## Thursday, December 26, 2013

### Distinguished Royals

I've spent the past few weeks walking through the strategy table for full-pay jacks or better video poker.  This week, we are in the heart of the table, which is to say we are in the middle of the really bad hands.  But, bad hands are the harder hands to play correctly and are just as important as the good hands in achieving the theoretical strategy of any particular game.  We finished up last week with the 4-Card Straights.  From here the hands only get uglier.  The next several entries are:

·       3-Card Double Inside Straight Flush with 2 High Cards
·       3-Card Inside Straight Flush with 1 High Card
·       3-Card Straight Flush with 0 High Cards
·       2-Card Royal Flush - "V3"

If you're relatively new to strategy tables, this part of the table has problem just given you a doozy of a headache!  The good news is that it is not nearly as confusing as it might appear.  Many of the hands listed above cannot co-exist - meaning that you can't have more than one in a particular hand.  Thus, remembering the exact order may not be as important as it might appear to be.  For example, you can't have a 3-Card Double Inside Straight Flush with 2 High Cards in the same hand as a 3-Card Straight with 0 High Cards.  You'd either have to have at least 4 cards of one suit or have 2 sets of 3-Cards of different suits - rather difficult with 5 cards.

There are still important things we can learn from this section of the table.  We separate the hands the way we do because in some games the impact of the subtle differences and the order of the hands will be different and thus become pertinent.  The first thing you might notice is the relationship of the top three hands.  We get a sense of the importance of a High Card.  A 3-Card Double Inside Straight Flush with 2 High Cards has a higher expected value than a 3-Card Inside Straight Flush with only 1 High Card.  Essentially, the value of the extra High Card is greater than the value of the additional Straight Flushes (and Straights) that may occur as a result of having an Open vs. Inside vs. Double Inside Straight Flush.  What this should also tell us is that you shouldn't hold your breath for those Straight Flushes.  They will occur, but not often.  At the same time, I have written at length over the years about how the Straight Flush is the forgotten hand of video poker.  Playing 3-Card Straight Flushes correctly is very important to drawing them in proper abundance.  While their pays are far short of the Royal, they still pay double what Quads pays so their value should not be dismissed.

Next up is the 2-Card Royal Flush - "V3".  2-Card Royals are given 4 different designation from "V0" to "V3".   We need to do this because the expected values of many of the hands in this part of the table differ by only 0.01 or 0.02.  As not all 2-Card Royals have identical expected values, we need to distinguish between them.  V3 means that the 2-Card Royal contains neither a 10 nor an Ace.  An Ace in a 2-Card Royal essentially makes it a Double Inside Royal.  All 2-Card Royals have the same number of ways to make a Royal - one.  But, with the Ace, we eliminate all ways to make a Straight Flush.  While a '10' doesn't have this problem, it does have the problem that it is not a High Card.  So, Aces are worth less than Jacks, Queens and Kings and 10's are worth less than Aces.  Thus, a 2-Card Royal with neither an Ace nor a '10' is the one with the highest expected value.  A V2 2-Card Royal means that the 2-Card Royal has an Ace, but no 10.  A V1 2-Card Royal is one that has no Ace, but does have a 10.  Lastly, a V0 2-Card Royal consists of a 10 AND an Ace.

For the moment, I'll jump to below the strategy table - to the V0 2-Card Royal.  We are 'below' the strategy table because this hand does not exist on the strategy table for full-pay jacks or better.  This means that we DO NOT PLAY an A-10 2-Card Royal.  Barring the other three cards forming an otherwise playable hand, we would simply hold the single Ace.  More on that in a couple of weeks.

