Your way blackjack

Quote: gordonm888

The stuff about checking for Low Hands of 17 or 18 in the middle probably only applies to 1 or 2 possible hands. In fact, I have been unable so far to identify a single 4-card hand that is affected by that rule -they all seem to be affected by higher level/priority rules

I just found it. These rules cause the dealer to set the hand T988 and 9988 as T-8|9-8 and 9-8|9-8. Without these rules these hands would be set as T9|88 and 99|88 respectively - forcing the dealer to hit a hard 16 in the Low hands.

I also see a few others where the dealer doesn't make Blackjack - A T T 9 (20-20) A T T 8 (20-19) A T 9 9 (20-19) A T 9 8 (19-19). If my memory serves me right when playing Switch you nearly always made BJ (except making two 18s vs 7 and similar), but then BJ beat 21.

For 2 nights, I watched the game for about 10 min; each time it had 2 players who were tourist type. Here are the issues for casinos:

1) slow: players have to wait for all 4 cards and dealer has to shuffle after every round of play - drop will be low

2) everyone bets $5 minimum, no one played even $10 - drop will be low

3) no one tips - because no big winnings (1 to 1 only) - dealers, the best sales, won;t be happy

4) less seats (5 or 4, less than 6 or 7) meant less tips for the dealers because players do not double-tipping

5) different rules set by different dealers and / or floor - for example, whether the dealer gets the rotating pock or not? only players get to rotate?

6) one player and the dealer each had a work sheet with wordy rules; from time to time, they even studied together

7) I heard because the developer knew the boss of the boss of the boss...

Good luck.

Quote: Ibeatyouraces

Me and my friend must be blind. :-)

it didn't have any sign on it

Quote: charliepatrick

I also see a few others where the dealer doesn't make Blackjack - A T T 9 (20-20) A T T 8 (20-19) A T 9 9 (20-19) A T 9 8 (19-19). If my memory serves me right when playing Switch you nearly always made BJ (except making two 18s vs 7 and similar), but then BJ beat 21.

Charliepatrick: Thanks for these catches. I am sitting down with my list of all the combinations and reworking them per the dealer way. It is tricky and tedious.

Here is an odd one: 9887 is set as 16-16 rather than the more intuitive 17-15, due to the last rule about setting the lowest scoring high hand. I think that is probably correct for the dealer though because the dealer's inability to hit a 17 in the HIGH hand costs him about 0.10 to 0.20 in EV - making a hard 17 in the HIGH hand an unusually terrible hand for the dealer.

There were many more changes to the dealers hand than I expected. I have recalculated the strategy for hard 9-21 and this new strategy appears below. Please remember that I have taken a few calculational shortcuts, so the EV numbers are "best effort" and not authoritative to 3 digits. However, I expect that the general features and trends of the calculations are reasonable.





In general, I found that use of the correct Dealer Strategy as provided by the Wizard seemed to lower the average EV of the hands by 0.05 to 0.08. I'll be curious to see what the Wiz calculates for House Edge if he does analyze this game. I was previously expecting the House Edge to be large and unattractive to players and I would now double down on that bet. :)

There is only one hand dealt from the single deck prior to reshuffle, so there is no possibility of bet variation. Even so, I would expect this game will be uncountable if it were dealt from multiple decks or when considered from the viewpoint of using a count to alter HIT/STAND/DOUBLE decisions.

I've probably made a mistake, especially as the House Rules seem a bit confusing, but it looks as if the dealer tries to make good hands (lo 19+, hi20+, 11, 10) and after High 17+ the low hand is fairly unpredictable. As I said I'm using infinite deck so that may alter things quite a bit, but it broadly agrees with yours (and given you've taken cards into account, I would trust yours over mine)!

My feeling is the low hand never gets good enough to double 9 or split low pairs, so you never have a good opportunity on the low hand. We know the High Hand is up against it, so my feeling is the House Edge is going to be pretty terrible.

