J. A. Nairn false claims about first to calculate split evs for finite decks

[ericfarmer]

Can you just specify a little more on this example? Greater understanding comes with greater responsibilities. I have received several dislikes and try to learn a little more into this.

There are several interesting wrinkles here, I'll just mention a few. First, the phrase "maximizes expected value" almost always requires clarification. There are two senses in which we can rarely actually claim to *maximize* achievable expected return. One is when restricting attention to total-dependent (vs. composition-dependent) strategy. That is, although composition-dependent strategy is more complex to *specify*, it's generally easier to actually *optimize*, i.e. to compute a strategy that actually maximizes the expected value of some appropriately-defined random variable. That's because composition-dependent expected value can be defined and computed recursively, while total-dependent strategy can have circular dependencies that require a "global" optimization approach that no CA I know of actually implements. For example, should I stand or hit with 10,3 vs. 10? That's a total of hard 13 vs. 10; resolving to hit requires ensuring that the EV of hitting is greater than that of standing... but the EV of hitting depends on making EV-maximizing decisions after the hit as well, e.g. 10,3,3 vs. 10, and the *probability* of even encountering that hand depends on having already resolved all decisions that *could* lead to that hand.

(Granted, I'm willing to let Nairn pass on this here, since he sort of sidesteps the issue by explicitly specifying a fixed playing strategy in his Appendix A. It simply would have been clearer/more correct to remove the claim of EV maximality altogether.)

Computing truly EV-maximizing total-dependent strategy is technically hard even if pair splitting were not a thing. But here we're focusing on pair splits, where things get even more complicated. This second problem is that any explicit strategy, whether total- or composition-dependent, computed by almost any CA (including mine), does not actually maximize the expected value of any easy-to-describe random variable, because of the complexity of accounting for pair splitting. For example, what should you do with 2,6 vs. dealer 5? If we temporarily ignore that pair splitting is a thing, we might resolve that the best strategy is to hit... but what if we encounter this hand as the result of splitting 2s against the dealer 5? Knowing that we're in half of the split might suggest that doubling down is optimal... but worse, if we now "retroactively" change our "zero memory" strategy to *always* double down 2,6 vs. 5, whether we are splitting or not, can *raise* the overall EV for the round.

(Various CAs expose various options for configuring how sophisticated our splitting strategy can be to deal with this sort of thing. For example, my CA's "default" strategy option, referred to elsewhere in this forum as CDZ-, is to stick with the "split-agnostic" strategy of hitting, applied after pair splits as well. There are other options like CDP1 or CDP, that allow us to "know" if we are in a split hand to modify strategy... but the holy grail referred to as CDZ (without the minus) that computes the "overall" EV-maximizing strategy that depends only on the dealer's up card and the composition of the player's *current* (possibly-split) hand, is beyond our current ability to compute AFAIK.)

There is more, but that's hopefully a conversation starter.

[Norm]

There is a like problem with risk-averse index generation. As you know, to gen RA indices, you first must determine variance and EV by count. Then you run the index generation and use these numbers to determine RA indices. Problem is, once you have done so, your original variances and EVs are now all wrong. So, I added something I call Multi-Pass iRA (iterative RA indices). Turn this option on, click Sim, and while you find something better to do with your time:

1. A sim is run to determine EV and var by count
2. RA indices are created
3. Another sim is run to find new vars and EVs using the new indices.
4. New, improved (as the ads for detergent say) RA indices are generated.
5. Then you can call CVCX and generate optimal Kelly betting.

Now, I just did this for completeness. The gain is very small and I advise not wasting time as I’m not trying to find five decimal EVs. I just want an optimal betting and playing strategy – not theoretical perfection which is, frankly, mythological due to constraints in a real casino. Then again, if you play for the long haul, nice to squeeze out what you can.

[aceside]

There are several interesting wrinkles here, I'll just mention a few. First, the phrase "maximizes expected value" almost always requires clarification.