The proper play of 2-Card Royals is critical to learning how to master video poker strategy.  Unlike the prior 3 hands, 2-Card Royals overlap with EVERYTHING.  You'll have 2-Card Royals with High Pairs, Low Pairs, 4-Card Straights, 4-Card Flushes, 3-Card Straight Flushes, 3-Card Inside Straight Flushes, etc...   If you blindly go after every 2-Card Royal, you'll hit more than your fair share of Royals, but you'll lower the payback of your play.  If you ignore 2-Card Royals, you'll miss you fair share of Royals AND lower the payback of your play.  The only answer is to play them when you are supposed to.  Based on the portion of the strategy table shown, you can easily have a 3-Card Double Inside Straight Flush with 2-High Cards AND a 2-Card Royal - V3.  For example, you could have 8-J-Q suited.  From the strategy table, we learn that we keep the 8 in this case.  You might also have a 3-card Straight Flush completely apart from a 2-card Royal.  For example, 3H, 4H, 5H, JD, QD.  In this case, we discard the 2-Card Royal in favor of the 3-Card Straight Flush with 0 High Cards.

I've now covered about 2/3 of the strategy table for jacks or better video poker.  Next week, we continue through the rest of the messy hands.

## Thursday, December 19, 2013

### Stroll Through the Strategy Table

This week we continue our walk through a video poker strategy table.  Specifically, the strategy table for full-pay jacks or better video poker.  Last week we left off at the 4-Card Flush which was the last of the hands with an expected value of greater than 1.0.  These are the hands that result in net wins in the long run.  The rest of the strategy table have expected values below 1.0.  This means that in the long run we will not get back our entire wager.  But, that doesn't make them any less important.  When playing video poker, playing every hand correctly is critical if you want to achieve the theoretical payback.

It could be argued that playing the hands below 1.0 correctly is more important than playing the ones above this line correctly.  First, the hands below make up the lion's share of hands.  Second, the hands below are by far more complex than the ones above.  You don't have to worry about confusing a Two-Pair with a 4-Card Straight Flush as this is an impossibility.  But a 4-Card Straight vs. a 3-Card Straight Flush might leave you shaking your head.

Without further ado, here are the next batch of hands on our strategy table:
·       4-Card Straight with 3 High Cards
·       Low Pair
·       4-Card Straight with 2 High Cards
·       4-Card Straight with 1 High Card
·       3-Card Inside Straight Flush with 2 High Cards
·       3-Card Straight Flush with 1 High Card
·       4-Card Straight with 0 High Cards

The first thing you'll probably notice is that with the exception of the Low Pair, the number of High Cards is specified.  When the inventor of video poker decided to pay on Jacks or better, he added an incredible layer of complexity to the strategy.  Simply put, in any hand without a Pair or better, any card that is a Jack or higher is worth considerably more than any other card.  The reason should be fairly obvious.  We have the opportunity to win with High Pairs.  For each High Card in the hand, we have three additional cards that we can draw that will turn our hand into a winner.  These three cards add just over 0.06 to the expected value of the hand.

Sometimes, this 0.06 means nothing and sometimes it means everything.  We just need to look at the first three hands to see the impact.  If you have a 4-Card Straight with 3 High Cards and a Low Pair, you keep the partial Straight.  If you have a 4-Card Straight with 2 High Cards and a Low Pair, you keep the Lower Pair.  These hands are not very common, but they illustrate the impact of the High Card.   So, if you have 10-10-J-Q-K (assuming no 3-Card Royal), then you keep the Straight.  If you have 9-10-10-J-Q then you keep the Low Pair.  Having a 9 instead of a King lowers the expected value so that it falls just below that of the Low Pair.

If we keep moving down the chart, we find that the next entry is the 4-Card Straight with 1 High Card.  As this is adjacent to the 2 High Card version, there really is no impact in this case.  We could lump these two hands together if we want to remove 1 hand from the strategy table.  We keep them separate because there are versions of video poker where it is relevant and we want to make sure the Player doesn't get 'lazy'.

In between a 4-Card Straight with 1 High Card and a 4-Card Straight with 0 High Cards we find 2 other hands.  They are both variants of a 3-Card Straight Flush.  The first is an Inside Straight Flush with 2 High Cards and the second is a 3-Card Straight Flush with 1 High Card.  This can start looking more confusing than it really is.  Most of these hands CANNOT occur in a single hand.  It is not possible to have both a 3-Card Straight Flush with 1 High Card and a 3-Card Inside Straight Flush with 2 High Cards.   But, you can have a 4-Card Straight with 1 High Card with a 3-Card Straight Flush with 1 High Card.  (8C 9D 10D JD 4H).  We learn from the strategy table that the right play is the 4-Card Straight.

