Blackjack - do the parameters of this particular game rig it in the player's favour?

The Game Room
1

I recently rediscovered a silly little handheld game (of the sort that would have been advanced in about 1991, though I acquired it - I forget how - only a few years ago) called “Casino Ace - 7in1”. Playing around with the Blackjack game, I noticed some unusual things about it (so unusual that the usual online Blackjack calculators are of limited use) that I think may tilt the odds in the player’s favour. If I lay out the parameters below, perhaps someone can tell me if this is indeed the case, and if so, what the modifications to normal basic strategy should be to maximise this? If I’m right, I assume this was done deliberately, because players would rather win more of the time than lose, but they’ve kept it close enough to real life that you wouldn’t immediately notice anything amiss. On the other hand, one could argue that this is the wrong approach for what is essentially a toy, as it could give kids the wrong idea about gambling. Anyway, I’m not really here for a moral debate, here are the parameters:

Single deck, shuffle after around 42 cards have been dealt (see spoiler box below).
No splitting, ever.
Double only on 11.
Dealer stands on soft 17.
Insurance available against Ace only.
Blackjack pays 2:1.
Player Blackjack wins against any dealer upcard (the program does not check the other card).
Player with 5 cards totalling 21 or less wins against any dealer upcard (the program does not check the other card).
Dealer hand with 5 cards totalling 21 or less wins against any live player hand (so everything except a Blackjack).

My intuition is that the biggest factor here is Blackjack paying 2:1 - I don’t think any of the other factors that go in favour of the dealer are enough to outweigh this (the splitting and doubling restrictions, basically).

Perhaps the biggest change to basic strategy is if you as the player have accumulated 4 cards totalling 11 or less, you can always hit for a 100% guaranteed win on the fifth card. So even if you have A-2-3-4 (for a total of soft 20) against a dealer 6, you should hit again, which is quite a radical departure (I don’t think the program lets you hit if you get to soft 21). On the other hand it is disappointing if you double on 11 against a dealer 2, get a 10 for a 21, and then see the dealer get to 14 in 5 cards which beats you :).

Here are a few test deals showing each hand and the cumulative total of cards used before the program displays its ‘shuffle’ animation. As you can see, it seems slightly inconsistent - I can only assume the program knows what cards are coming next and its cut off for shuffling is based on that somehow, as opposed to a more naive system.

1 6
2 12
3 17
4 22
5 27
6 32
7 36
8 42

1 5
2 9
3 15
4 20
5 26
6 30
7 35
8 39
9 46

1 7
2 13
3 18
4 23
5 30
6 36
7 42
8 47

1 5
2 12
3 18
4 23
5 29
6 34
7 41

1 6
2 14
3 19
4 25
5 31
6 36
7 42

1 5
2 10
3 14
4 21
5 27
6 32
7 37
8 42

Any thoughts, card sharps?

2

I have to dig out my copy of Scarne on cards where he went through the math, but 2-1 on natural Blackjack is huge. Per this related discussion at a gambling message board,and uncited or backed with math, though made by an admin,(Whew, all the disclaimers.):: Simply making Blackjack pay 2-1, with additional rules also in the case they were discussing, like double down on any two, or after splits, is enough of an edge while playing basic strategy to have about a 2% positive expectation.

Another poster broadens this: …

It’s kind of a big deal.

Ignoring dealer blackjacks, the 4 cards make 11 situation would arise for you fairly rarely, right? About 1 every 450-500 deals or so? I need to look at this in Excel—doubtlessly there’s a better program for this, but Excel’s what I know how to use—and get a more rigorous number

3

When we used to do Casino Night for charity, our rules were Blackjack paid 2-1, but dealer won all ties. It’s for charity, you know. The biggest of your rule changes there was the 2-1 payout; everything else but the 5-Card-Charley could be done with self discipline in Vegas if they were advantageous to the player. Playing with a single deck should make counting more advantageous.

Oh, and we had the rules posted at every table including in big capital letters THERE’S NO SUCH THING AS A 5-CARD-CHARLEY.

4

And going that deep into a single deck before shuffling would make counting a huge advantage.

5

Not being allowed to split is a pretty significant disadvantage. About 2.5% of all blackjack hands call for a split. I don’t think that takes away the entire advantage of 2-1 blackpack payoffs, but it’s a good chunk of it.

6

Yes that particular situation would be rare, but it’s just an extreme case - a guaranteed win whenever you hit a total of 10 or less with your first 4 cards seems like a decent advantage. With a single deck, I calculate this to be the case with the following possible hands (order doesn’t matter, of course):

A-A-A-A x1
A-A-A-2 x16
A-A-A-3 x16
A-A-A-4 x16
A-A-A-5 x16
A-A-A-6 x16
A-A-A-7 x16
A-A-2-2 x36
A-A-2-3 x96
A-A-2-4 x96
A-A-2-5 x96
A-A-2-6 x96
A-A-3-3 x36
A-A-3-4 x96
A-A-3-5 x96
A-A-4-4 x36
A-2-2-2 x16
A-2-2-3 x96
A-2-2-4 x96
A-2-2-5 x96
A-2-3-3 x96
A-2-3-4 x256
A-3-3-3 x16
2-2-2-2 x1
2-2-2-3 x16
2-2-2-4 x16
2-2-3-3 x36

This (if I have done my combinations correctly - quite likely not) represents 1,522 hands out of a possible 270,725 total 4 card combinations, or just over 0.56%.

