April 19th, 2016 at 10:47:34 AM
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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.

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.

So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.

April 19th, 2016 at 3:26:26 PM
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It was done by the computer cycling through all the combinations of the remaining cards and choose the play with the greatest return.

It's not whether you win or lose; it's whether or not you had a good bet.

April 19th, 2016 at 3:36:20 PM
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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.

"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

April 19th, 2016 at 8:20:17 PM
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Quote:MathExtremistI 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.

So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.

April 20th, 2016 at 6:51:04 AM
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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):

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.

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.