This is the promised follow-up to the earlier thread about SCORE for optimal play. As usual, let's set the stage: all results here are for 6 decks, S17, DOA, DAS, SPL1, no surrender, 4.5/6.0 pen, floored indices with 1/2 deck resolution, spread 1-16.

First, some house-keeping to clean up the table of SCOREs for the various complexities of betting and playing strategies: I re-formatted to make it a bit easier to read, changed the Hi-Opt II insurance index to +5, and computed accurate variance (resulting in slightly lower SCOREs) and exact corresponding risk of ruin (resulting in slightly lower RORs).

Code:
Bet strategy  | Play strategy  |  $/hr | SCORE | ROR % | <=0  +1  +2  +3  +4  +5  +6  +7  +8  +9 >=10
=====================================================================================================
Hi-Lo         | Basic TDZ      | 14.46 | 15.03 | 12.49 |   6, 20, 57, 93, 96, 96, 96, 96, 96, 96, 96
Optimal       | Basic TDZ      | 15.90 | 17.22 | 11.45 |   6:1:96

Hi-Lo         | Hi-Lo I18 BJA3 | 21.35 | 21.30 | 13.45 |   8, 23, 63,102,128,128,128,128,128,128,128
Hi-Lo         | Hi-Lo I18 Cac  | 21.51 | 21.59 | 13.29 |   8, 23, 63,102,128,128,128,128,128,128,128
Hi-Lo         | Hi-Lo full     | 22.22 | 22.47 | 13.08 |   8, 23, 63,102,128,128,128,128,128,128,128

Optimal       | Hi-Lo I18 BJA3 | 23.75 | 24.71 | 12.33 |   8:1:128
Optimal       | Hi-Lo I18 Cac  | 24.00 | 25.06 | 12.24 |   8:1:128
Optimal       | Hi-Lo full     | 24.67 | 26.01 | 12.00 |   8:1:128
Optimal       | Hi-Lo full     | 26.23 | 26.08 | 13.52 |   9:1:144

Hi-Opt II ASC | Hi-Opt II full | 27.01 | 26.90 | 13.48 |   9,  9, 25, 48, 72, 96,119,144,144,144,144
Optimal       | Hi-Opt II full | 28.25 | 28.59 | 13.04 |   9:1:144

Hi-Lo         | Optimal CDZ-   | 30.07 | 31.10 | 12.47 |   9, 31, 74,118,144,144,144,144,144,144,144
Hi-Opt II ASC | Optimal CDZ-   | 33.18 | 33.12 | 13.39 |  10, 10, 30, 55, 80,105,129,155,160,160,160
Optimal       | Optimal CDZ-   | 34.90 | 35.38 | 12.97 |  10:1:160
A few comments on this table: first, as has been pointed out earlier, any row without the word "optimal" somewhere in it isn't really news. That is, for example, a Hi-Lo player can significantly improve win rate performance by switching to Hi-Opt II playing strategy, in this particular case about 20% improvement. Nothing new there.

What I think is interesting new information, however, is the observation that perfect betting yields much less improvement than perfect play. The above table is sorted by increasing win rate and SCORE. At the low end is the basic strategy player betting Hi-Lo... and only slightly better is the basic strategy player betting perfectly. At the high end is the optimal strategy player, who does better than all of the rest of the crew, no matter what betting strategy he uses. (It is interesting that Hi-Lo index play seems to be the exception here; that is, full indices aren't worth as much as perfect betting with Illustrious 18.)

Also, note there are two entries for Hi-Lo full index play with perfect betting: one with minimum bet $8 and the other with $9. The former is included as a "level field" comparison with the Illustrious 18 players, but the latter is the best we can do subject to the 13.5% ROR constraint. The point in including both is to again highlight the cost of estimating (vs. exact computation of) ROR.


As mentioned at the end of the earlier thread, I think it's useful to be able to compare these various complexities of strategy when constrained to at least moderately more realistic betting ramps, and at more comfortably lower risks of ruin. To that end, the following figure is in essentially the same format as the original, but with all of the updates over the last month or two:

score_ror.jpg

Here is the idea: as with SCORE, let's assume we start with a $10,000 bankroll... but are constrained to bet a minimum of $10 on each round, bet the maximum of $160 on true counts of 10 or higher, and multiples of $10 in between. Each point in the above figure represents one possible such ramp, where the x-coordinate is the corresponding risk of ruin, and the y-coordinate is the corresponding win rate. The vertical red line indicates the SCORE-targeted risk of ruin of 1/e^2, or about 13.5%.

(For reference, I included the SCORE-optimized win rates from the above table on the figure, indicated by the individual points clustered around the red line.)

Subject to this $10-multiple betting constraint, we can use this figure to determine achievable win rate for any acceptable risk of ruin. For example, using the red line to target the same 13.5% ROR that we used to compute SCORE, we find the point to the *left* of that vertical line that has the highest win rate. The following table shows the results. Note, for example, that for fixed basic TDZ strategy, we simply can't do it: there is no point far enough to the left, i.e., the lowest risk of ruin is 14.90%.

Code:
Bet strategy  | Play strategy  |  $/hr | SCORE | ROR % | <=0  +1  +2  +3  +4  +5  +6  +7  +8  +9 >=10
=====================================================================================================
Practical     | Basic TDZ      |  8.59 |       | 14.90 |  10, 10, 30, 50, 70, 90,110,130,150,160,160
Practical     | Hi-Lo I18 BJA3 | 19.76 |       | 13.50 |  10, 20, 50, 80,110,150,160,160,160,160,160
Practical     | Hi-Lo I18 Cac  | 20.16 |       | 13.53 |  10, 20, 50, 80,110,160,160,160,160,160,160
Practical     | Hi-Lo full     | 21.18 |       | 13.46 |  10, 20, 50, 90,110,150,160,160,160,160,160
Practical     | Hi-Opt II full | 26.51 |       | 13.53 |  10, 10, 20, 40, 70, 90,110,120,150,160,160
Practical(HO2)| Optimal CDZ-   | 33.09 |       | 13.49 |  10, 10, 30, 60, 80,100,130,150,160,160,160
But more importantly, this figure shows how win rate degrades as we tolerate less and less risk of ruin. For example, suppose that we want to limit ourselves to, say, 5% risk of ruin. Constrained to integral bets starting with 1000 units, it can't be done! The lowest feasible risk of ruin is about 5.6%, realized only by the optimal strategy player (the left-most blue point in the figure), with a corresponding win rate of only 16.45/hr.

This is getting long, so one final thought: I'm not sure how best to compute these performance options for optimal play. In the above figure, I cheated: the blue optimal play points actually reflect betting behavior using the ramp for the Hi-Opt II ASC true count. This yields close to, but not quite, the optimal-betting performance. The problem is, I'm not sure how to "optimally" utilize any particular $10-multiple betting ramp, since we are constrained to a smaller number of distinct wagers (11, up through true count 10) than the bet spread (1-16) would otherwise allow.