Not only that, there will be a larger number of 4-card totals of 11 or more where hitting is still a much better play than it usually is. For example, holding 4 cards totalling 14, you would always hit against a dealer 10, but in this situation it’s actually odds-on to win for you (because any card of 7 or under is an automatic win for you). It also means that hits that would be wrong or marginal (e.g. hitting hard 13 against a dealer 2) probably become EV+. But figuring all that out is way beyond me.

Well, I assume it is ‘honestly’ shuffling 52 cards - the suits of the cards are displayed, so it wouldn’t take long for me to verify that the same card wasn’t coming up twice in the same deal. But this doesn’t guarantee that something funky isn’t going on in the background, unintentionally or otherwise.

I’ll look up some basic counting strategies against a single deck and see how that plays out (the program allows one to vary one’s bet between $1 and $100, so there’s a decent range to play with. Starting stack is $2,000).

7

In one case, machine shuffled after only 41 cards and failed to shuffle after 42 cards in another case!? Perhaps it looks ahead to see if there will be enough cards? If so, that means the optional final hand, if dealt, will be ten-rich — good for you!

I’ve prepended the effect on player advantage shown in Griffin’s Theory of Blackjack, to each rule change.
(+Big) Single deck, shuffle after around 42 cards have been dealt
-0.37% No splitting, ever.
-0.78% Double only on 11.
+0.19% Dealer stands on soft 17.
(Zero) Insurance available against Ace only.
+2.32% Blackjack pays 2:1.
+0.35% Player Blackjack wins against any dealer upcard [I computed this myself; it would be less if BJ paid 3-2.]
>> 0.57% Player with 5 cards totalling 21 or less wins against any dealer upcard
NEGATIVE Dealer hand with 5 cards totalling 21 or less wins against any live player hand (so everything except a Blackjack).

The late shuffle is a huge advantage if you’re counting but of course doesn’t affect basic odds. Insurance doesn’t affect basic odds either but … Does anyone insure when Dealer shows Ten??

Griffin doesn’t show — nor did I try to estimate — the five-card variations. How urgent is your need to know? BTW, many Asian casinos have a five-card rule, but when you exercise it you get paid 1-2, not 1-1. Griffin shows +0.57% for one such case.

On balance, the rules should give you a big advantage. Go for it!! (But do not bet more than you can afford to lose; have the Gambling Addiction hotline phone number handy! :cool: )

8

Thanks, that’s exactly the sort of thing I was looking for. How soon do I need to know? I don’t really need to know at all, it’s a cheap plastic game that occasionally affords a few minutes of diversion, with no prospect of ever winning (or losing) any real money. But I thought it might also make for a few minutes of diversion here, and it has.

By the way, the reason for the five card rule in this game is simple and obvious - the 1"x1.5" LCD display only has space to show 5 cards! I suspect no splitting is for similar reasons, though that could be more easily worked around had they wished.

9

Order does matter though. For instance, AAA6 can only come up in that order to count, because a hand of 6AAA would usuallyy end at 6AA; it wouldn’t make any sense to hit again unless the dealer was showing a ten. AAA7 in the order 7AAA or A7AA will usually end after two cards, and always end at three, because even in this odd version of the game hitting a soft 19 is very unwise. A lot of your combinations in certain orders never get to the fourth card. If you do, then of course you’re right - at that point you hit no matter what and take the dough. But we are presumably assuming logical play at all points.

I’m not sure how much that alters the odds though. Likely not a huge amount.

10

The fact that you can spread your bet between $1 and $100 with no pit bosses kicking you out, and the fact that the deal goes that far into a deck, a simple count would be trivial to implement to have a + win expectation.

11

Yes, good point. Although, we would also need to factor in that maybe, in this system, hitting on soft 18 with three cards is the right play against (say) a dealer 9, because if you draw one of the remaining 2 Aces or 4 2s you have a guaranteed win, a 3 gives you 21, a 4 gives you 12 but you can now draw any card but a 10 for a guaranteed win, etc.

12

Further comments:

(1) I assume that after doubling on Eleven, if Dealer ends up with Blackjack you lose Double. (In Nevada casinos where dealer doesn’t check hole card, you get half your money back. But some Asian casinos would take the full doubled bet.)

(2) The Dealer also has an automatic win with five cards, giving him a sizable advantage. I forgot to include this in my list of variation effects.

