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Feb/15/2008
The St. Petersburg Paradox

You're playing a coin flipping game. If the coin lands on tails on the first toss, you receive $1 and the game ends. If it instead lands on heads, the game goes on and the payoff doubles until the first tails. The structure is as follows :

$1 if T
$2 if HT
$4 if HHT
...and so on.

What would you pay to enter this game?

Looking at the expected payoff may be an apt method. So by summing every probability weighted state, you get

E = $1*1/2 + $2*1/4 + $4*1/8 + ...
= 1/2 + 1/2 + 1/2 + ... = ∞

This result clearly makes no sense, and the paradox is named after Daniel Bernoulli who published a solution in the Commentaries of the Imperial Academy of Science of Saint Petersburg in 1738. He introduced a utility function with a diminishing marginal utility of money. This basically means that in the equation for expectancy above you give each payoff a utility value [ $X -> utility($X) ] and that for each new coin toss the utility from the payoff is less than double that of the previous. This way the expectancy will be much more meaningful, as you now can take expected utility from this game and pay a corresponding amount [ EU = U(what you pay to enter) ].

Some economists have argued that the game cannot last until infinity. Let's just see how fast the payoff increases:

Toss# - $ Payoff
01 - 1
02 - 2
...
18 - 131,072
19 - 262,144
20 - 524,288
21 - 1,048,576 > A million
22 - 2,097,152
...
30 - 536,870,912
31 -1,073,741,824 > A billion
32 - 2,147,483,648
...
36 - 34,359,738,368
37 - 68,719,476,736 > Bill Gates
38 - 137,438,953,472
39 - 274,877,906,944
40 - 549,755,813,888
41 - 1,099,511,627,776 > A trillion
42 - 2,199,023,255,552
43 - 4,398,046,511,104
44 - 8,796,093,022,208
45 - 17,592,186,044,416 > US GDP
46 - 35,184,372,088,832
47 - 70,368,744,177,664 > World GDP

Back to expectancy formula; even with a potential gain of the World's GDP you'd not pay more than $23.5. If this game teaches us anything it is not so much about diminishing marginal utility, but about the abstractions of mathematics.

Mathematics is in your head only
I've earlier philosophized about all mathematics being based on an abstraction; the number one. Infinity, it seems, is independent of this base unit [ X/2 ≠ X would be true for any X except ∞ and zero ]. And this is what makes it so un-understandable. At least for me it is. I may know the definition of infinity, but I find it very hard to obtain an intuitive impression of it.

For instance; do you agree with me when I tell you that 0.999... (with an infinite number of 9s) is equal to 1?

It is not less than 1. It IS equal! The easiest way of making a skeptic agree is by asking if 1/3 = 0.333... .

This is true, and then it's obvious that since
1/3 * 3 = 1
and
0.333... * 3 = 0.999...
it must be the case that
0.999... = 1

What I want to come to here is that mathematics is an artificial tool for thought. There is no such thing as infinity in the universe. Except for inside your head. But as the mental tool it is, it can improve your actions. Just think of humankind. Mathematics exists only in our minds, but its impact through our actions (and the scientific method in particular) has become so immense we conquered the world.

While the good of mathematics comes from it improving the understanding of our surroundings, the St. Petersburg Paradox does the opposite. If we do not understand the mathematical tools we are using, its potential negative impact is immense. And in this case it is not a mathematical mistake, but a mistake in understanding the mathematics itself.

Try to make the coin a little bit unfair so that the chance of tails is > 1/2. Now calculate with an infinite maximum payoff, and see that the expected payoff is much more graspable. This comes from the probability weighted payoffs will converge toward zero, and then the infinity abstraction is close enough to the real result for the simplification in mathematical calculation to be worthwhile.

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