From: hall@kronos.arc.nasa.gov (Michael Hall)
Subject: MATH: Kelly criterion for simultaneous independent (BJ) wagers
Organization: NASA/ARC Information Sciences Division
Date: Wed, 11 Nov 1992 21:01:29 GMT

A word problem that you never saw in your math courses...

You and 999 of your card counting friends form a blackjack team.  You play
a single or double deck game with a 1-8 spread where you can earn .034 units
per round with a standard deviation of 3.97 units.  What percent more units
should the bankroll have to let you play one hand simultaneously with all
your friends (at different tables), as opposed to just alone?  (You are, of
course, a firm believer in the Kelly criterion as the One True Utility
Function...)

Griffin gives an approximate equation in "Gambling Ramblings" to compute f,
the fraction of bankroll to bet as per the Kelly criterion, given p (the
probability of winning), q (the probability of losing), and n (the number
of simultaneous independent bets).  In LISPese:

(defun f (p n)
  (let ((q (- 1.0 p)))
     (/ (- p q) (+ (* 4 p q) (* n (sqr (- p q)))))))

One can approximate blackjack with a given win per hand (wph) and variance
(s2) as a coin tossing game where you bet the square root of the variance
on a coin with the probability of heads given by:

(defun p-heads (wph s2)
  (+ 0.5 (/ wph (* 2 (sqrt (+ s2 (sqr wph)))))))

Oh yeah, sqr is of course:

(defun sqr (x) (* x x))

And the solution to the problem is given by:

(defun solution (wph s n)
  (let ((p (p-heads wph (sqr s))))
     (- 1.0 (/ (f p n) (f p 1)))))

The equation is simplified by some cancellation not shown.

Plugging in, we get:

(solution .034 3.97 1000) => .068

So, 6.8% more units are needed for you and your 999 card counting
friends to play one hand compared to if you were to just play one
hand yourself.

Here's a handy table:

SIMULTANEOUS   EXTRA KELLY
  HANDS          BANKROLL
(hundreds)      (percent)
    1              0.7
    2              1.4
    4              2.8
    6              4.2
    8              5.5
   10              6.8
   15              9.9
   20             12.8
   25             15.5
   50             26.8
   75             35.5
  100             42.3

It's reasonable to estimate about 100 hands per hour, so if a blackjack
team of 5 players plays 5 hours before regrouping to communicate changes
in bankroll, that's 25 man-hours, which requires the bankroll to be divided
into 15.5% additional units compared to playing alone.  This of course
will vary depending on your actual win-per-hand and its standard deviation.

Now the f equation is telling you to bet *more* than your whole
bankroll sometimes.  That's because the equation was an approximation.  But
in those cases, you should be betting *almost* all your bankroll anyway.  And
if you do bet more than your whole bankroll as it suggests, the probability
is minute that you will lose nearly all the bets and be forced to declare
bankruptcy after defaulting on your casino credit.

-- 
Michael Hall            \ You are here.
NASA Ames Research Center\ The Mirage is in Las Vegas.    - Billboard in
hall@ptolemy.arc.nasa.gov \ Hmmmm...                        Atlantic City
Newsgroups: rec.gambling