Evaluating Video Poker Strategies, Part 2: Dancer and Daily 8/5 Bonus Poker Card

This is the second post in a series evaluating published video poker strategies. The first post discussed the Dancer and Daily Jacks or Better strategy card. It can be found at https://blog.vidpoke.com/2021/08/evaluating-dancer-jacks-or-better-card.html.

The easiest Dancer and Daily card to evaluate next would be their 8/5 Bonus Poker card. The strategies on this card are relatively easy for my software to evaluate, because they are very similar to the Jacks or Better strategies. The game is also pretty important: as of this writing, it appears to be the best video poker paytable available in nineteen different casinos in the Las Vegas area. I have used this card a lot in casino promotions, e.g. for the "earn 1000 points in 24 hours, get $100 in free play" bonus offered at The D casino in Las Vegas using the coupons in the Las Vegas Advisor coupon book and the American Casino Guide. (Hint: most of American Casino Guide's 2020 coupons work for 2021 too. The details are here.)

Dancer and Daily's Beginner and Recreational strategies for 8/5 Bonus Poker are so close to their Jacks or Better strategies that you may at first miss the difference. Then, their Basic strategy for 8/5 Bonus Poker is letter by letter identical to their Basic strategy for 9/5 and 8/5 Jacks or Better. But their advanced strategy is pretty complicated, and it took me some time to code it in. They also provide an Advanced strategy for 8/5/35 Bonus Poker, defining it in terms of its difference with the 8/5 Advanced strategy.

The difference between the strategy cards' Beginner and Recreational strategies for Jacks or Better and 8/5 Bonus Poker only lies in how 3-card straight flushes are treated: In the Beginner strategy, "any 3-card straight flush" is moved below "KQJ all of different suits". In the Recreational strategy, "3-card straight flush with one gap [include 234] and/or one high card [except JT9]" (also known as SF3 -1) is moved below "KQJ all of different suits". That's it. There's no other difference.

The main problem I have with the Beginner strategy is a printing error: all the suits (♠♥♦♣) are missing from the strategy card:

I don't know how many printings have this problem, but this makes the strategy confusing and hard to grasp for a true beginner. Other than this, there is nothing wrong with the strategy itself: it's a pretty good beginner strategy.

I'm assuming you have read the first post in this series (No? Go read it now!). So you'd know about the issue of AH vs KH hands. The Beginner strategy on the 8/5 Bonus card has the same ambiguity. It tells the player to hold "two unsuited high cards" and doesn't say which one is preferred if more than one set exists in a hand. This makes the return of the strategy a range of numbers instead of a single hard number. If the player always holds the ace with one of the other high cards, they get the lower end of that range. If the player always discards the ace, they get the upper end. If they sometimes hold the ace and some other times discard it, their expected return is somewhere in the middle. But this can be assumed a reasonable sacrifice to keep a beginner strategy that has a pretty good return otherwise, simple.

A similar thing happens with the Recreational strategy. The body of the strategy itself doesn't say which of AH or KH is preferred. But fortunately there's a small footnote at the bottom of the strategy that says "When dealt three high cards of different suits including an ace, discard the ace." Because of this, for the sake of these posts the Recreational strategy is not considered ambiguous. So an exact return can be computed for the Recreational strategy.

Unfortunately, the ambiguity comes back in the higher-level Basic and Advanced strategies. There is no mention of KH being preferred to AH in these strategies. Indeed, the Basic strategy is letter-by-letter identical to the Jacks or Better Basic 9/5 and 8/5 strategy and has the same problematic "KH, AH" strategy line:

While this is fine at a Beginner or Recreational level, I think this should have been caught and corrected for the Basic and Advanced strategies. The authors could have simply written "KH > AH" and all would have been well. But the way the card is printed, the Advanced strategy is far from computer perfect, despite the hard work the authors have put in figuring out many of the edge cases.

(Interestingly, the strategy generated by the Video Poker by Winners software for 8/5 Bonus Poker has a very similar problem! It lists "AH; KQ" in a single line, while KQ is definitely more valuable than AH and they can indeed appear in the same dealt hand.)

As for all the exceptions mentioned in the 8/5 Advanced strategy, unfortunately they still don't make the strategy computer perfect. The strategy card itself admits that it has a simplification when comparing SF3 -1 to a jack. When comparing SF3 -1 to a Jack, it says:

2-low, 3-low, or 4-low < J when sp unsuited with J*

It adds a footnote clarifying that "This is a simplification of a complex relationship. For precise conditions, see Appendix to Dancer/Daily Winner's Guide to Jacks or Better."

