Re: Reasons for Kelly

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Posted by Pete Moss on July 14, 1997 at 17:10:05: In Reply to: Re: Reasons for Kelly posted by Bisser on July 14, 1997 at 11:17:26:

Both, I guess. You want to maximize the expected value of growth of capital,


        B(N)
       ------
       N B(0)

as N grows with out bound.

The bankroll B(N) after N hands is equal to the product,

            N
          B(0) prod {1+bi*Ri}
                i=1

where bi is the fraction of bankroll wagered on hand i, and Ri is the per-unit result of hand i. To maximize that product, it is sufficient to maximize its logarithm, because log() is a monotonically increasing function. Thus we maximize the limit as N grows of the expected value of


      N
     sum log(1+bi*Ri)
      i

which is maximized when


      E[log{1+b*R}]

is maximized for all situations. The reason this helps us is that if we know the "moments" of a random quantity Q (like 1+b*R), and we know the Taylor's series expansion of a function f, (like log), then we can easily calculate the expected value E[f(Q)], or approximate it with the first few terms, assuming the series converges in the range of interest. The powers in the Taylors series are simply replaced with the moments, and voila! The first moment is E[R], the second is E[R^2], the third E[R^3], etc. When we use only the first two terms, which give a good approximation for blackjack, we can solve for the maximum by differentiating and setting the derivative equal to zero. The result is the familar blackjack Kelly wager size.

Pete




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