EOR and BC Reexamined

[OMendoza]

Today I have posted to my blog ((Dead link: http://360.yahoo.com/orestes_mendoza)) an assessment of the EOR and BC methodologies that examines as a case study their extrapolation to Reno/Tahoe/H17 play. There are ramifications, in theory and in practice, that may surprise the card counter who has accepted these commodities on faith. I address some of these ramifications at length.

The common approach to “card-counting” involves a linearization of the effects of each card on player advantage. The “count” (or “running count,” technically) consists of the linear combination of system tags with the frequency of appearance of each rank of card. Additivity is not so much noted as installed in this approach. The “best” method of generating tags for such a linear combination is not uniquely defined.

Betting Correlation (BC), the measure of how closely correlated a given count is with Griffin’s “ideal” card tags for what was then the standard Las Vegas blackjack game, has been used widely to evaluate various counts for use in making betting decisions. Griffin used effects of removal (EOR) methodology, which looks tidy and productive at first and has, in fact, produced results that have been largely in accordance with output from computer simulations. Indeed, if the approach called “linearization” had little applicability to blackjack in practice, then the valid advantage play systems in use would be radically different from the current popular systems. But certain problems ensue from ascribing too much authority to BC, even when playing efficiency (PE), insurance correlation (IC), and other such measures are ignored.

 

[KenSmith]

I've often considered that the EOR approach overlooks the fundamental fact that blackjack is at its root combinatorial. A particular deck subset is not easily characterized merely by looking at single-card EORs. It's the exact combinatorial possibilities permitted by the remaining cards that create the playing situation. I'm sure there are some rich insights available somewhere in this line of thought.

 

[OMendoza]

Interaction effects are all over blackjack, with the ways in which Aces and ten-valued cards interact being only one case where they come into play. Griffin addresses the robustness of the linear assumption in the face of the obvious nonlinearity in the game more than once in Theory of Blackjack. At one point, he does a bit where he separates the player expectation into a "natural" component (i.e. the probability of being dealt an Ace and a ten-valued card and the accompanying payoff), which he assesses combinatorially, and an "everything else" component, upon which he performs the usual EOR routine. The idea is that approaching matters that way asks less of the EOR method.

I have often thought that pitch game players who keep Ace side counts could probably get a lot more power out of them by discontinuing the practice of adjusting linearly for Aces remaining and instead working out case-based strategies depending on the number of Aces remaining. The interaction effect suggests to me that for a given number of Aces plus ten-valued cards in the remainder, the maximum probability of a natural is realized when the number of Aces equals the number of tens. In practice, such maximization would almost always be about the number of Aces being maximized, I'd guess, since every BJ game starts out with four times as many ten-valued cards as Aces. It couldn't possibly be worth it to try this sort of stunt for a shoe game, though, only for pitch games (and perhaps only for single deck games).

Interaction effects get REALLY CUTE when examining the Lucky Ladies side bet. I have computed "ideal tags" based on EOR methodology, including a special tag for the Qh, for two-deck play. I have also computed "LLC's" for various promising counts; this latter group of computations is on my web site. However, my analysis of the LL shows that a linear tag for the Qh is a bit of an oversimplification given how its removal affects expectation not only depending on how many Qh's are left afterwards but when it is removed. (It's also an oversimplification for the other cards, although a less-interesting one.) It might be better, if the probabilistic work I have done on the side bet is indicative of anything, to keep separate "main count" indices for placing the bet, one each for each number of Qh remaining. But although linearization really is oversimplifying, it isn't completely worthless, especially in sims, and the "LLC" as it stands can serve as the same kind of "industry standard" as the BC does, with perhaps not a lot less applicability and rigor.


O.M.

