Rather than talking about p and b (your probability and net fractional odds*) doing things in terms of p and q = 1/(b+1) (your probability and implied odds probability) makes calculations easier and lots of the things you didn’t expect more intuitive. Specifically:

f=max(p−q1−q,0)

This means:

1. It is easy to see the fraction is piecewise linear 2. The answer to Q5 is obvious.

* aside: who talks in net fractional odds? None of the 3 major odds styles use it

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.

Rather than talking about p and b (your probability and net fractional odds*) doing things in terms of p and q = 1/(b+1) (your probability and implied odds probability) makes calculations easier and lots of the things you didn’t expect more intuitive. Specifically:

f=max(p−q1−q,0)

This means:

1. It is easy to see the fraction is piecewise linear

2. The answer to Q5 is obvious.

* aside: who talks in net fractional odds? None of the 3 major odds styles use it

I agree, and I’d put it in slightly different terms again.

Your

edgeis p−q, how much bigger your own probability estimate is than the one implied by the odds you’re getting. Themaximum possible edgeis 1−q, what your edge would be if youknewyou 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.

I used net fractional odds because that’s what the Wikipedia page used. I like your system better.