The game that stays huge even with a cap

Question

A casino flips a fair coin until the first heads. If the first heads is on flip $n$ , it pays $\$ 3^n $(not$ 2^n $). So heads on flip 1 pays$ $3 $, flip 2 pays$ $9 $, flip 3 pays$ $27$.

What is the expected payout? Then the house caps its maximum payout at $\$ 1$ million. What is the fair price now, and how does it compare to the capped St. Petersburg game?

Accepted Answer

Uncapped, the mean diverges even harder than St. Petersburg. The first heads lands on flip n with probability 2^{-n}, paying 3^n, so each term is P(n)\cdot 3^n = \frac{3^n}{2^n} = \left(	frac32ight)^n. These terms grow geometrically, so E[	ext{payout}] = \sum{n\ge1}(3/2)^n = \infty. In the classic game each term equalled exactly 1; here they explode, which foreshadows that a cap will not tame this one nearly as well. Now cap the payout at \3^L with 3^{12} = 531{,}441 \le \1	ext{M} < 3^{13}, so L = 12. For n \le 12 you still receive 3^n; for n 12 you receive the cap 3^{12}. E = \sum{n=1}^…

The game that stays huge even with a cap

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