Toward SCORE for optimal play

[ericfarmer]

Several weeks ago, I presented some very preliminary results of some updates to my CA software to compute the exact probability distribution of outcomes of a round for various playing strategies (see forum thread here). This allows us to get at not just exact EVs, but now also variances, for any depleted shoe and any playing strategy.

The eventual goal of those updates, and this ongoing analysis, is to compute SCORE for optimal play-- that is, if you knew the exact pre-deal distribution of outcomes of every round, how much (more) could you win, using SCORE as the yardstick to normalize for initial bankroll, hands per hour, etc.?

I have finally made some additional progress on that analysis. My more detailed notes are on my blog at the following link:
https://possiblywrong.wordpress.com/...gamblers-ruin/

But the gist is in the following figure copied from the post:

score_ror.jpg

I'm using the same setup (6 decks, S17, DOA, DAS, SPL1, no surrender, 75% penetration), and I'm comparing the same 5 strategies as in the earlier analysis, but this time considering betting variation as well as playing variation:

1. Fixed basic "full-shoe" total-dependent zero-memory strategy (TDZ), using Hi-Lo (floored) true count for betting only.
2. Hi-Lo Illustrious 18 indices.
3. Hi-Lo full indices.
4. Hi-Opt II full indices with ace side count.
5. "Optimal" betting and playing strategy, where playing strategy is CDZ- optimized for each pre-deal depleted shoe, and betting strategy is ramped according to the corresponding exact pre-deal expected return.

For each strategy (indicated by color in the figure), there are 1,307,504 points, one for each possible 1-16 spread betting ramp through TC=10 ($10 minimum, $160 maximum). The y-axis indicates win rate (assuming 100 hands per hour), and the x-axis indicates risk of ruin (assuming $10,000 initial bankroll).

This is pretty hot off the press, and I already have plenty of questions of my own, so any review is appreciated.

Happy New Year!
E

[bejammin075]

Is the red line full kelly?

Would the optimal bet ramp for the perfect player, or Hi-Opt II-Asc, change much by depth of the shoe?


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[bejammin075]

Looking at the left edge of the yellow and green curves, where they are perpendicular [meant to say parallel] to the Y axis, does that mean, for the same risk of ruin, the HOII player would make $14, and the HiLo full index player would make $9 ??


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[bigplayer]

Would the optimal bet ramp for the perfect player, or Hi-Opt II-Asc, change much by depth of the shoe?

The optimal bet is always Edge/Variance x Kelly Bankroll regardless of penetration.

[ericfarmer]

There have been a couple of interesting questions posed out-of-band, that suggest it is worth clarifying some details of methodology.

First, why so many data points in the figure, and how do we determine SCORE? I wanted to understand how the choice of betting ramp affected performance, so rather than just optimizing the ramp, it was feasible to evaluate and show performance for *all* possible "practical" ramps, where by "practical" I mean betting in $10 increments. Each data point (of a particular color, representing a particular playing strategy) represents one betting ramp. At one extreme is the ramp (10,10,10,10,10,10,10,10,10,10,160), indicating that we always bet $10 until the true count reaches +10; at the other extreme is the ramp (10,160,160,160,160,160,160,160,160,160,160), indicating that we bet $160 if the true count is positive, $10 otherwise. In between are ramps like (10,20,40,70,100,120,150,160,160,160), etc.

If we pick a particular threshold risk of ruin, such as the usual 13.5% shown in red (actually, this value is 1/e^2, or about 0.135335), then we can restrict our attention to those data points to the *left* of that line, and pick a point as high as possible on/near that line to maximize win rate subject to that risk of ruin. For example, the maximum win rate for Hi-Lo full indices for ROR=13.5% is $16.70/hour. But there's nothing particularly special about that red line; you can pick another target risk of ruin, and use the figure to determine the resulting new best win rate.

It would be very useful to compare this figure with similar data obtained previously via simulation. For example, consider Table 10.43 p. 236 in Schlesinger's BJA3; this is the closest thing to the Hi-Lo I18 setup described here, where the closest appropriate SCORE value is the "practical" (vs. "optimal", since I'm working in $10 increments) for a 1-16 spread. The value in that table is $23.79... compared with only $14.90 in the above figure!

Unfortunately, this isn't quite apples-to-apples, since I'm assuming SPL1 here (since full-distribution computation for SPL3 is currently intractable), and Table 10.43 is for SPL3. And rule variations certainly have a significant effect on SCORE, as the surrounding tables in BJA3 suggest: for example, add late surrender to the strategy options and the SCORE jumps from $23.79 to $36.37. Take away DAS and it drops to $17.19, etc.

[bejammin075]

The optimal bet is always Edge/Variance x Kelly Bankroll regardless of penetration.

Thanks. I just opened up CVCX and loaded the Halves full index canned sim to see about the tools there to look at this. I clicked on the "Bet Report" button and found what I was looking for. This is VERY interesting. So I'm looking at a 1:12 spread, and if I'm reading this right, the TC would need to be 6 1/2 at the beginning of the 6D shoe to make a max bet, but towards the end of the shoe, the max bet would come out at TC 3.

