Floating Advantage, and 'Off-Balanced' Counts
From Stanford Wong's BJ21
Posted by Brh on 9 Jul 1998, 5:39 a.m.
The recent discussion about floating advantage on the Misc page got me thinking.
If the advantage for a count of zero increases as the number of decks decreases, what if a constant offset is added to the running count before the true count is calculated? Mathematically this is equivalent to adding a factor of C/Ndecks to the true count, which will therefore increase as the deck is depleted.
If non-linear effects cause the count of zero to have a higher advantage deeper into the deck, then this extra term may make the true count a better predictor of the advantage at that point.
I decided to test this with Halves doubled, so that I could have more freedom with the constant term. Level one counts such as Hi-Lo could give a much larger weight to any integer addition to the true count.
The game is 6D, DAS, DOA, RSA, no surrender, dealt down to 1.5 decks.
Usually one true count for Halvesx2 is equivalent to about 0.25% advantage. Adding a term C to the running count will cause, for instance, a change of (1/2-1/6)*C=0.33*C to the true count from a full deck to 2 decks remaining. This is in effect a perceived increase of 0.25*0.33*C to the advantage.
I had no idea if the idea would work, so I ran four sims, with C=0,1,2, and 3, to see what would happen. The way I did it was keep the usual balanced count for playing decisions (hence for all sims identical cards and decisions were make making the results highly correlated and the differences significant), but the offset was added before true count was calculated for the spreadsheet-type output (ie that which is used in the bet postprocessing).
Here are the results for optimal 1-10 spreads for each value of C, and for four different Wonging situations. Note for the standard (C=0) case, the optimal Wonging point is TC=+2.
Play all :
C Avbet Ev Sd k Ev/Sd N0 Unit Bet EV SD %Pl
0 2.836 0.0210 4.387 913 0.480% 43333 $ 5.47 $11.53 $240.19 100%
1 2.826 0.0211 4.373 905 0.483% 42853 $ 5.52 $11.66 $241.53 100%
2 2.779 0.0209 4.324 893 0.484% 42687 $ 5.59 $11.71 $242.00 100% ***
3 2.811 0.0211 4.373 905 0.483% 42825 $ 5.52 $11.67 $241.61 100%
Play only TC >= 1 :
C Avbet Ev Sd k Ev/Sd N0 Unit Bet EV SD %Pl
0 0.974 0.0145 2.113 307 0.687% 21146 $16.26 $23.64 $343.84 36.2%
1 1.052 0.0152 2.228 324 0.686% 21212 $15.40 $23.57 $343.31 38.9%
2 1.129 0.0159 2.337 342 0.683% 21436 $14.61 $23.32 $341.51 41.7%
3 1.196 0.0164 2.427 358 0.677% 21783 $13.95 $22.95 $338.78 44.6%
Play only TC >= 2 :
C Avbet Ev Sd k Ev/Sd N0 Unit Bet EV SD %Pl
0 0.737 0.0122 1.760 252 0.698% 20523 $19.82 $24.36 $349.02 27.2%
1 0.798 0.0130 1.861 266 0.699% 20456 $18.77 $24.44 $349.59 29.4%
2 0.867 0.0137 1.971 282 0.696% 20590 $17.67 $24.28 $348.45 31.7%
3 0.928 0.0142 2.051 295 0.693% 20765 $16.91 $24.07 $346.98 34.1%
Play only TC >= 3 :
C Avbet Ev Sd k Ev/Sd N0 Unit Bet EV SD %Pl
0 0.550 0.0100 1.440 207 0.694% 20745 $24.09 $24.10 $347.15 20.6%
1 0.613 0.0108 1.558 223 0.698% 20486 $22.41 $24.40 $349.34 22.4%
2 0.671 0.0116 1.665 237 0.700% 20382 $21.02 $24.53 $350.23 24.2% ***
3 0.729 0.0123 1.764 251 0.700% 20400 $19.84 $24.50 $350.07 26.2%
Play only TC >= 4 :
C Avbet Ev Sd k Ev/Sd N0 Unit Bet EV SD %Pl
0 0.327 0.0064 0.940 137 0.684% 21315 $36.42 $23.45 $342.48 15.6%
1 0.374 0.0072 1.041 150 0.692% 20859 $33.24 $23.97 $346.20 17.1%
2 0.467 0.0088 1.266 182 0.695% 20700 $27.43 $24.15 $347.52 18.6%
3 0.511 0.0093 1.347 193 0.697% 20562 $25.87 $24.31 $348.69 20.2%
Clearly the main difficulty is getting the same percentage of hands played given the change in the distribution of the true count. However, for the play all case this is not a problem and for this case all the offset values give a positive result, with C=+2 is the best.For the other cases, the hand percentage problem makes comparison difficult, but one can find the overall best N0 by choosing the optimal Wong point. Again, C=+2 at TC>=+3 with N0=20382, compared to C=0 at TC>=+2 with N0=20523, is the best result
So adding +2 to the running count for Halves doubled, reduces N0 by about 500 for the play all case, and about 150 for the optimal Wonging case.