One last point that I should mention.  All the 4-Card Straights to this point have been Open Straight Flushes - meaning that they can be completed on both ends.   This means that as we have completed about 60% of the strategy table, we have accounted for all Pairs and for all 4-Card Straights (Open ended) and 4-Card Flushes.  The remaining 40% of the strategy tables contains very 'not pretty' hands.  It is a mish-mosh of 3-Card Straight Flushes, Inside Straight Flushes and even Double Inside Straight Flushes, along with 2-Card Royals and hands with just High Cards.  To make matters worse, these 40% of the entries account for nearly 50% of the hands.

Next week, we'll continue our stroll through the strategy table.

## Thursday, December 12, 2013

### Get the Inside Scoop

Last week I began reviewing the strategy table for full-pay jacks or better video poker.  I got about 20% of the way through the table by volume, but not very far in terms of useful information.  The top 8 hands were mostly of the no-brainer category as they were the pat hands with the exception of the 4-Card Royal.  This week, I'll keep moving down the table and provide some insight into the nuances of video poker strategy.  Please remember that this particular strategy is applicable ONLY to full-pay jacks or better.

After a Straight, we find the following entries on our table:

·       4-Card Straight Flush
·       Two Pair
·       4-Card Inside Straight Flush
·       High Pair
·       3-Card Royal Flush
·       4-Card Flush

The first thing you might notice about the above entries is that we have two entries for a 4-Card Straight Flush and a 4-Card Inside Straight Flush.  There is a big difference between the expected values for Straights that are open and those that are Inside (or Double Inside).  The common definition of Inside Straight is when the opening is in the middle and not on the ends (i.e. 5-6-7-9).  However, this leaves off some Inside Straights.  It is more accurate to define a 4-Card  Inside Straight as one that can only be filled ONE WAY.  So, an A-2-3-4 can only be filled with a 5 and thus is an Inside Straight.   With this definition you can see that an Inside Straight can be completed with only 4 cards while a regular Straight can be completed with 8 cards.  Straight Flushes are no different - except they have the possibility of being turned into Flushes as well.

In this particular case, there is really no benefit to splitting out the 4-Card Straight Flushes.  The one hand that lies between them can't possibly be a 4-Card Straight Flush (Inside or not).  We show them separately because in some version of video poker, the hands that appear in between may be able to overlap with them and we will find that in some cases we will want to keep a 4-Card Straight Flush ONLY if it is not an Inside Straight Flush.  Also, as we will see as we move down the table, this distinction becomes very important as we take a closer look at 4-Card Straights.

The 4th entry on the table is a critical one - High Pair.  It is the 4th most common hand.  Thus, playing it correctly is very important.  Looking at the entries above it and below it what we learn is that a High Pair is played OVER any 4-Card Straights and 4-Card Flushes.  We will, however, play all 4-Card Straight Flushes over a High Pair.  But, we will NOT play a 3-Card Royal over the High Pair.  So, if you have a suited J-Q-K along with another Queen, you stick with the sure winner - the Pair of Queens.

Below High Pair, we have a 3-Card Royal Flush and a 4-Card Flush.  There is much to learn here as well.  The most obvious is that if you have a 3-Card Royal and a 4-Card Flush, we hold the 3-Card Royal.  This can be a tough choice because the likelihood of hitting the Royal is still relatively small.  But, by holding a 3-Card Royal we give ourselves more chances for a Straight.  We might still hit a Flush and we have the longshot at the Royal.  Also, with a 3-Card Royal, we leave ourselves 2-3 cards that can be matched up for a High Pair.  The expected values are not really all that close with a 1.41 for the 3-Card Royal and 1.22 for the 4-Card Flush.  The decision is relatively clear.

From these entries we also learn that if the Player has a 3-Card Royal that is also a 4-Card Straight Flush (8-10-J-Q), we hold the 4-Card Straight Flush.  With the 4-Card Straight Flush, we still have many chances for Straights and Flushes so we don't throw away the extra card even if it gives us a chance to get the Royal.

