Ask the Slot Expert: What is a penalty card in video poker?

Question: I have seen some video poker pocket guides that talk about a penalty card. Can you explain what they mean by a penalty card?

Answer: Considering penalty cards in video poker strategy is the perfect example of the 80/20 rule. A simple strategy that does not take penalty cards into account gives you most of the benefits of using a strategy with relatively little effort to learn the strategy. More complicated strategies, that do take penalty cards into account, will get you a little more in long-term payback, but with a great deal more effort.

I first discovered penalty cards when I wrote a program to generate video poker strategies. The program analyzed each pre-draw hand to find the combination of cards with the highest expected value (EV). I wrote a pattern analysis function to describe the held cards in English (e.g., straight, RF4, SF3dih1) mimicking the terminology Dan Paymar uses on his strategy cards. The program then sorted the held cards by expected value, eliminated duplicate description-EV pairs, and displayed the resulting strategy hierarchy.

I tested the program with the 9/6 Jacks or Better paytable. I expected to get a simple list of descriptions and EVs like on Dan's strategy card and in Lenny Frome's books, but I found that one description could have many EVs and sometimes a hand that was lower than another on Dan's and Lenny's strategies was actually higher in my list.

My program generated accurate descriptions for the held cards in the anomalous hands, so the cause of the anomalies must be in the discards. When I looked at the other cards in the hand, I discovered that the cards I was dealt and discarded following the simple strategy sometimes affected the expected value of the cards I was holding and, in some cases, holding a different set of cards had a higher EV than the cards I held.

Here we are at the fifth paragraph and I still haven't said what a penalty card is. I guess I can't put it off any longer.

A penalty card is a card that affects the number of straights or flushes you can make with the other cards in your dealt hand.

An example from 9/6 Jacks. Your hand is three, ten, jack and ace of diamonds and queen of clubs (3d Td Jd Ad Qc). The obvious choices are the four-card flush and the three-card royal.

My (simple) strategy cards have the three-card royal over the four-card flush, so I would hold the three-card royal.

The Video Poker Hand Analyzer at the Wizard of Odds site however says the four-card flush is the better play. The EVs are 1.276 and 1.269 for the four-card flush and three-card royal, respectively.

How does discarding the queen affect the value of the three-card royal? Here's a hint. You get the same EVs if you replace the queen of clubs with the king of clubs.

Out three-card royal consists of a ten and ace with a jack stuck in the middle (with you — a 40+ year-old classic rock reference). The queen (or king) that we discard could have been used to make some straights if it were still in the deck. The queen is a straight penalty card because having it dealt to us and discarding it instead of having it remain in the deck eliminates all of the straights that could have been made with the queen and our three-card royal.

In our example hand, losing the queen eliminated enough straights to drop the expected value of the three-card royal to below that of the four-card flush. You'd rather have that queen in the pack and available to replace a discard than have the queen in the discard pile.

The rule given in the Wizard's optimal strategy for 9/6 Jacks is that a four-card flush beats a three-card royal when the royal contains a 10 and an ace and the unsuited card is a 10 or straight penalty card.

Why a 10? Even though we're concentrating on straights and royals, removing a 10 from the pack decreases the number of two pairs and trip 10s that the three-card royal could lead to and decreases its EV (1.275) to below that of the four-card flush (1.276).

What about a flush penalty card? Consider this hand: three, ten and jack of diamonds, six of hearts and king of spades (3d Td Jd 6h Ks). Should we hold the suited 10-jack or the unsuited jack-king?

This is an interesting hand. Both the Wizard's simple strategy and Dan's strategy say to hold two unsuited high cards over a two-card royal with a ten. The Wizard's intermediate strategy, however, has a suited 10-jack over an unsuited jack-king.

Plugging the hand into the Wizard's Hand Analyzer gives us the better play: hold the two high cards. Removing the three of diamonds from the pack eliminates enough flushes to make the king-jack have a higher EV (0.486) than the ten-jack (0.484).

In this case, the Wizard's simple strategy gave the correct play even though its long-term payback (99.46%) is lower than the intermediate strategy's long-term payback (99.52%). This example illustrates the trade-offs strategy designers make when they create their strategies. You mis-play a number of hands using the Wizard's simple strategy, but you'll misplay different hands when you graduate to the intermediate strategy. The key for the designer is to ensure that the cost of the mistakes in a more difficult strategy is less than the cost of the mistakes in a simplee strategy.

How much can you gain by learning penalty card situations? Well, the Wizard says that knowing the four-card flush versus three-card royal exception in our first example "will add 0.00000021 to the game return." I don't know if we can just add that number to the 99.46% for the simple strategy or we have to multiply it by 100 to get it to a percent before we add it. Either way, there are a lot of zeros to the right of the decimal point in that number.

The Wizard's optimal strategy includes many penalty card considerations. It's long-term payback is 99.54%. Remember that the intermediate strategy, which does not take into account penalty cards, has a long-term payback of 99.52%.

What does that mean in dollars? For every $10,000 you play, on the average you'll lose $48 playing the intermediate strategy and $46 playing the optimal strategy. Remember volatility too. It may take many hundreds of thousands or millions of hands to zero in on those long-term paybacks -- in other words, for the differences in strategy to have a greater effect on your results than luck.

Nearly 20 years ago, I wrote an article about penalty cards for Midwest Gaming and Travel. In the article, I described how I wrote a program to play 9/5 Jacks using the strategy in Lenny Frome's Winning Strategies for Video Poker and compared it with playing mathematically perfectly. Using Lenny's strategy, you misplay 14,928 hands out of the 2,598,960 pre-draw hands. Playing his strategy yielded a payback of 99.52% while playing mathematically perfectly yielded a payback of 99.53%. (I was a little lower than 99.54% due to rounding.)

I shared my results with Lenny. Lenny replied that his "concept of a winning strategy is a useable strategy that gets very close to optimum, not a mathematically perfect strategy. A perfect strategy would have 200+ rankings."

If you think you want to eventually learn penalty situations, get Bob Dancer's strategy cards and learn the simple strategy thoroughly. When you've mastered the simple strategy, move on to the more complicated strategies. You can also get strategies that take penalty cards into account on the Wizard's site.

It's true that the only way to get as close as possible to the maximum long-term payback possible from a paytable is to play as close to mathematically perfect as you can. You'll have to decide if you play enough hands or wager enough money to warrant putting in the extra 80% of effort to gain that 20% of benefit.