Evaluating Video Poker Strategies, Part 2½: Dancer and Daily 8/5 Bonus Poker Card, Bob Dancer’s way

Bob Dancer spent a generous amount of time going over my post on the Dancer and Daily 8/5 Bonus Poker strategy card, and kindly wrote a response to it. I feel honored and validated, as my efforts were not wasted. Even if no one else had read my work, I had found a very detail-oriented and meticulous reader, who has spent as much time going over the nuances of various video poker games as any other person may have. This is more than I could have asked for. I cannot be more grateful.

After I posted Mr Dancer’s response, I started to wonder how much of a return a player who had read his post and applied it would get. Imagine you start playing 8/5 or 8/5/35 Bonus Poker using one of the strategies on the card, with the extra explanation that came from his post. I’ll assume you haven’t had the time to go read the book yet, just the strategy card and Bob Dancer’s post.

So this post tries re-analyzing the 8/5 Bonus Poker strategy card, Bob Dancer’s way. Not the way a typical user of the card may read and interpret it, but the way I believe Mr Dancer would like a smart player read the strategies, as explained in his post.

I’m re-analyzing the 8/5 and 8/5/35 Bonus strategies step by step, with the issues I reported in my original post changed to Bob Dancer’s interpretation one by one. Here are the issues:

1. KH vs AH: In my original post, I assumed many players will read the card as saying KH and AH are equal. Now we’ll assume the player always holds KH over AH in hands that have AKH. Before this assumption, the Basic and Advanced strategies had a player choice, which resulted in a range of returns based on which of KH and AH the player may hold. With this clarification, the Basic strategy will now return the upper bound of that range: a fixed 99.1637% in 8/5 and a fixed 99.6589% in 8/5/35. The Advanced strategies will return a fixed 99.164986% for 8/5 and a fixed 99.661171% for 8/5/35.
2. The JT7 typo in QJ vs Jx7: The card has a typo in an Advanced strategy rule comparing QJ to Jx7, where JT7 is misprinted as J97. We’ll assume the player has read Bob Dancer’s response and has corrected the card to say:
           QJ ....... (<JT7 with no 8p; [...])
   If the player fixes this, the Advanced strategies will each return 0.000027% more, 99.165013% for 8/5 Bonus, and 99.661198% for 8/5/35 Bonus.
3. SF3: A-low vs A in 8/5: The card has a rule that says:
           any SF3 -1 < A when sp unsuited with A and no high card other than the A
   The rule, as printed, fails to choose the best option in hands such as A♣ 2♥ 3♣ 4♣ T♥, where the straight penalty is internal to SF3 due to the SF3 being ace-low (the strategy chooses the ace, but the correct hold is A34). I don’t know what exactly someone who has read Mr Dancer’s post would do about this hand, but let’s say they decide to never prefer an ace to an ace-low SF3. If we do that, we get a return of 99.165972% in 8/5, very very close to the computer perfect return of 99.165973%. The strategy will only miss the correct option in 48 hands, worth about 0.000001% (around one in eighty million).
4. AH vs QT in 8/5/35: The card has a rule that never applies if you read it too literally, as I had assumed some players may. After reading Bob Dancer’s post, I am convinced very few players will interpret the strategy that literally. We’ll assume the player has figured out ace penalties to the QT should be ignored (since otherwise the rule would make no sense). If we make this assumption, the Advanced strategy for 8/5/35 will return 0.000129% more, for a total of 99.661327%.
5. SF3 vs A and AH in 8/5/35: This one is a little complicated. The 8/5 strategy on the card has a comment saying “When either AH or KJ is present, it is the relationship of SF3 -1 with AH or KJ that is relevant, not the relationship with A alone or J alone.”
   
   
   
   
   I had assumed that this comment only applied to 8/5, and not 8/5/35. To confirm this, I had checked the book too, which had no such comment in its 8/5/35 strategy. (Mr Dancer assumes in his post that I missed the comment. No way! I had read the cards and their information sheets front to back and back to front several times. The book’s lack of such a comment for 8/5/35 had convinced me that the card’s comment is just for 8/5.)
   