	High				Low
UpCard	HIT	hit	Double	Split	HIT	hit	Double	Split
Ace	17	s19	-	-	13	17	10 11	A 8 7
Ten	17	s19	-	-	13	17	10 11	A 9 8 7
9	17	s18	-	-	13	17	10 11	A 9 8 7
8	17	s18	-	A	13	17	10 11	A 8 7
7	16	s18	-	A	13	17	10 11	A 8 7
6	16	s18	-	A	12	17	10 11	A 8 7
5	16	s18	-	A	13	17	10 11	A 8 7
4	15	s18	-	A	13	17	10 11	A 8 7
3	14	s18	-	A	14	17	10 11	A 8
2	15	s18	-	A	14	17	10 11	A 8

Nice job, CP! I assume you looked at doubling soft hands and it was not a go?

I think I'll assume your split pair decisions are correct and not try to duplicate your work. I had been hoping that the the player could arrange his hands differently than the dealer to take advantage of his ability to double and split such as:
AA54 --> AA-9 rather than s16-s15 (as the dealer would split it) to take advantage of the ability to split aces.
This would have been another way to reduce the house edge that is not available in standard BJ. But it is apparent to me that there are not many situations that are positive for the player, and the House Edge is likely to be too large to overcome by anything short of a shotgun.

Another factor that had occurred to me is that whenever you play an AA in the high or low hand, it is almost certain that neither of your other two cards will ever be a 9 or 10 -because if they were then you would play an AT or A9 in the high hand. Therefore, the split/no split decision on a pair of aces will always occur when your other two cards are 8 or lower. This would increase the EV of splitting AA, on average, by about 0.01 or so.

I'm working on my own analysis.

Can anybody tell me the number of hands the player may re-split to and whether re-splitting aces is allowed.

Thank you.

Quote: Wizard

I'm working on my own analysis.

Can anybody tell me the number of hands the player may re-split to and whether re-splitting aces is allowed.

Thank you.

Cards dealt to split aces are faced down, so probably no resplitting. For other splits, I was told by the dealer that I could only split once when I first played there, because "limited number of cards". But every time I played the game after that I was allowed to split to 4 hands.

I have an incomplete page on Your Way 21. As you can see, my basic strategies differs from those of gordonm888. Of course, I could be in error, as I have nothing else to compare my work against.

Quote: Wizard

I have an incomplete page on Your Way 21. As you can see, my basic strategies differs from those of gordonm888. Of course, I could be in error, as I have nothing else to compare my work against.

I have slightly different percentages as I have a Rule 3.5 which says the dealer makes a High Hand of 11 or 10 (before the lo 18s or hi 17s e.g. Hi:A5/T5 Lo:74) and also because I was using 6-decks to get to the dealer starting hands (i.e. what combos of two-card starts the dealer has) and then using infinite deck as to the ending total. I tried changing the first part to 1 deck and it didn't affect my strategy.

However I agree with all but three differences, you have three splits I wouldn't have done.
EVs of not splitting cf splitting
Split As vs Hi 9 -.2226 to -.2537
Split 9s vs Lo A -.0067 to -.0105 (I can quite believe this might flip on number of decks as it's very close)
Split 7s vs Lo 3 -.4262 to -.4468
I might look at 10s in more detail, since it's easier to exact the percentages, but as I said, I might have got the house way wrong.

Quote: charliepatrick

EVs of not splitting cf splitting
Split As vs Hi 9 -.2226 to -.2537
Split 9s vs Lo A -.0067 to -.0105 (I can quite believe this might flip on number of decks as it's very close)
Split 7s vs Lo 3 -.4262 to -.4468
I might look at 10s in more detail, since it's easier to exact the percentages, but as I said, I might have got the house way wrong.

Here are my figures. These should be considered rather rough. They are based on the final dealer probabilities on my site, which are derived by simulation. However, from there I used an infinite-deck and infinite re-splitting assumption to get at the player expected values and strategy.

Splitting aces vs 9 in the high: -0.160685.
Splitting 9s vs A in the low: -0.019868
Splitting 7s vs 3 in the low: -0.420061

I am still not sure what the actual re-splitting rule is.

After seeing your results, I studied my spreadsheets and your preliminary posted results.



1. OUCH! I realized that I mis-read House Way rule #6 for arranging cards "Otherwise, if a high hand of 17 or more can be made. . ." -I misread it as a " Otherwise, if a LOW hand of 17 or more can be made. . ."

