If there was a game with only a limited number of betting options each providing even odds. Given a player's probability of winning $p$ how can you determine the bet sizes to choose for each value of the portfolio to optimise the growth rate as per the normal Kelly criterion?
edit: bet size options are with a starting portfolio value of 1500
[50,100,500,2500,10000,50000,100000,250000,500000,1000000,2500000,4000000,5000000,10000000,15000000,25000000]
however I am also interested in the answer for the general case
my intuition says that the answer would be to just choose the bet size that maximises the expected log portfolio value for that turn however I am struggling to prove this.