Thursday, November 14, 2013

Table your Hunches

            Last week, I described how all casino game strategy is based on expected values.   You hit or stick in blackjack not because you hope the next card is of a certain value, but because there are certain probabilities as to what the next card will be and how it will affect your hand and your chances of winning or losing.   If you're dealt two face cards, you don't give much thought to strategy.  Hopefully, you're not one of those Players who even thinks about splitting 10's! 

            But, if you are dealt a 16 and the Dealer has a 7, you start giving thought to the strategy.  With a 16, you have 5 cards that will help you and 8 that will bust you.  The odds don't look to good and this is why a lot of people stick on this hand, albeit incorrectly.  You can stay put, but with a 16, the only way you can win is if the Dealer busts, which will happen only 26% of the time.  So, your choices are a 61% chance of busting right away or sticking and having a 74% chance of losing that way.  Of course, by hitting you also have an opportunity improve your hand.  All of the 5 possibilities improve your hand.  If you pick up an Ace, you'll be most likely to push.  Pick up a 5 and you'll win more than 92% of the time.  Don't get me wrong, it is not a strong hand and the decision to hit is not an overwhelming one, but it is still the right move.  In the simplest form, if you face this situation enough times - which you will if you play for a few hundred hours, you'll find that you do better by hitting than by sticking.

            In blackjack, you don't have to memorize all of the math behind the game.  You don't have to figure out how many cards will bust you or bust the Dealer.  To learn to play blackjack, many Players use a simple strategy table.  It is a simple little chart that shows every possible Player hand and each possible dealer upcard.  It then shows what to do - hit, stick, double, split, surrender, etc..  Guys like me have already done all the number crunching for you.

            Video poker is no different than blackjack except the decision making process is far more complex.  In blackjack, the result is essentially binary - you win or you lose (okay, you can tie also, so it is not really binary).  In video poker, you can have 1 of many results - ranging from a Royal Flush down to a High Pair or you can lose.  Since each of the different winning hands pays a different amount, the specific result must be taken into account.  If someone invented a game of video poker in which all hands above a certain rank paid a fixed amount, then we'd be able to lump all the hands into win or lose.  But, we need to know the probability of each final outcome with a different payout in order to appropriately determine the value of getting that hand.  Surely, it is more valuable to wind up with a Straight Flush than just a Straight.

            Video poker is also more complex than blackjack in that there is more than just a handful of different possibilities for each hand.  The Player can hold all 5 cards or discard all 5 cards or anything in between for 32 different possible plays.  Yes, most of these possibilities will be quickly discarded, but they still must be considered from a mathematical perspective.  They are only discarded because the human mind can quickly recognize possible draws that would clearly not be the best strategy. 

            Despite the extra complexity of video poker, the similarities are still stronger than the differences.  In the end the decision still comes down to the expected value.  Like in blackjack, you don't have sit there trying to figure out how many cards you need to complete a Straight or the like.  Again, guys like me have already done the job.  We have looked at every possible deal, every possible draw for every possible deal and summed up all of the final hands.  Using this distribution, each possible draw is assigned an expected value.  Whichever draw has the highest expected value is deemed the right play.  The last step in the process is too try and categorize the way each hand is played into a format that a human can use to play the hands.  We call this a strategy table.

            Unlike blackjack where the strategy is a matrix that crosses Player hands with Dealer hands and tells you what to do, a video poker strategy chart lists all the possible playable hands in order in a simple table.  The table usually contains the expected value of each hand too, but this is just for information.  To use the strategy table, you basically work from the top and find the first hand that your dealt hand can make and that is the way to play the hand.  So, if you are dealt a hand that is a 4-Card Straight and a Low Pair, you start at the top of the table and work downward.  If a 4-Card Straight appears first, you play that.  If a Low Pair appears first, you play the hand that way.  If you can't find any hand that matches the hand you were dealt, then you fall to the bottom of the table and find a RAZGU which means throw all five cards.

            Next week, we'll begin breaking down a strategy table for full-pay jacks or better.  You'll be on your way to becoming an Expert Player.

Thursday, November 7, 2013

The Advantage of Expert Play

            This morning, I had a discussion with a friend of mine about a game he is developing.  I explained that playing 'perfect' strategy would be nearly impossible due to some subtle complexities of the way the game is played.  As a result of this, the game would not likely play anywhere near its 'theoretical' payback.  Many games have this 'problem'.  Blackjack pays 99.5%, but very few players play anywhere near this.  Ultimate Texas Hold'em has a payback well into the 99% range too, but stats from the casinos make it clear that very few Players, if any, can manage this high of a payback.

            My friend stated that he thought that he would be able to play the game close to the theoretical because he is an accomplished Poker Player.   I asked him if he was an accomplished video poker Player and he said that he wasn't.  I told him that any table game against a Dealer was really nothing more than playing video poker and had no resemblance to poker even if the game resembles poker.   Poker is about reading Players, understanding their betting patterns and their tells.  Video Poker is about one thing - math.   There is no one to bluff.  All that matters is what is the probability of all final hands given what I choose to discard.   Let's take a look at a simple example:

5♠        5♦        6♣       7♥        8♦

            In theory, there are 32 ways to play this hand, but I think we can quickly rule out 29 of them.  I don't think anyone is seriously going to consider holding only the off-suit 6-8 or holding all 5 cards (which would result in an immediate loss).    There are really on 3 possibilities, 2 of which are identical.  The Player can either hold the Pair of 5's or the 4-Card Straight (hence, the 2 identical possibilities as it doesn't matter which 5 the Player keeps.)

            If the Player keeps the 4-Card Straight, 8 cards will result in a Straight and the rest will result in a loss.  So, if we add up the total payout, we'd have 8 Straights at 4 units each for a total of 32 units.  There are 47 possible draws.  We divide the 32 by 47 to get 0.68.  This is called the Expected Value (or EV) of this hand using this possible discard strategy.

            Calculating the Expected Value of holding the Pair is a bit more complex, but easy enough to calculate using a computer.  There are 16,215 possible draws if the Player holds 2 cards.  We look at these possible draws and look at the final hands.  The Player can wind up with a Four of a Kind, Full House, Three of a Kind or Two Pair.  We add up the total payout of all of these winning hands and divide by 16,215.  The result is an EV of 0.82. 

            This Expected Value is greater than that of the 4-Card Straight, so the proper play is to hold the Low Pair.  When Playing video poker (and virtually every other casino game), the proper play is to follow the one with the highest EV.  You don't go with a 'hunch' that a 5 is coming up or that you just feel a 4 or a 9 is going to fill out that Straight.  There is a distinct probability of each of these events occurring and we use those probabilities to our advantage.  This is what allows a Player go achieve the theoretical playback of a game. 

            It is an 'advantage' because most Players don't play this way.  Because of this, the casinos can off the games with a relatively high payback, knowing that they can rely on human error to pad their profits.  For the Players who play according to the math, they have the advantage of being able to play to the theoretical payback over the long run.

            Mastering video poker takes some significant effort.  The strategy is a complex one and learning whether to hold the Low Pair or the 4-Card Straight is merely one example of where a strategy where you play by what you think is right may in fact be quite wrong.  The good news is that thanks to guys like me, the toughest party of learning the strategy (creating it) has already be done for you.  The next step is learning that strategy and putting it to practical use.  We'll save more of that for next week.