This was a pretty egregious error and affects the arrangement of dozens of hands.

2. I have corrected this error in my spreadsheet and set about to confirm the features of your High hand strategy chart. Good news, my calculations now agree with all the features of your High Hand chart except one.

- I do calculate that standing on a 15 vs a 2 is better than hitting by about 0.02. Your chart recommends hitting 15 vs 2. I suspect that your calculation is correct.

3. There is another objection, on a more esoteric level to one specific feature of your High hand chart. It has to do with not doubling on an 11 vs. a 4 or 5. I actually believe your advice is wrong ( Player should Double 11 vs 4 or 5) even though your calculation is correct.
- First, I too now calculate that hitting an 11 vs a 4 or 5 provides a better EV than Doubling, although I do calculate that it is a close call. So, given your underlying assumptions, I am not challenging the integrity of your calculation.

- However, consider that a High hand of 11 must be accompanied by two additional cards, a Low Hand,which by definition must be in the range of 4-11. Thus, with very high frequency a High Hand of 11 will be associated with a Low Hand that consists of two small cards in the range of 2-6. I calculate that the effect of removing two low cards in the range of 2-6, makes Doubling an 11 in the High hand vs. a 4 or 5 the preferred option.
The underlying issue is that the high and low hands are not independent, they are highly correlated and form an ensemble of 4-card hands which the player may possibly be dealt. As another example, low hands of 14 will usually play a bit stronger than you calculate because they will always be accompanied by a High Hand that is in the range of 14-21 -which means that 2 high cards will usually be removed from the deck.
This is conceptually similar to the issue with Pai Gow Poker power ratings for front and back hands.

This is a nit-picky point and I only raise it because, quite unfortunately, it may turn the recommended strategy on 11 vs 4,5 upside down.

I'm not sure why we get different figures. This is for High 10.
To check the 21 figure of 24252.
Firstly consider all hands which have first card: there are 51*50*49 = 124950.
Second consider all hands which don't have an Ace = 47*46*45 = 97290.
So 27660 hands have an Ace. Most of these will play the Blackjack except:-
ATT9 (20/20) 1440
ATT8 (20/19) 1440
AT99 (20/19) 144
AT98 (19/19) 384
27660-1440-1440-144-384 = 24252.
The hands which make Hard 20 include TTT(T-2),TT9(8-2),TT8(8-A) etc which total to 68430.
However A9TT and A99T set a high hand of soft 20 (not the Blackjack) and this brings the total up to 70014.
I am wondering whether you have these hands wrong as they, nearly account for the differences in our 21 and 20 figures - the dealer does not always make a high hand of 21 if a low hand of 19 or 20 can be made.

Initial		Final	High	Hand
Total	Combos	21	20	19	18	17	Bust
21	24252	24252
20	70014		70014
19	8436			8436
18	4296				4296
17	3480					3480
16	2232	172	172	172	172	172	1 374	.076 923	.615 385
15	2088	173	173	173	173	173	1 223	.082 840	.585 799
14	2136	191	191	191	191	191	1 183	.089 213	.553 937
13	3096	297	297	297	297	297	1 609	.096 075	.519 625
12	4920	509	509	509	509	509	2 375	.103 465	.482 673
Total	124950	25 594	71 356	9 778	5 638	4 822	7 763
%		20.483%	57.107%	7.825%	4.512%	3.859%	6.213%
%	Wizard	21.694%	55.942%	7.816%	4.446%	3.842%	6.260%
Est	Total	27 107	69 900	9 766	5 555	4 800	7 822

^ Note about the above post - once I have got the dealer's initial four cards, the rest has been calculated using an infinite deck. Thus the cards that have gone do not affect the decisions or probabilities. That's why you see the chances of a 17 from a 16 as 1/13. Usually such approximations are near enough, but close decisions can flip.

The House Way is ambiguous on some hands such as:

AT32 --> s13, 13 Which is intended to be the high hand? I usually set the soft hand in the LOW Hand

Another issue not addressed in the Wizard's page is whether the House Way for arranging the cards is always optimum for the player.