   Anyway, this doesn’t affect the return by much in practice. Mr Dancer implies in his post that he considers the comment to apply to 8/5/35 too. I must take the master at his word. That would improve the return by another 0.000002%, for a final return of 99.661329%.
   

After these clarfications, the 8/5 Bonus Poker Advanced Strategy would only miss 48 hands:

• 4♥ 5♣ 6♣ 8♣ J♠ (and its 23 suit permutations)
• 4♥ 5♣ 7♣ 8♣ J♠ (and its 23 suit permutations)

In all these hands, the correct hold is the jack, but the strategy would recommend the SF3. The strategy card notes that its rules are simplified, and refers players to its companion book, Volume 1: A Winner's Guide to Jacks or Better. (The book covers these hands in its Appendix, and correctly recommends holding the jack.)

In 8/5/35, the Advanced strategy is still shy 0.000016%. Here are the hands that the improved strategy for 8/5/35 misses, ordered by the difference in EV between the correct hold and the strategy’s hold:

• 4♥ 5♣ 6♣ 8♣ J♠ (and its 23 suit permutations)
• 4♥ 5♣ 7♣ 8♣ J♠ (and its 23 suit permutations)
• 2♣ 3♣ 4♣ 6♥ J♥ (and its 11 suit permutations)
• 2♣ 3♣ 5♣ 6♥ J♥ (and its 11 suit permutations)
• 2♣ 4♣ 5♣ 6♥ J♥ (and its 11 suit permutations)
• 2♥ 3♣ 4♣ 6♣ J♥ (and its 11 suit permutations)
• 2♥ 3♣ 5♣ 6♣ J♥ (and its 11 suit permutations)
• 5♥ 6♣ 7♣ 9♣ J♠ (and its 23 suit permutations)
• 3♥ 4♣ 6♣ 7♣ Q♠ (and its 23 suit permutations) 
• 3♥ 4♣ 5♣ 7♣ Q♠ (and its 23 suit permutations)
• 3♣ 5♣ 6♣ 7♥ Q♠ (and its 23 suit permutations)
• 2♥ 3♣ 5♣ 6♣ Q♠ (and its 23 suit permutations)
• 3♣ 4♣ 6♣ 7♥ Q♠ (and its 23 suit permutations)
• 2♥ 3♣ 4♣ 6♣ Q♠ (and its 23 suit permutations)
• 2♣ 4♣ 5♣ 6♥ Q♠ (and its 23 suit permutations)
• 2♣ 3♣ 5♣ 6♥ Q♠ (and its 23 suit permutations)
• 2♣ 3♣ 4♣ 6♥ Q♠ (and its 23 suit permutations)
• 3♥ 4♣ 6♣ 7♣ J♥ (and its 11 suit permutations)
• 3♥ 4♣ 5♣ 7♣ J♥ (and its 11 suit permutations)
• 3♣ 5♣ 6♣ 7♥ J♥ (and its 11 suit permutations)
• 3♣ 4♣ 6♣ 7♥ J♥ (and its 11 suit permutations)
• 5♥ 6♣ 8♣ 9♣ J♠ (and its 23 suit permutations)
• 5♣ 6♣ 8♣ 9♥ J♠ (and its 23 suit permutations)

In all of the hands listed above, the correct hold is the single high card, but the strategy recommends the SF3. The book’s Appendix shows the correct hold for the hands with a jack. All that remains are 216 hands with a queen, and as Bob Dancer acknowledged in his post, he and the late Mr Daily missed them due to the game being too new at the time they were creating the strategies. This rule or some variation of it should cover those hands:

    SF3 -1 with external sp < Q with no sp and no fp

Here are the above numbers, in table form (I was really tempted to draw a chart too, but I was advised against it):

Strategy	Advanced 8/5	Advanced 8/5/35
My previous analysis	99.1557–99.164986%	99.6491–99.661171%
KH > AH	99.164986%	99.661171%
J97 typo fixed to JT7	99.165031%	99.661198%
SF3: A-low > A	99.165972%	N/A
A is not sp to QT	N/A	99.661327%
Only compare AH to SF3	N/A	99.661329%
Perfect play	99.165973%	99.661345%