Posted by: ardaayvaz | November 29, 2008

## St. Petersburg Paradox

The couple of posts below about the problem known as Sultan’s Dowry problem, or secretary problem encouraged me to write another post related to probability and decision theory. I want to talk about St. Petersburg paradox.

St. Petersburg paradox refers to a game where you toss a coin until the first tail appears. And the pot you win starts with 1 dollar, and gets doubled with every toss of heads until the first tail appears. The question is then how much you would be willing to pay to participate in this game, and we assume that the casino owner has unlimited resources.

The usual approach is to calculate the expected value of the game :

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

However, the series doesn’t converge, and the sum goes the infinity. So that means we should be willing to pay anything to play this game, since the expected value is infinitely large. Yet, it intuitively seems very hard to justify paying this sum to participate in the game, and there are some possible explanations to explain this paradox.

The first approach is to argue that the assumption of unlimited resources is a wrong one. Once we replace this assumption with a limited maximum prize that can be payed, the expected value of the game decreases drastically. Even when the maximum possible prize is equal to the world’s total GPA, which is around 54 trillion dollars, the expected value of the game is about only 23.7 dollars.

The second approach is to still keep the unlimited resource assumption, but change the way we make our decision about entering the game. Instead of calculating the expected monetary value of the game, we can assumed a utility function of our wealth U(w) and calculate the expected change of our utility caused by playing the game. We usually use a concave utility function, such as U(w) = ln(w), where the increases in wealth will have less effect on the utility as w increases. Once we use this approach in our decision making process, the expected value of the game becomes less than infinity.

Similarly, behavioral economists try to approach the issue as well, saying that people automatically discount very unlikely events. This was first proposed by Nicolas Bernoulli. Such views paved the way to prospect theory, created by Daniel Kahnemann, however through clinical experiments they found that people in fact overweight unlikely events in their judgment

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## Responses

1. Infinity is always a tricky subject. Back in ancient philosophical Greece, Zeno of Elea wanted to walk across the room. However, to do this, he had to first get halfway across the room first, which makes sense. Then, from there, he would half to walk halfways across the remaining half of the room… Since the sum 1/2+1/4… (similar to the series of the St. Petersburg paradox) equals one only as n approaches infinity, Zeno could never get across the room, could he? How frustrating, to be unable to traverse such a small, finite length! But that’s just math. In reality, I’m sure Zeno’s friends would have hurled him through the door themselves by this point.