss..... and so on.
>
>The Expectation value of the game is:
>
>E= 2^(1-1)*1/(2^1) + 2^(2-1)*1/(2^2) + 2^(4-1)*1/(2^4)+.......
>
>E=1 / 2 + 1 / 2 + 1 / 2 + 1 / 2 +......................= infinite
>
>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?????
>
>Since the even bet should be = the Expectation value = infinite
>the bettor should be willing to pay a infinite amount of money;
>but it happen that any reasonable person would pay not more
>than a very little amoumt of money.Ence the paradox.
The apparent paradox arises from confusing an abstract
mathematical model with the real world.
There's no paradox in the mathematics, because you
haven't given a mathematical proof of the statement
[*]
"any reasonable person would pay not more than a very
little amoumt of money".
Before you can do that you need a mathematical definition
of "reasonable person". _If_ by definition a reasonable
person always tries to maximize his expected gain then
[*] is simply not true, so there is no paradox. If the
definition of "reasonable" is something else then there
is also no paradox, because the apparent paradox depends
on the unstated assumption that a reasonable person
_does_ always try to maximize his expected gain.
About the real world:
First, in the real world it's simply not true that
a reasonable person always tries to maximize his
expected gain. For example, reasonable people buy
insurance, even though the fact that insurance
companies make money shows that your expected gain
(in the sense of probability theory) is negative
when you buy insurance.
Second:
In real life it will never happen that some casino offers
to play this game, because the casino would need to start
with infinitely much money. So the question of what an
actual real person should be willing to pay an actual
real casino to play this game simply doesn't come up.
>My thesis is that the S. Peterburg paradox in the above formulation
>is not a paradox but it contain inconsinstent Math premises.
>
>Let me know your opinions.
>
>best regards
>
>beda pietanza
>
>
>
>
>
************************
David C. Ullrich
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