- The St. Petersburg paradox was probably first discussed by Nicolas Bernoulli (one of a famous Swiss family of mathematicians in the 18th century) in 1713 in letters to another mathematician, Pierre Rémond de Montmort. It later appeared in a famous paper by his cousin Daniel Bernoulli submitted to the Imperial Academy of Sciences in Petersburg in 1738 [1], where he also discussed resolutions of the paradox. Here is another contemporary description from a letter by Gabriel Cramer, another 18th-century mathematician:
"For the sake of simplicity I shall assume that A tosses a coin in the air and B commits himself to give A 1 ducat if, at the first throw, the coin falls with its cross upward; 2 if it falls thus only at the second throw, 4 if at the third throw, 8 if at the fourth throw, etc. The paradox consists in the infinite sum which calculation yields as the equivalent which A must pay to B. This seems absurd since no reasonable man would be willing to pay 20 ducats as equivalent."
a. Assuming that you are allowed to play this game once in a casino exactly as stated above, what would you yourself be willing to pay to enter the game (reasoning rationally, and momentarily forgetting any personal objections against gambling)?
b. The formulation of the game and the estimate of its value as infinity makes several unrealistic assumptions, both from the player's point of view and from the casino's point of view. What are these assumptions?
c. One is that even a large casino does not have infinite resources, and would have to decide on a maximum number of rounds in the game to avoid going bankrupt. Assume that the maximum payout of the casino is X, and implement and perform a simulation that estimates the expected value of the game in this case.
First, choose a reasonable number of X (in SEK) for a large casino. As a guideline, one of the largest casinos in the world is The Venetian in Macau. Their yearly revenue prior to the pandemic was almost 3 billion USD. So, a maximum payout of 10 million SEK or larger seems quite possible.
Present the result of your simulations as a diagram that shows the average payoff calculated over the last n games as a function of n. Choose a suitable maximal number of games N yourself, repeat the simulation a number of times, and include the result of all simulations as well as a mean value in the diagram.