Pai Gow Poker
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Quote: jrthomas2Recently I was playing Pai Gow Poker. The dealer hand was 2,3,4,5,6,8,K. The King and the 5 were hearts and the 2,3,4,6,8 were spades. I thought that the hand would have been set with K,5 on top and the spade flush in the back. However, the dealer set the hand as a straight and moved the K,8 up top. IT seemed to me that this was giving the players an edge as it did not change the high card on the top hand I still only had to beat a King, but now I only have to get the third lowest ranking straight to defeat the back hand. When I questioned the Supervisor he was very sure that the hand was better that way but could not give me the numbers to back up his claim. Can you help me with these numbers? What is the math?
yes, that is the typical and standard house way, to play the best top between a straight and a flush decision, so it'll flub a few hands here and there.
Obviously, a generic house way that a typical dealer has to use cannot be expected to make "fine point" optimal play adjustments, so yes, it gives the sharp player an occasional edge.
Keep in mind that the casino has to use a consistent, non-arbitrary, and easy-to-deal house way, so you'll see some house way misplays from time to time.
The difference between optimal play and house way play is about 0.3%, so if you, as a Pai Gow player, play every hand like a tournament Pai Gow Pro, you'll pick of a third of a point in house edge.
My instinct is to say that the k-8 with the straight is better than the k-5 with the flush, but I haven't calculated anything. So my instinct is the house way is correct.
However, what you need to remember is that the house way is just a "good enough" strategy that lets the house get a reliable profit when combined with taking the commission. It's nowhere near optimal but the goal is to be simple enough for a dealer to remember and the floor/surveillance to supervise while still remaining profitable.
The casino has a house way in blackjack, too -- hit till 17. Of course that's not optimal, but it's good enough for the house to make a profit without having to worry about dealers making decisions.
A straight and a flush are both very strong hands. Both with beat most player hands. Both will also lose to a few very strong player hands. Therefore, create the best 2 card hand to improve the odds of beating both hands or at least pushing.
In other words, how often will a player have a five card hand that will beat the straight but not beat the flush? It's so rare that those few hands are ignored in favor of improving the two card hand.
Quote: sodawater
My instinct is to say that the k-8 with the straight is better than the k-5 with the flush, but I haven't calculated anything. So my instinct is the house way is correct.
The flush is considerably better.
K5/flush = .2813 EV
K8/straight = .2226 EV
King and Ace tops are very different in strength when it is Ace-face versus Ace-low, and KQ/KJ versus Kx,
but A7 versus A4, and K8 versus K5 are statistically equivalent.
Quote: PaigowdanThe flush is considerably better.
K5/flush = .2813 EV
K8/straight = .2226 EV
How are these EVs calculated? In other words, EV against what splitting strategy?
On an almost-relevant note, for heads-up PGP (where each player can set their hands however they want), is the optimal strategy fixed or mixed? Has this been calculated? I'd assume that there are two strategies, one for the banker (player who wins copies) and one for the player (player who loses copies)
Quote: AxiomOfChoiceHow are these EVs calculated? In other words, EV against what splitting strategy?
On an almost-relevant note, for heads-up PGP (where each player can set their hands however they want), is the optimal strategy fixed or mixed? Has this been calculated? I'd assume that there are two strategies, one for the banker (player who wins copies) and one for the player (player who loses copies)
There are slightly different strategies for banker vs. player on close hands (e.g., playing AJ/Straight versus A8/high flush), and is nicely outlined in Stanford Wong's "Optimal Play for Pai Gow Poker.".
The "standard strategy benchmark" ("splitting strategy") is something akin to Texas Station's [old] strategy.
Newer strategies account for keeping low pairs together (6's and less) with a king for the top. There is an optimized general strategy, (where, like the James L. Brooks movie, it's "As good as it gets"), and the advancements are:
1. Two pair splitting;
2. Straight Versus Flush (accounting for close tops to select the flush)
3. Two pair rule superseded when low pairs/crap top versus flush/straight good top (K+)
4. Full house (low pairs in full house keep with AK/AQ top)
5. Four of a kind handling.
Wow.Quote: PaigowdanThe flush is considerably better.
K5/flush = .2813 EV
K8/straight = .2226 EV
King and Ace tops are very different in strength when it is Ace-face versus Ace-low, and KQ/KJ versus Kx,
but A7 versus A4, and K8 versus K5 are statistically equivalent.
I guess my simple, non-math answer was way off.
The last thing I'd want to do is debate the finer points of Pai Gow Poker with the Pai Gow Poker master!
But my answer still makes sense in my head. I guess that's where it should stay...
