How Much Would You (Should You) Pay
For A Gamble?
St. Petersburg Paradox
You pay a fixed fee to enter the game. Then a fair coin is
tossed repeatedly until a tail appears, ending the game.
The pot starts at 1 dollar and is doubled every time a
head appears. You win whatever is in the pot after the
game ends.
The paradox is named from Daniel Bernoulli's
presentation of the problem and his solution, published in
1738.
17.1
How Much Would You (Should You) Pay
For A Gamble?
St. Petersburg Paradox
Thus you win:
1 dollar if a tail appears on the first toss,
2 dollars if a head appears on the first toss and a tail
on the second,
4 dollars if a head appears on the first two tosses and
a tail on the third,
8 dollars if a head appears on the first three tosses
and a tail on the fourth, etc.
2k−1 dollars if the coin is tossed k times until the first
tail appears.
17.2
How Much Would You (Should You) Pay
For A Gamble?
St. Petersburg Paradox
What would be a fair price to pay for entering the
game?
To answer this we need to consider what would be the
average payout:
With probability 1/2, you win 1 dollar;
with probability 1/4 you win 2 dollars;
with probability 1/8 you win 4 dollars etc.
Of course the probability of gaining a head on the kth.
play is 1/2k.
17.3
How Much Would You (Should You) Pay
For A Gamble?
0.5
Head
...
0.5
Head
0.5
Tail
0.5
Head
8
0.5
Head
0.5
Tail
4
0.5
Tail
2
0.5
Tail
1
But how much should you pay?
17.4
How Much Would You (Should You) Pay
For A Gamble?
The expected value (E) is thus
1
1
1
1
 1   2   4   8  ...
2
4
8
16
1 1 1 1
     ...
2 2 2 2

E
17.5
How Much Would You (Should You) Pay
For A Gamble?
The expected value (E) is thus, or notationally

E

1
k 1

2
k
2
k 1



1
2
k 1

where k counts the number of tosses.
17.6
How Much Would You (Should You) Pay
For A Gamble?
This sum diverges to infinity, and so the expected win
for the player of this game, at least in its idealized
form, in which the casino has unlimited resources, is an
infinite amount of money.
This means that the player should almost surely come
out ahead in the long run, no matter how much he pays
to enter.
17.7
How Much Would You (Should You) Pay
For A Gamble?
While a large payoff comes along very rarely, when it
eventually does it will typically be far more than the
amount of money that he has already paid to play.
According to the usual treatment of deciding when it is
advantageous and therefore rational to play, one
should therefore play the game at any price if offered
the opportunity.
17.8
How Much Would You (Should You) Pay
For A Gamble?
Yet, in published descriptions of the paradox, e.g.,
(Martin 2004), many people expressed disbelief in the
result. Martin quotes Ian Hacking as saying “few of us
would pay even $25 to enter such a game” and says
most commentators would agree. This is the paradox!
Martin, Robert (2004). “The St. Petersburg Paradox”.
In Edward N. Zalta. The Stanford Encyclopedia of
Philosophy (Fall 2004 ed.).
The Paper
17.9
How Much Would You (Should You) Pay
For A Gamble?
You will quickly discover, however, that you will not win
an infinite amount of money playing this game.
We seem to have a paradox.
The expectation value is infinite, but certainly no one
is going to put up an infinite amount of money, or even
a million dollars, for the opportunity to play this game
once. What is the solution to this paradox?
17.10
How Much Would You (Should You) Pay
For A Gamble?
The answer is that an expectation value is defined to
be an average over an infinite number of trials (or the
limit towards an infinite number).
But you are simply not going to play an infinite number
of games. In other words, the calculated expectation
value doesn’t agree with your experiment, because your
experiment has nothing whatsoever to do with the
precise definition of an expectation value.
17.11
How Much Would You (Should You) Pay
For A Gamble?
To be sure, if you did somehow play an infinite number
of games, then you would indeed have an infinite
average for your winnings.
The whole paradox arises from trying to make
“expectation value” mean something it doesn’t.
17.12
How Much Would You (Should You) Pay
For A Gamble?
Expected Utility Theory
The classical resolution of the paradox involved the
explicit introduction of a utility function, an expected
utility hypothesis, and the presumption of diminishing
marginal utility of money.
17.13
How Much Would You (Should You) Pay
For A Gamble?
Expected Utility Theory
In Daniel Bernoulli's own words:
The determination of the value of an item must not be
based on the price, but rather on the utility it yields….
There is no doubt that a gain of one thousand ducats is
more significant to the pauper than to a rich man
though both gain the same amount.
17.14
How Much Would You (Should You) Pay
For A Gamble?
Expected Utility Theory
A common utility model, suggested by Bernoulli
himself, is the logarithmic function u(w) = ln(w) (known
as “log utility”).
It is a function of the gambler’s total wealth w, and
the concept of diminishing marginal utility of money is
built into it.
17.15
How Much Would You (Should You) Pay
For A Gamble?
Expected Utility Theory
By the expected utility hypothesis, expected utilities
(EU) can be calculated the same way expected values
are. For each possible event, the change in utility
ln(wealth after the event)
.
- ln(wealth before the event)
17.16
How Much Would You (Should You) Pay
For A Gamble?
Expected Utility Theory
For each possible event, the change in utility will be
weighted by the probability of that event occurring.
Let c be the cost charged to enter the game. The
expected utility of the lottery now converges to a
finite value:

EU 



1
k 1

ln
w

2
 c  ln( w )
k
k 1 2
17.17
How Much Would You (Should You) Pay
For A Gamble?
Expected Utility Theory (Mathematical Detail)




1
k 1
EU 

ln
w

2
 c  ln( w )
k
k 1 2
Which is convergent. Essentially the general term is
k/2k and the ratio of the k+1th. term to the kth.
converges to ½ < 1 (d'Alembert's 1768 ratio test).
17.18
How Much Would You (Should You) Pay
For A Gamble?
Expected Utility Theory
This formula gives an implicit relationship between the
gambler's wealth (w) and how much he should be willing
to pay to play (specifically, any c that gives a positive
expected utility).
17.19
How Much Would You (Should You) Pay
For A Gamble?
Expected Utility Theory
For example, with log utility
a millionaire (106) should be willing to pay up to
$10.94,
a person with $1000 should pay up to $5.97,
a person with $2 should pay up to $2,
a person with $0.60 should borrow $0.87 and pay up to
$1.47.
17.20
How Much Would You (Should You) Pay
For A Gamble?
17.21
How Much Would You (Should You) Pay
For A Gamble?
We should take into account that a casino would only
offer lotteries with a finite expected value.
Under this restriction, it has been proved that the St.
Petersburg paradox disappears as long as the utility
function is concave, which translates into the
assumption that people are (at least for high stakes)
risk averse.
17.22
How Much Would You (Should You) Pay
For A Gamble?
Probability Weighting
Nicolas Bernoulli (he was older brother of Daniel
Bernoulli, to whom he also taught mathematics) himself
proposed an alternative idea for solving the paradox.
He conjectured that people would neglect unlikely
events.
17.23
How Much Would You (Should You) Pay
For A Gamble?
Probability Weighting
Since in the St. Petersburg lottery only unlikely events
yield the high prizes that lead to an infinite expected
value, this could resolve the paradox.
17.24
How Much Would You (Should You) Pay
For A Gamble?
Probability Weighting
The idea of probability weighting resurfaced much
later in the work on prospect theory by Daniel
Kahneman and Amos Tversky.
However, their experiments indicated that, very much
to the contrary, people tend to overweight small
probability events. Therefore the proposed solution by
Nicolas Bernoulli is nowadays not considered to be
satisfactory.
17.25
How Much Would You (Should You) Pay
For A Gamble?
Rejection Of Mathematical Expectation
Various authors, including Jean le Rond d'Alembert and
John Maynard Keynes, have rejected maximization of
expectation (even of utility) as a proper rule of
conduct.
Keynes, in particular, insisted that the relative risk of
an alternative could be sufficiently high to reject it
even were its expectation enormous.
17.26
How Much Would You (Should You) Pay
For A Gamble?
One Cannot Buy What Is Not Sold
Some economists resolve 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.
17.27
How Much Would You (Should You) Pay
For A Gamble?
Finite St. Petersburg Lotteries
The classical St. Petersburg lottery assumes that the
casino has infinite resources. This assumption is often
criticized as unrealistic, particularly in connection with
the paradox, which involves the reactions of ordinary
people to the lottery.
17.28
How Much Would You (Should You) Pay
For A Gamble?
Finite St. Petersburg Lotteries
Of course, the resources of an actual casino (or any
other potential backer of the lottery) are 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.
17.29
How Much Would You (Should You) Pay
For A Gamble?
Finite St. Petersburg Lotteries
If the total resources (or total maximum jackpot) of
the casino are W dollars, then L = 1 + floor(log2(W))
(floor(x) is the largest integer less than or equal to x)
is the maximum number of times the casino can play
before it no longer covers the next bet.
17.30
How Much Would You (Should You) Pay
For A Gamble?
Finite St. Petersburg Lotteries
The expected value E of the lottery then becomes:


1
E   k  m in 2 k 1 ,W
k 1 2
L

k 1
1
2k
2
L W
  L
2 2
L W
  L
2 2
k 1



1
 2k  W
k  L 1

1
 2k
k 1
17.31
How Much Would You (Should You) Pay
For A Gamble?
Finite St. Petersburg Lotteries
Mathematical aside

m in 2
k 1

2k 1 if
,W  
 W if
2k 1  W
2k 1  W
cut off at 2k-1 = W
that is floor(log2(W)) = k-1 since an integer solution is
required.
17.32
How Much Would You (Should You) Pay
For A Gamble?
Finite St. Petersburg Lotteries
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):
17.33
How Much Would You (Should You) Pay
For A Gamble?
Finite St. Petersburg Lotteries
Banker
Bankroll
Expected value of lottery
$100
$4.28
Millionaire
$1,000,000
$10.95
Billionaire
$1,000,000,000
$15.93
Bill Gates (2008)
$58,000,000,000
$18.84
U.S. GDP (2007)
$13.8 trillion
$22.78
World GDP (2007)
$54.3 trillion
$23.77
$10100
$166.50
Friendly game
Googolaire
17.34
How Much Would You (Should You) Pay
For A Gamble?
Finite St. Petersburg Lotteries
A rational person might not find the lottery worth
even the modest amounts in the above table,
suggesting that the naïve 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.
17.35
How Much Would You (Should You) Pay
For A Gamble?
Simulation
A typical graph of average winnings over one course of
a St. Petersburg Paradox lottery shows how occasional
large payoffs lead to an overall very slow rise in
average winnings.
After 20,000 game plays in this simulation the average
winning per lottery was just under 8 dollars.
17.36
How Much Would You (Should You) Pay
For A Gamble?
Simulation
The graph encapsulates the paradox of the lottery:
The overall upward slope in the average winnings graph
shows that average winnings tend upward to infinity,
but the slowness of the rise in average winnings (a rise
that becomes yet slower as game play progresses)
indicates that a tremendously huge number of lottery
plays will be required to reach average winnings of even
modest size.
17.37
How Much Would You (Should You) Pay
For A Gamble?
Time Series Plot of mean
9
8
7
Game /
Simulate
mean
6
5
4
3
2
1
0
1
2000
4000
6000
8000
10000 12000 14000 16000 18000 20000
Index
17.38
How Much Would You (Should You) Pay
For A Gamble?
Iterated St. Petersburg Lottery
The above might not be a very satisfying explanation, so let us get
a better feeling for the problem by looking at a situation where
someone plays N = 2n games.
How much money would a “reasonable” person be willing to put up
front for the opportunity to play these N games?
17.39
How Much Would You (Should You) Pay
For A Gamble?
Iterated St. Petersburg Lottery
Well, in about half games (2n-1) he will win one dollar;
in about one quarter (2n-2) he will win two dollars;
in about one eighth (2n-3) games he will win four dollars; etc., until
in about one game he will win 2n-1 dollars.
In addition, there are the “fractional” numbers of games where
he wins much larger quantities of money (for example, in half a
game he will win 2n dollars, etc.), and this is indeed where the
infinite expectation value comes from, in the calculation above.
17.40
How Much Would You (Should You) Pay
For A Gamble?
Iterated St. Petersburg Lottery
But let us forget about these for the moment, in order to just get
a lower bound on what a reasonable person should put on the
table. Adding up the above cases gives the total winnings as
2n-1×1+2n-2×2+2n-3×4+…+1×2n-1) = 2n-1×n.
The average value of these winnings in the N = 2n games is
therefore 2n-1n/2n = n/2 = log2N/2.
17.41
How Much Would You (Should You) Pay
For A Gamble?
Iterated St. Petersburg Lottery
