The St. Petersburg game is a 300 year old probability paradox that has confounded people for centuries. It's counterintuitive in that the mathematical expectation is infinite, but no one in their right mind would actually pay huge amounts of money to play this game. Where is the disconnect?
There's still research done on this problem, although a lot of the discussion is no longer mathematical, but more economic/philosophical. The focus is now more on modeling how a rational agent will behave and how they should value this game. A good overview from the philosophy point of view is:
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From the math side, I think the best reference is "An Introduction to Probability Theory and Its Applications" by William Feller. Feller proved that the St. Petersburg game converges in probability to log_2 and explains this problem in detail in his book.
Timestamps:
0:00 Intro to Problem
1:55 Example
4:16 St. Petersburg Simulations
5:24 Estimating the Average Payout's Growth Rate
9:17 Other Considerations