I've stopped at this particular point in the Strategy Table because the 14 hands I've listed (over the past 2 weeks) are the only ones with an expected value greater than 1.0.  That means these hands are net winners in the long run.  Some will be winners 100% of the time.  Some will not.  But in the long run, we can expect to get more back than we wagered.  These hands make up about 40% of the table and about 25% of the total hands dealt.  Beginning next week, we'll review the hands with an expected value below 1.0.  Even though these are losers in the long run, it doesn't make them less important.  In fact, they may be more important because they account for a larger percentage of hands dealt.

## Thursday, December 5, 2013

### Strategize

Every casino game that is more than pure luck has some strategy associated with it.  This goes beyond the basic strategy that simply says you're better off not playing at all.  For many games, the strategy can be summed up with a simple sentence or two.  For Three Card Poker, it is Play Q-6-4 or better.  Four Card Poker has a two sentence strategy that tells you when to fold and when to Raise.  Let It Ride's strategy takes a few sentences telling you when to pull down the 1 and 2 wagers.

As strategy gets more complex, it is helpful to try and put it into as easy as a format as possible to help a mere mortal to utilize it.  It is relatively easy to program a computer to play a game perfectly.   Very few humans can take every game to this level.  Also, expending that much energy on memorizing a very complex strategy can pretty much sap the fun right out of the game.  Blackjack utilizes a relatively simple matrix that crosses the Player's hand with the Dealer's upcard.

Creating a strategy for video poker is quite a challenge.  As said earlier, telling a computer which one of the 32 ways to play a hand is relatively easy.  But, there are 2,598,960 unique 5-card deals from a standard deck.  Coming up with a way to group these together in a way that a Player can use is a whole different story.  I believe it was my father, Lenny Frome, who was the first person who accomplished this.  He grouped hands together in a way that Players could easily understand and hopefully memorize.

A video poker strategy table consists of only two columns.  The first contains the hand rank as it was categorized by my father.  The second contains the expected value of the hand.  Ironically, this second column isn't even needed to play video poker properly.  It is there just for reference.  So, that means the video poker strategy table consists of a single column - usually with about 30-40 rows/entries in it.  To play video poker the correct way, you have to memorize the order of these entries.  This is not nearly as daunting as it seems.  About 10-15 of these entries are more than a little obvious.  So, you're left with about 25 hand types that you need to learn.

Let's start at the top of the strategy table which contain the most obvious hands:

·       Royal Flush
·       Straight Flush
·       Four of a Kind
·       4- Card Royal
·       Full House
·       Flush
·       Three of a Kind
·       Straight

We'd be having a great night at video poker if these were the only hands we were dealt.  These are all big winners, all with expected values of 4.00 or better.  In fact, only one of these hands is not a sure winner - the 4-Card Royal.  This is also the only hand that might overlap with any of the others, creating the only strategy decision in the bunch.  What do you do if you are dealt a Straight (or a Flush) that is also a 4-card Royal?  Well, now you know the answer.  You have to throw away the sure winner to go for the big winner.  The good news is that if you have a 4-Card Royal, you have a very good chance of still winding up a winner.  There are 47 possible draws, 1 of which will result in the Royal.  Another will give you a Straight Flush.  6 or 7 more (depending on whether you threw away a Straight or Flush) will result in a Flush.  5 or 6 will result in a Straight and a host more will give you at least a High Pair which will seem like small consolation.

While this decision might be agonizing, mathematically, it is very clearly the proper play.  The expected value of the 4-Card Royal is 18.66.  The expected value of the Flush is 6 and the Straight is 4.  Of course, don't expect to see this hand every hour.  A 4-Card Royal will show up once in about 2700 hands and only about a third of these will be a Straight or a Flush.   One other key point to note.  Do NOT throw away a Straight Flush to go for the Royal.  That Straight Flush has an expected value of 50 which far exceeds the 18+ of the 4-Card Royal.

Next week, I'll move down the strategy table to the hands that require a bit more thought.