From: sethb@fid.morgan.com (Seth Breidbart)
Subject: Re: Samuelsen
Organization: my opinions only
Date: Thu, 23 Jul 1992 00:08:57 GMT

In article <JACOBS.92Jul16133326@cells.cs.utah.edu> jacobs@cs.utah.edu
(Steven R Jacobs) writes:
>In article <1992Jul16.175655.19706@leland.Stanford.EDU> zwiebel@leland.Stanford.EDU (Jeffrey Zwiebel) writes:
>
>Summary of discussion: (please correct me if this is wrong)
>
>   Kelly isn't special, but only one of a family (CRRA) of utility functions
>   that might be useful for players who like to _optimally_ bet in proportion
>   to their bankroll.
>   
>   Kelly is the most risk averse of the CRRA utility functions, while
>   expectation is the most "risk aloof" of the CRRA utility functions.

Correct.

>So, since a=1 for the "expectation" utility function, this would seem to
>imply that using Kelly betting with basic strategy will be "far from
>optimal".  I assume this really means that the Kelly betting screws up
>the optimality when using expectation as a utility function (meaning that
>you give up a lot of expectation by using Kelly betting instead of a
>"bet it all" betting strategy).  The converse would not necessarily
>follow (meaning that using Kelly betting with basic strategy doesn't
>necessarily hurt much if your desired utility function is logarithmic).

The question here is how much your utility function is hurt by playing
basic strategy rather than Kelly strategy.  Actually, there are three
strategies to be considered:  (i) Basic strategy, as a non-counter
would play it.  (ii) Optimal strategy (linear utility), as a counter
would play it.  (iii) Kelly-optimal strategy.  Clearly, (i) is worse
than (ii) is worse than (iii).  The question is, by how much?

>>  One can reconcile the basic strategy together with betting only a 
>>small fraction of one's wealth if one is globally risk averse, but locally
>>(for small bets) virtually risk nuetral.  Hence choosing the amount to bet
>>based on ones utility function, and then playing the basic strategy may be a
>>reasonable approximation to one's optimal strategy.

For small enough bets, log(bankroll+bet) is proportional to bet, so
the linear-optimal strategy (ii) above isn't much worse than the
optimal strategy.

>OK, makes sense.  The implication for blackjack players is that betting
>strategy is probably more important than playing strategy for maximizing
>utility (assuming the playing strategy is reasonable, as in "no worse than
>basic strategy").

No, a simple non-optimal betting strategy might cost you less than
using (i) instead of (iii).  You have to specify the strategies to
determine how much each will cost you.

>OK, it appears that "be true to [your] own utility function" here probably
>means "_bet_ true to your own utility function".

No, you should _be_ true to it.  Betting and play both matter; it's
just that for blackjack, with small bets, the linear utility function
is a good approximation to others.

>|> The bottom line is, there really is no good theoretical justification for
>|> the Kelly betting criterion.  THe optimal decisions will depend on your
>|> own utility function (how risk averse you are, etc...).

Well, in some cases there is theoretical justification for Kelly.

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