A variation on computing optimal count-based strategy

This is a question for the math geeks in the group, since it has probably little effect on the bottom line of card counting, but it is addressed to those who want to have the most accurate statistics possible. I'd like to know if anyone has computed optimal strategy for a card counting system (or even the subset, the Illustrious 18), taking into account the fact that hitting when you have a good count affects not just the expected value of this hand but the number of high-count hands left in the shoe. Realize that I am not saying that your choice to hit or stand in any way affects the card count, but it does affect how many more rounds are left in the shoe.

As an extreme example to illustrate the point, suppose the count is high and you are in a situation where standing nets you 40% and hitting nets you 41% but will cause a reshuffle, whereas standing will get you one more round before reshuffling. If your expectation at the current count is 1.5% higher than a neutral shoe, then you should definitely stand, taking a 1% hit on that round but getting one more round of 1.5% higher return in the shoe. Conventional wisdom would dictate the opposite strategy. This idea generalizes to any depth in the shoe. In general, if you are playing heads up, then every card you take reduces the number of rounds in the shoe by about 1/5. So when you compute optimal strategy you need to take this into account, and sometimes take a little hit in terms of expected return on the current round in order to get as many high-count rounds in as possible. Conversely, when the count is bad, you might take a little penalty in hitting, in order to skip through to the end of the shoe faster, which is a similar effect to Wonging out of the shoe.

So I am tweaking the conventional wisdom that says that optimal count-based strategy is to choose the action (Stand,Hit,Double,Split,Surrender) that maximizes the expected return of the current hand, which is formulated simply as follows:

argmax ER(S)
S

where S is strategy and ER is expected return on this strategy. I think the more optimal strategy, assuming a particular number of other players, is:

argmax ER(S) + dR(S) * (EV(CurrentCount) - EV(NeutralCount)
S

where dR(S) is the change in the number of remaining rounds in the shoe if you perform strategy S,and EV is the expected value per hand for a specified count. dR(Stand) is 0, whereas dR(Hit) is about -1/5 for heads up games, and dR(Split) would be less than -2/5.

My suspicion is that although this is the more correct way to compute optimal strategy, the practical economic effect is probably negligable. However, there is a possibility that one of the Illustrious 18 thresholds might change by 1 if you compute the strategy this way. The only way to know is to try, so I was wondering if anyone has ever gone down this route.

- David

Hello David, and welcome to the forums! You clearly have a fairly good understanding of expectation (at least the idea of expectation), and I think you have a really interesting post that a actually touches on a topic many counters have already argued over. Usually it comes down to "do what you feel/want" but this is because people play different styles. Some are professionals, looking for longevity. So that 1% sacrifice comes to a +EV move because they're not drawing heat to themselves because they can come back and play tomorrow. Others, such as myself, are "weekend warriors" so when I go on a trip I don't care all that much if I get heat/backed off/etc.

Basic strategy, and advanced strategy (such as I18) are generally formed from an infinite deck approach. Even the Wizard's video shows an infinite deck model for calculating the argmax ER(S). I don't believe any of them take in to account the count at the last end of the shoe. This would be a rare situation, but you do have a point. What it comes down to is the player though, ultimately. Would I rather play this hand, stand, forfeit that 1% of expectation, and play the next? Or would I rather play this hand to the max and get that extra 1% now?

..How about a 3rd option? Why not stand on this hand, then play multiple hands on the next hand, since it's the last deal anyways. Then you're getting 2x your EV on the last hand. I think there are a lot of other options than the ones you've proposed, which will make calculating all of them quite difficult. Not only that you have other concerns which are quite variable, such as casino heat, etc. It can often be a tip off that you're spreading horizontally at the end of the shoe, especially if they review that it's in a positive count. Horizontal spreading is getting to the point where it's worse than vertical spreading.

After realizing there are a lot more factors that are quite situationally dependent, we then come to the same conclusions again that this will have a negligible effect on the economic outcome, per trial. I do however believe over the long run that playing to maximize your EV will add up. This is a debate many card counters have had though. I, personally, will do what I believe maximizes my EV.

So in your example above (assuming heat is not an issue, etc) I would stand, dropping 1% of that particular hands expectation, in order to assure my advantage on the next round, and I would spread to multiple hands to get to maximum EV (possibly even 3 hands or whatever I feel I can get away with). So standing here, instead of dropping 1% in expectation, overall adds a lot of EV because I'm adjusting my EV on the final round to make up for the 1% loss on the previous 1 hand. I hope you could see how my answer may change if I were a professional not looking to draw attention from this place though.

Thanks for the comment. I hadn't expected any actual players to be interested in this, so it's good to hear your viewpoint on this. I will make one clarification, because you mention that the calculation of advanced strategy does not take into account effects near the end of the shoe. Note that what I'm saying is that it should take into account this effect everywhere in the shoe, not just near the end. So my question is really towards the very few math types who have been involved in calculating count-based strategy tables. If you optimize count-based strategy taking into account the change in the number of remaining hands in the shoe, does the Illustrious 18 change? Mathematically there will be a very small difference, but does this result in any actual change in the strategy? Almost certainly the expected value will not be affected by enough to concern any real player, only math geeks.