Basic strategy is synopsized by “Standing Numbers” for a fresh single deck. Here are the Standing Numbers when Dealer shows Four or Ace; the three numbers are standing numbers when Player already has 2, 3, or 4 cards respectively. (Obviously these numbers are all the same.)
Normal Hard standing = 12, 12, 12 vs Dealer Four
Normal Soft standing = 18, 18, 18 vs Dealer Four
Normal Hard standing = 17, 17, 17 vs Dealer Ace
Normal Soft standing = 18/19, 18/19, 18/19 vs Dealer Ace
(Soft-18 vs Dealer Ace is so borderline, correct play depends on rule details.)

Here are the Standing Numbers I estimate by simulating the new Super-duper Game:
SDuper Hard standing = 12, 14, 16 vs Dealer Four
SDuper Soft standing = 18, 19, 22 vs Dealer Four
SDuper Hard standing = 17, 17, 18 vs Dealer Ace
SDuper Soft standing = 19, 20, 22 vs Dealer Ace
Obviously you always hit a four-card Soft 21!

(It looks to me that hitting a four-card Soft-19 is usually best strategy! I might repeat my simulations with greater care, precision and thoroughness.)

And, even in normal Blackjack, it’s more important to hit soft-18 when Dealer shows Nine than when Dealer shows Ten. (This is logical: Dealer 19 is easier to beat than 20.)

When I played in Nevada — long long ago — hitting Soft 18 against 9 or 10 was a tell: Skilled players always took this hit; ordinary players never did. Some dealers would even call out “Hitting Soft 18” to alert floormen to the unusual play!

As a compromise I might stand on Soft 18 versus Ten — why alert the management for only a tiny increase in expected earnings? But versus Nine, the high-advantage hit was irresistible!

13

I should learn not to post a complicated incomplete post, hoping to clean it up in the 5-minute edit window! :smack: Corrections in Red.

14

A four card soft 19 is a 100% win, since in the OP’s game the player automatically wins with any five card hand. The real question is whether a three card soft 19 is worth a shot. It might be. I don’t think so, but I’m not feeling up to doing the math.

15

You read #12, but obviously not my corrections in #13. “Four” was a typo for “three.” :stuck_out_tongue:

Even corrected, my sentence was badly phrased. Hitting three-card soft-19 appears correct against Dealer Four, but not against Dealer Ace. I didn’t check other Dealer Up-cards, and wasn’t too careful even with these.

16

I did pursue this little problem with greater care; here are my conclusions. I hope there’s a better-than-even chance that they are basically correct! In any event, no animals were harmed during the experimentation. (Strategies shown are for the initial deal of a fresh deck.)

Double down on Eleven EXCEPT if Dealer’s Up card is an Ace. Otherwise hit if your total is less than the standing number in the following table.

The first column is Dealer’s Up card; then hard standing numbers for the cases where you already have 2, 3 or 4 cards; then soft standing numbers for 2, 3 or 4 cards; then your expected winnings against that Up card as a percent of your initial bet.

Ace) 17 17 18 // 19 20 28 // -31.7%
Ten) 17 17 18 // 19 20 28 // -12.0%
9) 17 17 18 // 19 19 28 // -0.5%
8) 17 17 18 // 19 19 28 // +8.3%
7) 17 17 17 // 18 18 27 // +16.5%
6) 12 12 15 // 18 19 25 // +22.9%
5) 12 12 15 // 18 19 25 // +21.9%
4) 12 13 16 // 18 19 26 // +17.0%
3) 13 13 16 // 18 19 26 // +12.3%
2) 13 14 17 // 18 19 27 // +7.9%

Your Net Expectancy is +2.05%


Yes, I can see you all grimacing and wondering if you should recommend I seek psychiatric help.  But it happens that I enjoy working out such little problems.  Alternative hobbies — sky-jumping, dating strippers, running marathons — are no longer available to me.
17

I take it this was a Monte Carlo simulation, rather than a direct calculation?

And be honest-- Even when skydiving and stripper-dating was an option, this is still how you’d have been spending your free time.

18

Yes. Direct calculation would have been better and faster, but is more effort to program.

And yes again. My life has been tamer than most. Though not always quite as tame as it is now.

19

Yup, computer time is a lot cheaper than programmer time. And Monte Carlo also makes it a lot easier to avoid mistakes that would screw up the whole calculation, so even while it’s less precise, it can lead to greater confidence in the results.

And I don’t know if “tamer” is the word I’d use. Programming can, in its own way, be wilder than skydiving (says a guy who also doesn’t date any strippers or jump out of airplanes).

20

Great work! My only quibble is that although your table is clear, it would have been clearer still had you used the Bridge convention of A = Ace and T = 10 :).

Now, given there should be a positive outcome just for playing the correct strategy, should I just play the maximum bet all the time, or alternatively pursue a counting strategy that requires varying my bet size?