Evaluating Video Poker Strategies, Part 1: Dancer and Daily Jacks or Better Card

The first strategy we used when playing video poker was something I had written down on a piece of paper from the Wizard of Odds website. I think it was his simple Jacks or Better 9/6 strategy. My spouse and I had found a Jacks or Better 9/6 machine in the New York-New York casino in Las Vegas and were very pleased with ourselves. It was a negative play, since the machine didn't even take a player's card, but we were nevertheless enjoying it a lot!

In the years since, we first graduated to the Wizard's more advanced strategies, and after many overused printed strategies torn to pieces, we ended up buying Bob Dancer and Liam W. Daily's strategy cards which we've learned to love dearly. If we're in a casino, we are packing them.

A few years ago, I was playing NSUD using a Dancer and Daily strategy card in the high-limit room of the GSR casino in Reno, when I met one of the first professional gamblers I had ever met. He was playing maximum coin-in on the machine next to me to maximize his entries in a drawing, and he was kind enough to give me a few pointers on the challenges of the journey.

I don't remember his name unfortunately, but one thing he said (which at my peril I outright dismissed) was not to trust the published strategies 100%. He said he had found mistakes in them, but didn't remember any specific ones at that time. When I asked him how he makes his strategy decisions then, he told me he uses a hand analyzer (such as the Wizard's own) and determines his own strategy by going through all the various hands. That felt too overwhelming for me to even consider.

Fast forward to 2021, I found some interesting differences in the basic strategies generated by Video Poker for Winners and the Wizard's Video Poker Strategy Maker, so I started on a quest to figure out which is better and by how much. Each of them publish their numbers claiming they return a specific percentage, but are the numbers correct? Are their strategies the best basic strategy one could come up with? What about the returns of various Dancer and Daily strategies? How much are those?

I have a background in computer science, so it was time to write some code. The piece of software I wrote analyzes a given strategy for a given paytable and tries to figure out how good the strategy is by figuring out its return and comparing it to the return of the perfect play with that paytable. (It also outputs all the hands where the strategy is different from perfect play, so I can double check to make sure the software has understood the strategy correctly. I can also use that output to figure out how the strategy could be improved.)

(I posted some early findings on how to improve the Wizard's generated basic strategy for Jacks or Better 9/5 on the Wizard of Vegas forum here. Unfortunately it looks like the issues are not easily fixable because the original developer has left the effort.)

I plan to write a series of posts on the strategy returns and potential fixes to the strategies, and I'm starting with Dancer and Daily's Jacks or Better strategy card, which I assume is their most popular card.

The strategy cards are a little unclear sometimes. So whenever I run into ambiguities, I try my best to resolve the problem by looking elsewhere on the card where there may be further information. I also read the information sheet that comes with the cards thoroughly, to make sure I'm not missing anything. Although the cards refer to the authors' Winners Guide series of books too, I'm assuming most users of the cards have not read the books and will never read them, so I'm intentionally avoiding using the books to resolve the ambiguities.

Here are the main concerns I have with the Jacks or Bettter strategy card:

1. At least in my copy, the Basic 9/6 and 8/6 strategy has a typo, implying HIGH PAIR only means JJ-KK instead of JJ-AA by saying:

   HIGH PAIR [JJ-KK]

   
   In my analysis I have assumed that the card's users would figure out that it's a typo. If they don't, and thus never hold just two aces, they're in for an expensive play. They'd be giving back an extra 3.4% to the casino. (For the exact numbers, see the table below.)

2. The main source of ambiguity in the strategies is the comparison between KH and QJ hands on the one hand and AH hands on the other. In all Jacks or Better variations covered by the card, KH and QJ are better to hold than AH because they provide more opportunities for a straight, but some of the strategies are either not clear about the choice or leave the choice to the player.

   In the Beginner and Recreational strategies, the main body of the strategy just asks to hold "two unsuited high cards" and doesn't mention what to do if two or more sets of two unsuited high cards are in the dealt hand (which would happen if the machine deals an AHH). Fortunately, there is a small footnote in the Recreational strategy implying KH and QJ being preferred over AH by saying: "When dealt three high cards of different suits including an ace, discard the ace."

   

   But unfortunately, the choice between AH and KH is not mentioned in the Basic strategy for the 9/5 and 8/5 games. There's a line in the strategy saying "KH, AH", creating the illusion that perhaps in the flush-5 games these holds are equal (or never occur together without another higher hold), but they are not equal.

   

   (In the footnotes of that strategy, we see "Prefer HH to AHH", but there's no indication of HH being preferred to AH).

   For the returns I've calculated (see the table below), when the strategy card is not clear on which cards to hold when dealt AHH, I'm providing a range of returns. The lower end is for a strategy when AH is always held, and the upper end is for when the non-Ace cards are always held. I assume some players may vary the pair of cards they hold based on a hunch or other cards in the dealt hand, so their expected return would fall somewhere in the middle of that range.

3. The exceptions in the Basic strategy for 9/5 and 8/5 look like an advanced strategy, where a hold is to some degree determined by other cards dealt (as opposed to just the combinations to be held). For example, three consecutive lines read:

   AKQJ (always < QJ8)
   KH, QJ (always < KH9, QJ8)
   SF3 +0

   But of course, what is considered “basic” is quite subjective, and the authors are allowed to have their own definitions. Nevertheless, it should be possible to recreate an alternative basic strategy with a similar return without using such exceptions. (I'll gladly continue to use the exceptions though! They add about 0.0045% to the return in 9/5 and about 0.0047% in 8/5.)

Without further ado, here are the returns I arrived at:

Strategy	9/6	8/6	9/5	8/5
Beginner	99.4462–99.4748%	98.2952–98.3238%	98.3396–98.3682%	97.1886–97.2172%
Recreational	99.5375%	98.3862%	98.4449%	97.2936%
Basic with the typo	96.0976%	94.9775%	N/A	N/A
Basic	99.5429%	98.3917%	98.4356–98.4476%	97.2843–97.2964%
Advanced	99.5439%	98.3927%	N/A	N/A
Perfect play	99.5439%	98.3927%	98.4498%	97.2984%

Curiously, Dancer and Daily, in their Winner's Guide books, assume their published Beginner and Recreational strategies are worse than they actually are. Here is a comparison table of the returns they claim, compared to the actual calculated returns:

Strategy	9/6 Claimed	9/6 Actual
Beginner	~99.2%	99.4462–99.4748%
Recreational	~99.4%	99.5375%
Basic	>99.5%	99.5429%
Advanced	perfect	perfect (99.5439%)

(I should note that while their numbers undervalue some of the actual returns, the strategies published in their Volume 1: A Winner's Guide to Jacks or Better don't have the major issues I mention above. In the book, they have a footnote under their Beginner strategy saying "when dealt three high cards of different suits including an ace, discard the ace" and in their Basic 9/5 and 8/5 strategy they list AH below KH in a separate line. The book's Basic 9/6 and 8/6 strategy doesn't have the JJ-KK typo either.)

Conclusions? I'll continue to carry and use their Jacks or Better card, correcting the "KH, AH" line in the Basic 9/5 and 8/5 strategy to "KH > AH". I'll know that their (corrected) Basic 9/5 and 8/5 strategy is good enough if I need to play those games for a promotion, leaving only 0.002% on the table. But if I recommend the card to a beginner, I'll make sure to tell them about the AA typo in the Basic 9/6 and 8/6 strategy. I've also become quite confident that the Dancer and Daily's Advanced strategy for 9/6 and 8/6 is indeed perfect.

(I would appreciate any suggested corrections to this post, and will try to fix any errors I would find.)