simulating the St. Petersburg Paradox [duplicate]

Question

• from the following simulation it seems the fee I am willing to pay should be smaller than 10, instead of $\infty$, differs from the paradox, what is happening here?

Edit:

For those that aren't familiar with the St. Petersburg Paradox, the Stanford Encyclopedia of Philosophy describes it as follows: "A fair coin is flipped until it comes up heads the first time. At that point the player wins $\$2^n$, where n is the number of times the coin was flipped. How much should one be willing to pay for playing this game?"

As the article points out, if we try to solve this using the standard method for expected value we get $$ \frac{1}{2}\times2+\frac{1}{4}\times4+\frac{1}{8}\times8 + \dots = 1+1+1 + \dots = \infty $$ So the player should be willing to pay any amount, even though they are likely to win very little.

Answer 1

You are "only" running about $2^{20}$ simulations, so your average reward is about $\frac12\cdot20 = 10$. The average reward is infinite because you can win arbitrarily large rewards, albeit with adequately small probabilities.