The game is as follows: I put in a dollar and if I get heads, I double my money. I can then continue playing and double my $2. Basically, I'm always allowed to continue playing and double the previous amount. However, if it's the coin lands on tails, I lose whatever amount I'm currently playing for and have to restart the game (which still would be a net loss of -1 because that's what I paid to play).
Since I can stop the game at any point and cash my winnings out, when should I do that?
Another assumption is that the casino has an infinite amount of money so it can play forever. However, although I'm very very rich and can play the game for a long time, I can't play it forever.
The starting price is $1 and the winnings double after each turn. A loss only results in me losing the initial dollar and the potential of receiving more if I would've cashed out instead.
My question is whether there would be an "optimal" strategy playing. That means, should I play the game and hope for, let's say, 5 in a row, then cash out (resulting in me receiving $32) and then start a new game? Or should I always cash out after 3 wins in a row? Perhaps 10 wins?
Never cashing out is not an option since I can't play the game forever and at some point I would have no money left to play.
At which point should I decide to collect my winning and then restart the game? Will I eventually go bankrupt or would I become infinitely rich at some point?
EDIT: It is basically this question (When to stop in this coin toss game?) but the reward is not +100 but instead the double of the pot.
EDIT 2: I have thought more about this problem and it seems for me that the expected payoff should be zero. Let's assume that on the third round, I would win \$8 (2 -> 4 -> 8). For that to happen, I would need to double my bet three times. The probability of that happening is $\frac{1}{8}$, so in theory it should happen one out of eight games.
That would mean, that I would need to play 8 games and therefore pay a total of \$8 to receive my winnings of \$8. The same applies to $16 and every other amount.
Is that correct or am I missing something?
Best Answer
In the St. Peterburg Paradox a player is given a coin and simply flips the coin until they get a tail. Their money doubles everytime they get a head. The paradox is that you would have an infinite expectation if the casino has infinite money.
That is
$$ E(X) = \frac{1}{2}\cdot 2 + \frac{1}{4} \cdot 4 + \frac{1}{8}8 + \frac{1}{16}16 + \cdots \\ = 1 + 1 + 1 + 1 + \cdots \\ = \infty $$
as you stated you don't have an infinite amount of time but if they have an infinite amount of money then your optimal strategy is to simply sit there until you pass away
you don't actually go bankrupt. you take whatever is in the pot. There is a really high probability you simply double your money a ton of times. You'd basically live in a casino however it goes against human principles.