5 card Pai Gow Poker?

Now that charliepatrick has tortured me into debugging my hand analyzer, here are the corrected values for the high hands. The revelation remains the same: that the presence of low cards in player's hand causes the dealer's hand to qualify so much more frequently that a Ts9s8s-TT hand is much more profitable than a AsKsQs-AA.

Player Hand	EV
Ts9s8s-TT	0.79642802
TsThTd-9h9d	0.784934081
Ts9s8s-AsAd	0.758028188
AsKsQs-AA	0.667928125

QUADS with corrected values

Repeating a previous post with corrected entries in the table and some new commentary.

We aren't dealt quads very often, but from a game theory point of view it's a surprising and interesting hand category.

The usually irrelevant singleton kicker in a Quads hand can actually be quite significant because it affects the strength of the two card hand when playing Trips in the 3-card hand (when the kicker is a higher rank than the quads.)

Here's a table of calculated EVs, where the columns headed by 6 to A denote the singleton kicker in the player's quad hand. I've kept the entries to only 4 digits to make it easily readable

Player Hand	6	10	J	Q	K	A
AAAA	0.6402	0.6434	0.6094	0.6089	0.6083
KKKK	0.5422	0.5456	0.5103	0.5091		0.5453
QQQQ	0.4634	0.4650	0.4292		0.4602	0.4815
JJJJ	0.3984	0.3973		0.3904	0.4061	0.4342
TTTT	0.4730		0.4713	0.4834	0.5037	0.5383
9999	0.4355	0.4497	0.4377	0.4531	0.4770	0.5180
8888	0.3952	0.4125	0.4012	0.4180	0.4470	0.4946
7777	0.3645	0.3831	0.3708	0.3913	0.4255	0.4799
6666		0.3523	0.3391	0.3633	0.4029	0.4643
5555	0.3186	0.3224	0.3080	0.3361	0.3812	0.4497
4444	0.2880	0.2908	0.2756	0.3077	0.3587	0.4350
3333	0.2665	0.2681	0.2525	0.2889	0.3459	0.4296
2222	0.2344	0.2359	0.2216	0.2624	0.3254	0.4170

The revelation here is the enormous difference that the quads kicker can make! The EV of 2222-J is 0.2216; the EV of 2222-A is 0.4170. That's greater than a 19 percentage point swing in EV as the kicker to quad 2s goes from a Jack to an Ace. Wow.

Once again, we notice the influence of having a high card in your hand on increasing the DNQ frequency of the dealer's hand. Jack is the worst kicker possible because it lowers the chances of the dealer's hand qualifying and a J-x in the two card hand isn't much better than two low cards.

This probably took a lot of work, well done :)
Have you gotten close to finding the expected RTP given optimal strategy?

Quote: harris

This probably took a lot of work, well done :)
Have you gotten close to finding the expected RTP given optimal strategy?
link to original post

I expect that that is the house advantage claimed by the manufacturer.

I cannot easily calculate the expected House Advantage. I imagine Charliepatrick or one of the other programming wizards can verify the published number.

I have no idea whether the logic is correct and haven't checked it, although I'm glad the DNQ figure is close to what I worked out a while ago, but this doesn't prove my Player hand analysis logic, and got the following:-
Trial 1
DNQ :2313845
Win :1911620
Tie :3661175
Lose :2113360
Trial 2
DNQ :2312980
Win :1911876
Tie :3660177
Lose :2114967
Trial 3 (longer!)
DNQ :23 127 128
Win :19 105 243
Tie :36 608 304
Lose :21 159 325
I have assumed, after getting rid of any DNQ hands, a Win is where you can create a result where both hands win, a Tie is where you can create a result where one hand wins, and otherwise Lose.
You can see you roughly lose just over 2m more hands than you win, this suggest a House Edge about 2%.
Does anyone know what the published HE is?

Quote: charliepatrick

I have no idea whether the logic is correct and haven't checked it, although I'm glad the DNQ figure is close to what I worked out a while ago, but this doesn't prove my Player hand analysis logic, and got the following:-
Trial 1
DNQ :2313845
Win :1911620
Tie :3661175
Lose :2113360
Trial 2
DNQ :2312980
Win :1911876
Tie :3660177
Lose :2114967
Trial 3 (longer!)
DNQ :23 127 128
Win :19 105 243
Tie :36 608 304
Lose :21 159 325
I have assumed, after getting rid of any DNQ hands, a Win is where you can create a result where both hands win, a Tie is where you can create a result where one hand wins, and otherwise Lose.
You can see you roughly lose just over 2m more hands than you win, this suggest a House Edge about 2%.
Does anyone know what the published HE is?
link to original post

Between 2.7-2.8%

^ Thanks.
My long sim seems to suggest you lose $2,054,082 when considering the 76,872,872 hands excluding the DNQs. That may just be a coincidence at 2.672%! Alternatively it's possible I'm allowing too many Player hands to win, for instance those which have set a good, but invalid, Low hand, to stand off.

Quote: charliepatrick

^ Thanks.
My long sim seems to suggest you lose $2,054,082 when considering the 76,872,872 hands excluding the DNQs. That may just be a coincidence at 2.672%! Alternatively it's possible I'm allowing too many Player hands to win, for instance those which have set a good, but invalid, Low hand, to stand off.
link to original post

I'm not aware of any requirement for players 2-card hand to qualify. I'm certainly not analyzing the game like that. When I play the demo game, it has no trouble giving me a push for a winning 3-card hand even when my 2-card hand is 3-2 and a total loss.

It seems that the major impact of being able to see dealer's hand is to allow player to turn losses into pushes by shifting high cards into the 3-card hand while giving up on the 2-card hand.

Jargon:

Push = when dealer and player each win one of the two hands

Tie = when dealer and player both have the same hand. Ex: The 2-card hands tie when both dealer and player have KQ in those hands. Note: in this game (and in 7 card PGP) a tie means that dealer wins that comparison and player loses it.

^ If 2.7% is correct then I'm letting too many hands win, or not enough lose!! What I was wondering whether I had a bug in my program. Suppose the dealer has KT A73 and I have AK964 then I could work that setting my hand A9 K64 so my low hand would win, however this is an invalid way to set one's hand, so it should be a loser.
I agree that the advantage you have in this game is being able to turn a losing hand into a Tie. The disadvantage is the dealer does not qualify, or pay out, on bad Low hands.

Are you saying that you might not be checking for "fouling" i.e., for having a 3-card hand that isn't higher than the 2-card hand?

^ I had a bug when the Player had two pairs and mistakingly allowed the larger one in the Low Hand! When I fixed that I got a House Edge of about 2.93%, which for me is close enouigh to the range they give. (The difference by calling Win-Ties a Win, rather than a stand-off, would be about 0.5%, and a Tie-Tie doesn't happen very often!)