Lets say $2$ players $A$ and $B$ make a bet, who can have more money at the end after playing the following game:
a coin is flipped:
with $51\%$ probability it lands tails, with $49\%$ probability it lands heads
you win if it lands heads, where you get back your bet $\times 2$.
e.g. you bet $\$1$ and it lands heads, then you get back $\$2$
e.g. you bet $\$2$ and it lands tails, then you get back $\$0$
here are the rules to the bet between A and B (the winner of the bet wins $\$100000$):
- you both start with $\$100$ (given to you for free, you're not allowed to cash this out nor the money you make from the coin game)
- each player may play the game as many times as they want and bet as much as they want for each time they play the game
- player A must go first (player A plays the casino games as many times as he wants then decides to stop, after that, A can't play the game anymore)
obviously the optimal strategy for player $B$ involves playing until $B$ either goes bankrupt or has more money than $A$ (although it's not obvious what bet sizes to use).
what would be the optimal strategy for $A$?