Posted by Steve Heston on July 08, 1997 at 10:15:18: In Reply to: Thanks, but...... posted by Mike Lea on July 08, 1997 at 08:27:08:Obviously maximizing a function is the same as maximizing the logarithm of that function. This is trivial and has nothing to do with blackjack. But people have often confused maximizing actual growth with maximizing expected growth.
Note that G = (1+f)^W*(1-f)^L. The logarithm of G divided by N equals (W/N)*ln(1+f) + (L/N)*ln(1-f). Since the expected value of W/N equals p, the expected logarithm of G equals p*ln(1+f) + (1-p)*ln(1-f).
If we substitute W = pN we would say G = (1+f)^(pN)*(1-f)^((1-p)N). But this is not expected growth! Maximizing (the logarithm of) expected G is not the same as maximizing the expected logarithm of G.