Re: This is the million dollar question!

Posted by Bisser on July 15, 1997 at 12:22:53: In Reply to: Isn't log($) arbitrary? posted by Thicko on July 15, 1997 at 04:02:09:


WHAT IS THE VALUE OF MAXIMIZING THE EXPECTED LOGARITHM OF GROWTH RATE???????

If, in fact, we agree that maximizing the expected logarithm of growth rate is a reasonable goal, then we should bet according to Kelly's rule. If we just wish to maximize the expected growth rate, then we should bet everything on every bet. If we wish to maximize something in between, then we should bet in between.

If we set "utility = log(money)", then we can argue that Kelly is the way to go. But you are right, it is arbitrary. Using different utility will give you a different rule.

There are two points here -- the utility function we choose and what we wish to maximize (expectation in Kelly). We could try to maximize the median of the growth rate, or its 10th percentile or the 30th percentile of log growth rate. Maybe the way to go would be to look at the distribution of your Nth bankroll B(N).

If you bet all your money on every positive expectation bet, then you will go broke and Grimmy will laugh at you. Yes, you have maximized your expected growth rate, but B(N) has a strongly skewed distribution and you will get broke with probability 1-P^N and be filthy rich with probability P^N (P is the probability that you win one bet). Thus as N --> infinity, you have 0 chance to get rich and will be broke.

Kelly's rule gives you some balance. You will not get broke, you will be ahead if you play long enogh. There are also other betting schemes that will imply an almost sure win.

So Kelly gives a reasonable bet size, but if somebody wants to take more risk or be more conservative, he could use a different (also reasonable) betting scheme.

Bisser


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