Utility of Money and the St. Petersburg Paradox

Consider the following game:

We will flip a fair coin, until a tail appears for the first time.

• If the tail appears in the first throw, you win $2^1=2$ dollars.
• If the tail appears in the second throw, you win $2^2=4$ dollars. 
• If the tail appears in the third throw, you win $2^3=8$ dollars.
• ...
• If the tail appears in the $n$-throw, you win $2^n$ dollars.

What is the amount of money that someone should risk to enter this game? (This question works best when given to a person that claims to never play a lottery, roulette, or any gambling game, because the expected return is lower than the bet.)

Computing the expected return of this game, we have:

$E=\frac{1}{2}\cdot 2+\frac{1}{4}\cdot 4 + \frac{1}{8}\cdot 8 + \cdots = 1+1+1+ \cdots =\infty$

In other words, the expected utility is infinity, and a rational player should be willing to gamble an arbitrarily large amount of money to enter this game.

Can you find anyone willing to bet \$1,000 to play this game? Or \$10,000? Or even \$100? Yes, I did not think so.

This paradox is called the St. Petersburg Paradox, posed in 1713 by Nicholas Bernoulli and solved in 1738 by Daniel Bernoulli. Since then, a number of potential explanations appeared.

The most common approach is to use expected utility theory. In this case, we introduce a utility function $U(x)$, which describes the "satisfaction" that someone would get by having $x$ amount of money.

Utility of Money: The basic idea is that people do not bet based on the absolute amounts of the return but rather based on the utility of the award. The value of an additional \$100 when I have \$100 in the bank is much higher compared to the case when I have \$1,000,000 in the bank. This means that the "utility of money" function is a concave function of the available funds.

Just for demonstration, below you can see such a concave utility-of-money function that we have computed as part of a research project:

This concavity also partially explains the "risk aversion" that most people have: they prefer certainty over uncertainty. This means that they will reject even a reasonable bet with positive expected return. Why? Notice that the utility gained by winning is smaller than the decrease in utility that results from losing the bet. The higher the concavity, the higher the risk aversion.

If you want to read more about utility of money and its applications to portfolio management, insurance, and analysis of other cases, take a look at this book chapter.

So, next time that someone claims never to engage in any bet with a negative expected return, give the setting of the Bernoulli paradox with the positive expected return and observe the reactions...