Introduction
The St. Petersburg Paradox, also known
as the St. Petersburg Lottery, appeared in the Commentaries of the Imperial
Academy of Science of Saint Petersburg in 1738. It was presented and resolved
by Daniel Bernoulli who resided in the eponymous city at the time. Nonetheless,
the problem was first introduced by Daniel’s brother Nicolaus Bernoulli in a
letter to a French mathematician in 1713.
The Bernoulli Family
The Bernoulli family generated many
prominent mathematicians and Daniel and Nicolaus were one of the most
prominent. They were Swiss although Daniel was actually born in the Dutch
Republic. Nicolas was the older brother born in 1695, but he died early in 1726.
He actually taught mathematics to Daniel who was born in 1700 and lived a long
life dying in 1782. Daniel was most known for his work in fluid mechanics and
in probability and statistics. Leonhard Euler was actually a student of their
father.
The Paradox
The problem arises from a game of
chance where a coin is tossed with equal ½ odds of yielding heads or tails. The
payout starts at 2 dollars and every time a head appears, the pot is doubled.
The expected value of the game is thus
the following:
EV = ½ * 2 + ¼ * 4 + 1/8 * 8 + 1/16 *
16 + …
= 1 + 1 + 1 + 1 + …
= ∞
Assuming the “casino” where the game is
played has unlimited resources, the expected payoff for playing the game is
infinite. Due to this unlimited theoretical payoff, one should be willing to
pay any sum of money to play the game. However, research showed that most
people were not willing to pay even small sums to play the game. The paradox is
the discrepancy between the infinite payoff and the people’s willingness to pay
to play.
Bernoulli’s Take
His resolution had to do with the
difference between expected value and expected utility. He introduced a utility
function, the concept of expected utility (which was referred to moral
expectation versus mathematical expectation back then) and the concept of
diminishing marginal utility of money.
The utility function is the function
that take into account people’s preference. What Bernoulli put forward with his
expected utility theory is there is something other than expected value which
people look at in situations with uncertain outcomes. This something is the
impact of the outcome of the gamble on the person taking the gamble, or the
impact on his happiness. Finally, the concept of diminishing marginal utility
of money means that the more money you have, the least impact an extra dollar
will have on you. If we think about it, this is very intuitive. Let’s look at
two situation:
11) Suppose that it’s the first week of the
semester and you had a sick job this summer which compensated you handsomely
and you have $3000 dollars saved for the semester. Then someone offers you to
fill in for his shift as a line judge at the football game. You’ll get paid $30
dollars.
22) Suppose that it’s the first week of December,
you are back from Thanksgiving break. You have had a blast of a semester so
far, but you had expensive taste and you have only $150 dollars left from your
summer. Then someone offers you to fill in for his shift as ball boy at a
tennis tournament in the fieldhouse. You will work for 3 hours on a Sunday
morning and will make $25 dollars.
It is very likely that you would not fill in for your friend
in situation 1, but probably would in situation 2, and that is despite the fact
that the payoff in situation 2 is less than in situation 1. However, that $25
has a much bigger impact on your holdings when you have $150 than $30 when you
own $3000.
Here we can see a graph of a relationship where utility is marginally
decreasing as X increases.
Bernoulli himself put forward a common utility model which
is the logarithmic function U(w) = ln(w) where U is utility and w represents a
gambler’s wealth. His utility function attempted to determine the cost someone
would be willing to pay to actually play the game aforementioned. The expected
utility of the game, given a cost to play of c would be the following:
Following this formula, we learn that someone with a million
dollars would only be willing to pay up to $10.94 to play the game and someone
with a wealth of $2 would be willing to pay up to $2.
Conclusion
Bernoulli’s take on the paradox was studied and many
mathematicians improved his work. However, he pioneered utility theory and its
application in mathematical modeling of behaviors of people in situations where
the outcome is uncertain.
Looking forward to our discussion in class tomorrow!
Cheers,
Sam
Works Cited:
Hi Sam, nice post! Would someone with a wealth of 2 be willing to pay up to $2 since 2 is not much more than 0 so they might as well try to make more?
ReplyDeleteSam, I can see youre all about the money making based on your two recent blog posts. But this theory is so funny to me because its so true. But im right there with Lauren, because i guess you would always want to try to make the most possible amount right?
DeleteUtility theory is key in a lot of areas of game theory, economics, and mathematical finance! Daniel Bernoulli solved the St. Petersburg Paradox in 1738 with this revolutionary construction, but the first very important use of this wasn't until the 20th century (!), when von Neumann and Morgenstern proved the the von Neumann-Morgenstern utility theorem. Really interesting stuff abounds with utility theory. Is this partially a plug for Math 321? ...Maybe.
ReplyDeleteVery interesting stuff! It never occured to me that gamblers need to take their personal utility of an extra dollar into account. While it makes sense to accept or reject a job offer based on how much money you currently have, I never imagined gambling as having a cost other than the absolute value of the money you put down. Clearly, each person's dollar could be worth a different amount (it felt weird typing that), and so different people would play the coin flip game a different number of times. Again, very interesting stuff.
ReplyDeleteI never realized the breath of Bernoulli's work. I've encountered Bernoulli's equation of partial pressure in physics, and Bernoulli differential equations, but I didn't know he also played a hand in game theory (and indirectly economics and utility theory). I guess he really had his shit together.
ReplyDeleteI love spending money I don't have... so I find myself walking the fine line between poverty and payday all the time. It's amazing how much my value of money fluctuates!