From: sethb@fid.morgan.com (Seth Breidbart)
Subject: Re: Samuelson (was Re: Yeah... ANOTHER variance vs expectancy posting.)
Organization: my opinions only
Date: Wed, 22 Jul 1992 23:37:32 GMT

In article <JACOBS.92Jul17153726@cells.cs.utah.edu> jacobs@cs.utah.edu
(Steven R Jacobs) writes:
>In article <1992Jul17.201334.26393@kronos.arc.nasa.gov> hall@kronos.arc.nasa.gov (Michael Hall) writes:
>> In article <1992Jul15.235907.17186@leland.Stanford.EDU> zwiebel@leland.Stanford.EDU (Jeffrey Zwiebel) writes:

>>>What is true, is that maximizing the expected log (Kelly) will, with
>>>probability approaching 1 as the number of gambles grows large, yield
>>>a higher return than any other betting strategy.

As I showed in an earlier post, this is only true under certain
circumstances.

>>>The reconciliation of these two apparently contradictory facts is actually
>>>quite simple.  While it is true that if one plays long enough, with a very
>>>high probability one will do better with Kelly, in the (very unlikely)
>>>states where one does worse, one does much worse, enough so to outweigh
>>>doing better most of the time.  
>>
>> With probability exactly *1.0*, one will do better with Kelly, for any
>> final state utility function that holds that the more money the better,
>> assuming one has unlimited time (and a few other less constraining
>> assumptions.)
>
>Jeez Michael, that _exactly_ is only reached after an infinite number of
>bets.

That's the key point.  The probability of having more money by using
Kelly approaches 1.  Since the probability <1 after finite time, if
the excess utility in that fraction is large enough (e.g. linear
utility), Kelly is not optimal.

>>It is not the case that a Kelly bettor would *ever* do
>>worse than any other kind of bettor in the limit; at some number of bets
>>it *always* surpasses any other betting function and keeps on going.

Consider linear utility, and the 60% bet with even money payoff.
Optimal is to bet it all.  How many bets does it take for Kelly to
surpass this?  (With probability 1 it will happen after a finite
number of bets, but for any given finite number, the probability it
happens within that many bets is < 1.)

>Sigh.  You're bucking Paul Samuelson on this one, and I suspect he's
>a _tad_ more qualified to talk about possible limitations of Kelly.

Argument by resorting to authority isn't very useful.  You're much
better off giving examples and explanations (as I did, above).

>>Again, the only assumption I need to make about one's utility function
>>of money is that it is nondecreasing in the limit; this forces you to
>>choose the Kelly criterion.
>
>Kelly betting will optimize the expectation of log(bankroll).  Other
>betting schemes will be optimal for other utility functions.  Your
>problem is that you don't want to _allow_ other utility functions.
>You are a Kelly bigot.

Steve is right.  Kelly will also optimize in one other case: if you
put a sufficiently high utility on bragging rights.

>>However, Kelly betting is near optimal even for not-all-that-many bets (I
>>can't seem to track down the reference, though.)

>It is near optimal _only_ if your utility function is log(bankroll),
>or something close to it.  When are you going to realize that there is
>more than one way to approach gambling?  You're "how can you argue
>against more money" attitude is rather short sighted, it just doesn't
>tell the whole story.

I suspect he means that the probability of Kelly exceeding some other
method is close to 1 after a reasonable number of bets.  (This is the
"bragging rights" case I mentioned before.)

>It doesn't take anything that weird to make Kelly betting suboptimal.

Another simple example, which I believe I gave much earlier:

You have $100.  Your utility function for money is log(bankroll);
utility($100)=100.  If you have $200 you can take the hot babe (tm)
standing next to you out for dinner, a show, and whatever.
utility(whatever)=10,000.  You have 2 minutes, and you're at a
roulette wheel.  I claim that the optimal strategy is to bet it all on
{odd, even, black, red}.

>>I personally plan to play, or have played for me, millions of blackjack
>>hands in my lifetime.  Can you show me a reasonable utility function of
>>money that would make Kelly betting suboptimal for this case?  (You may
>>use mean .021 units and variance 7.66 units^2 per hand of blackjack, if
>>you need some numbers.)

Sure.  Linear utility of money.  The optimal strategy is to bet it all
whenever the odds are favorable.  You have a very small chance of
ending up with a very very large amount of money.


I just realized that there is another situation in which Kelly might
be optimal.  Under all admissable strategies, I think Kelly has the
highest probability of ending up ahead after a large number of bets.
(An admissible strategy is one where the percentage of bankroll to bet
is a function of the odds and payoffs.  Otherwise, a strategy like
"Play Kelly until you're ahead, then play 0" will obviously dominate
Kelly.)

Seth		sethb@fid.morgan.com
Newsgroups: rec.gambling