Table of the Closest Decisions in Blackjack

Tell me if this table is interesting to you.

Here is a table of the closest decisions in Blackjack.
Composition dependent decision making,
6 decks,
dealer hits on soft 17,
player may re-split to four hands,
no drawing to split aces,
double after split allowed.

The first row says that with a dealer ten against a player 9,7 it is marginally better to hit than it is too stand.
The rest of the decisions are ordered by how small the improvement in expected value is between best choice and next best. These decisions are the ones most likely to change by the cards on the table, or by changes in rules of games (number of decks, DAS not allowed, etc.

I re-edited this post to include a column if the "best play" and "next best" results in a positive or negative expected value I think that this has some pedagogical value since it shows how difficult it is to go from a negative to positive expectation.

No	Player	Dealer	Best	Next Best	Best Expectation	"Next Best" Expectation
1	9,7	10	Hit	Stand	Negative	Negative
2	7,A	2	Double	Stand	+	+
3	2,A	5	Double	Hit	+	+
4	4,A	4	Double	Hit	+	+
5	6,A	2	Hit	Double	Negative	Negative
6	10,2	4	Stand	Hit	Negative	Negative
7	7,2	2	Hit	Double	+	+
8	10,6	10	Hit	Stand	Negative	Negative
9	5,4	2	Hit	Double	+	+
10	6,3	2	Hit	Double	+	+
11	8,4	3	Hit	Stand	Negative	Negative
12	7,5	3	Hit	Stand	Negative	Negative
13	9,2	A	Double	Hit	+	+
14	3,3	2	Split	Hit	Negative	Negative
15	8,A	6	Double	Stand	+	+
16	9,3	3	Hit	Stand	Negative	Negative
17	8,3	A	Double	Hit	+	+
18	3,3	8	Hit	Split	Negative	Negative
19	8,4	4	Stand	Hit	Negative	Negative
20	9,3	4	Stand	Hit	Negative	Negative
21	3,A	4	Hit	Double	+	+
22	7,7	8	Hit	Split	Negative	Negative
23	7,5	4	Stand	Hit	Negative	Negative
24	7,4	A	Double	Hit	+	+
25	5,A	3	Hit	Double	+	Negative
26	6,5	A	Double	Hit	+	+
27	10,2	3	Hit	Stand	Negative	Negative
28	10,3	2	Stand	Hit	Negative	Negative
29	5,3	6	Hit	Double	+	+
30	2,2	8	Hit	Split	Negative	Negative
31	5,A	4	Double	Hit	+	+
32	9,4	2	Stand	Hit	Negative	Negative
33	9,9	A	Stand	Split	Negative	Negative
34	3,A	5	Double	Hit	+	+
35	8,8	A	Split	Hit	Negative	Negative
36	6,2	6	Hit	Double	+	+
37	6,3	3	Double	Hit	+	+
38	5,4	3	Double	Hit	+	+
39	7,2	3	Double	Hit	+	+
40	8,A	5	Stand	Double	+	+
41	10,2	5	Stand	Hit	Negative	Negative
42	6,A	3	Double	Hit	+	+
43	7,5	2	Hit	Stand	Negative	Negative
44	7,3	9	Double	Hit	+	+
45	9,3	2	Hit	Stand	Negative	Negative
46	8,4	2	Hit	Stand	Negative	Negative
47	8,5	2	Stand	Hit	Negative	Negative
48	7,6	2	Stand	Hit	Negative	Negative
49	6,4	9	Double	Hit	+	+
50	7,3	10	Hit	Double	+	Negative
51	5,5	10	Hit	Double	+	Negative
52	4,A	3	Hit	Double	+	Negative
53	8,2	9	Double	Hit	+	+
54	4,4	5	Split	Hit	+	+
55	6,6	7	Hit	Split	Negative	Negative
56	6,4	10	Hit	Double	+	Negative
57	5,5	9	Double	Hit	+	+
58	8,2	10	Hit	Double	+	Negative
59	7,5	5	Stand	Hit	Negative	Negative
60	8,4	5	Stand	Hit	Negative	Negative
61	7,A	3	Double	Stand	+	+
62	2,A	4	Hit	Double	+	+
63	9,3	5	Stand	Hit	Negative	Negative
64	2,2	2	Split	Hit	Negative	Negative
65	9,6	10	Hit	Stand	Negative	Negative
66	9,9	7	Stand	Split	+	+
67	10,5	10	Hit	Stand	Negative	Negative
68	4,4	4	Hit	Split	+	+
69	8,7	10	Hit	Stand	Negative	Negative
70	7,A	10	Hit	Stand	Negative	Negative
71	10,2	2	Hit	Stand	Negative	Negative
72	2,A	6	Double	Hit	+	+
73	9,7	9	Hit	Stand	Negative	Negative
74	10,6	9	Hit	Stand	Negative	Negative
75	4,A	5	Double	Hit	+	+
76	10,3	3	Stand	Hit	Negative	Negative
77	9,4	3	Stand	Hit	Negative	Negative
78	5,A	2	Hit	Double	Negative	Negative
79	10,2	6	Stand	Hit	Negative	Negative
80	3,A	3	Hit	Double	+	Negative
81	8,5	3	Stand	Hit	Negative	Negative
82	7,6	3	Stand	Hit	Negative	Negative
83	7,5	6	Stand	Hit	Negative	Negative
84	8,4	6	Stand	Hit	Negative	Negative
85	5,3	5	Hit	Double	+	+
86	5,A	5	Double	Hit	+	+
87	9,3	6	Stand	Hit	Negative	Negative
88	9,7	A	Hit	Stand	Negative	Negative
89	3,A	6	Double	Hit	+	+
90	9,2	10	Double	Hit	+	+
91	6,2	5	Hit	Double	+	+
92	10,6	A	Hit	Stand	Negative	Negative
93	5,4	7	Hit	Double	+	+
94	8,3	10	Double	Hit	+	+
95	9,7	8	Hit	Stand	Negative	Negative
96	3,3	3	Split	Hit	Negative	Negative
97	6,5	10	Double	Hit	+	+
98	8,8	10	Split	Hit	Negative	Negative
99	5,5	A	Hit	Double	+	Negative
100	6,3	7	Hit	Double	+	+

