I agree, and I’d put it in slightly different terms again.
Your edge is p−q, how much bigger your own probability estimate is than the one implied by the odds you’re getting. The maximum possible edge is 1−q, what your edge would be if you knew you would win. (This also equals your counterparty’s probability that you lose.) Kelly says: the fraction of your funds to bet equals the fraction of the maximum possible edge that you’ve got.
Equivalently: you bet nothing if you’ve got no edge, bet all your funds if you know you’re going to win, and interpolate linearly in between. (That’s exactly lsusr’s observation, of course.)
I don’t really anything to add to this except to acknowledge that I really like gjm’s way of thinking about this equation. I expect to use gjm’s labels in the future.
I agree, and I’d put it in slightly different terms again.
Your edge is p−q, how much bigger your own probability estimate is than the one implied by the odds you’re getting. The maximum possible edge is 1−q, what your edge would be if you knew you would win. (This also equals your counterparty’s probability that you lose.) Kelly says: the fraction of your funds to bet equals the fraction of the maximum possible edge that you’ve got.
Equivalently: you bet nothing if you’ve got no edge, bet all your funds if you know you’re going to win, and interpolate linearly in between. (That’s exactly lsusr’s observation, of course.)
Your explanation is fantastic!
I don’t really anything to add to this except to acknowledge that I really like gjm’s way of thinking about this equation. I expect to use gjm’s labels in the future.