Coin flipping Expected value and Probability

Question

Consider the following game:

• A game costs €100.
• A balanced coin is tossed.
• If it comes up HEADS, the game ends.
• If it's TAILS :
• The player receives a bonus (initially 1€).
• The next bonus is doubled (2€, 4€, etc.).
• Raise the coin as long as it's TAILS.

eg. TAILS - TAILS - TAILS - HEADS => -100 + 1 + 2 + 4 = -93

Questions :

• What is the player's expected payout?
• What is the smallest number of games the player has to play for the probability of winning to be greater than 1/2?

Answer 1

This game is referred to as the "St. Petersburg Paradox". See this wiki for a full explanation:

https://en.wikipedia.org/wiki/St._Petersburg_paradox

Answer 2

This set-up is similar to the lottery game considered by Bernoulli. The expected payout is infinity. That is because the sum of the expected payout under all lengths of the game is always a sum of ones: $$ -100 + \sum_1^{\infty} 2^i \cdot \frac{1}{2^i} = -100 +\sum_1^\infty 1 = \infty. $$ For your second question, you should clarify what 'winning' is supposed to be. Is it to end up with a positive net payout? In that case, find $i$ such that the partial sum exceeds 100. That should be something you can easily do. (You must win $1$ how many times for you to justify paying 100?)

Answer 3

I think I've managed to write a program that calculates the probability of making a net profit as a function of the number of games played. Unfortunately, memory fails after 1000 games... The graph looks very strange and, according to my calculations, it would take more than 10^50 games to reach 50% equity. Does this sound plausible?