I start with a certain amount of cash $C$ and I play a game where on each win, I double my money.
The game is structured such that the first time I play, I have a 50% chance of winning.
Let's say I win the first game, then on the second game I have a 51% chance of winning, if I win the second game, then I have a 52% chance of winning etc all the way up to 70%.
If at any point I lose, then on the next game my chances reset to 50%.
e.g.
Game 1 with cash 10 is a win so I earn 10 (and I have a total of cash 20)
What is the best way to play such that I earn the most?
My intuition is telling me that you should just put all of the cash in each time, and then half the amount on each win i.e if you start with 100 an win then you play 100 again, then 100 again etc until a loss, but I'm not sure if this is optimal. Any help would be appreciated :)
Your intuition is completely at variance with mine. If you bet everything you have a $50\%$ chance of losing all your money in the first game. Surely it would be more advantageous to bet conservatively at first and wait for your odds to go up.
By the Kelly criterion even if the probability of winning were always $70\%$ the best strategy would be to bet $40\%$ of your current bankroll at each turn. Since the actual situation is less advantageous to the player than this, some lesser bet is indicated.
I don't know if the reasoning underlying the Kelly criterion can be extended to this case. I would try it with just two probabilities to start. One further complication is that when the probability of winning is $\frac12,$ the Kelly criterion says not to bet, so perhaps you ought to start with two probabilities that are each greater than $\frac12.$