I don't recall doing a Kelly problem before, so be sure to check my work to see
that I've interpreted your question correctly.
I think the proper way to do this comparison would be to use 30 final hand
combinations rather than taking "the additional frequency of each ending hand."
That allows the ending hands to have the normal frequency total of 1, rather
than zero. It also allows you to distribute the varying results from the
straight draw of 0 bets, 1 bet, and 2 bets among all the final hands from the
deuce draw rather than apportioning them in some other manner or than just using
its expected value of the return of 1 bet.
30 is the product of the 3 final hand possibilities drawing to the straight and
the 10 final hand possibilities drawing to the deuce. If you can assume that
the results of the different draws would be independent, you can set up your
frequencies and difference in results with one example being a probability of
(81560 * 19) / (178365 * 47) for the combination of drawing no scoring hand from
the deuce and a straight from the W987. This would have a result of a loss of 2
bets. Another example would be determining the probability and difference of
quad deuces vs. drawing trips from the 4-card wild straight draw: probability
of (44 * 9) / (178365 * 47) with a resulting gain of 199 bets. Do this for all
30 combinations and then do the calculus required by Kelly.
I cheated and used Excel's Solver rather than doing an analytical maximization
by setting the derivative of log of the product of the 30 possibilities equal to
zero. If I did it right, I came up with an answer of a required bankroll of
81,272 bets.
________________________________
From: Tom Robertson <007@embarqmail.com>
To: vpFREE@yahoogroups.com
Sent: Mon, March 21, 2011 3:29:31 AM
Subject: [vpFREE] Math Help Needed
I'm analyzing something having to do with a hand in
1-2-3-4-4-9-15-25-200 Deuces Wild. The hand is 973s8W, in which the
"best" play is to keep the deuce, but only by a tiny margin over the
straight draw. My question is the bankroll required, according to the
Kelly Criterion, to draw to the deuce instead of the presumably lower
fluctuation straight draw. I've come up with an answer of 35,624
units, which seems reasonable, but I'm not sure I've done it right.
My method was to take the additional frequency of each ending hand
that drawing to the deuce results in over what drawing to the straight
results in, but that required the total frequency of ending hands be 0
instead of the normal 1 that I'm used to in working with the Kelly
Criterion and that the frequency for the straight be negative, neither
of which I'm sure works correctly. Is there a better way of going
about it?
[Non-text portions of this message have been removed]
