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

[ericfarmer]

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.

Thanks for providing this example. Nairn's value here appears to be correct; all of my, k_c's, and even *your* CA gives the same value (-0.11230494397 for a few more digits).

I took a closer look at this to see if there might be confusion about the intended post-split strategy (note that he provides the assumed strategy explicitly in his Appendix A). I cheated and just used CDZ-, but fortunately this is a "nice" case where the only relevant composition-dependent strategy variation is to hit 10,3, which he calls out in his appendix.

E

[ericfarmer]

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.

I should clarify/emphasize that I'm not trying to claim, nor do I much care whether, I did anything "first." And I've tried to be diligent about giving credit where appropriate (e.g., is "London Colin" around here, maybe under a different handle?). And I haven't cared much about attribution in the other direction. For example, your kind words from 20 January 2003 in a bjmath.com thread (subject "Re: Improved CA (version 5.0)"): "First of all a very big thank you to Eric for sharing your c-a program. I relied on it heavily to create mine. There are a couple of places where you lost me so I had to modify them for my own. One place was the falling factorial function - I have no idea what that is or how it works... so I made a similar array to look up dealer probs..."

I do recall the discussion with Steve, Don, etc., about the distinction between *specifying* a particular type of strategy-- and nailing down terminology for those various types, as you describe-- and *computing* EVs for any particular specified type of strategy. What I was referring to here, however, was the bjmath.com thread that I started on 18 May 2003 (subject "Finite deck calculations?"), where I was trying to work out the details of generalizing the previously-worked-out formulas for infinite deck limited resplits to finite decks, and we were disagreeing about the equality of expected values of individual split hands in various situations. Note that we're talking about SPL3 here, not just SPL1: the context was discussion about outcomes of the form that you denoted "NN", where if we split, say, a pair of 10s, and resplit at every opportunity up to a maximum of four hands, then conditioned on the situation where we *don't* resplit (i.e., we draw non-10s to both halves of the split), then the expected value of both split hands are equal. This is critical to the accuracy of the formula that I was trying to clarify.

E

[MGP]

No worries. I didn't make a big deal out of it then, but this thread was initially about Nairn claiming to be first. He wasn't. As you have pointed out, even in my post 2 above yours, I have always given you credit both publicly and in emails for your CA and how I learned a lot from it. In fact it was because of your CA that I emailed you when we figured out splits lol. Thank you for it. The reason I am pointing it out now though is because when you publish something in a scientific paper form like the one you linked to, it is customary to give credit because, even if you're not claiming to be first, not doing so implies no previous work was done. That is not the case here, previous work was done. That's why anyone that wrote a CA also has to give credit to Griffith for his book if they write it up. I wasn't talking about listing every contributor every post.

I missed that about the CD for Nairn. Then I agree it seems like he's getting the numbers correct. What he doesn't say though is how he knows they're correct. By the time he put his paper out our exact numbers for split EVs were all over the forums and the only one I know of who confirmed them by brute force was Ken. So that would be interesting to know.

What you're talking about is the fact that there are 2 ways to get the exact same EVs and what I was referring to earlier about the hand order that you spelled out vs the one that I used originally using the effects of removal. The only time the difference becomes apparent is with Australian rules otherwise they are equivalent. Here are the SPL2 differences in the hands used, their probs differ too but the net calculated result is exactly the same:

SPL2 EOR Recursive
NN EV(N) + EV(N-N) 2*EV(N-N)
Pxxx 3*EV(x-P) 3*EV(x-P)
NPxx EV(N) + 2*EV(x-PN) EV(N-P) + 2*EV(x-PN)

I guess in a way it's not critical to have them be the same. I never figured out why but that's a whole other discussion!

[MGP]

This is again another discussion, but to me this isn't the same thing as a global optimization problem like we see in neural nets with gradient descent [who uses simulated annealing anymore ?] and multiple possible local minima, except maybe with very customized shoes. The differences are:

1) The probabilities are fixed and decreasing so there is a guaranteed significant reduction in effect of a hand on an overall strategy as you get more cards.

2) The possible paths of a given strategy are well defined and very limited (Hit or Stand) and the effects of changing that strategy are very small at the edge cases.