[ZenMaster_Flash]

"TC would need to be 6 1/2 at the beginning of the 6D shoe to make a max bet,
but towards the end of the shoe, the max bet would come out at TC 3."

Will someone please explain the above ?

[ericfarmer]

Looking at the left edge of the yellow and green curves, where they are perpendicular to the Y axis, does that mean, for the same risk of ruin, the HOII player would make $14, and the HiLo full index player would make $9 ??

I assume by "perpendicular" you meant to say "parallel"? At any rate, I think you have the right idea, but the values end up being slightly different (the points are big enough that it's hard to read the level of detail we need for this question from the figure): for Hi-Lo full indices, the minimum realizable risk of ruin is about 0.102355, for a win rate of $8.87/hr. We can't actually realize that small a risk of ruin with Hi-Opt II; the minimum in that case is about 0.10492, for a win rate of $13.98/hr.

Because, as you point out, the (yellow) boundary for Hi-Lo full is so nearly vertical in this region, we can improve our win rate quite a bit with just a marginal increase in risk of ruin. If we "level the field," so to speak, by allowing the Hi-Lo full player the same risk of ruin of 0.10492 as the Hi-Opt II player, then he can win $10.63/hr.

[bejammin075]

Will someone please explain the above ?

I think I got something backwards. The problem with this sim information in CVCX is that the column of the bet report on the far left appears to be cards, but doesn't say if it is cards dealt or cards remaining. The column on the far right has SCORE by depth, and this gave me a clue I got it backwards. The score should be better for hands played towards the end I think. I'll need to think about this some more, but there is definitely a difference in optimal bet ramp by depth.


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[DSchles]

"It would be very useful to compare this figure with similar data obtained previously via simulation. For example, consider Table 10.43 p. 236 in Schlesinger's BJA3; this is the closest thing to the Hi-Lo I18 setup described here, where the closest appropriate SCORE value is the "practical" (vs. "optimal", since I'm working in $10 increments) for a 1-16 spread. The value in that table is $23.79... compared with only $14.90 in the above figure!

"Unfortunately, this isn't quite apples-to-apples, since I'm assuming SPL1 here (since full-distribution computation for SPL3 is currently intractable), and Table 10.43 is for SPL3."

No, unfortunately, it isn't remotely possible that splitting to four hands, instead of two, accounts for an increase in SCORE of 60%!! Nothing could possibly be further from the truth, so I'm afraid the problem lies elsewhere.

Don

[bejammin075]

These are the kinds of tables I was looking at. Load CVCX, pick a canned sim of choice (I chose Halves, full indexes). Click on Bet Report. The optimal ramps here vary by penetration, as I just learned by clicking the help button, it says "Click on Bet Report from the Viewer to reach this page. On entering, the optimal bets will be displayed for every available penetration for the currently displayed number of decks and rules."

Bet Ramp 1.jpgBet Ramp 2.jpgBet Ramp 3.jpg

[DSchles]

"but there is definitely a difference in optimal bet ramp by depth."

Of course there is. What made you think otherwise? Just look on any page of Chapter 10 and go from the top chart to the bottom one to see the obvious differences.

Don

[ericfarmer]

"It would be very useful to compare this figure with similar data obtained previously via simulation. For example, consider Table 10.43 p. 236 in Schlesinger's BJA3; this is the closest thing to the Hi-Lo I18 setup described here, where the closest appropriate SCORE value is the "practical" (vs. "optimal", since I'm working in $10 increments) for a 1-16 spread. The value in that table is $23.79... compared with only $14.90 in the above figure!

"Unfortunately, this isn't quite apples-to-apples, since I'm assuming SPL1 here (since full-distribution computation for SPL3 is currently intractable), and Table 10.43 is for SPL3."

No, unfortunately, it isn't remotely possible that splitting to four hands, instead of two, accounts for an increase in SCORE of 60%!! Nothing could possibly be further from the truth, so I'm afraid the problem lies elsewhere.

Don

This may very well be the case... but is there any actual data (e.g., simulation results) to back this up? Note the earlier observation that the single rule change of allowing surrender-- all other rules in this context held fixed-- causes an increase in SCORE of 53%. Changing from disallowing DAS to allowing it causes an increase in SCORE of 38%. This is from the corresponding tables in BJA3. In light of this range of values, I'm not yet sure how disturbed I should be about the SPL1 vs. SPL3 difference.

You are absolutely right that, as I said, this data is pretty hot off the press, and certainly warrants review/confirmation/etc. I would like to compare with simulation results if they exist, or work on running my own if they don't. (Norm or others, I've looked at archived sim results on this site, but SPL1 doesn't seem to be one of the options; is anything like this available?) But with respect, "Don said nothing could possibly be further from the truth," without accompanying numbers, is not a sufficiently convincing argument.

E