"Single 21" - Colusa Casino

I recently played a strange game at Colusa Casino in California. They did not offer regular blackjack when I was visiting, so I was forced to play this. I'm wondering how the house edge is altered pursuant to these rules.

Bad for player:

Blackjack pays 1:1 and suited blackjacks pay 3:2. Both are automatic winners. For insurance purposes, even money is only offered on the suited blackjack.

Good for player:

Full Single Deck.

Surrender is allowed at any time on any number of cards, even after a player doubles down.

Splitting is allowed at any time on consecutive equal cards, even after a player doubles down.

6 cards is an automatic winner.

Example Hand:

Let's say I have 23 against a 7. I hit and receive a 5. I double down at that point and receive another 5. I have 15 and have the option of staying, surrendering, or splitting. If I split, I still have 10 on the first hand and the 5 on the other hand. I now can double down on the 10 again.

I saw this at Bally's Tunica. I would also like to see the strategy.

Quote: bthekuta

I recently played a strange game at Colusa Casino in California. They did not offer regular blackjack when I was visiting, so I was forced to play this. I'm wondering how the house edge is altered pursuant to these rules.

Bad for player:

Blackjack pays 1:1 and suited blackjacks pay 3:2. Both are automatic winners. For insurance purposes, even money is only offered on the suited blackjack.

Good for player:

Full Single Deck.

Surrender is allowed at any time on any number of cards, even after a player doubles down.

Splitting is allowed at any time on consecutive equal cards, even after a player doubles down.

6 cards is an automatic winner.

Example Hand:

Let's say I have 23 against a 7. I hit and receive a 5. I double down at that point and receive another 5. I have 15 and have the option of staying, surrendering, or splitting. If I split, I still have 10 on the first hand and the 5 on the other hand. I now can double down on the 10 again.

The game is terrible.

The slaughtering of the payouts on blackjacks is what does it. You have a 4.8% chance of being dealt a blackjack. A blackjack will be suited 1/4 of the time. So 3/4 of the time, you get shorted, making the average payout on a blackjack 1.125:1. Since 4.8% of the time, you are being shorted 0.375 bets, that would be a loss of 1.8% right there. (Everybody, feel free to check my math on all this.)

You get back a bit of this in that the dealer also gets a BJ 4.8% of the time, and now you win rather than tie. The dealer will get a blackjack less often than that when YOU have one, because there are only three Aces left in the deck. So let's give her a 3.6% chance of getting a BJ. So about once every 600 hands, you win when you would otherwise have tied. 1.125 extra bets every 600 hands is a little less than +0.2%. So we're now up to -1.6%.

Surrender after double isn't all that valuable. There are only a few situations where it would be worthwhile to surrender, such as 16 vs. 9 or 10, 15 vs. 10 (and 9, I THINK), and a couple of others. And the gain from surrendering in those situations is very small--basically, you need to be winning less than 25% of the time to make surrender a good move. So that rule doesn't do much for you--charitably, +0.1%. We're up to -1.5%.

The re-split would only be available when consecutive cards are dealt, so a 1 in 17 chance, all other things being equal. And you might not want to use that option all the time. Even when you do, it might not be that big of a gain. So I can't guesstimate this rule as being worth more than 0.05%. We're down to -1.45%.

6 cards being an automatic winner is worth virtually nothing since you will very likely have a strong hand anyway if you manage to get six cards without busting. And, of course, it will happen very rarely.

So if they hit soft 17 (I assume they do), the base game is worth -0.2%, the BJ short payouts knock it down to -2.0%, and the favorable rules kick it back up to -1.65%. Yucko.

Thank you for your analysis. I have a few questions/comments.

I think surrendering at any time is more valuable than you portray. 1) You can surrender after a double. The reason you don't surrender a 12 or 13 v. a 9 or T is because you can improve a 12 or 13 by hitting. But if you completely whiff on a double (e.g. getting a 2 or 3 on an 11 against a 9), I'm sure surrendering would be the appropriate move. 2) You can surrender at any time. In a typical game you can only surrender with two cards, but now you can surrender with 3 or more. So if I hit a 12 v. an A and receive a 4, I now have the luxury of surrendering. I wouldn't have that in a normal game.

Also, you didn't address the doubling on any amount of cards. I'm sure this would give the players some odds as well.

Here is an example hand from my night that I could not have performed in regular blackjack.