3) If you aren't counting, then the previously removed cards are unknowns and don't effect the math. So you'd have to be counting or keeping track of every card for this to come into play.

So you may be correct, but you'd need to find a case where cycling ends up changing a strategy back and forth to prove there is such a thing as a local minima in this situation.

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

[iCountNTrack]

This is again another discussion, but to me this isn't the same thing as a global optimization problem like we see in neural nets with gradient descent [who uses simulated annealing anymore ?] and multiple possible local minima, except maybe with very customized shoes. The differences are:

1) The probabilities are fixed and decreasing so there is a guaranteed significant reduction in effect of a hand on an overall strategy as you get more cards.

2) The possible paths of a given strategy are well defined and very limited (Hit or Stand) and the effects of changing that strategy are very small at the edge cases.

3) If you aren't counting, then the previously removed cards are unknowns and don't effect the math. So you'd have to be counting or keeping track of every card for this to come into play.

So you may be correct, but you'd need to find a case where cycling ends up changing a strategy back and forth to prove there is such a thing as a local minima in this situation.

I have been away from BJ and these discussions for a long time and dont quite remember what i did. But with optimal post-split strategy, the strategy are recomputed for every possible playing decision after a split taken into account all cards visible on all split hands. It's computationally very expensive for resplits, and also for hands such as 9,9 vs 10 because unlike 9,9 vs 6, the optimal strategy is to keep hitting so more possible outcomes and more calculations.

[ericfarmer]

No worries. I didn't make a big deal out of it then, but this thread was initially about Nairn claiming to be first. He wasn't. As you have pointed out, even in my post 2 above yours, I have always given you credit both publicly and in emails for your CA and how I learned a lot from it. In fact it was because of your CA that I emailed you when we figured out splits lol. Thank you for it. The reason I am pointing it out now though is because when you publish something in a scientific paper form like the one you linked to, it is customary to give credit because, even if you're not claiming to be first, not doing so implies no previous work was done. That is not the case here, previous work was done. That's why anyone that wrote a CA also has to give credit to Griffith for his book if they write it up. I wasn't talking about listing every contributor every post.

I've updated my write-up to reference all of the participants in that discussion, at least those that I have archived. I tried finding the actual bjmath.com posts in the wayback machine, but unfortunately it only has limited snapshots, and even those only have the views of the thread trees, not the content of the actual posts. All that I have copies of are my posts, with the surrounding "in reply to" contexts.

[ericfarmer]

What you're talking about is the fact that there are 2 ways to get the exact same EVs and what I was referring to earlier about the hand order that you spelled out vs the one that I used originally using the effects of removal. The only time the difference becomes apparent is with Australian rules otherwise they are equivalent. Here are the SPL2 differences in the hands used, their probs differ too but the net calculated result is exactly the same:

SPL2 EOR Recursive
NN EV(N) + EV(N-N) 2*EV(N-N)
Pxxx 3*EV(x-P) 3*EV(x-P)
NPxx EV(N) + 2*EV(x-PN) EV(N-P) + 2*EV(x-PN)

I guess in a way it's not critical to have them be the same. I never figured out why but that's a whole other discussion!

You'll have to elaborate on what you mean by "their prob[abilities] differ too"? I'll re-iterate my question: consider being dealt an initial pair (of, say, 10s vs. dealer 6), that we split and repeatedly re-split at every opportunity, up to a maximum of *four* hands (SPL3 in our now-reasonably-established notation). Repeat this experiment in Monte Carlo fashion, ignoring all outcomes *except* those where we end up with exactly *two* split hands (i.e., when we draw a non-10 card to each half of the split). Let the random variable X1 be the final outcome of the first half of this split, and X2 be the outcome of the second half. Then is it true that E[X1]=E[X2]?

[ericfarmer]

This is again another discussion, but to me this isn't the same thing as a global optimization problem like we see in neural nets with gradient descent [who uses simulated annealing anymore ?] and multiple possible local minima, except maybe with very customized shoes.

