about st peterburg paradox

1. St PETERSBURG PARADOX
PREPARED BY :
MEGHANA
PRAKHAR
CHASHWIN
RAJAH

2. What is St.
Petersburg
Paradox???

3. The St. Petersburg paradox or St. Petersburg lottery is a paradox related to
probability and decision theory in economics.
 It is based on a particular lottery game that leads to a random variable with
infinite expected value but nevertheless seems to be worth only a very small
amount to the participants.
 The St. Petersburg paradox is a situation where a naive decision criterion
which takes only the expected value into account predicts a course of action that
presumably no actual person would be willing to take.
The paradox takes its name from its resolution by Daniel Bernoulli, one-time
resident of the eponymous Russian city, who published his arguments in the
Commentaries of the Imperial Academy of Science of Saint Petersburg
(Bernoulli 1738).
However, the problem was invented by Daniel's brother Nicolas Bernoulli who
first stated it in a letter to Pierre Raymond de Montmort on September 9, 1713
(de Montmort 1713)

4. THE PARADOX
 The St Petersburg paradox involves around a simple gambling game, in which
the gambler pays a fixed cost $x to enter the game, and then you receive some
payoff $y based on the results of the coin flips. The idea is that there is a “pot”
that starts out at $1, which doubles every time that a (perfectly fair) coin
appears heads.
 The game stops as soon as the coin appears tails, and the gambler takes home
the current value of the pot. For instance, if the coin flips are HHHT, then the
gambler takes home a pot of y = $8. On the other hand if the result is T (i.e.,
the first flip comes up tails), the gambler takes home the initial pot y = $1.

5. Resolutions of the St. Petersburg Paradox
There are resolutions of the St. Petersburg Paradox in which
people try to put a fair price on this game.
They are not exactly resolutions in the traditional sense, in that
even today there is not an exact figure on a fair price for this
game, and new articles pertaining to a resolution for the St.
Petersburg Paradox still appear periodically.
Some particularly well-known Resolutions have been given.

6. The simplest way to try and resolve the paradox to and a fair
price for the game was done by Georges-Louis Leclerc,
Comte de Buffon, a French mathematician.
He simply ran the experiment himself a total of 2,048 times.
We can calculate the expected winnings based of this
experiment. Be Careful when calculating the probabilities.
For example, the probability of rolling a tails on the first flip
based on this experiment is 1061/2048 , not ½ because
Buffon got this result 1,061 out of the 2,048 trials he did.
(A)Resolution by experiment

7. Trial (k) where first tails
appears
Frequenc
y
Payout
1 1061 $2
2 494 $4
3 232 $8
4 137 $16
5 56 $32
6 29 $64
7 25 $128
8 8 $256
9 6 $512
So the expected winnings based on this experiment is
($2)(1061/2048 ) + ($4)( 494/2048 ) +($8)( 232/2048 ) + ($16)(
137/2048 ) +($32)( 56/2048 ) + ($64)( 29/2048 )+ ($128)( 25/2048 ) +
($256)( 8/2048 ) +($512)( 6/2048 ) =$9.82.
One should expect to pay about $10 to play this game, based on this
experiment

8. (B) Resolutions based on utility
 Some resolutions use the concept of utility to derive a fair price for this game.
Utility can be thought of as how much practical use you gained.
 Surely going from $0 to $1,000,000 is more dramatic than going from $1,000,000 to
$2,000,000, even though they are both increases of a million dollars. In the former, you
went from nothing to millionaire, and in the latter, you essentially went from rich to rich
again.
 So it can be said that if you won $2,000,000 then you would gain more utility in your
first million than you would gain in your second million.

9.  Daniel Bernoulli used the equation Utility = log10(w), where w is the
winnings in dollars (Bernoulli actually used a different currency than U.S.
dollars, but the currency is arbitrary).
So if you win $2, then by this equation, you have won log10(2) _ 0.301 in utility.
 A new table can be constructed for the St. Petersburg game now:

10.  Now instead of expected winnings in dollars, we can calculate the
expected utility, which turns out to be about 0.602, or about $4.
 This resolution is not satisfactory however. All one must do is change
the
payouts to $102k and the result is an infinite expected value in dollars and
an infinite expected utility, so the paradox returns.
 In fact, it turns out that any unbounded utility function would prove to
be unsatisfactory in the same way.
 Based on of articles in past centuries mentioning the St. Petersburg
Paradox, it took quite a while for the need for a bounded utility function
to be recognized.