Consider: 4332 House Way: 6,6 Possible Basic strategy: 7:5

Or: AT99 House Way: 20 (A9), 19 (T,9) Possible Basic Strategy: 21 (AT), (99) 18 or split 9s depending on upcard

Actually it gets more confusing deciding on hands such as A743 because I read it as creating A3 and 74, but as A3 is 14 it goes in the high hand. This may be the reason they have rule 4 (make Hi hand 11/10) to stop you making A7 first, as in my tables rule 4 only seems to apply to hands such as 7642 (74 62) or 5532 (55 32) rather than a hand that can make 17+s.

The usual definition of an Ace is that it is 11 unless it causes the hand to bust, when it is a 1. This is mentioned in the last rule, that soft totals should be used and [an exception is] AA=12. Therefore an Ace must be being considered as being 11 and there is a need to clarify that AA is 12 [by implication A2=13 etc.].

Wizard's table (at time of writing) does not define the strategy for how to split your hands - technically this requires similar logic to Pai Gow Tiles, but in practice, I would use an infinite deck assumption. His tables show the strategy of whether to Hit/Stand/Double or Split once you've set your two hands.

Quote: gordonm888

3. There is another objection, on a more esoteric level to one specific feature of your High hand chart. It has to do with not doubling on an 11 vs. a 4 or 5. I actually believe your advice is wrong ( Player should Double 11 vs 4 or 5) even though your calculation is correct.
- First, I too now calculate that hitting an 11 vs a 4 or 5 provides a better EV than Doubling, although I do calculate that it is a close call. So, given your underlying assumptions, I am not challenging the integrity of your calculation.

Here are my exact figures:

11 vs. 4 in the high:

Double: -0.045045143
Hit: -0.003066316


11 vs. 5 in the high:

Double: -0.044429848
Hit: 0.007909841

As you can see, hitting has the higher EV in both cases. I admit that my infinite-deck based strategy could cause marginal errors.

Quote: gordonm888

The House Way is ambiguous on some hands such as:

AT32 --> s13, 13 Which is intended to be the high hand? I usually set the soft hand in the LOW Hand

I assume the player can put the soft 13 in either hand. The rules just say the high hand must be equal or greater to the low hand. Looking at your exact example, playing the soft 13 in the low is the right play against any dealer up card.

Quote:

Another issue not addressed in the Wizard's page is whether the House Way for arranging the cards is always optimum for the player.

Consider: 4332 House Way: 6,6 Possible Basic strategy: 7:5

Or: AT99 House Way: 20 (A9), 19 (T,9) Possible Basic Strategy: 21 (AT), (99) 18 or split 9s depending on upcard

I'm quite sure it is not. For example, two aces are much more powerful for the player than dealer, especially in the low.

Quote: charliepatrick

Wizard's table (at time of writing) does not define the strategy for how to split your hands - technically this requires similar logic to Pai Gow Tiles, but in practice, I would use an infinite deck assumption. His tables show the strategy of whether to Hit/Stand/Double or Split once you've set your two hands.

The way my program works is like my old Blackjack Switch strategy, where the player has to add expected values and go with the greatest sum.

Quote: Wizard

Here are my exact figures:

11 vs. 4 in the high:

Double: -0.045045143
Hit: -0.003066316


11 vs. 5 in the high:

Double: -0.044429848
Hit: 0.007909841

As you can see, hitting has the higher EV in both cases. I admit that my infinite-deck based strategy could cause marginal errors.

Thanks, could you provide similar info for 11 vs 3 and 11 vs 2 in the high? That would be helpful to me.

That should be adequate for the vast majority of the hands.

There is a theoretical issue on hands where the decision is close -say, a margin of 1 or 2 %.

The players are really dealt 4 cards and there are (I think) 715 unique 4 card hands.

For each of the 715 4-card hands, the best possible calculation would be to look at the possible arrangements and calculate the EVs of the possible arrangements with all four cards gone from the deck.

