by Winston Yamashita.

Posted by Winston Yamashita on June 08, 1997 at 23:49:57:

brh's method for solving for T is incorrect. It does not apply to discrete functions. His equation 4):

" b(T) = EV(T)/VAR(T) = M / K(T), where Ks = K(T) . (4)" applies only for continuous funtions.

The easiest way to determine the Optimal Betting Strategy (as I described in my Kelly Generalization) is to determine the minimum and maximum betting fractions for the given bet spread.

They can be determined just by maximizing the expected log growth rate:

From the log expectation equations it can be shown that the expected log growth rate is :

S(i):x(i)a(i)f(i) - x(i)[var(i)f(i)^2] /2

Where i=true count, S(i):=summation over all (i), var(i)= variance(or average squared outcome) at count (i), a(i)= advantage at true count (i), x(i)= fraction of counts at count (i), f(i)= fraction of bankroll bet at count (i)

As long as the betting fraction, f(i), is between the minimum and maximum betting fractions, the expected log growth is maximized at each true count when f(i) =a(i)/var(i) [the regular Kelly fraction at each true count].

The log expectation at the minimum and maximum betting fractions is :

S(imin): x(i)a(i)f(min)-x(i)[var(i)f(min)^2]/2 + S(imax):

x(i)a(i)f(max)-x(i)[var(i)f(max)^2]/2

With S(imin): = summation over all true counts (i) at the minimum betting fraction, S(imax): = summation over all true counts (i) at the maximum betting fraction, f(min) = minimum betting fraction, f(max) = maximum betting fraction, also let r= f(max)/f(min) = betting spread

When f(max)=rf(min) is substituted above, the maximum occurs at:

f(min)= [S(imin):x(i)a(i)+S(imax):rx(i)a(i)]/[S(imin):x(i)var(i)+S(imax):(r^2)x(i)var(i)]

If the calculated f(min) and f(max)=rf(min) fall outside the intermediate bets (bet at their regular Kelly fraction,f=adv/var at each true count) and are less than or greater than respectively the regular Kelly fractions of the minimum and maximum bets, you are done. The intermediate bets are just their regular Kelly fraction and the minimum and maximum bets are the f(min) and f(max)=rf(min) that were calculated.

To systematically find the f(min) on a simulation program or spreadsheet:

1) assign the maximum bet (r) to all counts with a positive advantage.

2) assign the minimum bet (1) to all counts with a negative advantage

3) calculate f(min)=expectation/variance (of the game) and f(max)=rf(min)

4)If none of the true counts has a regular Kelly bet(f=adv/var at the true count) that falls between the calculated f(min) and f(max), you're done.

5) If any of the true counts has a regular Kelly bet(f=adv/var at the true count) that falls between the calculated f(min) and f(max) , include that count(s) in the intermediate bets and for the f(min) calculation set the bet at that count(s) equal to zero.

6) Go back to step 3) and recalculate f(min)

Once you have calculated f(min) you can find the betting indexes by dividing all the f's by f(min). The same procedure also works for running counts.