?Maximizing logarithmic increase

From Stanford Wong's BJ21

Posted by ML on 13 Dec 1997, at 2:40 p.m.

A post on another page reminded me of a problem in understanding I continue to have.

It is said over and over that the "Kelly" sizing is the "best" "function" to determine bet sizing because it "maximizes logarithmic increase of bankroll." Maximizing other utility functions of bankroll are discussed but it is said Kelly deals with a "log utility function."

Bankroll increase when proportionally betting a simple game is mathematically known.

Where B(n)=expected bankroll after number of plays.

N=number of plays.

p=probability of winning each time.

f=fraction of instant bankroll bet.

B(n)=((1+f)^(N*p))*((1-f)^(n*(1-p)))*B(0).

The equation can be generalized for a compound game by adding each level and inserting the correct expected wins and losses for each level given the various frequencies at which advantage levels occur and multipying f by the multiple bet at that level. To get the correct answer, number of levels-1 must be subtracted from the final answer and multiplying by B(0).

The point is that if either the simple or compound game equation is manipulated by manipulating the value of f to find where maximum growth exists, (Where B(N) is largest, it will be largest at the "Kelly" value for a simple game or what I have been calling the "Yamashita" value for any defined compound game.

Since the bare bankroll, unmodified by a log function or any other utility function which might be talked about, is being maximized by determining the f that maximizes the bare bankroll, what does utility function theory have to do with the calculation? I just do not see it.

I would appreciate some of the more knowledgeable people on this board try to explain this to me one more time. I know some have tried before but the explanations have not gotten through this thick head.

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