We use simulated annealing, SPSA, and various other gradient descent algorithms in my work all the time. You're right that this is different in that the problem is discrete-- we're moving around in a graph, where vertices have finite neighborhoods, instead of moving around on a continuous surface. But we have the same basic problem: the strategy with maximum EV might not be reachable by steps of single-decision changes to strategy.

The differences are:

1) The probabilities are fixed and decreasing so there is a guaranteed significant reduction in effect of a hand on an overall strategy as you get more cards.

2) The possible paths of a given strategy are well defined and very limited (Hit or Stand) and the effects of changing that strategy are very small at the edge cases.

3) If you aren't counting, then the previously removed cards are unknowns and don't effect the math. So you'd have to be counting or keeping track of every card for this to come into play.

I don't understand (2) and (3). But (1) gets at the heart of the problem-- because this isn't true! That is, the probability of encountering a particular subset of cards in our hand is *not* monotonic in the number of cards in that hand.

I thought I would have to make up some pathological "custom" shoe subset and wonky playing strategy to show this, but it turns out this comes up dozens of times (180, to be precise) in actual single-deck (S17) play using CDZ- strategy. For example, consider the probability of encountering the hand (A,4,8,T) vs. dealer ace. This probability is about 0.000190435... but the probability of encountering the hand (A,8,T), which is a *subset* of this hand, is only about 0.000157602.

What's happening here is that drawing a 4 to (A,8,T) isn't the only way we can encounter the larger hand (A,4,8,T). And that *total* probability of encountering that drawn hand *depends* on our choices of stand/hit strategy for those "earlier" hands. And that total probability of encountering (A,4,8,T) *affects* the "impact" of our stand/hit decision for that hand.

So you may be correct, but you'd need to find a case where cycling ends up changing a strategy back and forth to prove there is such a thing as a local minima in this situation.

That's not how proving optimality works. I can't say, "My algorithm does a pretty good job in the cases I've tested. The burden is on you/others to find an example of some other case where my algorithm's output is explicitly suboptimal... and until then, we'll assume that its output is *always* optimal."

E

[iCountNTrack]

I have noticed a problem with J.A Naim numbers. I am not sure whether because he is using singles instead of doubles. But his EV value for Split3 are always less than mine, which means he's not actually generating optimal values. For example, he reports +0.424280 for splitting 9,9 vs 6. I get +0.4243275335. Higher EV indicates more optimal play. Also interestingly for this hand, 36% of the time you will win 2 units.

[ericfarmer]

I have noticed a problem with J.A Naim numbers. I am not sure whether because he is using singles instead of doubles. But his EV value for Split3 are always less than mine, which means he's not actually generating optimal values. For example, he reports +0.424280 for splitting 9,9 vs 6. I get +0.4243275335. Higher EV indicates more optimal play. Also interestingly for this hand, 36% of the time you will win 2 units.

His value is correct... and so is yours , I'm sure. That is, note that he's not claiming to compute *optimal* expected values, he's computing *exact* expected values assuming a particular zero-memory strategy (similar to our CDZ-) that can't vary from one split hand to the next, like your CA allows.

[iCountNTrack]

His value is correct... and so is yours , I'm sure. That is, note that he's not claiming to compute *optimal* expected values, he's computing *exact* expected values assuming a particular zero-memory strategy (similar to our CDZ-) that can't vary from one split hand to the next, like your CA allows.

Thanks Eric. Like I said I have been away for a long time so please forgive me for any stupidities . I remember once during our private chats, discussing you possibly adding to your CA optimal post-split strategy. i am not sure if you have ever gotten to it.

[DSchles]

What's happening here is that drawing a 4 to (A,8,T) isn't the only way we can encounter the larger hand (A,4,8,T).

Guess I'm missing something here, Eric. Can you explain under what circumstances someone would be drawing to A,8,T? Or even how he would GET to A,8,T vs. A, given that he would stand with A,8, with A,T, or with 8, T?

Don

[iCountNTrack]

Guess I'm missing something here, Eric. Can you explain under what circumstances someone would be drawing to A,8,T? Or even how he would GET to A,8,T vs. A, given that he would stand with A,8, with A,T, or with 8, T?

Don

Eric is talking about a depleted deck where there will be significant departures from Basic Strategy. Please see example below



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