Computing an optimal strategy in pai gow poker
June 23rd, 2015 at 11:33:53 AM
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Hi, Michael, and others,
On the Wizard Of Odds main page for pai gow poker,
... Michael shows the house edge when playing house way or the "optimal" strategy. I've got a math-y question about the optimal strategy.
Seems to me, in order to truly compute an optimal strategy, one would have to take every possible hand, of which there are (53,7)
(that's how I write "the number of combinations of 53 items taken 7 at a time)
and for *each* hand you might receive, compare the possible ways you could play that hand against the (46,7) different hands that the dealer could have, and then see which way of playing your cards worked out best overall.
But (53,7) = 154,143,080
and (46,7) = 53,524,680
and there's no way to iterate through all of those possibilities with current computers. I don't see how anyone could compute a true optimal strategy for pai gow poker.
Maybe "optimal" doesn't mean what I think it means in the context used on that page.
Can you set me straight? (no pun intended!).
Thanks,
--Steve
On the Wizard Of Odds main page for pai gow poker,
... Michael shows the house edge when playing house way or the "optimal" strategy. I've got a math-y question about the optimal strategy.
Seems to me, in order to truly compute an optimal strategy, one would have to take every possible hand, of which there are (53,7)
(that's how I write "the number of combinations of 53 items taken 7 at a time)
and for *each* hand you might receive, compare the possible ways you could play that hand against the (46,7) different hands that the dealer could have, and then see which way of playing your cards worked out best overall.
But (53,7) = 154,143,080
and (46,7) = 53,524,680
and there's no way to iterate through all of those possibilities with current computers. I don't see how anyone could compute a true optimal strategy for pai gow poker.
Maybe "optimal" doesn't mean what I think it means in the context used on that page.
Can you set me straight? (no pun intended!).
Thanks,
--Steve
June 23rd, 2015 at 1:31:56 PM
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53c7 * 46c7 = 8.25E15, which is the denominator you see in the charts.
You have to evaluate every possible hand, but you don't have to evaluate them individually. There are several articles out there about how to do suit-folding or use equivalence classes.
Welcome to poker software. :)
You have to evaluate every possible hand, but you don't have to evaluate them individually. There are several articles out there about how to do suit-folding or use equivalence classes.
Welcome to poker software. :)
"In my own case, when it seemed to me after a long illness that death was close at hand, I found no little solace in playing constantly at dice."
-- Girolamo Cardano, 1563
June 23rd, 2015 at 1:41:26 PM
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You can compute a very strong and effective house way/player way strategy from the 5-card and 2-card side hand probabilities.
Beware of all enterprises that require new clothes - Henry David Thoreau. Like Dealers' uniforms - Dan.
June 23rd, 2015 at 1:53:15 PM
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You can also look at boundaries - for instance with Two Pair you can look at Pair a Pair b with kickers c d e. In theory the suits might matter, but you'll get an idea of what Q-high is good enough to play 3s and 2s together. While you might miss some obscure exceptions, it's a start.
June 23rd, 2015 at 2:04:28 PM
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Hi, MathExtremist,
Can you get me some links to the articles you mentioned? I'd like to do some reading!
Can you get me some links to the articles you mentioned? I'd like to do some reading!
June 23rd, 2015 at 2:04:52 PM
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I have a spreadsheet model that evaluates "every" possible PGP hand that the dealer can have, taking into account the removal from the deck of the cards in your own hand plus any other cards you see.
To do this, I represented all the possible PGP hands as approximately 250,000 groups of hands and calculated the probability of each of the 250,000 groups as a function of the composition of the remaining cards in the deck. Within each group, all hands yield the equivalent "front" and "back" hand, in terms of winning power of the front and back hands.
A PGP hand like KKK-JJ-32 will be played:
Front: JJ
Back: KKK-32
The suits of the cards are irrelevant in this hand. If modeling the player's hand, the ranks of the two unimportant cards, 3 and 2, should be modeled explicitly, IMO, because their absence from the deck affects the probabilities of the hands that the dealer can get. However, when modeling the Dealer's hand, one can simply model JJ-KKK-xx where xx are any two unpaired singletons -as long as one correctly writes the equation for the probability of KKK-JJ-xx.
It took me over a year to write my PGP hand evaluator code (with 250,000 hands) and about 3 years to debug it! I believe that the Wizard's posted Pai Gow Strategy was accomplished with a Monte Carlo model that uses a random number generator and deals out hundreds of millions of hands and records the results. My results generally agree with the Wizard's results with high accuracy (with some exceptions due to different House Ways) but my code allows me to rapidly evaluate a lot of scenarios that his MC code does not. However, an MC code is a superior method if you wish to calculate House Advantage over the complete universe of all possible hands.
To do this, I represented all the possible PGP hands as approximately 250,000 groups of hands and calculated the probability of each of the 250,000 groups as a function of the composition of the remaining cards in the deck. Within each group, all hands yield the equivalent "front" and "back" hand, in terms of winning power of the front and back hands.
A PGP hand like KKK-JJ-32 will be played:
Front: JJ
Back: KKK-32
The suits of the cards are irrelevant in this hand. If modeling the player's hand, the ranks of the two unimportant cards, 3 and 2, should be modeled explicitly, IMO, because their absence from the deck affects the probabilities of the hands that the dealer can get. However, when modeling the Dealer's hand, one can simply model JJ-KKK-xx where xx are any two unpaired singletons -as long as one correctly writes the equation for the probability of KKK-JJ-xx.
It took me over a year to write my PGP hand evaluator code (with 250,000 hands) and about 3 years to debug it! I believe that the Wizard's posted Pai Gow Strategy was accomplished with a Monte Carlo model that uses a random number generator and deals out hundreds of millions of hands and records the results. My results generally agree with the Wizard's results with high accuracy (with some exceptions due to different House Ways) but my code allows me to rapidly evaluate a lot of scenarios that his MC code does not. However, an MC code is a superior method if you wish to calculate House Advantage over the complete universe of all possible hands.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
June 23rd, 2015 at 5:07:25 PM
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Quote: TucsonSteveHi, MathExtremist,
Can you get me some links to the articles you mentioned? I'd like to do some reading!
Here are a few from the VP world:
https://wizardofodds.com/games/video-poker/methodology/
https://web.archive.org/web/20141203235121/http://www.vpgenius.com/articles/deck-simplification.aspx
https://web.archive.org/web/20141208125514/http://www.vpgenius.com/articles/analyzing-video-poker-hands.aspx
(VPGenius.com is gone?)
"In my own case, when it seemed to me after a long illness that death was close at hand, I found no little solace in playing constantly at dice."
-- Girolamo Cardano, 1563