To belabor my point: an 88 in the low hand will occur, by definition, only when the player also holds a High hand that is one of these:

High Hand	Freq.
TT	43.7%
T9	18.7%
T8	4.1%
99	1.3%
98	0.8%
88	0.2%
AT	25.0%
A9	5.0%
A8	1.1%

Thats it! The 8-8 pair will be accompanied 43% of the time by a high hand of T-T. The absence of two tens from the deck will improve hitting a hard 16 and weaken SPLIT 88 pair.

The 8-8 pair will be accompanied 25% of the time by a high hand of A-T. Again, the absence of a ten will improve HIT hard 16 and weaken SPLIT 88 pair; the absence of an ace will very slightly weaken HIT HARD 16, and will significantly weaken a SPLIT 88 pair.

And so on. The existence of a HIGH Hand definitely improves the prospects of a HIT HARD 16 and definitely degrades the prospects of SPLIT 88 Pair.

Given all that, are you sure that splitting a pair of eights in the low hand is always the mathematically best move?

Quote: Wizard

Here are my figures. These should be considered rather rough. They are based on the final dealer probabilities on my site, which are derived by simulation. However, from there I used an infinite-deck and infinite re-splitting assumption to get at the player expected values and strategy.

This just sunk in. Each time I examine any hand I have been re-calculating the dealer probabilities (via combination math) to reflect the two cards in the player's hand . One of the shortcuts I have been taking is that I usually only look at one combination of ranks for the players hand: say, for a player 15, I use T5. However, when the decision is a close call, I then also look at 96, and 87 and average them to get the correct strategy call. So, I guess I'm a little more impressed with what I am doing then I was previously.

My Methodology
The players as-dealt hand is defined first and then reflected in two ways when calculating the dealer probabilities:
(1) the probabilities of each of the 715 4-card dealer hands are recalculated accounting for the absence of the players cards and,
(2) for each of the 715 4-card hands, the dealer probabilities of 17, 18, 19 20, 21 and Bust are calculated for the High and Low hands

Next, probablistically weighted values of Dealer 17, 18, 19, 20, 21 and Bust are compiled for Dealer 2-A scenarios.

Then, given the players starting hand, Player values of Stiff, 17, 18, 19, 20, 21 and Bust are calculated for both high and low hands. These algorithms recognize the absence of the 2 cards in the players hand from the deck as well as the absence of the Dealers up-card from the deck, but not the universe of cards that might have been consumed by the Dealer as Dealer hit his hand. Like everyone else, I start with Player 21, then 20, 19, 18 and work my way down to establish the HIT/stand breakpoints -which are used on the algorithms for lower Player hands.

Lastly, I convolve the Dealer and Player probabilities to calculate EV for the hand match-up.

My composition-dependent algorithms for BJ dealer hitting a single card A - T are comprehensive and cover all possibilities, no matter how many cards are used to hit the hand. However, for this application I needed to jury-rig algorithms for what happens when dealer starts with a soft 2-card hand (as opposed to a single Ace). I was lazy, so I wrote composition-dependent algorithms to address a dealer starting with an A6, A5, A4, A3, A2 and AA that only extend to hitting these soft 2-card hand with 3 additional cards. So when I say I have been taking some calculational short-cuts, that is the major one! Basically, my calcs of dealer probabilities are compromised because they doesn't include 6-card, 7-card and higher card number hands when Dealer is hitting AA-A6. Lol.

Quote: gordonm888

Thanks, could you provide similar info for 11 vs 3 and 11 vs 2 in the high? That would be helpful to me.

11 vs. 2 -- high:

Hit: 0.014506891
Double: -0.006769887

11 vs. 3 -- high:

Hit: 0.002110371
Double: -0.019171497

Okay, this program is 1125 lines long, not counting code for the RNG. It is practically all custom code. I think the only copy and pasting were simple shuffling and sorting functions. After checking about the first dozen hands it seems to be working correctly.

That said, I'm getting a house edge of 2.62%. I would consider this figure rather soft. It would be nice to have anything to compare it against. Just going off of industry norms, I think a house edge that high will kill the game, given I expect a huge rate of player strategy errors. If the game owner consulted me I would have advised putting the theoretical house edge between 0.0% and 0.5%, because I think player errors will add 2%-4% to that.

If anybody goes back to Orleans, please ask about the re-splitting rule. My figure assumes the player may re-split anything as much as he wishes.