A reasonable person should therefore expect to win at least
log2N/2 dollars per game.
By “expect”, we mean that if the player plays a very large number
of sets of N games, and then takes an average over these sets, he
will win at least 2n-1n dollars per set.
This clearly increases with N, and goes to infinity as N goes to
infinity. It is nice to see that we can obtain this infinite limit
without having to worry about what happens in the infinite number
of “fractional” games.
Remember that this quantity, log2N/2, has nothing to do with a
17.42
true expectation value, which is only defined for N → ∞.
How Much Would You (Should You) Pay
For A Gamble?
Feller’s Law Of Large Numbers
More than 200 years after Daniel Bernoulli’s paper the
case of the St. Petersburg problem was reopened by
William Feller (1945).
Feller insisted that the question for the fair price of a
game of chance only made sense in connection with the
law of large numbers.
17.43
How Much Would You (Should You) Pay
For A Gamble?
Feller’s Law Of Large Numbers
According to this law, the average payoff in a long
series of independent repetitions of the same game
converges towards the expected value. Hence if the
price charged per game is only slightly smaller than the
expected value, the game in the long run becomes
profitable for the player.
17.44
How Much Would You (Should You) Pay
For A Gamble?
Feller’s Law Of Large Numbers
The traditional law of large numbers still holds for the
St. Petersburg game and states in this case that the
average payoff in a series of repeated games will
converge to infinity:
(E1+…+EN)/N → ∞
Ei being the expected payoff in the wth. game.
17.45
How Much Would You (Should You) Pay
For A Gamble?
Feller’s Law Of Large Numbers
However, this convergence to infinity can be extremely
slow.
If you bet a fixed amount on every instance of the
game, you are guaranteed to win in the long run.
However you might have to wait until the end of your
days before a large payoff washes away all your
previous losses.
17.46
How Much Would You (Should You) Pay
For A Gamble?
Feller’s Law Of Large Numbers
This gave Feller the idea that one should consider a
variable price, dependent on the number of repetitions
you are allowed to plan.
Feller supported this idea by a Law of Large Numbers
essentially stating that the average payoff per game in
the long run is about log2(N)
(E1+…+EN)/Nlog2(N) → 1
17.47
How Much Would You (Should You) Pay
For A Gamble?
Feller’s Law Of Large Numbers
Leading to the conclusion that the value of the St.
Petersburg game ought to be log2(N) for someone who
may play the game N consecutive times:
Number of Games 10 100 1000 10000
Prices per Game 3.32 6.64 9.97 13.29
17.48
Facebook - IPO Purchasers May Face A Paradox
(IPO - Initial Public Offering)
Facebook and the St. Petersburg Paradox
by Jason Zweig, The Wall Street Journal,
February 4, 2012
A high-growth stock like Facebook is a lot like
that St. Petersburg coin. The potential payoffs
are enormous, although not infinite—and the
game might peter out all too soon.
17.49
Facebook - IPO Purchasers May Face A Paradox
At the end of 2010, 608 million people actively
used Facebook every month; by this past Dec.
31, 845 million people did. If Facebook keeps
growing that fast, more than 22 billion people
will be using it 10 years from now, or three
times the estimated population of the planet
today.
17.50
Facebook - IPO Purchasers May Face A Paradox
The main difference is that the warm glow
Facebook users get from its services may blind
them to the St. Petersburg Paradox. Because