 

[sagefr0g]

I've often considered that the EOR approach overlooks the fundamental fact that blackjack is at its root combinatorial. A particular deck subset is not easily characterized merely by looking at single-card EORs. It's the exact combinatorial possibilities permitted by the remaining cards that create the playing situation. I'm sure there are some rich insights available somewhere in this line of thought.

even though this thread is way over my head i'd like to ask a question.
to me when i ponder the nature of the advantage card counting affords it seems as if one glaring short comming stands out. that being there is no way of knowing the order inwhich the cards (how ever advantageous they may be) are going to come out. so one may lay out a big bet in hopes of reaping the advantage only to find that one has a hard 16 against a dealers most promising up card. is the situation described illustrative of what your talking about with regards to combinatorial possibilities permitted by the remaining cards that create the playing situation?

best regards,
mr fr0g

 

[KenSmith]

Card order isn't what we're discussing. Instead, we're talking about the way some cards interact with each other in ways that aren't conveniently represented by EOR (effect of removal).

I'll try to fill in some details...

Peter Griffin in his book The Theory of Blackjack presents the effect of removing a single card of a particular rank. That is, he looks at how much the house edge changes when you take a single card out of the deck. For example, say you are playing a single deck S17 game and you see the dealer's burn card after the shuffle. It's a five. Griffin's numbers show that the removal of that single five reduces the house edge by 0.69%, making the game nicely positive for the player right away.

What if the dealer were to burn a second card, and show it to you as well? Let's say it is another five. Using the EOR approach, we would reduce the house edge by another 0.69%. That is, a five is a five is a five, and the EOR methodology values them all the same. That's a simplification, and since counting systems are based on this idea, it is an interesting question to consider just how skewed the results of this idea become in typical deck subsets.

If I get a chance, I'll calculate that particular example just for fun.

 

[sagefr0g]

Card order isn't what we're discussing. Instead, we're talking about the way some cards interact with each other in ways that aren't conveniently represented by EOR (effect of removal).

I'll try to fill in some details...

Peter Griffin in his book The Theory of Blackjack presents the effect of removing a single card of a particular rank. That is, he looks at how much the house edge changes when you take a single card out of the deck. For example, say you are playing a single deck S17 game and you see the dealer's burn card after the shuffle. It's a five. Griffin's numbers show that the removal of that single five reduces the house edge by 0.69%, making the game nicely positive for the player right away.

What if the dealer were to burn a second card, and show it to you as well? Let's say it is another five. Using the EOR approach, we would reduce the house edge by another 0.69%. That is, a five is a five is a five, and the EOR methodology values them all the same. That's a simplification, and since counting systems are based on this idea, it is an interesting question to consider just how skewed the results of this idea become in typical deck subsets.

If I get a chance, I'll calculate that particular example just for fun.

hmmm, yes that is definately interesting. i never thought about EOR having such significant consequences in that way. so for example if a clumsy dealer lets us see the bottom card before the cut is made and say we see a five and that five ends up behind the cut card we know going into the game the house edge has been reduced by 0.69% for a single deck game. yep thats interesting alright. thank you for the explaination.

best regards,
mr fr0g

 

[KenSmith]

OK, I worked out the details on the example, with a single deck S17 NDAS game.

For a basic strategy player, this game has an 0.0022% player advantage off the top.

If you remove one five, and still use regular basic strategy, that player advantage increases to 0.6823%.

If you remove two fives, still use regular basic strategy, the player advantage is now 1.3457%

So, removing the first five added 0.6801%, while the second five was worth less, only 0.6634%. The effect is not strictly additive, though this example is quite close. These small differences are the point of OMendoza's essay. In some cases, they may not be so small.

 

[Automatic Monkey]

Well, sure. If you were playing a shoe game and there were nothing but 8's left, the player would have a 100% advantage (assuming he knew there were only 8's left).

Card counting is just an approximation, and the relationship of sum EOR to advantage is good for the most common sets of removed cards, but not all. Hence the added volatility of SD games.

 

[xengrifter]

this game has an 0.0022% player advantage off the top.

Incorrect! The subject game has a .022% PA OTT.