Illustrious 18 Expected Returns

I was recently reviewing the Wizard’s Blackjack Appendix 1: Expected return for every play, with the idea of adding a few new index plays to my game. The appendix makes intuitive sense of various index plays, i.e. if two decisions have the same expected return, the index is 0. I understand that indices do not correlate exactly to difference in expected return because of countless factors, but I have found a loose approximation, and I thought y’all would be interested in seeing the breakdowns for Schlesinger's Illustrious 18 indices presented below (source). The table shows the expected return for close decisions for player and dealer hands listed in the I18. A negative difference signifies that the left column (Hit) is the BS play, and a positive difference indicates that the right column (Stand, Double or Split) is basic strategy.



A few things that pop out at you. Firstly, that each change of 1 in TC corresponds to about 0.15 change in difference of expected return. Of course, some decisions rely more heavily and in different ways on deck composition than others. The most notable deviations from this ‘rule of thumb’ are:



Going off of difference in expected return alone, splitting 10s rather than staying loses 3 times as much return on your original bet as any other decision in F18. Hard 20 v 5 or 6 is one of the more commonly dealt hands at 1.5% frequency. To clarify, the table shows that, out of an infinite deck, you sacrifice 29 and 24 cents of original dollar bet when splitting tens against 5 or 6 respectively. Most of the other indices only show a difference of a few cents. Additionally, from this table alone, it looks like +5 is too high for 16 v 9 and +4 and +3 a little low for doubling 10 v A and 9 v 7 respectively.

Each index comes from simulating that situation while varying the deck composition until one decision outperforms the BS play. Here is a helpful graphic from the awesome site blackjackincolor.com. I do not presume to challenge these well-established indices. Again, I know that there are other factors than difference in expected return, and my hope is to present food for thought and start a stimulating discussion about what these other factors might be.

Thanks for the interesting post stud! Definitely some good info to think about.

I might be miss-reading the chart, but when you say you sacrafice 29 and 24 cents to split tens to a 5 or 6 respectively... The chart appears to show a positive difference for the "hit/split" column. Wouldn't this indicate that you gain 29 and 24 cents to split tens to a 5/6 respectively? Again, I might be missreading the chart as I eat lunch =P.

I appreciate it Romes! I was hoping you'd see this because your great 'A to Z' thread inspired me to look more closely at the numbers and try to explain strategy instead of accepting it. I think that the confusion is due to my limited column labels. The third column shows the expected return for 'Hit' and the fourth column shows the expected return for 'Stand or Double' for every situation listed except for the pair of 10s. For 10, 10 versus dealer 5 or 6, the third column shows return for standing and the fourth column shows splitting returns.

So, according to cited Wizard Appendix 1, standing on 20 yields a return of $0.67 (versus dealer 5) or $0.70 (versus 6) where splitting 10s yields $0.37 (versus 5) or $0.48 (versus 6) off of the top of the shoe. These differences are far larger than anything else in I18, which is why I am so surprised that Don Schles found splitting 10s to be profitable at somewhat common TCs of +4 and +5, in which a 10 or A is only 4% more likely.

I'm really glad the A-Z thread inspired you to learn more about the why! Thanks for sharing that, it makes me happy to hear.

Ah, I believe I see how your charts are set up now. The difference in standing/splitting, from the Wizard's Appendix 1, doesn't take counting in to question... which is why it's better to stand a hard 20 instead of splitting it all the time =p. Also, as you stated already, there are other factors that go in to making these plays more profitable. Not only does the changing of the composition of the deck change the hit/stand (or split/stand) expected values (I think you stated by about .15 per TC), but it also of course changes the frequency of the large/small cards.

As I'm sure you're aware (but simply tossing out for discussion) most all of a players edge in blackjack comes from the ability to split, double, and receiving 3/2 on blackjacks. When the TC is higher, the dealer is more likely to bust than normal. When it's at a certain point (splitting 10's to a 4, 5, 6 happens symmetrically enough at +6, +5, +4 respectively) this will encourage more splits and doubles (double 8v6, 8v5, A-3v5, etc, etc). As I said I'm sure you're aware of most of this. I'm just trying to relate how these other topics could factor in without actually doing the math =P.

Thus, really I'm just rationalizing why the I18 is correct, while being lazy and not doing the math =D. If they were wrong, or didn't work, we'd have enough part time / full time counters over the numerous forums available sharing their results showing that they didn't work. I'll take the I18 at face value until that time lol.

I had a little fun and have a spreadsheet to work out the strategy against different "counts". I'm having to assume it's an infinite deck, in a ratio based on removing N cards from the initial 312. So 1 corresponds to removing 1/5th Ace 4/5th Ten and adding back in 1/5th 2-6. Here's the close decisions (note it's based on s17 no-peek UK, so the normal action is NOT to double 11 vs 10).
What I find interesting is thinking about doubling 8 vs 6, I've seen it done and always assumed it's stupid, but it comes in fairly early.