Quote: SOOPOOGood pickup, jrthomas. You intuitively noted that the dealer did not set the hand optimally, but as Pai Gow Dan pointed out, the dealer did set the game properly as per the casino's house ways. There are these rare instances when the house ways are clearly not the best way, and you noticed one of them! One of my favorites is to get a full house, say three 3's, two 2's, and an AK. I will always play the full house, and be ready for the stares. The number of times you lose to the copy AK is less than the number of times the dealer has a higher 3 of a kind, straight, or flush....
Scott - very sharp about FH with low pair and AK (even AQ) top. Keep in mind that 10% of the time the dealer has a straight or better, and that the difference between AK and 22/33 is less than 10%, so it IS the better play to AK (even AQ) on top, with the 3's or 2's down with the Full House together.
Let me say (or actually announce) that:
1. There DOES exist a "basic template" or Reference Pai Gow House Way that the Gaming Industry uses to mathematically analyze Pai Gow poker as a reference point (7-card with bug joker, 'Nevada' style, not So-Cal with wild joker) for Pai Gow variants that are developed. They are NOT going to re-program a General Pai Gow testing system from scratch; they use the reference house way, and then modify from that based on the submitter's specs; most times the reference house way is good-to-go, and speeds development time, and time-to-market. Gaming Labs like GLI and BMM, as well as gaming mathematicians like Charles Mousseau, Steve How, and Cindy Liu use a "reference point" house way, and apply programming patches to the supplied developer's house way. Cindy Liu and I developed a reference "strong but generic" House way for game development use, (descendant and related to the Station Casino Group's House way, a truly "median" house way) to speed development, in working with her and BMM compliance labs, back when Cindy was an independent mathematician. (She now works for slot developer company 'X'). This "General reference" House Way should be posted by Mike on WOO (and I will supply), I feel, if he's going to post the capricious and endlessly-changing house ways of local casinos.
I have two other PGP house ways:
2. A "close-as-you-can-get" to tournament Pai Gow "Player's Way," - a Player's Optimized method for facing the median-reference-standard house way that a casino would use, as described above. This REALLY should be posted on WOO.
3. A "Tiny" house way that load and run very quickly in an App. A stripped version that'll play well within ~ 0.3% of optimal. Here it is:
• No pair: Place the highest card in the high hand and the next two highest cards in the low hand.
• One pair: Place the pair in high hand and the next two highest cards in the low hand.
• Two pair:
o Ace and any other pair - always split.
o Face card pair and pair 7's or higher - always split
o With any other 2 pair, split unless the hand contains an ace for the top
• Three pair: Always play highest pair in low hand.
• Three of a kind: Always play three of a kind in high hand except break up three aces.
• Straight, Flush, or Straight Flush/Royal:
6 card long - use the highest card in the low hand.
5 or 6 card long with a pair - use the pair as the low hand.
With 2 pair - play the 2 pair rule.
A flush and a straight with no pair - play the combination that results in the highest 2 card hand.
• Full house: Split except with pair of 2's or 3's and an AK can be played in low hand.
• Four of a kind: Play according to the rank of the four of a kind:
2 through 6: keep together with a King or better, else split.
7 through 10: keep together with an Ace or better, else split.
Jack through king: Split unless hand also contains a pair of 5's or higher.
Aces: Split unless a pair of 7's or higher can be played in low hand.
• Five aces: Split off two Aces for the low hand.
The "reference" house way is this, except slightly more detail in two pair handling, Straight/flush handling, and four of a kind handling.
The optimal Player's house way breaks it out to Ax versus Ace-face and King-x versus KQ/KJ in two pair, S/F, and four of a kind handling, and one pair of S/F in some situations, also Straights versus flushes with detailed two-card sides. Even includes splitting two pairs 10's and 8's and 10'sand 9's with various ace-high two-card sides. It even has recommendations to play four of a Kings or Aces as AK/AAAx, keep QQQQx with AK-A10 top, but play Trip Queens/AQ as top with an 9 or less kicker instead of splitting QQ/QQAxx, etc. It got nuts for the extra 0.0000002% edge, etc., I felt I was becoming like an obsessive card-counter over Pai Gow poker. Very Strong but very hairy for the extra and last little bit of 0.1% Edge, -and I don't think it's worth it.
Quote: PaigowdanThe flush is considerably better.
K5/flush = .2813 EV
K8/straight = .2226 EV
King and Ace tops are very different in strength when it is Ace-face versus Ace-low, and KQ/KJ versus Kx,
but A7 versus A4, and K8 versus K5 are statistically equivalent.
