Let's call the starting bankroll S and the current bankroll C.
suppose you play with biased coin which has 51% for heads and 49% for tails. So, you keep betting on heads and your edge is 2%.
Kelly says that you should always bet 2% of your current bankroll. The fraction f of the current bankroll to bet, is constant, and it is f=0.02*C
Now how about all the possible formulas that suggest to bet e.g:
f=0.02*C when C=S,
f<0.02*C when C<S,
f>0.02*C when C>S.
Or more generally, where X any given number,
f=0.02*C when C=X*S,
f<0.02*C when C<X*S,
f>0.02*C when C>X*S.
C=((1+f)^W)((1-f)^L)S
(also see the relevant previous post of mine where I explain this equation)
So we shape the graph of the above function, where at the axis of Y we put the values for C, and at the axis of X we put the values for f. And we see that the curve which is shaped at this graph, gets its highest for the value of f which is equal to your edge.
Do you get it now? This proof says that IF you decide to bet a CONSTANT fraction of your current bankroll, then the growth of the current bankroll is maximised for f=edge. Therefore it has nothing to say mathematically for all other betting systems that do not suggest to bet a constant f. If it HAS something to say, then this needs additional mathematical proof.
Well, here's my take on it. Your formula above is exponential growth (G), right?
Maximizing your growth for logarithmic growth for that forrmua would be ln(G) = ln(1+f)*W + ln(1-f)*L. It's growth is maximized by f=1 but the expected growth is maximized by f=.02 in your example. Logarithmic growth per hand = ln(1+f)*W/N + ln(1-f)*L/N. But a Kelly better does not maximize actual growth, he maximizes the log of his expected growth. He can't maximize actual growth becasue he doesn't know if he will win.
While W/N will converge to expected value p over time, and so your actual results will converge to your expected logarithmic growth per hand, also W/N will be normally distributed with a mean of p so the median growth per hand = the mean growth per hand. So a Kelly better is maximizing both mean and median growth per hand. That's the Kelly fraction 1. I think mean growth is higher than median growth.
Some people assume W=p*N and maximize G. That's not right.
People like to maximize different things. A square-root better would bet about twice Kelly, a risk-neutral better would bet everything. They may maximize actual growth but not expected growth.
I think Kelly takes into account risk vs reward - I think that's why an AP guy betting to a spread, sometimes even playing -EV hands, obviously not betting a constant fraction of current roll, can still achieve the max log growth of his expectation. He cannot have a risk of 0 because he could lose every hand while still betting in a way that maximizes his expected log growth. It turns out that the risk that optimizes the log of expected growth is 13.53%. (the log of -2/Kelly fraction 1).
Not sure if this answers your basic question. But this is what I think Kelly tries to do. Other betting systems actually can have higher actual growth rates than Kelly. Whether betting a constant fraction of current roll or, like an AP guy, betting to a spread at different advantages, even in -EV situations, and different variances at each count, there's only one way that will maximize the logarithmic of growth of your expectation.
I know what your're gonna say Blackajck Avenger, lmao. I said one way and you said two lmao. I'm thinking about it lol.
But I think I may have been making the mistake that Kelly maximized actual growth rather than expected growth.
