Null Result II

Posted by Brh on July 31, 1997 at 04:36:26:

Okay, here is the second stage.

I wanted to see if the full Kelly distribution is altered if one is unable to resize for a
finite number of hands. Grimy suggested that a shoe is a good set to consider.

Now, there are some problems doing this, I hope the approach I have used is reasonable. I
will try to explain it in some detail, as it was difficult for me to figure out at the time.

The main problem is that once we have more than one consecutive hand, we no longer have a
well defined value of the True Count. That is, it is no longer possible to factorize the
distribution of outcomes into the form P(TC,U), and as such it also does not make sense
to minimise

  J(i) = SUM(U) P(i,U) LOG(1 + b(i)*U)

in order to find the optimal bet b(TC). Basically, before we start the shoe, we do not
know how many hands we will play at each TC, so that we cannot pick a prior betting level
the way we can before we play an individual hand knowing the TC.

So after long thought, I came up with this approximation, and I emphasize it is an
approximation. What I did was collect the statistics, by true count, of the outcomes U, for all
the hands which were played at each true count, during the course of the shoe. If this sounds
confusing, it is, I will try to explain it with some examples.

For all my sims I have used a 6 deck shoe, 1.5 cutoff, one player playing heads up. For the
first study, all hands were played, I will address the case of wonging later. With this
scenario, the distribution of total hands played per shoe is

   40   41   42   43   44   45   46   47   48
  0.1  1.2  6.2 18.5 31.1 27.3 12.2  3.0  0.3

Take a true count of zero, for every shoe that is played, there is at least one hand with a TC
of zero, namely the first hand. In all likelihood, there will be many more. Just say, that for
a particular shoe, the true count of zero (or for my stats, exactly zero and all TC's between
0 and +1, I call those between -1 and 0 "TC=-1") occurs 30 times with a combined outcome of
+4 units for those 10 hands. This is what I record as the 'shoe' outcome U for a TC of zero.

Now take a TC of +4, it may have occured twice in the shoe, for a combined outcome U of 4 units,
say there were two doubles and both won. This will be recorded as a TC=+4 shoe outcome of +4.

Now what about a TC of +10 that did not occur in this shoe? Well, it is recorded as an outcome
of U=0, but in a different category than if the TC did occur and there was a real outcome of
zero, ie one hand that was a push. It does not actually effect the results either way, since
LOG(1+b*0)=LOG(1)=0, and neither case can contributes to the utility function sum J.

With this approximation, the distribution at each true count tends to be larger and more spread
near zero, since these happen more often resulting in a larger range of outcomes {U}. As one
goes to the high values of TC, past the point where the average number of hands per shoe is
much less than one, the null outcome dominates, but the others will tend to the single hand
values. Basically, if a TC of +15 ever occurs, it may occur only once per shoe, so that the
set of outcomes per shoe, will be very close to a single hand outcome.

The consequences of this are that for very high TC's, the single hand result of the previous
post will hold, namely that EV/VAR is a good approximation. It is in the intermediate TC's
in the range of +4 or so, which could perhaps exhibit anomalous behaviour. It is
the "lose 5 big bets in a row" story which this will investigate.

So we have the same utility function as before, except that P(TC,U) is now the probability
of total outcome U, for all the hands of true count TC, which occured during the shoe,
and we minimize the expression of the form above.

