From: sethb@fid.morgan.com (Seth Breidbart)
Subject: Re: Samuelsen
Organization: my opinions only
Date: Wed, 22 Jul 1992 23:56:43 GMT

In article <JACOBS.92Jul16083656@cells.cs.utah.edu> jacobs@cs.utah.edu
(Steven R Jacobs) writes:
>In article <1992Jul16.023514.2819@leland.Stanford.EDU> zwiebel@leland.Stanford.EDU (Jeffrey Zwiebel) writes:
>>In article <JACOBS.92Jul15190723@cells.cs.utah.edu> jacobs@cs.utah.edu (Steven R Jacobs) writes:
>>>In article <1992Jul15.235907.17186@leland.Stanford.EDU> zwiebel@leland.Stanford.EDU (Jeffrey Zwiebel) writes:
:

>This sounds similar to the problem with Martingale systems -- they look
>good to the casual observer because they win most of the time, but when
>they do lose, they lose big time.

>Let me back up and ask a couple more questions here.  Are you saying that
>the utility function must be CRRA, or only that the utility function must
>be "close" to logarithmic in order for the strategy to just copy the
>one period strategy?  Also, if having a close to 0 part of the iff.

If the utility function is CRRA, then the optimal bet is a function of
the odd & payoffs, times the size of the bankroll.  If the utility
function is not CRRA, then the optimal bet depends in a more complex
way on the bankroll.

>>No.  If utility is CRRA, the optimal strategy will be static (given
>>by the one period strategy).
>
>Is the optimal strategy also static for a logarithmic utility function?

Yes.

>Is the optimal strategy static whenever risk aversions is zero (as in
>utility equals expectation, betting strategy is "bet it all", playing
>strategy is basic strategy).

Playing strategy still depends on the count :-)  When risk aversion is
zero, you have linear utility, and the strategy is "bet it all
whenever the bet is favorable".  (Actually, that's not true for a game
like blackjack, where you might want to retain the ability to increase
your bet by splitting or to take insurance (obviously, you'd rather
have bet twice as much at the start than to double down).)

>  If not, then this would seem to imply
>that basic strategy doesn't really optimize expectation if the player
>continues to play for a long sequence of bets.  This would be a bit
>disturbing, especially to Bill :-)

It doesn't, unless you're playing in a favorable game and can't count,
in which case you want to bet Kelly (if you have log utility) and play
basic (which is optimal, in the sense of "the best you can").

>Makes perfect sense to me -- isn't this just saying that if you want
>optimal results, the betting strategy and playing strategy need to optimize
>the same utility function?

Actually, the betting strategy and playing strategy can't be uncoupled
that easily: is the decision whether to take insurance betting or
playing?  What about doubling and splitting?  Once you have placed a
bet, the optimal strategy optimizes your utility function.  Given that
you are going to play a certain strategy, the optimal bet optimizes
your utility function.

>>>> (The paper also shows that if utility is bounded above and below, the Kelly
>>>> rule will asymptotically outperform other uniform rules; but 1) this 
>>>> assumption on utility is quite restrictive and 2) under this assumption,
>>>> utility is not CRRA, hence uniform rules are not optimal anyways).   

>This would seem to apply when utility is pure expectation, or am I
>misinterpreting what is meant by "utility bounded above and below".

I think so.  Pure expectation is not bounded above, since you can (in
theory) get unlimited amounts of money.  It is bounded below by 0.
Log utility is unbounded both above and below.  A function like
"arctan(bankroll)" is bounded above and below.  Kelly wins because
with high probability it will get near the upper bound.

>>Basically, (and Samuelson doesn't get credit for this - these are well
>>known economic results) if utility is CRRA, one doesn't have to worry about
>>the fact that one plays repeatedly (ignoring counting, covering and other 
>>aspects that make the gamble different each hand); one only needs to solve 
>>for the single-hand optimal strategy.  This would involve evaluating the 
>>distribution of outcome under a given strategy (where strategy both
>>includes proportion of wealth bet and how the cards are played) under the
>>given utility function.  This might be a pain to solve for in closed form,
>>but fairly easy to simulate.  

>Depends on what you mean by "closed form".  It is possible (but not easy)
>to precisely compute the expected utility for each possible play, and
>pick the best option for form an optimal play (at least, for plays other
>than pair splitting).  Finding actual _equations_ to describe the optimal
>strategy would be a much more formidable task -- probably woulnd't even
>fit on a 640K PC.

The optimal strategy is given by a table.  I don't think even pair
splitting would be that tough; resplitting makes life much harder, of
course.  "Closed form" means that you have used equations rather than
simulation to calculate the optimal table entry at each point.

>Finding a "non-static" strategy would likely be much more difficult.
>Probably much easier to ignore their existence :-)

Yup, except when that hot babe (tm) is standing there :-)

Seth		sethb@fid.morgan.com