Assume we are a player in a fair game with expectation value $0$. He starts the game with a total capital of $M_0$. The Player is always betting the exact same amount $B$ and wins $2B$ with probability $50\%$ or looses $B$ with probability $50\%$. He starts the game with a total capital of $M_0$.
How to determine the best value for $M_0$ as function of $B$, such that the number of games played is maximized.
Sure the bigger $M_0$ the better. But is there something like a sweet-spot? It is a one-dimensional random walk so it would make sense to choose a starting capital $M_0$ that is greater than the variance of the random walk $\sqrt{n}$, with a given number of steps $n\in\mathbb{N}$.
How does casinos or vending-machine operators with a slight advantage in every game manage this problem? Potentially they always have the risk of back luck or not? How does the minimize this risk?
Any assistance or thoughts or explanations would be much appreciated.