thanks, Joao. I think you are using the same rules as mine, but instead of stopping when I'm 1 dollar up, I stop when I'm 2 dollars, 3 dollars ... 26 dollars up. Then the question is: which number between 1 and 26 gives the biggest expected payoff per game. Is that it?
If so, there must be a theoretical solution, and this might be where the 'difficult' warning comes in.
I'll think about this more. If k is the number of dollars chosen as a quit level (i.e. I stop the game as soon as I'm k dollars ahead), then I think we need to consider the probability that in a sample of n cards, n/2 + k (if n is even) or n/2 + k/2 (if n is odd) are red, given equal numbers of red and black cards. Must be a binomial distribution. But 'must be' needs to be justified... Hope to be back on this one.