Added more material to my Your Way 21 table. I welcome all comments.

Quote: Wizard

That said, I'm getting a house edge of 2.62%. I would consider this figure rather soft. It would be nice to have anything to compare it against. Just going off of industry norms, I think a house edge that high will kill the game, given I expect a huge rate of player strategy errors. If the game owner consulted me I would have advised putting the theoretical house edge between 0.0% and 0.5%, because I think player errors will add 2%-4% to that.

I was absent from the forum for 24 hours due to real life. I will work to give you a comparison basis using my composition-dependent model. My guess is that the actual House Edge may be significantly higher than you have estimated.

You have a typo in the title of your web page that needs fixing.

I am going to announce my page on Your Way 21 in a few days. Has anyone done any further analysis of it? I was hoping to get a copy of the official math report but never did.

Quote: Wizard

I am going to announce my page on Your Way 21 in a few days. Has anyone done any further analysis of it? I was hoping to get a copy of the official math report but never did.

So I'm reading the WoO page without reviewing the thread above; sorry for any duplication.

The title of the page and the URL (tab in the window) both read "You Way" rather than "Your Way".

Under Rules, 6. the following 2 questions have not been resolved. IMO, you need these answers before publishing the page.

•Player may re-split to ? hands.
•Re-splitting aces is/is not (?) allowed.

Under Rules, 7. "if the dealer busts, then that hand shall immediately lose". I think you mean "player".

Under House Way, 5. typo "dealer makes bets possible hand" should be "best".

Under House Way, 6. typo "dealer places on Ace in the high hand" should be "one".

In your probabilities tables, the tables are based on a dealer exposed card. You don't mention that there are one or more dealer exposed cards in the Rules (I figured the dealer cards were all face-down until the player had set their hand; guess not). So one exposed card of the dealer's 4? I think it's worth mentioning in the rules.

Displaying my ignorance to ask (I'm sure), but in the high hand EV table, 11 is +EV except on 4, 9, 10, A, so I'm wondering why there's no double indicated in the high hand strategy chart? Is there a +EV threshold to meet, not just +EV?

You mention the sidebet "P" but no paytables or strategy on that yet.

Hope this is more helpful than nit-picky. FWIW, I would definitely play this game, despite the 2.6 HE. I think it compares much more to pai-gow tiles than PGP in reading how it's laid out. Wish I'd thought of it; was trying to work something similar, but did not come up with the BJ overlay.

Quote: beachbumbabs

In your probabilities tables, the tables are based on a dealer exposed card. You don't mention that there are one or more dealer exposed cards in the Rules (I figured the dealer cards were all face-down until the player had set their hand; guess not). So one exposed card of the dealer's 4? I think it's worth mentioning in the rules.

.

So, reading from the beginning, you must set your hands before you see ANY dealer cards. Then you get to see 1 card of 4. THEN you make your hit/stand/etc. decisions. I think you definitely have to explain that sequence of events in your Rules writeup.

Just left the Orleans. That was the only table with no action. Talked the pit boss into letting me play a match play on the low hand side with half bet on the high side. ;-)
Dealer broke on both ends.

Wizard, these are some comments on the (preliminary) Your Way 21 page on WOO that you posted.


Structural

1. The name of the game is misspelled in the title of the page; It is "Your Way 21", not "You Way 21."

2. In the page's section titled "Splitting Strategy", EV tables are provided for "2-card BJ hands vs the dealers upcard" for use in analyzing optimum strategy for splitting the player's 4-card hands. These tables are not appropriate for this use because the splitting decision must be made by the player before the dealer's card is revealed.

3. In the tables for High Hand Basic Strategy, it is recommended to hit a hard 4 vs a 2 and a (2,2) pair vs a 2. In Your Way 21, these hands do not exist. To set a pair of 2s or a hard 4 (i.e., a 2-2) in the High hand requires that the low hand also be a 2-2. For this to occur versus a dealer 2 would require five 2s. That cannot occur because this is a single deck game.


Issues with Mathematical Strategy

1. The table for Low Hand Basic Strategy recommends that the player split a 9,9 pair vs a dealer 9, T or Ace. I believe this is incorrect; the player should stand on the 9,9 pair vs 9,T, A.

