Ins and outs of casino game analysis
September 19, 2017 3:00 AM
by Elliot Frome
To date, I don’t think I’ve ever been asked to analyze a game that I found impossible to analyze. Some are more difficult than others. The number of cards dealt plays a significant part in the complexity of the analysis based on the sheer number of combinations.
A traditional Omaha game with nine cards dealt would be a challenge. Games with community cards shared between player and dealer add to the complexity as well. This is because it becomes harder to define either’s hand without doing it relative to the community cards. All the more reason why a 9-card Omaha game with five community cards would be a nightmare.
On the other end of the spectrum are paytable games. Something like Let It Ride is very easy to analyze. Only five cards are dealt out to the player and there is no dealer hand. Three Card Poker has a dealer hand, but with only three cards to each it also becomes very easy to analyze. It also helps that there is only one relatively simple decision to make – Play or Fold.
As complex as the strategy for video poker is, in the end, it sits on the “easy” part of the spectrum as well. The complex part is it is a draw poker game. This leads to a lot of combinations to look at. Thankfully, for today’s computers it really isn’t that hard. Of course, we don’t actually go through every possible hand one by one. We utilize combinatorial math to help us figure out how many of each type of final hand will result from each possible hold/draw combination.
To begin with, we have to realize there are 2,598,960 five-card deals from a 52-card deck. However, we can actually reduce these hands down by quite a bit when we realize many hands are identical. The hand that contains 2-5 (Hearts) and 7-8-J (Spades) is really the same as 2-5 (Diamonds) and 7-8-J (Clubs). By putting these hands together, we can greatly shrink the number of hands that need to be reviewed.
For each of these deals, there are 32 different ways the hand can be played. These range from keeping all five cards to discarding all five and every possible combination in between. Now, the real fun. Excluding the case where we hold all five cards, these draw combinations can result in anywhere from 47 to over 1.5 million hands. In the end, billions of different deal/draw combinations are analyzed – all to figure out which of the 32 ways for each of the nearly 2.6 million deals is the best way to play the hand.
Once we know which way we will play each of the 2.6 million hands, we sort those results to figure how best to define each hand. Besides the obvious Pair, Two Pair, we also rank the partial hands such as 4-Card Flush, 3-Card Inside Straight Flush with 2 High Cards, etc. The result is what is known as a strategy table. This tells the player how to play each hand. Basically, you start at the top and work your way down. If your five-card deal can be defined by the row on the strategy table, that’s how you play it.
The strategy table will usually also contain the expected value of that hand, which is the amount of coins you can expect to be returned, on average, for that hand type. This expected value is really just for information purposes. All that matters is the order of the strategy table, which should be sorted based on expected value from high to low.
The bottom line of all this work is we have a 100% accurate picture of how every hand should be played. The analysis has the details down to each and every hand. The strategy table does take some liberties and can sometimes categorize a hand ever so slightly incorrectly. The overall impact is likely to be only 0.01% or 0.02% difference in the payback. But this is to provide a human with a relatively easy strategy table.
Video poker is a game of mathematical certainty in the long run. You are playing against a paytable; there is no bluffing going on. Unlike a live table game, there is no potential to see extra cards and attempt to use that information to change what you do. You are dealt five cards and have one decision to make – how to draw. The decision is based on the probability and payout of every possible hand. It is that simple.