Since this strategy appears to have the greatest benefit if the magnitude of C is of the order of the level of the system, it is natural to see what happens to a level one count like Hi-Lo. All is the same as the above except this time the constant C is added to the running count of Hi-Lo before computing the true count for betting purposes.
Play all :
C Avbet Ev Sd k Ev/Sd N0 Unit Bet EV SD %Pl
0 2.787 0.0193 4.331 967 0.447% 49897 $ 5.16 $10.02 $223.84 100.0%
1 2.824 0.0197 4.377 970 0.451% 49126 $ 5.15 $10.17 $225.59 100.0%
2 2.703 0.0193 4.284 948 0.451% 49028 $ 5.27 $10.19 $225.81 100.0% ***
3 2.759 0.0193 4.296 951 0.451% 49048 $ 5.25 $10.19 $225.77 100.0%
4 2.791 0.0193 4.336 969 0.447% 50019 $ 5.15 $ 9.99 $223.56 100.0%
Play only TC >= 0 :
C Avbet Ev Sd k Ev/Sd N0 Unit Bet EV SD %Pl
0 1.412 0.0169 2.721 436 0.623% 25764 $11.44 $19.40 $311.50 53.0%
1 1.528 0.0174 2.850 465 0.612% 26615 $10.75 $18.78 $306.48 59.4%
2 1.704 0.0183 3.065 511 0.598% 27874 $ 9.76 $17.93 $299.48 65.5%
3 1.872 0.0190 3.261 559 0.582% 29424 $ 8.93 $16.99 $291.49 71.0%
4 1.989 0.0191 3.397 603 0.563% 31508 $ 8.29 $15.86 $281.68 75.9%
Play only TC >= 1 :
C Avbet Ev Sd k Ev/Sd N0 Unit Bet EV SD %Pl
0 0.713 0.0114 1.736 262 0.662% 22799 $19.07 $21.93 $331.14 26.5% *
1 0.829 0.0126 1.899 285 0.664% 22680 $17.48 $22.04 $332.01 31.1%
2 0.965 0.0137 2.098 319 0.657% 23145 $15.65 $21.60 $328.66 36.2%
3 1.123 0.0151 2.341 361 0.647% 23889 $13.81 $20.93 $323.50 41.7%
4 1.249 0.0158 2.505 396 0.631% 25074 $12.60 $19.94 $315.76 47.6%
Play only TC >= 2 :
C Avbet Ev Sd k Ev/Sd N0 Unit Bet EV SD %Pl
0 0.334 0.0063 0.980 150 0.650% 23607 $33.18 $21.18 $325.42 14.7%
1 0.383 0.0069 1.059 160 0.660% 22914 $31.18 $21.82 $330.31 17.7%
2 0.530 0.0092 1.395 209 0.665% 22612 $23.82 $22.11 $332.51 21.0%
3 0.655 0.0108 1.632 245 0.666% 22527 $20.40 $22.19 $333.13 24.7% ***
4 0.793 0.0123 1.866 283 0.659% 22988 $17.66 $21.75 $329.78 28.8%
Play only TC >= 3 :
C Avbet Ev Sd k Ev/Sd N0 Unit Bet EV SD %Pl
0 0.237 0.0053 0.881 144 0.608% 27043 $34.48 $18.48 $304.05 8.2%
1 0.177 0.0037 0.601 95 0.631% 25095 $52.51 $19.92 $315.63 10.1%
2 0.274 0.0055 0.859 133 0.645% 23983 $37.54 $20.84 $322.86 12.3%
3 0.305 0.0058 0.898 137 0.653% 23404 $36.35 $21.36 $326.83 14.8%
4 0.381 0.0069 1.052 160 0.656% 23222 $31.16 $21.53 $328.11 17.6%
This time, it seems that C=2 or C=3 is the best figure, which is interesting since Hi-Lo is a level one count with about 0.5% per true count.Cheers,
Brett.