Posted by David D'Aquin on June 28, 1997 at 11:44:27: In Reply to: Confidence levels for interval resizing posted by David D'Aquin on June 27, 1997 at 13:21:58:I'm posting this last one simply because people define "long run" differently. It is sometimes said that Kelly betting will always outperform fixed bet sizes in "the long run". This is again a new definition, unlike the one which defines "long run" as the number of bets needed to have a certainty of winning within 3 standard deviations.
What this equation will do is take the two very extreme cases. The Kelly bettor has results so poor that the number of wins is 3 standard deviations to the left. The fixed size bettor has very good results, such that the number of wins is 3 SDs to the right. Under these conditions, how many hands must be played (bets made) before the Kelly bettor passes the fixed bettor. Here it is:
[ (7.2 * SD per hand) / exp win per hand ]^2
This is a very big number, and it is the number of hands needed to assure Kelly comes out on top. The number of hands needed for Kelly to exceed the expected win for fixed bets, with results this poor, is only a little less. Use 7 instead of 7.2. The number of hands needed for Kelly to do better when both have results this poor is found usingg 6.96 in the equations. The numbers are all very close for the simple reason that once a sufficient number of hands are played to overcome these very poor results, the Kelly bank will quickly surpass any possible win by fixed bet sizing.
Proof? You can see it if you know the procedure to find the standard deviation for Kelly betting. That is complicated itself.