WOO High Card Flush Calculation Methodology

I have a question for the Wizard regarding the WOO page on the High Card Flush game. There is a calculation of House Advantage under optimal strategy, with a statement that the mathematically optimal strategy is unknown. So was the "HA with optimal strategy" a Monte Carlo calculation? Assuming it was, how were player decisions to FOLD vs Call made? Is modeling a 7 card hand vs a dealer 7 card hand a major production, a la simulations of Pai Gow, or did you use a short cut like ignoring the ranks of the non-flush cards? (or ignoring the ranks and suits of the non-flush cards?)

I realize that most of the High Card Flush situations can be calculated analytically. i.e. player has 4-7card flush. A potentially tricky aspect is realizing that the either (or both) the player and dealer can be dealt a hand with two 3-card flushes (in different suits) and that the higher of the two flushes plays. I'm just wondering if the hands with two 3-card flushes were treated explicitly.

It was done by the computer cycling through all the combinations of the remaining cards and choose the play with the greatest return.

I think it's a misinterpretation -- the optimal strategy is known, just not by any people. It's a line-by-line optimal play for each hand combination rather than a succinct, human-readable strategy. I've got the same issue right now: I have optimal player strategies for two of my games but they're in the form of thousands of rows of spreadsheet, not 10 rows of English.

Quote: MathExtremist

I think it's a misinterpretation -- the optimal strategy is known, just not by any people. It's a line-by-line optimal play for each hand combination rather than a succinct, human-readable strategy. I've got the same issue right now: I have optimal player strategies for two of my games but they're in the form of thousands of rows of spreadsheet, not 10 rows of English.

I understood that. I've actually been cranking through the calculations on a spreadsheet myself. Its just that I've found in the past that the calculations underlying the WOO strategies for less widely played games occasionally have short-cuts which make the calculations "approximate." So, given that this is a 7-card hand vs a 7card hand which involves tens or hundreds of thousands of rows on a spreadsheet, I thought I would ask.

But, I read into Shackleford's response that he (or J.B.) went through all the 7-card combinations for the dealer while randomly selecting hands for the player. That is what I had wanted to know. Thanks, all.

There are 15,584,868 different "unique" player hands (two hands are unique if one cannot be converted to the other simply by switching the suits):

Suits	Hands
7	1716
6,1	22308
5,2	100386
5,1,1	217503
4,3	204490
4,2,1	725010
4,1,1,1	1570855
3,3,1	1063348
3,2,2	1740024
3,2,1,1	3770052
2,2,2,1	6169176
Total	15584868

For each one, there are combin(45,7) = 45,379,620 different possible dealer hands.
That is a total of 707,235,387,590,160 deals.
If you can calculate 100 million of them per second, which would require a really fast computer, that's almost 3 months of work. Of course, if you have multiple computers working on different sets of player hands simultaneously, it can be done faster.
Also, such a brute force method shouldn't be necessary; that is, it shouldn't require you to check all 45 million dealer hands for each player hand.