(Almost) Null Result III

Posted by Brh on August 03, 1997 at 09:20:22:

Hi again,

This is the third installment of this shoe-Kelly betting series. In some ways, it is also
the simplest, as most of the hard work has already been done.

What the first two posts have shown, was that the EV/VAR approximation for the intermediate
Kelly bets, and as a consequence, for the formulation of an optimal betting strategy, was
perfectly valid in the single hand case. I then went on to demonstrate that it remained valid
for the number of hands likely to be played at each TC for the duration of the shoe.

This third post, assumes an optimal betting spread has been chosen, but no attempt at
resizing will be made during a shoe. Basically, we will have a single 'shoe' game, with
a specified EV, VAR, for a given unit bet. All that is required is to keep a record of
the number of outcomes U for each shoe, to build up a distribution. It is just the plain
win/loss distribution for each shoe, and it is this distribution which I have already
said is not exactly normal. For comparison, I also produced the single hand outcome
distribution.

Since we are only dealing with total outcomes, it is not necessary to do as many hands in
the sim, I settled on 150 million which seemed sufficient.

I looked at three cases, the first two of which incorporated Wonging.

In the Wonging cases, the player leaves the table completely, ie a new shuffle is done, if the
Hi-Lo true count ever dips to -1 or below. For spread 2, the player backcounts and then
jumps in at a TC of +2. If the TC then drops below +2, he goes back to backcounting, but
remains in case he gets a chance to jump in again, unless of course the TC dips below
-1 in which case he 'wanders off'.

                     Spr 1          Spr 2          Spr 3
        TC < -1    :    leave          leave          1 unit
  -1 =< TC <  0    :    1 unit         0 units        1 unit
   0 =< TC < +1    :    1 unit         0 units        3 units
  +1 =< TC < +2    :    2 units        0 units        7 units
  +2 =< TC < +3    :    4 units        1 unit        10 units
  +3 =< TC < +4    :    7 units        2 units       10 units
  +4 =< TC < +5    :    9 units        2 units       10 units
  +5 =< TC < +6    :   10 units        3 units       10 units
  +6 =< TC < +7    :   10 units        3 units       10 units
  +7 =< TC < +8    :   10 units        4 units       10 units
  +8 =< TC < +9    :   10 units        4 units       10 units
  +9 =< TC <       :   10 units        5 units       10 units

Note the very flat spread for the backcounter, this has been noted before, most recently
in a post by Mike Lea. In reality, since the bankroll fraction for spread 2 is that much
lower than for spread 1 or 3, an optimal bettor using spread 2 would actually be betting more
money at each level than a bettor using the other spreads.

Once the distribution was accumulated, the same full cost function minimization can be
done as previously described, except in this case we only have one betting fraction to
evaluate. That is, for the full (shoe) distribution P(U), we can minimize

 J = SUM(U) P(U) LOG(1 + f*U) =  SUM(U) P(U) LOG(1 + U/K)

where K=1/f is the optimal bankroll in units. This is to be compared to what one would
obtain simply by evaluating K'=VAR/EV.

Results :

Spread 1 :

           Ev        Sd       N0      Var/Ev     1/f     (1/f)/(Var/Ev)    Total Samples
Hand :   0.0290    3.9844   18906     547.85    546.62      0.99775        150000000
Shoe :   0.6713   19.1085     810     543.92    550.24      1.01162          6475055

Spread 2 :

           Ev        Sd       N0      Var/Ev     1/f     (1/f)/(Var/Ev)    Total Samples
Hand :   0.0088    1.0945   15528     136.39    135.90      0.99641        150000000
Shoe :   0.2035    5.2546     667     135.70    138.12      1.01783          6475055

Spread 3 :

           Ev        Sd       N0      Var/Ev     1/f     (1/f)/(Var/Ev)    Total Samples
Hand :   0.0189    4.2270   49978     944.97    943.59      0.99854        150000000
Shoe :   0.8369   28.0327    1122     939.00    942.81      1.00406          3388946

Note, the number of samples for Spread 3 is lower, since more hands are played per
shoe without Wonging. The Ev, Sd and N0 figures for the shoe, are in units of
per shoe, or shoes respectively.

All the shoe P(U) distributions show the Exp(-|U-Ev|*Sqrt(2)/Sd) behaviour. If
anyone wants the data to look at themselves, just let me know.

One can see that there is a difference between the values of Var/Ev and (1/f) for all
cases considered. Firstly for the single hand resize case, the Kelly fraction f is
actually slightly higher (1/f smaller), than that given by Ev/Var. This may another
case of the true Kelly fraction being slightly greater than given by Ev/Var as seen
in the previous two posts. This may be a general feature of the discrete blackjack
distribution.

However, for the shoe cases things are a little more interesting. Firstly though,
the player can be assured that there isnt a radical difference between Var/Ev and (1/f), so
that what they may have been using up to now is perfectly Ok, hence the title of this post.

As one might expect, the biggest difference is for the spread with the smallest bankroll
fraction, Spread 2. Due to the strict Wonging, most of the bets made are 'big' bets,
and so there is the most potential for large swings in the course of a shoe. But even
in this case, the bankroll required is only 1.7% greater than for single hand resizing.

For the play all case, the difference is almost negligible, due to the large ratio
between big bet and bankroll.

Nevertheless, there is an effect which would probably increase if one were to delay resizing
for a number of shoes.

Cheers,
Brett.

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