Ed wrote:
>Betting a fixed fraction of your bankroll repeatedly on a +EV event causes
>your bankroll to experience exponential growth. As the number of trials
>approaches infinity, the rate of exponential growth is the only thing that
>matters.
>
>Kelly maximizes the rate of exponential growth.
>
>You have maximized average bankroll growth for a finite, N, number of
>trials. Eventually Kelly betting will overtake repeated full bankroll
>betting because Kelly's exponential growth rate is higher.
No it won't. B x ((.505 ^ T) * (2 ^ T)) > B x (1.01 ^ (T * .505) x
.99 ^ (T * .495)) (B being original bankroll and T being the number of
trials) for any positive T or B. The probability of the entire
bankroll bettor losing must be 1 in order for the Kelly bettor to
outperform him or her, but that's not the case for any finite number
of trials.
>By choosing a small number of trials, you're allowing a fixed term, the
>initial bet size, to dominate. But over an infinite number of trials, only
>the exponential growth rate matters.
Since it doesn't maximize average bankroll, for what purpose does it
matter?
>Kelly addresses your specific question in his original paper:
>
>http://www.racing.saratoga.ny.us/kelly.pdf
>
>Though this discrepancy between exponential growth rates at infinity and
>average bankroll growth over a finite number of trials is one of the many
>reasons I think Kelly is utterly unsuited to enter most average gambler's
>decision-making.
>Ed
Yes, a theory that deals only with an infinite number of trials has
limited use.
I'll try to understand his formulas later, but he does, as you say,
address my question. He agrees with me that average bankroll is
maximized by betting all of it on any advantage, no matter how many
trials there are, and he agrees with you that, eventually, the Kelly
bettor will outperform any other. At one point, he incorporates
utility function to explain why Kelly betting is preferred, but in his
conclusion, he seems to contradict that, which confuses me, since I
think the diminishing marginal utility of money is necessary for the
Kelly Criterion to be optimal. I assume that no one's utility
function is exactly logarithmic, but that everyone's approximates it,
which I believe is required for the Kelly Criterion to be in
contention as the optimal approach to gambling.