This is interesting to me.

I suspect a lot of card counters already know this info and use it for cover.

I like this kind of info so I can tell players who ask me for advice, like whether they should hit sixteen, that "it doesn't make much of a difference."

I was playing recently and one of the gentlemen at my table almost had a conniption when another guy stood on sixteen. Of course, it doesn't matter what other players do but the guy who stood had by no means made a bonehead play.

I would like to see something like this for a S17 game, or even better, Spanish 21.

My uncle enjoys playing blackjack. Once I watched him play, and he was pretty bad. I suggested he learn the basic strategy, and replied, and I'm paraphrasing, "I'm sure that would improve my game, but then it wouldn't be any fun, because I would lose my free will."

Perhaps a chart like this would be helpful to people like him, who want to play their gut sometimes. I wonder if it would be a public service to create a modified basic strategy, with "free will" squares for borderline plays. Nah, it would violate everything I stand for.

Nice chart Paco!

Similar to what the wizard mentioned above, I think this chart would be useful to some players who for whatever reason insist upon playing with a certain style. For example, a player who likes to double a lot could look at the table and pick out the 'double' entries in the 'next best' column and see where his additional doubles are the least detrimental. A player who hates to draw cards could do the same by looking for 'stand' under this same column.

I think the existence of "basic strategy", "modified basic strategy" and "simplified basic strategy" is predicated on the same concept of "marginal advantage of optimal decision over the next-best decision".

Someone who has just bought a Basic Strategy card in the gift shop and is sitting there trying to figure it out doesn't need to make optimal decisions. A strategy that is easy to remember can be far more beneficial than a complex, but optimal strategy which gets mis-applied.

The neophyte benefits from simplicity. The card-counting ultra sharpie can benefit from simplicity too. If the next-best deision is only insignificantly less advantageous for him, it might make good cover. So too will making an occasional bonehead play. Its a matter of having the information and making the choice. Some people prefer to be an automaton that sits there making the optimal choice even if he has to carry out the math to six decimal places. I've heard of one player who used to not even look at his two cards. Each represents freedom, but common sense and fun seems to be somewhere between the two extremes.

Paco,

Excellent table; and truly helpful.

Just a quick thought, it may be instructive if you could illustrate how this relates to Appendix 9 (and possibly a column to show just how close the decision is)?

Thanks for posting the table.

And, not for nothing, this is just yet another example of the high quality of posts on this forum that you just don't see elsewhere.

I did try and reproduce some of the expected values in Appendix 9 on a spreadsheet. I discovered that even if the player stands, there are at least 150 possible outcomes for dealer play. I was able to reproduce a dozen numbers, and then I decided that I would trust the Wizard's analysis. But it got me thinking about how close some of the decisions were. I made the table from his appendix 9, results.

