The paradox against a real casino's bankroll

Question

In the St. Petersburg game a fair coin is tossed until the first head; the payout on toss $n$ is $\$ 2^n $, giving an infinite expected value. A real casino cannot pay unbounded sums, so suppose the house caps its maximum payout at$ $1{,}000{,}000$.

With that cap, what is the game actually worth?

Accepted Answer

Find the toss beyond which the raw payout 2^n would exceed the cap. Since 2^{19} = 524{,}288 and 2^{20} = 1{,}048{,}576 1{,}000{,}000, the payout hits the cap starting at toss n = 20. For n \le 19 you receive the full 2^n; for n \ge 20 you receive the capped \1{,}000{,}000. Split the expectation at n = 19: E = \underbrace{\sum{n=1}^{19} \frac{1}{2^n}\cdot 2^n}{	ext{full payouts}} + \underbrace{\sum{n=20}^{\infty} \frac{1}{2^n}\cdot 1{,}000{,}000}{	ext{capped tail}}. The first sum is 19 terms each equal to 1, so it contributes 19. The tail is 1{,}000{,}000 \sum{n=20}^{\infty} \frac{1}{2^n} =…

The paradox against a real casino's bankroll

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