Ah, I think I have a fair understanding of what you're getting at. You might have trouble finding someone to do the actual math as it's my educated guess (as a math geek myself) that it would be a tremendous amount of work to do it properly for something that is more/less negligible. The only reason to do the math is to do the math, which can be fun if the topic peeks your interest (I did all the math for every bet and some combinational bets in craps when I was learning the game purely because it excited me). But here, it's not practical as you could pretty much never use it, for numerous reasons.

Yes, I do believe it would change basic strategy and I18. This would, in my opinion, be almost exclusively at the end of a shoe, where the number of hands you can play would come in to greater affect. While I understand why you'd want to apply this throughout the entire shoe, to see if you could max more hands, that may not be a good thing. You don't know at the beginning, or middle, of a shoe what the count will be at the end. Perhaps it's negative and you want to draw extra cards to end the shoe faster? It's just as likely to go positive as negative, so they cancel each other out until you can control them in a particular situation to take advantage of what it currently is. RE: The end of the shoe would be the only time this would come in to affect. Thus, the functions/calculations used to determine this strategy dynamically would be 90% useless during the shoe.

One major 'practical' reason this would never work or be used is because of it's complexity. There is no way anyone could remember every situation whether or not they should sacrifice 1% of EV to get 1 hand later in the shoe and which plays to make. One would almost certainly need a device to determine the proper play, which is illegal. If anyone was ever to attempt this not using a device, I would be willing to bet my bankroll that they would make more mistakes, costing them more EV, than the entire "system" would be worth. So without even doing any math, I believe I've proven this system would lose money based on practicality.

I apologize if you're just looking for the math, but as I mentioned above, it will probably be quite difficult to get someone to do what appears to be a very large amount of work for something we can practically rule out as ever affecting the game. It's an interesting thought and I love that you bring up a topic I've never heard/thought of (attempting this for more than just the 'end game' of the shoe). Keep on that imaginative thinking, sooner or later you'll get on to something quite interesting and I'll be much more willing to lend any math skills I have your way =).

Your question raises an interesting point in that optimal play assumes you want to maximise the profit (minimise the loss) for this particular hand - thus for a given hand (and presumably deck combination) there is a best way to play.

Sometimes people mention close plays, typically doubling 11s, and say that while it might be best to hit, you might be better off doubling since this the odds on your double decision, while not optimal, is better than making a second Blackjack bet.

I have never seen it discussed the other way round where it might be better to hit (not double, or not split) because making another Blackjack bet is a better proposition. Taken to its extremes this might mean you stand on more 12s when it's good or hitting more 12s when it's bad - I don't know the counts where these decisions flip, but this factor might affect the switchover point.

Romes, good points about practicality. Just one minor point. When you point out that you don't know at the beginning or middle of a shoe what the count will be at the end, it doesn't matter. What matters for a decision part way through the shoe is the knowledge that on average the count will be exactly the same throughout the rest of the shoe. If later on in the shoe, the count changes, your strategy will change too, so in general your strategy will be biasedto use slightly less cards through positive counts versus negative counts. The strategy will be the same for any point in the shoe, not just the end. Adding a special strategy for the second-last round would be, as you say, too much work.

You give me much to think about =p. I don't know if I agree with the thought that the count will be exactly the same throughout the rest of the shoe. The whole idea of counting is to track the shoe. For balanced counts (like hi/low) it starts and ends with zero, but in between is unpredictable (just as likely either way). If the TC is +10 after 3 decks, I would certainly NOT expect "the count will be exactly the same throughout the rest of the shoe." I would expect the count to fall back to 0 by the end, because I know it will for certain.

The special strategy for 2nd to last hand I actually don't think is too much work. It's called "end game" and the idea of it has been around since Thorp even discussed it. While he was mostly discussing knowing the last card (for places that played down to the last card) and how to use that in end game, his ideologies on how many hands to play and how to bet on the 2nd to last hand to trap this card or push it away I think apply to this situation.

To me, you have a fantastic idea/theory that unfortunately doesn't change how we've been doing things at all =). Any part of the shoe before "end game" (near the cut card) is negligible, in my opinion. It's not practically quantifiable and any affect it would have would be negligible. So the only part that actually has an impact is the end of the shoe. You're looking for the math behind something that is (in my opinion) fairly intuitive after you have a little experience. Every counter, at the beginning of their career, hates when the count is finally high and you hit the cut card. You slowly learn why Penetration is the most important aspect of the game and then you wish you bet 2 hands since the cut card was coming out so you could at least have gotten another hand in on the high count. Eventually you learn to watch closer for the cut card and make a more educated decision... Can I get 2 rounds out of what's left? Then I'll play 1 hand and bet 2 hands on the next (last) round. Can I only get 1 round? Then I'll play 2, or more, hands right now to get extra hands/EV. This idea is something blackjack players have been doing for a very long time.