  TC       Freq        Ev          Var       Ev/Var       b(TC)
 -17     0.00012    -0.00002     0.00022    -0.10043    -0.09972
 -16     0.00031    -0.00003     0.00062    -0.04855    -0.04799
 -15     0.00072    -0.00012     0.00145    -0.08378    -0.08191
 -14     0.00151    -0.00016     0.00299    -0.05468    -0.05460
 -13     0.00355    -0.00036     0.00769    -0.04697    -0.04687
 -12     0.00725    -0.00082     0.01644    -0.04999    -0.04976
 -11     0.01368    -0.00143     0.03245    -0.04410    -0.04389
 -10     0.02526    -0.00279     0.06520    -0.04284    -0.04262
  -9     0.04484    -0.00408     0.12187    -0.03348    -0.03331
  -8     0.07832    -0.00728     0.23727    -0.03068    -0.03055
  -7     0.12392    -0.01099     0.41513    -0.02648    -0.02638
  -6     0.19847    -0.01747     0.76335    -0.02289    -0.02282
  -5     0.28587    -0.02356     1.29069    -0.01826    -0.01819
  -4     0.42043    -0.03498     2.35408    -0.01486    -0.01482
  -3     0.56696    -0.04660     4.03213    -0.01156    -0.01153
  -2     0.75990    -0.05902     7.41574    -0.00796    -0.00794
  -1     0.93219    -0.04942    11.26933    -0.00439    -0.00438
   0     1.00000    -0.01645    15.68726    -0.00105    -0.00105
   1     0.73904     0.02258     7.00927     0.00322     0.00322
   2     0.54461     0.03024     3.84045     0.00787     0.00788
   3     0.39708     0.02509     2.15985     0.01162     0.01163
   4     0.27044     0.01919     1.24658     0.01539     0.01541
   5     0.18333     0.01457     0.71681     0.02033     0.02037
   6     0.11358     0.00920     0.39617     0.02321     0.02329
   7     0.07070     0.00639     0.21876     0.02922     0.02936
   8     0.04108     0.00419     0.11619     0.03605     0.03629
   9     0.02266     0.00225     0.05802     0.03872     0.03901
  10     0.01201     0.00127     0.02814     0.04506     0.04550
  11     0.00629     0.00060     0.01400     0.04319     0.04341
  12     0.00295     0.00031     0.00612     0.05004     0.05061
  13     0.00153     0.00013     0.00315     0.04275     0.04305
  14     0.00060     0.00006     0.00117     0.05265     0.05315
  15     0.00027     0.00003     0.00046     0.05836     0.05937
  16     0.00012     0.00001     0.00021     0.06789     0.06890

Note that now, 'Freq' must be given a different interpretation. It is the frequency for which
true count TC occurs at least once during the shoe. For TC=0 is 1.0, as this always occurs.
It is not a measure of how many hands at the TC actually happened, just the frequency that
at least one hand at this TC will happen during the course of the shoe. Note that the Ev
and Var automatically have 'Freq' factored into them, since if a count occurs more often, it
will win or lose more, and have a larger corresponding variance. The many null events at the
high TC's have the effect of suppressing the mean values. Note, though that this cancels out
in the ratio of Ev/Var, and as I explained before, also does not contribute to the utility
function. Note the peak of unit Ev at TC=+2.

Note that the values of Ev and Var are the actual unit expectation and variance per shoe
per true count, and the largest unit variance occurs around zero, since this is where
most hands are played.

Again, we see that there is a small tendency for the exact Kelly fraction to be a little
more than Ev/Var at high true counts, but we do not see any large anomalies in the
values.

Conclusion:  The approximation Ev/Var still holds in the computation of optimal betting
             levels, for entire shoes.

The third stage, is to investigate the effect on bankroll requirement for a player using
an optimal spread, where resizing is restricted to after every shoe. Parts I and II show
that the choice of min, intermediate, and maximum betting fractions are the same as for
single hands, but it remains to demonstrate whether the overall bankroll fraction f, or
equivalently, the optimal unit bankroll, needs to be decreased or increased respectively,
due to the much larger 'system' Ev and Var for the whole shoe.

That will be the subject of the next post 'Almost Null Result III' ;-)

On a final note, I do not believe that various modes of Wonging changes the above.
It may reduce the number of negative hands per shoe, but it does not increase the
absolute number of large Tc hands per shoe, which is what is being tested here.

It may impact upon the subject of the final post, I am still investigating this possibility.

Cheers,
Brett.

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