How Much Would You (Should You) Pay For A Gamble?

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



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



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.

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17.4But how much should you pay?

0.5Head

0.5 . . .

Head

0.50.5 TailHead 8

0.5 0.5Head Tail

4

0.5Tail

2

0.5Tail

1


The expected value (E) is thus

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...

...E

2

1

2

1

2

1

2

1

816

14

8

12

4

11

2

1


The expected value (E) is thus, or notationally

17.6

1

1

1

2

1

22

1

k

k

kk

E

where k counts the number of tosses.


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.

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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.

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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

http://plato.stanford.edu/archives/fall2004/entries/paradox-stpetersburg/


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


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.

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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.

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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.

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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.

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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



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)

.

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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:

17.17

)wln(cwlnEUk

kk

1

122

1


Expected Utility Theory (Mathematical Detail)

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).

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)wln(cwlnEUk

kk

1

122

1



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).

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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.

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17.21


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.

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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.

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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.

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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.

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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.

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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


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



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.

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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.

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The expected value E of the lottery then becomes:

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L

kkL

L

k Lkk

k

k

kk

WL

WL

W

W,minE

k

22

2

1

22

2

12

1

22

1

1

1 1

1

2

1

1



Mathematical aside

cut off at 2k-1 = W that is floor(log2(W)) = k-1 since an integer solution is required.

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WifW

WifW,m in

k

kkk

1

111

2

222



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):

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17.34

Banker Bankroll Expected value of lottery

Friendly game $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

Googolaire $10100 $166.50



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


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.

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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.

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17.38

Game / Simulate

20000180001600014000120001000080006000400020001

9

8

7

6

5

4

3

2

1

0

Index

mean

Time Series Plot of mean


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?

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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



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.

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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 true expectation value, which is only defined for N → ∞.

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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



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



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



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



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) → 117.47



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:

17.48

Number of Games 10 100 1000 10000

Prices per Game 3.32 6.64 9.97 13.29

Facebook - IPO Purchasers May Face A Paradox

(IPO - Initial Public Offering)

Facebook and the St. Petersburg Paradoxby 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.

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http://online.wsj.com/article/SB10001424052970204662204577200862677176998.html?KEYWORDS=jason+zweig

http://online.wsj.com/article/SB10001424052970204662204577200862677176998.html?KEYWORDS=jason+zweig


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.

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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.

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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."

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The slightest stumble in high growth rates can lead to enormous changes in value in fast-moving companies. Since today's stock prices are highly sensitive to projections of long-term future profits.

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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.

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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.

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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.

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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.

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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.

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In this bullish scenario, what would happen to Facebook's stock?

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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.

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Like most companies planning a public offering, Facebook declined to comment.

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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

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How Much Would You (Should You) Pay For A Gamble?

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