I bet $10 and got a 54 v. a 7. I hit and received a 2. I then doubled (not allowed in regular BJ.) I then received another 2. I was able to split the 2 and use the doubling $10 on the second hand. I then doubled again on the first 11 and received a 21. I forgot what happened the second hand.

Overall, I know that the terrible blackjack payouts are tough to overcome, but I'm still curious about the overall house edge.

Quote: bthekuta

Overall, I know that the terrible blackjack payouts are tough to overcome, but I'm still curious about the overall house edge.

The reason that I didn't address the double on any number of cards is that your original post didn't list that as one of the rules. That rule, per the Wizard, adds 0.23%.

The six card auto-winner adds a bit more than I originally thought. The Wiz quotes a gain of 0.16% in an 8-deck game. I would imagine the figure for this single deck game, as it is harder to make a six-card non-busting hand, would be slightly less. Call it 0.15%.

Normal late surrender, that is, you can only do it AFTER the dealer has checked for a blackjack, is only worth 0.08%. The enhanced surrender isn't going to add all that much. That's because your chances of winning the hand have to be TERRIBLE for surrender to be a worthwhile option: specifically, they have to be less than 25%. Even in the worst case scenario, if you have a 16 vs. a 10, you still win almost 23% of the time by hitting. This means that the advantage from surrendering 16 vs. 10 is that you gain about 2% (vs. hitting). This isn't that much help. And if you double down vs. a 10, and make a stiff, it isn't all that much of a benefit to ba able to surrender--the dealer will still bust over 21% of the time, so you gain about 3.5% by surrendering. What all this means is that you are only gaining two or three hundredths of a bet when you are able to three-card surrender, and those situations aren't going to come up that often anyway.

So I slightly underestimated this game's return by underweighting the six-card Charlie, and the additional rule that you originally didn't disclose adds another 0.23% to the equation. So this game is returning about -1.27%.

(By the way, the six card Charlie advantage would only be realized if you made the appropraite adjustments to Basic Strategy, such as always hitting a five-card 12, 13, 14, or 15.)

My fault on omitting the doubling down rule, and thank you for your analysis. It appears that any game that offers 6:5 or worse blackjack cannot be overcome by player friendly rules. This was very helpful, thanks!

I think, you are still underestimating it.

Quote: mkl654321

And if you double down vs. a 10, and make a stiff, it isn't all that much of a benefit to ba able to surrender--the dealer will still bust over 21% of the time, so you gain about 3.5% by surrendering. What all this means is that you are only gaining two or three hundredths of a bet when you are able to three-card surrender, and those situations aren't going to come up that often anyway.

You will surrender after doubling down with any hand <17 against any dealer card >7
With basic strategy, you'd double down against 8,9 with 10 or 11, and against a 10 with 11 only.

I'll go with an infinite deck, and only consider doubling down on the first two cards, because I am lazy.
The probability you'll get <17 if you have a 10 is 5/13, and if you have 11 it is 4/13.
The probability dealer has >7 (but no blackjack) is 2/13+4/13*12/13+1/13*9/13 = 0.4142.

The probability you get 10 on the first two cards is 9/169, and the probability of 11 is 8/169.

So, getting it all together, the probability you will surrender after a double is
0.4142*(9/169*5/13+8/169*4/13) = 77*0.4142/13^3 = 0.01452, and it will save you your full original bet in this case.

So, it looks like this rule alone is worth about a full 1% of HE.
Unless, of course, I screwed up some math here as I usually do :)

How this SHOULD read is:

BAD FOR THE PLAYER:
You're at an Indian Casino

GOOD FOR THE PLAYER:
Reno is only a couple hours away.

End of story.

Single21 HE is ~ 0.66%... Game is also present at Bally in Tunica..

Quote: bthekuta

I recently played a strange game at Colusa Casino in California. They did not offer regular blackjack when I was visiting, so I was forced to play this. I'm wondering how the house edge is altered pursuant to these rules.

Bad for player:

Blackjack pays 1:1 and suited blackjacks pay 3:2. Both are automatic winners. For insurance purposes, even money is only offered on the suited blackjack.

Good for player:

Full Single Deck.

Surrender is allowed at any time on any number of cards, even after a player doubles down.

Splitting is allowed at any time on consecutive equal cards, even after a player doubles down.

6 cards is an automatic winner.

Example Hand:

Let's say I have 23 against a 7. I hit and receive a 5. I double down at that point and receive another 5. I have 15 and have the option of staying, surrendering, or splitting. If I split, I still have 10 on the first hand and the 5 on the other hand. I now can double down on the 10 again.

