Okay so I ran my program and I didn't manage to find the exact optimal median but I narrowed it down a lot. I'm assuming each bet is a whole number of cents. I

Okay so I ran my program and I didn't manage to find the exact optimal median but I narrowed it down a lot. I'm assuming each bet is a whole number of cents. I found that there was a strategy that achieved $6.84 with probability 1/2 + 1/2^10. So the median is at least $6.84 (which is 3.8 times the median achieved by Kelly criterion!). I found that there was no strategy that achieved $6.87 with probability 1/2 or greater, so the median is less than $6.87. I found that $6.85 and $6.86 could both be achieved with probability 1/2 but no greater. So the median is somewhere in the range [$6.84, $6.855]. The optimal starting bets of these strategies show some interesting behaviour. For example if you want to have a greater than 1/2 chance of achieving $6.84 then your starting bet can be $0.22, $0.26, $0.31, $0.36, $0.40 or $0.50, but no other value. Make of that what you will.