Win this poker game

52 poker cards (half red and half black). Every time you draw a card, if you draw a red card, then you win one dollar. If you draw black card, then you lose one dollar.

Say you start with $n$ dollars, if you can stop/quit this game anytime you want, how much are you willing to pay this game?

What would be your optimal play strategy?

Attempt:

I think this is very similar to the gambler's ruin problem; We can let each state denote the amount of money that the gambler has. However, the transistion probability would be dependent/varied from state to state.

• Do you put back the card into the 52 card piles after u draw it? – Siong Thye Goh Mar 26 '17 at 5:43
• if I can put it back, then it will become gambler's ruin problem. Let us assume that's the case. Even with this assumption, I still have trouble to figure out "how much you are willing to pay for this game", if we can stop anytime we want. – wrek Mar 26 '17 at 5:45
• what does this have to do with markov chains/processes ? – mercio Mar 26 '17 at 8:38
• stopping when $R < B - \sqrt(B)$ seems to be pretty close to the optimal strategy. – mercio Mar 26 '17 at 9:02
• "Let us assume that's the case" ?? The question is interesting when one does not put back the cards. – Did Mar 27 '17 at 7:17