Exact distribution of outcomes for index play

[ericfarmer]

A few weeks ago I made some updates to my CA to efficiently compute not just the exact expected value for various playing strategies, but the exact probability distribution of possible outcomes of a round, from which EV, as well as variance, may be derived. Computing EV allows evaluation of playing strategies; computing the distribution and variance allows evaluation of betting strategies as well. (Ref: algorithm description, software with source code, and additional analysis results.)

As an initial quick look, I have some resulting data to share, evaluating the following 5 playing strategies, intended to span the range of complexity from (1) being the simplest reasonable thing a player can do, to (5) corresponding to the best possible strategy assuming (incorrectly and illegally) a laptop at the table:

1. Fixed basic total-dependent strategy (TDZ).
2. Hi-Lo Illustrious 18 (I18) indices.
3. Hi-Lo full indices.
4. Hi-Opt II full indices with ace side count.
5. "Optimal" composition-dependent zero-memory strategy (CDZ-), re-calculated for the depleted shoe prior to every round. ("Optimal" is qualified to emphasize the zero-memory-ness of the post-split strategy.)

First, the following figure shows a comparison of all 5 strategies' exact standard deviation vs. exact expected value (per unit wager), prior to each round over 100,000 simulated shoes played to 75% penetration (6 decks, S17, DOA, DAS, SPL1). Each dot represents a particular depleted shoe, with the coloring indicating concentration of density (i.e., a lot of dots in that area). The x-coordinate is the corresponding exact pre-round EV, and the y-coordinate is the exact pre-round standard deviation.

stddev_vs_ev_all.jpg

It's interesting that there appear to be two distinct types of trend behavior. In the case of basic TDZ and I18 strategies, the more favorable the shoe, the lower the variance, while for the full index and optimal strategies, more favorable shoes tend to correspond to higher variance. (I will likely mix the terms standard deviation and variance, depending on context for disambiguation, since the former is simply the square root of the latter, and "variance" is easier to type.)

Since we don't actually know the exact EV prior to a given round, let's instead look at the same data, but instead scattering the standard deviation vs. the true count:

std_dev_vs_tc_all.jpg

(Aside: note the vertical "smear" of points at any given true count, indicating that, if all we know is the true count, we don't know the exact standard deviation for the current depleted shoe, either. We have seen this before in past discussion; see this plot of EV vs. true count, for example; even with a TC as high as +5, you can still actually be in a negative-EV situation.)

To compare with similar results from other sources, the yellow curve indicates the overall standard deviation vs. true count, aggregating all of the sampled shoes comprising that true count. These results agree nicely with Norm's simulation results here, for example; this despite a not-quite-apples-to-apples comparison, since (1) the rules are not quite the same, SPL1 vs. SPL3; and (3) the true counts are not quite the same, floored in my case vs. truncated in the linked results. (Norm, please correct me if I misspeak here.) The interesting additional view here is of how the trends continue into more extreme true counts, particularly negative TCs (although the "jaggies" at the endpoints are not to be trusted, due to the small sample sizes).

As I said, this is just an initial look-- eventually the goal is to use this data to evaluate betting strategy, SCORE, etc. All of the raw data here is available on request.

Thanks,
Eric

[Gramazeka]

Hi Eric! Very good work! What the mail you have? I send for you one program.

[ZenMaster_Flash]

Extraordinary work. Mega Kudos to Eric Farmer !

I found this to be most interesting:

• Hi-Lo Ill 18 playing efficiency PE = 0.309.
• Hi-Lo with full indices has PE = 0.470.
• Hi-Opt II with full indices has PE = 0.639.

P.S.

In DD, I use Aces and Sevens
side-counted. How big is MY P.E.?

[seriousplayer]

Extraordinary work. Mega Kudos to Eric Farmer !

I found this to be most interesting:

• Hi-Lo Ill 18 playing efficiency PE = 0.309.
• Hi-Lo with full indices has PE = 0.470.
• Hi-Opt II with full indices has PE = 0.639.

P.S.

In DD, I use Aces and Sevens
side-counted. How big is MY P.E.?

What? I ran simulations with 70 indices in Hi-lo and compared it to Hi-lo full indices. Performance wise Hi-lo with 70 indices is about the same as Hi-lo-full. With very marginal differences. So how does that affect PE?

[Three]

Very nice Eric. You always limited the number of cards when using CA. It was always interesting but not exactly what we tend to play. Now you can do simulations using CA of 6 deck shoes. That is a big step forward.

I hope people think some when looking at these graphics. A few observations that may give those something to chew on.

It's interesting that there appear to be two distinct types of trend behavior. In the case of basic TDZ and I18 strategies, the more favorable the shoe, the lower the variance, while for the full index and optimal strategies, more favorable shoes tend to correspond to higher variance.

The difference between the two types of strategy is the ones that use more indices (mostly more negative TC indices) have a very low SD in negative EV situations or negative TC's and those same counts have a much better worst case EV. All the methods have close to the same SD at high TC's. Both sets of graphs combined show the power of negative TC indices to increase EV and reduce SD for your smallest bets if you play those TC's. Also the higher SD for full indices at high EV or high TC is likely due to more doubling and splitting when the additional double and split indices for full indices are applied. This would result in higher EV and higher SD. The question is how does that affect optimal bets and overall monetary results. Not enough here to say, but sim results say the combination of the negative and positive additional indices when using full indices adds to EV and for more advanced counts there is an increase in optimal bet but for simpler approaches optimal bet increase from full indices is often less significant.