There are only 3 player hands in which there is a 9,9 pair in the low hand: T-9-9-9, T-9-9-8 and 9-9-9-9. I have calculated the EVs of the low and high hand for each of these three player hands, taking into account the absence from the deck of the 4 cards in the player hand and the dealers upcard and using a composition-dependent spreadsheet for hitting the High and Low hands of both the player and dealer. It is better to STAND.

In all these cases, the players hand are composed of Tens and Nines and the dealers card is 9, 10 or Ace - all of these are desirable cards when hitting a split pair of nines. I think your use of an infinite deck assumption to model the players hands has led you astray here. Remember this is a single deck game and the player's hand is 4 cards -not 2 cards- so an infinite deck assumption can result in large errors.

2. Your strategy for playing a pair of 7s in the low hand is to hit the 14 (7,7) vs a 2 and split the 7,7 pair vs a dealer 3-A. This is a reasonable approximation but is not the exact strategy. I calculate that it is always correct to split 7,7 vs a Dealer 4-Ace, but the strategy is more nuanced vs a 2 or 3. Here is a table listing all the player hands where a 7,7 pair is in the low hand and my calculated best strategy against a dealer 2 or 3.

Player hand	vs. 2	vs. 3
AT77	Hit	Split
A977	Hit	Split
A877	Hit	Split
A777	Stand	Split
TT77	Hit	Hit
T977	Stand	Stand
T877	Hit	Split
T777	Hit	Split
9977	Stand	Split
9776	Stand	Split
9775	Split	Split
8777	Split	Split
8776	Stand	Split
7777	Stand	Split

3. In the low hands, I found that the player's Hit/Stand decisions on stiff hands, particularly 12, 13 , 14 , 15, depends upon the ranks of the four cards in his hand. For example, a 12 vs. 6 in the Low hand:

If the player's hand is 9975 vs. 6, it is best by a large amount to Stand on the 12 in the low hand. All four cards in the player's hand are desirable when hitting a 12 -but the absence of the two 9s is a major factor, in particular, because 12 + 9 = 21 and there is a higher than normal premium on getting a 21 in this game.

Conversely, with a T775 vs a 6, the player should Hit the 12 in the Low hand. In this hand, the player has no 9's and has a TEN which is a bust card when hitting a 12.

4. In the High hands, the HIT/Stand decisions are similarly sensitive but with higher stiff hands (or 17). For example, with 9855 you should HIT the 17 in the High hand, but with 9844 you should STAND on the 17 in the High Hand. The difference is those two 4s in the latter hand -when you hit a hard 17 you are obviously hoping for a 4.

There appear to be more than 100 exceptions to your Hit/Stand strategies when you take into account the knowledge of all four cards in the player's hand.

By the way, in my calculations I did assume there was no re-splitting of pairs. I agree there is some ambiguity about the rules but:

- my educated guess is that No Resplitting is the correct rule, so that the dealer does not run out of cards
- and honestly, I do not beleive that resplitting or "no resplitting" of pairs will change any of the optimal strategy decisions, especially given the reduced importance of resplitting in single deck games.

In my calculations I considered the players hands to be:

AAAA
AAAT
.
.
8765
8764
.
.
2222

I count 715 unique player hands when using this system. Of course, each player hand can face a dealer upcard of 2->A (except for the 9 combinations that require 5 cards of a single rank other than 10s (such as 6666 vs 6) so there are 7141 unique as-dealt situations to analyze. And each analysis requires analyzing two Blackjack-like confrontations between player and dealer. Of the 7141 cases, I have analyzed 5627 to date; in probability space these cases are approximately 90.7% of all the dealt hands.

With optimal play, the aggregate House Edge is about 0.9 % in the cases I have analyzed. I think that optimizing player decisions with the knowledge of all 4 cards has the effect of diminishing the House Edge by several tenths of a percent.

I have not yet investigated the optimal player strategies for splitting the 715 as-dealt player hands, so my calculations assume the player uses the House Way to arrange the hands. I can see that there are at least a handful of Player's hands that will have a higher EV if they are set differently.