I liked your example of being uncertain of your probabilities. I note that if you are trying to make an even money bet with a friend (as this is a simple Schelling point), you should never Kelly bet if you have discounted rate of 2⁄3 or less of your naïve probabilities.
The maximum bet for kp−((1−kp)+(1−k)p) is when p is 1, which is 3k−2, which crosses below 0 at k=23
I liked your example of being uncertain of your probabilities. I note that if you are trying to make an even money bet with a friend (as this is a simple Schelling point), you should never Kelly bet if you have discounted rate of 2⁄3 or less of your naïve probabilities.
The maximum bet for kp−((1−kp)+(1−k)p) is when p is 1, which is 3k−2, which crosses below 0 at k=23