(Aside: note the vertical "smear" of points at any given true count, indicating that, if all we know is the true count, we don't know the exact standard deviation for the current depleted shoe, either. We have seen this before in past discussion; see this plot of EV vs. true count, for example; even with a TC as high as +5, you can still actually be in a negative-EV situation.)

This is problematic for betting accuracy. Your optimal bet is proportional to EV and inversely proportional to SD. Ideally you want to size your bet based on the EV and SD. The farther off your bins average (SD and EV) to specific situations within the bin the worse the betting accuracy (not BC). This will cause smaller optimal bets. Those that find it hard to push out their biggest bets should note that certainty of results increases both with higher EV's and lower SD.

[ZenMaster_Flash]

"The illustrious 18 falsely inflates SCORE for real life play. 12vs5,6 not a realistic play 100% as factored in Sims".

Standing on 12's at negative counts is generally faulty play.
We know that YOUR belief is that playing 12's correctly
will make you subject to severe scrutiny. I stridently disagree !

12 is the most common hand ~ this error approaches egregiousness.

[Norm]

Moses, this is a serious study. Please keep this crap out of this thread.

[ericfarmer]

Extraordinary work. Mega Kudos to Eric Farmer !

I found this to be most interesting:

• Hi-Lo Ill 18 playing efficiency PE = 0.309.
• Hi-Lo with full indices has PE = 0.470.
• Hi-Opt II with full indices has PE = 0.639.

P.S.

In DD, I use Aces and Sevens
side-counted. How big is MY P.E.?

As with the quoted PEs, I'll include another disclaimer from the same post: "A word of caution: before anyone runs off quoting this as “the” formula for playing efficiency, note that these particular constants depend on all of the rule variations, number of decks, and penetration assumed at the outset of this discussion."

That is, answering your question would require more information, and a chunk of simulation time/resources to answer. For the former, you would need to provide more details about the particular game rules (S17/H17, resplits, surrender, penetration, etc.), as well as the complete list of indices used in your playing strategy, specified in "machine-readable" form-- see here for examples, and the accompanying readme for an explanation of the format.

[ericfarmer]

What? I ran simulations with 70 indices in Hi-lo and compared it to Hi-lo full indices. Performance wise Hi-lo with 70 indices is about the same as Hi-lo-full. With very marginal differences. So how does that affect PE?

For those wondering where these numbers are coming from, I should emphasize that (1) they are from this three-year-old study, and (2) the setup there differs slightly from the results posted in this thread (SPL3 vs. SPL1). (I intentionally mirrored that earlier setup as closely as possible for comparison, but the exact distribution algorithm is currently only tractable for SPL1.) The corresponding changes in PE are small change: for I18 it's 0.310, for Hi-Lo full it's 0.469, and for Hi-Opt II it's 0.642.

To answer your question, I would ask two questions in response. First, "Which 70 indices?" Using individual lines in the machine-readable file format as a rough metric, I18 involves 27 "playing situations," while full indices require 760 to specify. There is a lot of room in between, and it isn't clear to me how linearly or otherwise the increase in PE behaves as we add more and more complexity to our strategy.

Second, when you say, "performance-wise," what metric are we talking about? That is, what metric's value is "about the same" with 70 indices vs. full indices?

[ericfarmer]

Very nice Eric. You always limited the number of cards when using CA. It was always interesting but not exactly what we tend to play. Now you can do simulations using CA of 6 deck shoes. That is a big step forward.

Thanks! I may misunderstand your comment here; can you explain what you mean by, "limit the number of cards when using CA"? I think my confusion arises from the subsequent sentence, since the ability to "do simulations using CA of 6 deck shoes" isn't really new. That is, since the earlier 2013 study, we have been able to efficiently optimize and/or evaluate exact *EV* for arbitrary shoe subsets, using playing strategies ranging from simple total-dependent basic strategy to "perfect" composition-dependent per-shoe strategy, to (most recently) index play. The number of decks isn't really the hard part.

The only thing that is really new here is to replace "EV" with "entire probability distribution of outcomes"... and even then only without resplitting.

[Three]

Sorry about my ignorance Eric. It is just that in your studies that I am familiar with you used less than 1 deck to manage the number of combinations in the CA so it looks like I made a faulty assumption based on that observation. Perhaps I only remember the limited number of combinations or maybe I am out of touch with your research. I love your research so the latter would be a shame.

[Freightman]

I suppose I should comment on the sims, and the effort behind them. Not withstanding my prior comments of the theoretical vs the practical - this is REALLY INTERESTING SHIT.

I'll reread this a few times before commenting further. Thanks.

[seriousplayer]

f
second, when you say, "performance-wise," what metric are we talking about? That is, what metric's value is "about the same" with 70 indices vs. Full indices?

The metric of SCORE I am talking about. Some indices in Hi-lo Full increase variances therefore decreases expected value which I remove when I did the simulation with each individual indices starting with the Ill 18 and fab 4 then I expanded from there. With increase in variances the SCORE will go down.

What I've done is add indices one by one and then simulate it to see if it increases SCORE if it doesn't the index is not used. Quite a bit of work.