Mathematics of Blackjack Discussion
Moving on, the general equation for exponential growth is: g = aent. If you don't agree with this last equation, then we really don't have any common ground and in that case would suggest that you take a bit of time to rethink your position. I can provide text references while you re-think if you want them. But I expect you have probably already checked it out and found that what I'm saying is correct.
So moving on, in the specific case we were discussing,
g = Bn/Bo,
a = 1 and,
e = a well known value, approx 2.7183...
t = loge(Bn/Bo)1/n
This gives us Equation 4 from my previous post.
Equation 4) Bn/Bo en*loge(Bn/Bo)1/n
Because the base of the exponential equation [Equation 4] is the fixed value "e", there is no way that it can be a growth rate. After all, we both know that "e" is a constant.
So, let's take a look at a couple of your previous statements,"To me, it's as clear as can be that y is the growth rate. How could it be otherwise?"
Let' me be clear. The "y" you're referring to is the base of the exponential function, xyz, which in the specific case is the constant "e". "e" is fixed and cannot grow. So that is precisely how it is otherwise.
[ET Fan Responds:]
Richard, when you start writing, there's no stoppin' ya, is there!? Hey: THIS AIN'T FRENCH! :-}
I'm afraid you are confusing the thing with an attribute of the thing. I think Steve Heston may have caused some of this confusion. (I was just reading over an old archive you have from him on this subject.) I don't doubt for a millisecond that he is a better mathemetician than I am, but please remember that mathemeticians are often lax in their language. Bertrand Russell once said that (paraphrasing) Mathematics is the subject where you never know what you are talking about, and you never know whether or not what you're saying is true. I don't think Steve meant his post to be taken QUITE so literally.
I want to try and untangle this knot (with gusto), not because I'm a demon for details, (or a nit picking cotton picker, as Tenessee Erny Ford once put it), but because it really seems to me that your consistent choice of words, in this case, betrays a slight blind spot in your understanding of Kelly betting.
Let's forget about dusty old formulas, for a second, and look at the concept of growth. I promise to get back to your formulas in a minute.
Suppose we're driving along in our Mercedes at 25 mph. Suddenly we exit the city limits and decide to test the effect of exerting a little extra pressure on the gas pedal. One minute after the start of our experiment we are going 30 mph. One minute after that we're going 35 mph. Then 40, 45 etc. in one minute increments until we decide to call it quits when we approach the speed limit.
Clearly this is a linear growth rate of the speed of the vehicle, commonly referred to as "acceleration." We write simply: Speed(t) = 25 + 5t, where t stands for time from the start of the test.
Now what is the rate of growth in this simple equation? Since we are looking at the growth of speed, in this case, the rate of growth is the same as the rate of acceleration, namely: 5. Note that the rate of growth is a constant, for this experiment. For a different experiment, you'd get a different rate, of course, which shows that when you broaden the universe of discussion, constants can become variables.
You wouldn't argue, would you, that Speed(t) itself is the growth rate? I'm not sure about French, but in English, the rate of growth (in this case linear) is an attribute we assign to Speed(t). It makes no sense to call the attribute of a thing the thing itself!
Switching to bankrolls, suppose we had a "sure thing" where we could win x dollars per minute. We have: B(t) = B(o) + xt. Clearly, x is the growth rate, so I'll just go ahead and rewrite it:
B(t) = B(o) + gt
Now, as blackjack players, we rarely have a sure thing, so we generally speak in terms of expected bankroll as a function of rounds instead time. But let's go on pretending we have a sure thing to make things simple.
Now exponential growth, aka geometric growth, is simply the growth exhibited by any geometric sequence (as MathProf once had to explain to me in excruciating detail). Here are some examples:
3, 6, 12, 24, 48, ...
2, 6, 18, 54, 162, ...
100, 102, 104.04, 106.1208, 108.243216, ...
In each case, to get the next member of the sequence you multiply by some constant factor. The constant for the first series is 2, for the second it's 3, and for the third, 1.02. (Since we're still using time, instead of rounds, it's not inconceivable to have a 2 or 3 to 1 "sure thing" payoff over some given period of time.)
We may write:
B(t) = B(o)*g^t
Now, we can divide both sides by B(o), or use rounds instead of time, or Expected bankrolls or limiting fractions, or whatever else you like. To me (and I hope to you now, too) it's clear that g is the growth rate. It's no longer the "linear" growth rate, and it's become a dimensionless quantity, but clearly, it's still the thing that defines "growth" in this equation.
Now to turn to your equation:
g = a*e^(nt)
I have seen this before. But I have not seen the g on the left, except in that post by S.H. This is not how Thorp, or Kelly wrote it, and it's not what you'll find in the textbooks, unless g is defined for some other purpose. What you'll find is something like:
1) F(t) = a*e^(nt)
Note that t is the basic unit of measure, here, not n. t may be taken to stand for "time," or "trials," or rounds of blackjack, etc. It may interest you to know that the "n" in this equation, is known as the "Malthusian parameter," and is not, in general, an integer.
Also note that mathematicians like to put things in the base e to simplify analysis. The expression can instantly be converted to any base. The most useful base probably being g=e^n, which gives the simplification:
F(t) = a*g^t
Now, F(t) is the function which exhibits exponential growth, therefore it can NOT be the growth rate itself.
In our case we have:
F(t) = B(t)
a = B(o)
and we're done!
Now, I truly admire your creativity in getting 1) to fit your definition of growth, but won't you please admit that my way is a little simpler, and perhaps, just a little more likely?
Now g is not g (or G) to everyone. (Please reread the quote by Russell.) The g above is what I have consistently called the "geometric growth rate." It is also, in the limit as t approaches infinity, what Wilson calls the "growth rate." Kelly (and Thorp after him) have chosen to call (log g) the "exponential growth rate." Note again the base e is not required. Check Kelly's paper to confirm he uses base 2 for his analysis, to conform to his two way fuzzy communications model.
Truly, I believe you would get a better handle on this if you stopped thinking in terms of e. e really has very little to do with the matter. Exorcise it from your system. You don't need it!
Note that choosing g' = (log g) is a matter of convenience, since if log g is at a maximum, g is also. The choice was made simply to ease analysis of the distribution, as we switch from a "sure thing" bet to uncertainty and expectations. If we had a linear model:
Exp[B(t)] = B(o) + gt
corresponding to a flat bettor, then we would study g itself without taking the log, since B(t) is now normally distributed without transformation.
I hope we are clear now that the growth rate does not have to grow in order to be a growth rate. In fact, if we define g your way, g = B(n)/B(o), then it would never reach a maximum value, since this g grows without limit right along with n. It could only be maximized (in a trivial sense) by betting the entire bankroll at every opportunity - but we didn't need J.L. Kelly to tell us that!
Hope this puts a light at the end of the tunnel.