I just briefly read the conclusion of this paper but still haven't got the exact difficulty for this calculating. Basically the paper considers two scenarios separately:
#1. when re-splitting is not allowed
#2. When re-splitting is allowed
In scenario #1, the paper says it is easy. However, in scenario #2, the paper says it is hard because it requires a more powerful computer. Is the difficulty existing entirely in scenario #2? Thank you for your work.

[ericfarmer]

I just briefly read the conclusion of this paper but still haven't got the exact difficulty for this calculating. Basically the paper considers two scenarios separately:
#1. when re-splitting is not allowed
#2. When re-splitting is allowed
In scenario #1, the paper says it is easy. However, in scenario #2, the paper says it is hard because it requires a more powerful computer. Is the difficulty existing entirely in scenario #2? Thank you for your work.

I may have confused things a bit-- there are a couple of ways in which I think Nairn's work is "behind" the state of the art. One is not addressing just how many different kinds of playing strategies we can specify for playing split hands; this was my rant in the earlier comment.

But your question here is another good-- and separate-- point. Before digging in here, let's first consider phrasing the problem slightly differently, as Nairn does, where we let the parameter H be the maximum total number of hands we can end up with in a round as a result of splitting pairs. Then your scenario #1 is just H=2, while scenario #2 is, well, usually H=4, but my point is that we can in principle generalize the problem to consider arbitrary H>=2.

Okay, then Nairn's analysis suggests that the problem is relatively easy when H=2 (scenario #1), but it gets much more difficult when H=4. This is true... but only when computing expected value *recursively* as he is doing, effectively "visiting" every possible completion of the round (i.e., every ordered sequence of cards dealt that finish the round). Tackled this way, the amount of computational work grows roughly *exponentially* in H; that is, for each additional allowed split hand, we need to do X *times* more work than before (for some scaling factor X).

Granted, Nairn speeds things up by caching previously computed results, both probabilities of dealer outcomes and previously encountered *unordered subsets* of cards in the player's (split) hand. This helps... but it doesn't help much, since the approach still "visits" all possible completions of the round multiple times-- it just gets to "lookup" already-computed results for all but the first of those repeat encounters. As detailed in the paper, this remains complex enough that it was only feasible to execute for H=4 for single deck, since for more decks the "scaling factor" X mentioned above (essentially the *base* of the exponential expression, where H is in the exponent) is even bigger.

So, the second sense in which this isn't really anything new is that we can do much better than this: we can evaluate exact split EVs, for arbitrary number of decks, for arbitrary number of allowed split hands H, with an amount of computation time that only increases *linearly* in H instead of exponentially. This takes some mathematical machinery to accomplish; the details of how this can be done are described here.

E

[MGP]

So T Hopper doesn't know him so he didn't copy from him, and digging in he didn't get it right anyways. He's really close but not exact. Thanks.

aceside: When we say we are calculating the exact expected value of a split it is based on a full shoe, finite or infinite. This isn't a guess, it's not an estimate, it's not "a plausible calculation"; it is exact. If you are talking about during game play without a full shoe then that is a whole different story. What you said about 26 card doesn't make any sense as k_c is explaining, if you want the exact calculations for a 26 card shoe you can use my CA, enter the cards and get them.

Gronborg: The infinite deck calculations are exact based on an infinite deck. There isn't such a thing except on computers as you point out, so they are an estimate of finite decks but that doesn't negate the exactness for what is calculated.

ericfarmer: Good point that it may have been difficult to find our work. My CA has been exact since 2002 and I'm not aware of any bugs - are you referring to another CA? It wasn't on Ken's site yet until a bit after that though. Btw we don't need a Monte Carlo approach since we confirmed our methods with brute force calculations (which took Ken days).

[MGP]

About TD Strategies, ericfarmer, this is just like our previous discussions You did a great job summarizing it thank you.

My CA actually goes through the TD strategy twice in order to find the EV maximizing TD strategy for exactly the reason you describe. Doing that found one strategy change compared to what's normally published. I think is was R,S instead of R,H for 12,3 but can't remember (maybe the opposite?). I didn't find any cases where going through a 3rd time would make a difference so didn't do it as a "while no changes".

