Suppose there is a magical slot machine at a casino. You can start with however big of a bet you want, and for each pull of the slot machine, you have a 99% chance of doubling your money and a 1% of losing your money.
The game has the following rules:
1) You can only play this game once in your life. You keep pulling until you quit the game or lose your money.
2) You must bet all the money that you have gained since you started plus your starting bet. For example, if you start with a \$10 bet, then on the first pull, you have a 1% chance of losing the money, and a 99% of getting to \$20. If you want to pull again, you must bet \$20, in which case you would have a 1% chance of losing the money and a 99% chance of getting to \$40. This continues until you lose or quit the game.
This seems to be like a very winning bet, where your expected value is almost double your starting bet. Game theory would tell you that when your "return on investment" is greater than 0, you should theoretically take the bet. However, if you always take the bet, then you would just keep playing the game until you eventually lose all your profit and your original bet, making your return on the investment negative for the whole game.
It also doesn't make sense to quit after any pull because the pulls are independent events from each other.
I tried calculating the expected value after n pulls, and found that this diverges, meaning you should never stop pulling.
Here is what I got when I tried to calculate the expected value:
Starting with an initial bet of $x$,
EV = $(.99 \cdot 2)^nx$
after n pulls on the machine. To maximize the expected value (the goal of the game), it seems like you would let $n \rightarrow \infty$. This obviously isn't right because after infinity pulls you are guaranteed to lose all your money and have a negative ROI.