Suppose the host borrows one dollar from you and place it in a pot. You may take the money in the pot away or toss a coin. If Tail appears the pot becomes empty and you get nothing unless you start the game afresh. If Head appears the host doubles the money in the pot. Again it's your decision to quit the game taking away the money in the pot or toss a coin. If Tail appears host takes away the money in the pot and if Head appears host doubles the money in the pot. This is known as Double Or Nothing betting game.

The question is could we develop a strategy to make some earning from the game? In the n-th toss you earn dollar 2ⁿ having probability 1/2ⁿ. So, the expected earning from the game is 1 + 2 x 1/2 + 4 x 1/4 + 8 x 1/8 + ... which is infinity.

I devised a simple ppt macro model to simulate the game. I start with zero bank balance and each time I start a game I pay dollar one from my account which may hold negative balance. I go on tossing the coin and transfer the money from the pot whenever that total of my bank balance and the pot money exceeds K. Any time in the game I get to set the value of K to be 0, or 10, or 50, or 100, or 1000.

My first strategy fixes the value of K to 0. My second strategy sets different values of K depending on my bank balance. Higher is the bank balance larger is the value of K. Below is a comparison chart for 500 tosses.

Strategies are fine. Depending on the runs of Heads and Tails my bank balance may shoot up towards positive infinity or may shoot down to negative infinity. 

QUESTION I: For a fixed value of K could we find the distribution the earning E?
QUESTION II: Could we develop a two-player game on this idea?

REFERENCE: Double Or Nothing Betting Game Wikipedia, Martingale Betting System Wikipedia