We actually could compute a CDZ- strategy and it shouldn't take that long using the conditional split calculations however how do you display the strategy? You can do it in real time though (again with my ca with the Realtime option) very easily. For SPL1 you would have 1 extra strategy per hand, for SPL2 you would have 3 possible strategies per hand based on the split state, and for SPL3 you would have 8 possible strategies per hand. By hand I mean like Eric's example of 2,6 v 5. So it would just take that many times + 1 to compute CDZ compared to CDZ- strategy. As Norm points out though, the utility of the calculation would be pretty much zero though as counting is way easier and yields much greater rewards.

[ericfarmer]

So T Hopper doesn't know him so he didn't copy from him, and digging in he didn't get it right anyways. He's really close but not exact. Thanks.

I would be curious to learn what Nairn's paper describes that is "close but not exact"? I admittedly only read the paper through once, and did only spot-check comparisons with my results, but the description of the approach looks reasonable (if inefficient) to me.

Good point that it may have been difficult to find our work. My CA has been exact since 2002 and I'm not aware of any bugs - are you referring to another CA? It wasn't on Ken's site yet until a bit after that though. Btw we don't need a Monte Carlo approach since we confirmed our methods with brute force calculations (which took Ken days).

This was back in 2011, when I was working on changes to make my algorithm faster, and several of us were comparing notes on blackjackinfo.com on splitting algorithm assumptions, implementations, etc., using "crafted" shoe subsets as test cases to compare/verify results. By "crafted" I mean pathological subsets where it was reasonable to actually compute exact EVs by hand. For example, consider a shoe with a single 2, a single 6, and thirteen 10s, playing S17/DAS/SPL3, with CDZ- strategy. The correct overall EV for a round is 38.095238095%. (That's one of the examples we were using to compare results; I have some archived notes from those discussions, let me know if it's helpful to dig around and find some more of these.) It's worth noting that, at the time, almost *all* of our CAs initially got this wrong (mine, yours, k_c's-- only iCountNTrack produced correct results in this case).

[ericfarmer]

My CA actually goes through the TD strategy twice in order to find the EV maximizing TD strategy for exactly the reason you describe. Doing that found one strategy change compared to what's normally published. I think is was R,S instead of R,H for 12,3 but can't remember (maybe the opposite?). I didn't find any cases where going through a 3rd time would make a difference so didn't do it as a "while no changes".

This is what I mean by requiring global optimization-- what you're describing are "local" departures from the strategy in each pass. In graph terms, with each pass you're only evaluating candidate strategies in the "neighborhood" consisting of *single* changes to strategy for a single hand/up-card combination. Even if you continued this search for more than just two passes, there is no guarantee that you haven't simply gotten stuck in a local optimum, and that there isn't a strictly better, but more "distant" strategy out there, requiring more radical departure from the currently-known-best strategy.

This is why simulating annealing is a thing, for example. But that's also just a heuristic that performs well in practice. My point is that we don't *know* that there isn't a better strategy out there.

(Along these same lines, the minus sign in the CDZ- notation denotes our similar acknowledgment that the algorithm for computing this strategy is not known to be actually optimal among all possible strategies that depend only on dealer up card and player current hand, independent of whether it's involved in a split or not. It "probably-usually" *is* optimal... but we don't know if or under what conditions it is. Indeed, I only accidentally stumbled on a single example that *proves* that minus sign deserves to be there; see this past discussion.)

E

[MGP]

I'm saying that because the values he's getting are off after after 3 or 4 decimal places. It should be accurate to 15.

Actually, if you remember, I emailed you how to calculate the split values exactly on January 21 of 2003. I had exact results using a my burn card probabilities. I then figured out how to calculate the splits using conditional probabilities and shared that method with you on April 27, 2003 which I had figured out in March. It was after that you wrote it up on May 26 (without crediting for the last equation in your paper).

That's an interesting shoe. Maybe there is a bug in my CA handling really small shoes and running out of cards since for that one in the sense that it's giving NaN for SPL2-3 for upcards 2 and 6.