My software found quite a few hands where the Advanced strategy doesn't mention an exception or mentions it incompletely (or incorrectly?). Here are some examples:

• A♣ 2♥ 3♣ 4♣ T♥: Correct hold is A♣ 3♣ 4♣ (SF3 -1), but the strategy card recommends A♣, since there is a straight penalty (2♥) unsuited with the ace and no high cards other than the ace. (The book's strategy is identical here. In the detailed explanations for the strategy, the book claims "the straight penalty must be external to the SF3 -1 to avoid creating a more valuable 4-card open-ended straight”. In this case, that 4-card straight, A♣ 2♥ 3♣ 4♣, is never held, since it's still an inside straight with just one high card. I believe that "must" is a part of the analysis, not the strategy, so I don't consider the book's strategy to be correct in this case either.)
• Q♥ J♣ 10♣ 7♣ 6♥: Correct hold is J♣ 10♣ 7♣ (SF3 -1) but the card recommends Q♥ J♣. This is probably because of a typo in the card, since the book's strategy is different.
• Q♥ J♣ 9♣ 7♣ 6♥: Correct hold is Q♥ J♣ but the card recommends J♣ 9♣ 7♣ (SF3 -1), since there is no 8-penalty. This is probably because of a typo in the card, since the book's strategy is different. Here is what the card says:
  
  
  
• J♠ 8♣ 6♣ 5♣ 4♥: Correct hold is J♠ but the card recommends 8♣ 6♣ 5♣ (SF3 -1). This is one of the simplifications the authors refer to when comparing SF3 -1 to a jack. The table in the book's appendix mentions the correct hold for this case.

Similar hands exist for the 8/5/35 Advanced strategy. Here are some example hands where the Advanced strategy specified for the 8/5/35 game is different from the best cards to hold:

• J♠ 8♣ 6♣ 5♣ 4♥: Correct hold is J♠ but the card recommends 8♣ 6♣ 5♣ (SF3 -1). This is one of the simplifications the authors refer to when comparing SF3 -1 to a jack. The table in the book's appendix mentions the correct hold for this case.
• Q♥ J♣ 9♣ 7♣ 6♥: Correct hold is Q♥ J♣ but the card recommends J♣ 9♣ 7♣ (SF3 -1), since there is no 8-penalty. This is probably because of a typo in the card, since the book's strategy is different.
• A♣ Q♥ 10♥ 7♣ 6♠: Correct hold is Q♥ 10♥ but the card seems to imply that A♣ Q♥ may be the correct hold. It says: "AH < QT with neither fp nor sp" but since the ace is a straight penalty to the QT, the exception can never hold:
  
  The detailed explanation in the book mentions that such a straight penalty can only be a 9 or an 8. The authors probably had every intention of excluding the ace as a potential straight penalty to QT, but literal interpretation of the card would consider the ace a straight penalty. The strategy should probably say something like “AH < QT with no 8p, 9p, or fp”. (I wondered if a “penalty card” is defined as any card that is not mentioned in any of the options. But the information sheet that comes with the card says: “A penalty card for one option may also be part of an alternative option”.)
• J♥ 6♣ 5♣ 3♣ 2♥: Correct hold is J♥ but the card recommends 6♣ 5♣ 3♣ (SF3 -1). This is one of the simplifications the authors refer to when comparing SF3 -1 to a jack. The table in the book's appendix mentions the correct hold for this case.
• A♥ J♠ 5♣ 4♣ 2♣: Correct hold is 5♣ 4♣ 2♣ but the card recommends J♠, since the ace is a straight penalty unsuited with the jack. This is one of the simplifications the authors refer to when comparing SF3 -1 to a jack. The table in the book's appendix mentions the correct hold for this case.
• Q♠ 7♣ 6♣ 4♣ 3♥: Correct hold is Q♠ but the card recommends 7♣ 6♣ 4♣ (SF3 -1). The book is also incorrect here. Neither the card nor the book mention that a queen could ever be better to hold than SF3 -1.

Here are the returns I have arrived at for the 8/5 Bonus Poker strategy card:

Strategy	8/5	8/5/35
Beginner	99.0597–99.0831%	99.5504–99.5791%
Recreational	99.1617%	99.6565%
Basic	99.1545–99.1637%	99.6468–99.6589%
Advanced	99.1557–99.1650%	99.6491–99.6612%
Perfect play	99.1660%	99.6613%

The return of the various strategies is not mentioned on the card or its accompanying information sheet. The authors' Winner's Guide series, which covers 8/5 Bonus Poker together with Jacks or Better flush-5 games in the later part of Volume 1: A Winner's Guide to Jacks or Better, does not mention the return of the various 8/5 Bonus Poker strategies either. Although there is some language implying that the authors thought the Advanced strategies to be computer perfect or close to it. (Note that there are some differences between the strategies published in the book and those published on the card.)

Conclusions: The Beginner and Recreational strategies are pretty good simple strategies and only leave a little more on the table than perfect play (0.08–0.11% for the Beginner strategy and about 0.004% for the Recreational strategy). For the Basic and Advanced strategies, which I assume most of the readers of this post would be using, the card should be corrected to say "KH > AH". With that correction, the Basic strategy would only leave an extra 0.002% on the machine. The Advanced strategies need some fixes to the exceptions, apart from the AH vs KH issue. But after correcting "KH, AH" to "KH > AH", the 8/5 strategy would only differ from computer perfect by 0.0010% and the 8/5/35 strategy only by 0.0002%.

(I've tried my best to be as accurate as possible in my analysis. I would appreciate any suggested corrections to this post, and will try my best to fix any errors reported.)

(Update: Bob Dancer has generously written a response to this post, which you can find here. I have posted an updated analysis based on his feedback.)