Expected log(growth) < log(expected growth)

Posted by Steve Heston on July 03, 1997 at 22:00:49: In Reply to: Exponential Growth !! and Logarithmic Growth ?? (long) posted by Richard Reid on July 01, 1997 at 21:33:57:

"Exponential growth" = growth = G = Bf/Bi = ((1+f)^W)*((1-f)^L). Logarithmic growth = ln(G) = ln(1+f)*W + ln(1-f)*L. Logarithmic growth per hand = ln(1+f)*W/N + ln(1-f)*L/N. The Law of Large Numbers says W/N will converge to the expected value p. So your actual results will converge to your expected logarithmic growth per hand. The stronger Central Limit Theorem says W/N is normally distributed with a mean of p. This means median logarithmic growth per hand equals mean logarithmic growth per hand. In contrast mean growth is a lot higher than median growth.

Some people erroneously assume W = p*N and maximize G. But you can't maximize actual growth, because you don't know whether you will win! But you can maximize the expected utility of G. "Kelly" betting maximizes the mean (and median) of logarithmic growth. Since proportional bettors will converge to their expected logarithmic growth per hand, the Kelly bettor eventually ends up richest. But a square-root bettor will bet roughly twice Kelly and a risk-neutral bettor will bet everything.


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