nightoftheiguana2000 wrote:
>--- In vpFREE@yahoogroups.com, 007 <007@...> wrote:
>> 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.
>
>
>One trick is that it's the geometric mean that counts, not the arithmetic mean. Taking the arithmetic mean assumes you can just go on forever averaging a series of outcomes, but in the real world you can not. Once you bust out, that's it, you're busted, game over, no more chances and it doesn't matter how well you were running before you busted out. Having a zero in a list that is arithmetic meaned just lowers the mean. Having a zero in a list that is geometric meaned sets the mean to zero, irregardless of how great the other results were. Kelly optimizes the geometric mean of bankroll growth, which is why a Kelly better would never bet it all, unless there was no risk of losing. Under the Kelly system, if you bust out once, that's it. You have not only not optimized bankroll growth, you have in fact committed bankroll suicide.
>
>http://en.wikipedia.org/wiki/Mean
>http://en.wikipedia.org/wiki/Kelly_criterion
I like how the latter article points out that a valid alternative to
the Kelly Criterion is "utility theory" and that only if one's utility
function is logarithmic does it coincide with the Kelly Criterion.
That at least makes more sense, since I still don't see how, if
there's no diminishing marginal utility of money, betting one's entire
bankroll on any advantage, due to how it maximizes average bankroll,
doesn't outperform all other possible systems, including the Kelly
Criterion. A study of the extent to which human utility functions are
logarithmic might be interesting. I assume mine is. If you believed
your utility function were linear, so that you believed you had no
diminishing marginal utility of money, would you still prefer the
Kelly Criterion over betting your entire bankroll on any advantage? I
find the argument that betting one's entire bankroll on any advantage
might lose the entire bankroll to be an inadequate explanation of its
weakness, since that possibility is included in the calculation of
average resulting bankroll. It's like saying not to lay a big price
on a favorite in sports because it might lose when the possibility of
losing is already included in the estimate of expected value.
