I'm looking for an optimal strategy as well as the derivation of this strategy for a simple game.
- Let there be $N$ white balls and one black ball in a pouch.
- Upon entering the game, the player pays an entry fee $c_e$.
- The player makes a decision:
- end the game $\rightarrow$ the player collects all the gained rewards and the game ends. Therefore, the player has won the sum of these rewards minus $c_e$ (which she paid before the game started).
- The player draws one ball from the pouch, randomly.
- If the ball is white, she gains a reward $c_i$ where $i$ is the number of the round. Repeat from step 3.
- If the ball is black, the player loses all the gained reward and the game ends. Therefore, the player has won nothing and lost the entry fee $c_e$ (which she paid before the game started).
All parameters - $N$, $c_e$, $c_i$ for each $i$ - are known to the player.
The strategy (i.e. the merit of this question)
I would like to construct a strategy (i.e. a mapping from the game state to the decision whether continue/end) which maximizes the expected total winnings from the game. What is such strategy and how to derive it? Possible de-generalization is that all $c_i$ are equal.
Note: it's not a homework question, I actually want to conduct this game and would like to know some properties of it but I'm not very good at probability and statistics.