Null Result I

by Brett Harris


Posted by Brh on July 30, 1997 at 12:39:20:

Hi All,

Grimy's post reinforced my concern about various Kelly issues to do with distributions, number of hands before resize etc..

I hope I am able to put everyone's mind (well mine at least) to rest with some of these things.

Firstly, I had to convince myself that for single hand resizing, that the EV/VAR approximation was valid, even at high counts. And it was a good test for the next stage. So I adapted my simulator to keep full statistics on the single hand outcomes. That is :


P(TC,U) : where U = {-3,-2,-1.5,etc...}

the probability of all possible outcomes at all values of the true count. This allowed the full minimization of the Kelly function, ie find b(i), such that


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

using a Newtonian minimization algorithm.

The result ? Virtually no difference from using EV/VAR,
which is very good to know.

A typical 6 deck game, 1.5 decks cutoff, 2 billion hand sim, gives the following comparison :


TC Freq Ev Var Ev/Var b(exact)
-15 0.00002 -0.09450 1.51363 -0.06244 -0.06158
-14 0.00005 -0.08443 1.48676 -0.05679 -0.05607
-13 0.00012 -0.07956 1.47344 -0.05400 -0.05338
-12 0.00025 -0.06744 1.46298 -0.04610 -0.04565
-11 0.00050 -0.06248 1.44923 -0.04311 -0.04273
-10 0.00102 -0.05469 1.43155 -0.03820 -0.03791
-9 0.00194 -0.04865 1.41874 -0.03429 -0.03406
-8 0.00379 -0.04201 1.40506 -0.02990 -0.02973
-7 0.00673 -0.03590 1.39111 -0.02580 -0.02568
-6 0.01244 -0.03068 1.37847 -0.02226 -0.02216
-5 0.02127 -0.02543 1.36733 -0.01860 -0.01853
-4 0.03907 -0.02060 1.35595 -0.01519 -0.01515
-3 0.06753 -0.01548 1.34643 -0.01150 -0.01147
-2 0.12445 -0.01041 1.34342 -0.00775 -0.00774
-1 0.19054 -0.00581 1.33590 -0.00435 -0.00435
0 0.26530 -0.00141 1.33563 -0.00105 -0.00105
1 0.11779 0.00474 1.34609 0.00352 0.00353
2 0.06472 0.01010 1.34156 0.00753 0.00754
3 0.03649 0.01563 1.33959 0.01167 0.01169
4 0.02069 0.02125 1.36221 0.01560 0.01563
5 0.01172 0.02844 1.37806 0.02064 0.02070
6 0.00636 0.03444 1.39891 0.02462 0.02471
7 0.00350 0.04097 1.40260 0.02921 0.02935
8 0.00187 0.04889 1.39139 0.03513 0.03534
9 0.00095 0.05376 1.38085 0.03893 0.03918
10 0.00047 0.05909 1.36723 0.04322 0.04353
11 0.00023 0.06703 1.36015 0.04928 0.04969
12 0.00010 0.07168 1.34602 0.05325 0.05371
13 0.00005 0.07901 1.33575 0.05915 0.05969
14 0.00002 0.08899 1.32558 0.06714 0.06798

While of course the negative b's are unphysical, they are there for the curious.

Note, if there is any sort of difference, it suggests that the correct Kelly bet is actually slightly greater, than that given by the Ev/Var approximation, but the difference is so slight that it is virtually irrelevant.

The next post will show that this result is unchanged even if the distributions are per shoe, rather than per hand.

Cheers,
Brett.