Quote: weaselman

So, getting it all together, the probability you will surrender after a double is
0.4142*(9/169*5/13+8/169*4/13) = 77*0.4142/13^3 = 0.01452, and it will save you your full original bet in this case.

So, it looks like this rule alone is worth about a full 1% of HE.
Unless, of course, I screwed up some math here as I usually do :)

The only question would be, what would be the expectation of the doubled bet when you CAN surrender, vs. the expectation of the doubled bet when you CAN'T?

Let's say that the original bet was $10. You doubled, and got a bad card---you want to surrender. The expectation of this total bet is now -$10. Alternatively, you decide to soldier on. If the dealer has a 10 showing, you will lose 78.5% (rounded) of the time, making your expectation about -$11.40. Thus, the option to surrender, in this instance, is worth $1.40, or about 1/7 of a bet. As the situation becomes less dire, you gain less and less from surrendering. You obviously want to surrender whenever you are stuck with a stiff vs. a dealer 8,9,10, or A, but you only realize a significant gain from surrendering vs. the 10 and Ace.

This all goes to the fact that if you decline to surrender, you may wind up with four times as much money as you will have if you surrender. Giving up that chance to win is a rather large hit to expectation--so not only is surrender not a good option very often, but it is only a small gain when you do use it. Yes, you will employ it more often in this game, but its effect is still relatively small.

Quote: mkl654321

The only question would be, what would be the expectation of the doubled bet when you CAN surrender, vs. the expectation of the doubled bet when you CAN'T?

The expectation of surrendering is better - that's why you want to surrender in the first place.
But you are right, we need to look at the expected value, not just probability (I knew, I was missing something :))

The probability that the dealer will bust when he has 8,9 or 10 is about 20%, making the EV of not surrendering 0.2*2 - 0.8*2 = -1.2, making the real benefit of being able to surrender after double about 0.3% - indeed, much less then I wrote initially, but still more significant than your estimate.

Quote: weaselman

The expectation of surrendering is better - that's why you want to surrender in the first place.
But you are right, we need to look at the expected value, not just probability (I knew, I was missing something :))

The probability that the dealer will bust when he has 8,9 or 10 is about 20%, making the EV of not surrendering 0.2*2 - 0.8*2 = -1.2, making the real benefit of being able to surrender after double about 0.3% - indeed, much less then I wrote initially, but still more significant than your estimate.

But my estimate was on the effect of the ability to surrender after doubling on the OVERALL GAME. Surrender-after-double is only significant on those hands where you a) double b) against a strong dealer upcard and c) you make less than 17. So let's use the 22% bust figure, which is fairly close to the overall chances of the dealer busting with upcards 8-A. Let's also, for shorthand, call the a)b)c) hand a Double Surrender hand, or a DS. If you can surrender a DS hand rather than holding your breath and hope that the dealer busts that 22% of the time, then that increases your expectation from 22% to 25% (by surrendering, you achieve the same expectation as if you had a 25% chance of winning). This means that you gain 3% of those bets where you double and make a DS hand, which increases the overall expectancy of doubling down against a strong dealer upcard.

So a 5% increase in the overall expectation of a bet that you don't make all that often (doubling against an 8,9,10, or Ace), and only of a SUBSET of the results (you make a total of less than 17) you achieve, is not all that significant. I don't know how often you get dealt a double down hand, but I would guess that you find yourself doubling down only once every fifteen hands or so. And you make less than seventeen (a DS hand) less than half of the time--for instance, when you double on hard 10 or 11, you only make 16 or less with five of the thirteen possible cards. So 1/15 of the time, you double, and 5/13 of that time, you make a DS hand, and you save 3% of your bet by doing so. 5/180 is about 0.03. So 3.0*0.03= 0.09, or about a tenth of one percent. Of course, this calculation assumes that my guesstimate of 1 double every 15 hands is correct. But even if it's really more like one in ten, that's still only a gain from DSing of 0.15%.

Quote: NightStalker

Single21 HE is ~ 0.66%... Game is also present at Bally in Tunica..

Does anyone know of this is really the house edge for this game? It been added to Fitz in Tunica as well as Bally's. What attracts me to the game is it being single deck and $5 minimums. I played it the last time I was in Tunica at Bally's and won a few bucks but I only played a couple of hands because I wasn't sure about the HE.

It seems this game is gaining some in popularity is there any chance of getting some info about it on the site Wizard?