While looking at this my CA looks like it calculates Post-split CD exceptions and that I called what I was describing as CDP and CDPN strategies. In the options you check "Include post-split exceptions" and then look on the analysis tab for the exceptions, but I haven't found any.

I did find the thread you're referring to though and will check it out later. I completely forgot about that discussion and it looks interesting:

Software update with new split EVs - Blackjack and Card Counting Forums (blackjackinfo.com)

[ericfarmer]

I'm saying that because the values he's getting are off after after 3 or 4 decimal places. It should be accurate to 15.

Can you provide an example or two of a value where his value is incorrect? My spot checks agreed with all of his provided significant figures. (I'm not sure what you mean by "should be accurate to 15"? If you're just referring to double precision floating point, then arbitrary-precision calculations yielding exact rationals would require 17, not 15, decimal digits to "round trip" from 64-bit double to decimal string back to 64-bit double.)

Actually, if you remember, I emailed you how to calculate the split values exactly on January 21 of 2003. I had exact results using a my burn card probabilities. I then figured out how to calculate the splits using conditional probabilities and shared that method with you on April 27, 2003 which I had figured out in March. It was after that you wrote it up on May 26 (without crediting for the last equation in your paper).

Hmmm. I don't remember email, but I do remember bjmath.com forum discussions about this, that I have archived and can exchange details if desired. That exchange includes discussion right up to 25 May, involving a significant disagreement that I don't have memory or record of being resolved. With hindsight, I now don't doubt that you were consistently able to produce correct overall expected values-- but at the time, that wasn't at all clear, because of those disagreements, and without your software (although everyone had mine) to verify or reproduce any results other than those individual example cases discussed in the forum.

(Looking back on that discussion, the issue was the equality of expected values of individual hands of a split. For example, against a dealer 6, split and resplit a pair of 10s. Conditioning on exactly two split hands, the expected values of the first and second resolved hands are equal. Or at least that was my claim, and is implicit in the equations in the paper that you reference, despite at the time claiming that these two expected values are *not* equal.)

That's an interesting shoe. Maybe there is a bug in my CA handling really small shoes and running out of cards since for that one in the sense that it's giving NaN for SPL2-3 for upcards 2 and 6.

Yep, or at least that's what I think/guess is all that's happening; I've never seen disagreements with my or other results when the shoe contains at least one of all ten card values.

E

[MGP]

3,3 vs 2 DAS. Nairn: SPL1 -112304, SPL3 107650; Correct SPL1 -0.11746, SPL3 -0.113294. There are others but it's not worth the time going through them.

I have all the emails. I can forward them to you if you need to refresh your memory. It was right after I sent the "how to" that you caught up and posted your paper with my conditional calculation for the CDNs from the CDPs.

The disagreements weren't actually with the numbers if I remember. Cacarulo, Ken and I all agreed. Like I said, Cacarulo actually calculated them first based on the idea I had that we were figuring out in an email thread (he brute-forced the next card removal), Ken confirmed, then my burn-card programming caught up and I confirmed them. The discussions were about terminology and what to consider TD vs CD vs CDP vs CDPN vs CDZ- vs CDZ+ etc and calculating those. My CA in 2011 was already calculating all those exceptions.

I'm not sure why you're saying about "the issue was the equality of expected values of individual hands of a split." For SPL1 I have always made it clear the EV is exactly the same as is the strategy. I read my replies in the 2011 thread and don't see any disagreement about that.

[iCountNTrack]

Norm, Eric, k_c, MGP, this certainly brings back memories . I can safely claim that my CA was the first published CA to generate optimal post-split strategy, general probabilities of possible outcome and standard deviation of a hand. I cant seem to find the source anymore, but here is the link to the latest update https://code.google.com/archive/p/bl...yzer/downloads

[MGP]

Mine might have beat you to post-split strategy but definitely not the other stuff. Eric's was the first really public CA with source code that I found. But I am positive Cacarulo and I were the first to figure out splits and I was the first to figure out the conditional nature of the hands. Steve Jacobs pointed out the various hand types which was a key point, and as I said Ken had the brute-force algorithm that allowed us to confirm everything. It was a lot of fun!