Pai Gow Poker House Edge

I'm trying to determine if there is a higher house edge when I play Pai Gow Tiles or when I play Pai Gow Poker.

In tiles, I've pretty much got simple strategy down, which makes the house edge when I play around 2.0%

In PGP, I'm much more skilled, and even know that keeping Full Houses together makes sense sometimes, even if other players yell at me. The Wizard writes that the house edge on PGP is right around 2.5% when using the house way. Do we have any idea how much optimal strategy reduces the house edge?

I don't know if anyone knows the real house edge. PGP is a bitch to analyze without a LOT of computing power, and without taking illegal shortcuts such as assuming that the dealer's hand is dealt from a different deck than the player's hand.

In the worst case scenario analysis, you've got combin(53,7) player hands * 21 ways to split it * combin(46,7) dealer hands = 173,259,639,655,502,400 pairs of comparisons to make (pairs, because you need to compare the 5-card hands as well as the 2-card hands).

Now, in the same way that the number of starting hands in video poker can be reduced from 2,598,960 to 134,459 (for a 52-card deck), the number of starting hands for PGP can be reduced from 154,143,080 to 6,972,147. That brings our total down to 6972147 * 21 * 53524680 = 7,836,820,678,847,160 scenarios to test.

And while there are 21 ways to split a 7-card hand into a 5-card hand and a 2-card hand, not all of them are valid (some are foul). I come up with the following totals for the number of valid ways to split PGP hands:

Valid Ways To Split	Total Hands	Unique Hands
1	4,150,964	200,829
2	4,651,952	236,574
3	3,312,668	158,533
4	30,046,884	1,363,252
5	2,223,576	111,104
6	7,186,780	366,431
7	662,864	33,302
8	1,244,576	62,349
9	2,597,100	129,899
10	56,888,820	2,506,548
11	669,384	33,674
12	521,312	27,000
13	143,564	9,253
14	94,284	5,238
15	26,162,652	1,121,390
16	18,904	1,107
17	18,612	1,034
18	13,440,184	595,591
19	0	0
20	0	0
21	108,000	9,039

(I calculated those totals a long time ago, so I'm not entirely sure if they are accurate, but for now let's assume they are.)

If you take the product of the leftmost and rightmost columns, and sum that product for each row, you come up with 64,977,799 unique hand/split scenarios that need to be tested against every possible dealer hand. This brings the total down to 64977799 * 53524680 = 3,477,915,898,579,320 player vs. dealer scenarios to test.

So in order to calculate the actual house edge, you need to be able to run through those 3.5 quadrillion scenarios before the power goes out... and you better hope that you didn't make a mistake in your house way code or in the handling of the Joker (or forget to make your program display the results when it's done processing 30 years later).

I think it would require a mainframe - or a freakin' cluster of them - to properly calculate this game in a reasonable amount of time without taking 'lossy' shortcuts.

About 0.1% to 0.2%. Not that much. Calculating the improvements by an exhastive search/simulator is not the way to go, though a lot of tuning can be done with an algebraic approach and an accurate power ranking table that includes the effect from two-side copies. Also, searching on select multi-path hands (hands that can be played well in either setting, such as two pairs, straight or flush with one or two pairs, and four of a kinds, etc.) can prune the simulator process. It's not that bad.

1. Most of the edge increase can come from properly playing two pair hands properly and very accurately (keeping 7's and 2's together with a KJ or KQ instead of requiring an ace, and keeping All two pairs together with an AK, etc.);
2. Some comes from properly handling straights and flushes with one pair (Play a pair of Jacks or better with and A-10 or better if the straight or flush has a queen or less top); straight or flushes with two pairs (Play the straight or flush if it has an AK top, or if the two pairs are 6's and 5's or less with a Q or less top); the hand 6543322 is better played as 32/65432 than as 65/33224 or 22/33654, Playing a high straight or flush with an AK top is always better than playing it as any two pairs, even a "strong" two pair hand.
3. and a tiny amount of improvement comes from properly handling full houses with an AK top (don't split if the pair part are 4's or less) and properly handling four of a kinds, but they are so rare a misplay here means very little. Next to no improvement can come from worrying about how to handle full houses that also have a straight or flush with a pair (hands that always include the joker, and where either play is really never a misplay).

There is a slightly different strategy for banking, than for playing against a banker.
Also important to consider is to eliminate the wins and loses that cancel each other out. In this sense, it is better for the house or banker to win two hands and push four hands, than to win four hands and lose two back - for the same net two wins, especially on a commission-free version where no trivial income is made by "churning hands." In one case, two net wins comes from two transactions, where in the other, the two net wins comes from six transactions, four of which were useless.

Getting the extra 0.2% while using a strategy of one page, and is still easy to deal is the holy grail in PGP. For the house, the extra 0.2% can add up to about 2% extra in table hold.

And then there is 'real world' pai gow strategy. Seated in the middle I usually can see 28 to 35 cards before making a decision. Knowing all 5 aces are not availabe to the dealer allows the playing of a k up top in situations where you would not normally do so. This helped me last trip turn a push into a win... I didn't announce why i did it, but I loved the 'gosh he doesn't know how to play' looks I got as I payed the commission.

I think there are more opportunities in Pai Gow Tiles. For one thing, the banking rules are often more lenient, at least in commercial casinos. Also, a lot of time when you are banking the opposition will discuss their hands before you set yours. I bet you could do very well if you knew Chinese and were a white person, too.

Oops, doing the wrong kind of thinking, again. Sorry, Dan :)

Your confession is accepted. Three Our Fathers and two hours of Pai Gow Poker at $25 a hand, no peeking....and a massage at Spring Valley and Valley View with the winnings. Viva Las Vegas.