Facebook connects people so powerfully, the
people who use it may feel powerfully
connected to the company, too—and to its
stock.
17.51
Facebook - IPO Purchasers May Face A Paradox
Nevertheless,
"it's an incredibly difficult thing to forecast
the future cash flows of this kind of company,
even for quantitative investors,"
says Charles Lee, a professor of accounting at
Stanford University's business school and a
former head of equity research at Barclays
Global Investors (since acquired by BlackRock).
Thus, says Mr. Lee,
"once your projections go out beyond two or
three years, you're in very murky waters." 17.52
Facebook - IPO Purchasers May Face A Paradox
The slightest stumble in high growth rates can
lead to enormous changes in value in fastmoving companies. Since today's stock prices
are highly sensitive to projections of long-term
future profits.
17.53
Facebook - IPO Purchasers May Face A Paradox
Just as the coin-flipping game in the St.
Petersburg Paradox can end on any toss, even
the fastest-growing company's upward
trajectory can flatten in a flash. Just ask
shareholders in Amazon.com, who this week
lost 8% in one day on the company's warning
that it might lose money next quarter.
17.54
Facebook - IPO Purchasers May Face A Paradox
If Facebook comes out at the high end of the
valuation range proposed for the stock in its
first sale to the public, the company would
have a total market value of around $100
billion.
17.55
Facebook - IPO Purchasers May Face A Paradox
Now, let's say Facebook will be as successful in
the future as Google already has been,
suggests Jay Ritter, a finance professor at the
University of Florida and an expert on initial
public offerings.
"Facebook is basically on Google's trajectory,
so I think that's a very reasonable scenario,“
he says.
17.56
Facebook - IPO Purchasers May Face A Paradox
Facebook, at $3.7 billion in revenues and $1
billion in profits in 2011, already has nearly
three times the sales and 10 times the profits
that Google had when that company first listed
in 2004.
17.57
Facebook - IPO Purchasers May Face A Paradox
Now, imagine that Facebook continues its
torrid growth and expands over the next 10
years until it has grown as big as Google is
today—with annual revenue of nearly $40
billion and net income of almost $10 billion.
That would imply that Facebook will grow
roughly tenfold over the coming decade—an
average annual growth rate of about 26%,
which is seldom sustained by big companies.
17.58
Facebook - IPO Purchasers May Face A Paradox
In this bullish scenario, what would happen to
Facebook's stock?
17.59
Facebook - IPO Purchasers May Face A Paradox
At today's valuations, Google's shares trade at
a total market value of just over $190 billion.
If Facebook's shares rose from a total initial
value of $100 billion to $190 billion 10 years
from now, they would deliver a 90% cumulative
gain, for an average annual return of 6.8%.
Bottom line:
"The valuation is so high today that the upside
potential is limited,"
Mr. Ritter says.
17.60
Facebook - IPO Purchasers May Face A Paradox
Like most companies planning a public offering,
Facebook declined to comment.
17.61
Facebook - IPO Purchasers May Face A Paradox
There certainly is a chance that Facebook's
first public shareholders will be richly
rewarded over the years to come—the same
way someone taking the bet in the St.
Petersburg Paradox could become fabulously
wealthy. But, with Facebook, the odds would be
a lot better if the price to make this particular
bet were a lot lower
17.62