# When to quit a coin toss doubling game?

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?

• The optimal strategy is to never leave the game until you have to, I.e. until the coin shows tails. At every point in the game, your expected benefit is infinite, so it would never be ideal to leave voluntarily. – David Oct 17 '18 at 0:41
• Yes, that's true. But if I would play the game in real life and would have, let's say, a billion dollars, I would lose all my money if I would keep playing. I understand that it's correct to always continue playing but it's not applicable to real life. – OldMcDonald Oct 17 '18 at 0:46
• How would you lose all your money if you keep playing? That's not how the game works. – littleO Oct 17 '18 at 0:56
• I confused how the question of the St. Petersburg paradox and have clarified my original question. Sorry for that. – OldMcDonald Oct 17 '18 at 1:00

## 1 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

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?

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.

• Thanks, perhaps I confused something about the St. Petersburg paradox. My question is different, I've changed the question to further reflect what I want to find out. – OldMcDonald Oct 17 '18 at 0:56
• I'd have to think about the other one. – Shogun Oct 17 '18 at 1:05
• The answer of when to quit should be determined by Kelly Criterion. en.wikipedia.org/wiki/Kelly_criterion – irchans Oct 17 '18 at 1:39
• @irchans Can it though? I'm not familiar with Kelly Criterion but it seems to me that it's more about how much you should bet in comparision to the payoff, while this problem involves a constant bet amount. – OldMcDonald Oct 17 '18 at 1:47
• " Kelly bet is a formula used to determine the optimal size of a series of bets in order to maximize the logarithm of wealth." In your case there is no varying bet size, but you can determine a strategy that maximizes the expected value of the log of your wealth. If I remember correctly, following such a strategy gives the maximal exponential growth rate for your wealth. – irchans Oct 17 '18 at 4:47