As pointed out earlier, one possibility is to allow you to successfully hide your play if you are card counting, but without making too many serious errors. Another is to point out which plays are very close, and the best play is very likely to change by making small rule changes, or changing the number of decks, or dependent on previous cards played.


Now for the flip side. This table show the 40 strongest plays of a blackjack player can make in the same game (excluding pat hands and splitting Aces). The table has more of a pedagogical use. About two thirds of the strongest plays in blackjack still result in a hand with a negative expected value. Slightly fewer change from negative to positive.

I think that to a novice it is difficult to understand the purpose of basic strategy. I sometimes tell people that they will be dealt the hands that will win them the most money, and they just need to stand. The primary purpose of basic strategy is to improve negative hands. But more often than not, the hand will still have a negative expectation even with the best play. So don't expect any empirical evidence at the table from playing basic strategy. That's why casinos are not afraid of cheat sheets. The serious players simply memorize basic strategy, while the players with cards will usually grow frustrated that they seem to be losing despite playing strategy.

For example splitting 8's against an 8 is the seventh strongest play of the basic strategist. But it turns one negative hand into two hands, that both have negative expectation.

No	Player	Dealer	Best	Next Best	Expectation	Change
1	8,8	7	Split	Hit	Positive	Change
2	8,8	6	Split	Stand	Positive	Change
3	2,A	A	Hit	Double	Negative
4	8,8	5	Split	Stand	Positive	Change
5	3,A	A	Hit	Double	Negative
6	8,8	4	Split	Stand	Positive	Change
7	8,8	8	Split	Hit	Negative
8	4,A	A	Hit	Double	Negative
9	2,A	9	Hit	Double	Negative
10	5,2	7	Hit	Stand	Negative
11	2,A	10	Hit	Double	Negative
12	4,3	7	Hit	Stand	Negative
13	8,A	8	Stand	Double	Positive
14	8,8	3	Split	Stand	Positive	Change
15	6,2	8	Hit	Double	Negative
16	5,3	8	Hit	Double	Negative
17	5,A	A	Hit	Double	Negative
18	3,A	9	Hit	Double	Negative
19	7,7	6	Split	Stand	Positive	Change
20	3,A	10	Hit	Double	Negative
21	2,A	8	Hit	Double	Positive	Change
22	8,8	2	Split	Stand	Positive	Change
23	3,2	7	Hit	Stand	Negative
24	6,5	6	Double	Hit	Positive
25	4,A	9	Hit	Double	Negative
26	4,A	10	Hit	Double	Negative
27	7,4	6	Double	Hit	Positive
28	8,3	6	Double	Hit	Positive
29	9,2	6	Double	Hit	Positive
30	6,2	9	Hit	Stand	Negative
31	4,4	A	Hit	Stand	Negative
32	5,3	9	Hit	Stand	Negative
33	5,3	A	Hit	Stand	Negative
34	6,2	A	Hit	Stand	Negative
35	6,3	A	Hit	Double	Negative
36	7,2	A	Hit	Double	Negative
37	5,4	A	Hit	Double	Negative
38	3,2	8	Hit	Stand	Negative
39	7,7	5	Split	Stand	Positive	Change
40	3,A	8	Hit	Double	Positive	Change

Quote: Wizard

Perhaps a chart like this would be helpful to people like him, who want to play their gut sometimes. I wonder if it would be a public service to create a modified basic strategy, with "free will" squares for borderline plays.

Maybe something like what can be found in Lucky Ned's Incredisystem Winstitute





EDIT - Lucky Ned's system is for entertainment purposes only.

Quote: Croupier

Maybe something like what can be found in Lucky Ned's Incredisystem Winstitute

Lucky Ned's Seems to have come up with his system while drunk.


I think I looked at these tables partly because of camouflage, but also because I was curious about which moves had the smallest possible increases in expected value, but required you to double or split thereby increasing you risk of ruin. Basic Strategy assumes you are dealing from the top of a new deck, and it doesn't worry about bankruptcy. I would have a lot of trouble increasing my stake based on a move that offers me only a very slight increase in expected value, especially if I know that the two choices may have reversed based on the cards already played.

It is not particularly surpising that some of the strongest plays are splitting 8's against a 7, 6, 5, or a 4. However it is a surprise that splitting 8's against an 8 is one of the strongest moves you can do despite the fact you end up with two hands with negative expectation.