11. Finite St. Petersburg lotteries
The classical St. Petersburg lottery assumes that the casino has infinite resources.
This assumption is unrealistic, particularly in connection with the paradox, which
involves the reactions of ordinary people to the lottery. Of course, the resources of
an actual casino (or any other potential backer of the lottery acare finite.
More importantly, the expected value of the lottery only grows logarithmically with
the resources of the casino. As a result, the expected value of the lottery, even when
played against a casino with the largest resources realistically conceivable, is quite
modest.

12. If the total resources (or total maximum jackpot) of the casino are W dollars,
then L = floor(log2(W)) is the maximum number of times the casino can play
before it no longer fully covers the next bet. The expected value E of the
lottery then becomes:

13. The following table shows the expected value E of the game with various
potential bankers and their bankroll W (with the assumption that if you win
more than the bankroll you will be paid what the bank has:
Banker
Bankroll
Expected
value of
lottery
Friendly
game
$100 $7.56
Millionaire $1,000,000 $20.91
Billionaire $1,000,000,000 $30.86
Bill Gates
(2015)
$79,200,000,00
0
$37.15
U.S. GDP
(2007)
$13.8 trillion $44.57
World GDP
(2007)
$54.3 trillion $46.54
Googolaire $10100 $333.14

14.  A rational person might not find the lottery worth even the modest
amounts in the above table, suggesting that the naive decision model of
the expected return causes essentially the same problems as for the
infinite lottery. Even so, the possible discrepancy between theory and
reality is far less dramatic.
 The assumption of infinite resources can produce other apparent
paradoxes in economics. In the martingale betting system, a gambler
betting on a tossed coin doubles his bet after every loss, so that an
eventual win would cover all losses; in practice, this requires the
gambler's bankroll to be infinite.

15. Recent discussions
• Although this paradox is three centuries old, new arguments are still being
introduced.
(A)Samuelson
Samuelson resolves the paradox by arguing that, even if an entity had infinite resources,
the game would never be offered.
If the lottery represents an infinite expected gain to the player, then it also represents an
infinite expected loss to the host. No one could be observed paying to play the game
because it would never be offered.
As Paul Samuelson describes the argument:
"Paul will never be willing to give as much as Peter will demand for such a
contract; andhence the indicated activity will take place at the equilibrium level of zero
intensity."

16. Ole Peters thinks that the St. Petersburg paradox can be solved by using concepts
and ideas from ergodic theory .
In statistical mechanics it is a central problem to understand whether time
averages resulting from a long observation of a single system are equivalent to
expectation values
Peters points out that computing the naive expected payout is mathematically
equivalent to considering multiple outcomes of the same lottery in parallel
universes. This is irrelevant to the individual considering whether to buy a ticket
since he exists in only one universe and is unable to exchange resources with the
others.
It is therefore unclear why expected wealth should be a quantity whose
maximization should lead to a sound decision theory.
(B) PETERS

17. Indeed, the St. Petersburg paradox is only a paradox if one accepts the
premise that rational actors seek to maximize their expected wealth. The
classical resolution is to apply a utility function to the wealth, which reflects
the notion that the "usefulness" of an amount of money depends on how
much of it one already has, and then to maximise the expectation of this.
The choice of utility function is often framed in terms of the individual's
risk preferences and may vary between individuals: it therefore provides a
somewhat arbitrary framework for the treatment of the problem.
An alternative premise, which is less arbitrary and makes fewer
assumptions, is that the performance over time of an investment better
characterises an investor's prospects and, therefore, better informs his
investment decision.

18. The St. Petersburg paradox and the theory of marginal utility have
been highly disputed in the past. For a discussion from the point of
view of a philosopher.
Further discussions

19. References
Wikipedia
COMPSCI 3016: Computational Cognitive Science Dan Navarro & Amy
Perfors
University of Adelaide
C. Ariel Pinto and Paul R. Garvey. Advanced Risk Analysis in Engineering
Enterprise Systems. Boca Raton, FL: CRC Press, 2012. pp. 415-419.
Donald Richards. "The St. Petersburg Paradox and the Quantification
of Irrational Exuberance." Power Point Presentation. Penn State University.