The same game, but the prize is capped

Question

Take the same doubling coin game: a fair coin is flipped until the first tails on flip $k$ , paying $2^k$ dollars. But now the casino caps any prize at $\$ 2^{20} $(about a million dollars): if you would win more than that, you simply get$ $2^{20}$.

What is the fair value (expected payout) of the game now?

Accepted Answer

Split the sum at the cap. For an uncapped game each term \left(	frac12ight)^k \cdot 2^k equals \1, and that infinite pile of \1s is what blew up. The cap tames the tail. For k = 1 through 20, the payout 2^k is at or below the cap, so those terms are unchanged: 20 terms, each contributing \1, for \20. For k 20, the prize is frozen at \2^{20}. The total probability of the game running past flip 20 is \left(	frac12ight)^{20}, and it always pays the cap, contributing 2^{20}\cdot\left(	frac12ight)^{20} = \1. Adding up, E[	ext{payout}] = 20 + 1 = \21. A prize cap of a million dollars turn…

The same game, but the prize is capped

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