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.
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.
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
On Tue, Sep 20, 2011 at 11:18 PM, 007 <007@embarqmail.com> wrote:
> **
>
>
> Ed wrote:
>
> >If you bet your whole bankroll on any edge, then X percent of the time you
> >will bust and then it's impossible to grow the bankroll. That severely
> >limits your average bankroll growth.
> >
> >Kelly seeks to maximize the average rate of growth of a bankroll. In other
> >words, if you bet according to Kelly at a +EV game then you will
> experience
> >a greater average bankroll growth than if you bet any other way.
>
> Maybe an example would clarify. Let's say there's a gambling
> proposition, similar to blackjack, that pays even money and has a
> 50.5% chance of winning and a 49.5% chance of losing. That's a 1%
> advantage and the Kelly Criterion says to bet 1% of one's bankroll on
> it. Let's say, for ease of use, that one's bankroll is $10,000.
> After 1 $100 bet, one's average bankroll has grown by 1% of 1%, or $1,
> and is now $10,001. If one bets $10,000, after 1 trial, one's average
> bankroll is (maximized at) $20,000 x .505 = $10,100. How does the
> Kelly Criterion maximize average bankroll growth? The same comparison
> can be made after the next trial, and so on, forever, the fact that
> betting one's entire bankroll on any advantage runs the risk of losing
> all of it notwithstanding.
>
>
>
[Non-text portions of this message have been removed]
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