Just noticed this older thread. I am wondering whether Paigowdan's EVs were calculated with power ratings?
I have a composition-dependent analytical model of PGP. It has almost 250,000 lines of code. Assuming the dealer is playing Trump Casino rules:
K5/ flush (86432)
Win= O.2136 Push = 0.6682 Lose =0.0122 EV= 0.2913
K8/6-high straight
Win = 0.2732 Push = 0.6808 Lose= 0.0459 EV=0.2136
Quote: gordonm888Just noticed this older thread. I am wondering whether Paigowdan's EVs were calculated with power ratings?
I have a composition-dependent analytical model of PGP. It has almost 250,000 lines of code. Assuming the dealer is playing Trump Casino rules:
K5/ flush (86432)
Win= O.2136 Push = 0.6682 Lose =0.0122 EV= 0.2913
K8/6-high straight
Win = 0.2732 Push = 0.6808 Lose= 0.0459 EV=0.2136
Yes, it is basically by power rankings, but it is both composition dependent and very subtle. Most players miss this, though it matters little in these cases in terms of EV.
When given a choice between straight versus flush with essentially the same top, either a K-"x" top or an A-"x" top, the flush over straight differential is greater than, let's say, a K-8 versus K-5, ignoring composition. Not meaning to split hairs, play the flush over the straight when your two-card is "essentially" the same: Q-x, K-x, A-x, or King-face or Ace-face versus Kx/Ax.
But the difference between Ace-face or King-face is more considerable: AQ versus A-x, K-face versus K-x, play whichever has the better top when with a "face-kicker" versus Ace-low or King-low. I group Ace-face/King-face above Ace-low, King-low; it's when the two-card side's kicker is the same face or "low" that makes this "fine-point" rule of thumb.
Quote: offTopicI recently had the "honor" of losing with 2 pair/AA to another 2 pair/AA hand. From a probabilistic perspective, where does this rate against, say, being dealt a royal or 5 aces? This was the first time this has ever happened to me, and I've been playing for a pretty long time.
As I understand it, you had a three pair hand with AA or (A-jok) as your high pair, and the dealer also had 3 pr with AA or A-Jok as the high pair? The odds of that are 0.0000191 or slightly more than 1 in 50,000. The odds of losing the 5-card back hand match-up would be 50%, so the odds of what happened to you were slightly more than 1 in 100,000.
The odds of being dealt a 3 pair hands with Aces-Up (counting the joker as an ace) are 0.0062. Given that you have been dealt that hand, the odds of the dealer also having a 3 pair AA-xx-yy-z hand are 0.0031. The product of those two numbers is 0.0000191.
Quote: gordonm888As I understand it, you had a three pair hand with AA or (A-jok) as your high pair, and the dealer also had 3 pr with AA or A-Jok as the high pair? The odds of that are 0.0000191 or slightly more than 1 in 50,000. The odds of losing the 5-card back hand match-up would be 50%, so the odds of what happened to you were slightly more than 1 in 100,000.
The odds of being dealt a 3 pair hands with Aces-Up (counting the joker as an ace) are 0.0062. Given that you have been dealt that hand, the odds of the dealer also having a 3 pair AA-xx-yy-z hand are 0.0031. The product of those two numbers is 0.0000191.
Score! Thanks!
Quote: PaigowdanYes, it is basically by power rankings, but it is both composition dependent and very subtle. Most players miss this, though it matters little in these cases in terms of EV.
There IS a clearly more composition-dependent situation in PGP, and it occurs with strong four of a kinds.
The hand KKKKA72 is better played as AK/KKK72, while AAAAK72 is better played as AA/AAK72. Why?
Having One Ace with KKKK is better played as AK/KKKxx, because you will beat any ace top like Ax/xxyyz hand with four kings out in your hand, as you'll more likely face and beat an Ace-Q or less top with one pair or better, but with AAAAK72, you'll risk facing two pairs routinely split against you when two pairs occur without an available Ace for your opponents' two-card side to beat. Holding almost all the aces makes any two pair hand more likely split against you.
With KKKKA72, you'll see something more like AJ/77225 to beat, but with AAAAK72 played as AK/AAA72, you see 8844K32 played against you as 44/88K72 - to deny you the win, whereas AA/AAK72 will give you the win.
Quad Kings, a rare hand, makes any Ace-King the best low hand (short of a pair top against you), but holding quad Aces makes any pair top against you more likely to be split against you, so split best aces against split pairs on both sides against an opponent.
That's the only composition play I know in PGP.
Edit Not true. I do know a few more....
