Brian Alspach
Poker Digest Vol. 1, No. 9, November 20 - December 3, 1998
I trundled into the club and saw Bib Ladder sitting by himself at a table busily working on the construction of a sandwich. I took the seat opposite him and opened with, ``Hi, Bib. I haven't seen you in a while. They have a kitchen here, you know, so what are you doing making your own sandwich?''
Looking up he replied, ``Good to see you too, professor. As far as the sandwich is concerned, this place is very tolerant of bringing in food from other places. Now take a look at this sandwich. You go out on the street here and see several Italian delis. Well, the French like to brag about their baguettes, but the Italians with their crusty bread don't have to take a back seat to anyone. Down the road is a supermarket with cervelat and havarti cheese. Let me make a sandwich for you and you can share this wonderful simple food.
By the way, some time back I asked you about 3-card flushes being judged as better than 3-card straights at a home game I played in years ago. You showed me why they were wrong. Well, I have another example of something they did which I also suspect was wrong.''
This piqued my interest and I replied, ``Let's hear it.''
``One of the players always dealt 5-card stud with sousem rules in force. You know, for-card straight beats a pair, four-card flush beats a four-card straight and, get this, a four-card straight flush beats two pair. Now I played in that game for four years and saw only one or two four-card straight flushes in all that time. I think it is a much stronger hand than they judged.''
``This is rather intriguing, Bib. When people begin introducing variations on standard hands, they sometimes create monsters without being aware of it. Let me give you a hypothetical situation first and then return to sousem.
When one is trying to introduce a ranking to a collection of objects, there are certain principles to which we normally adhere. Often these principles are not verbalized, but lurk in the background as common sense. For example, one principle is the rarer an object, the higher it is ranked. So let's consider a collection of 650 objects. Suppose 200 of them have property A by itself, such as being a four-card straight, 190 have property B by itself, such as being a four-card flush, 40 of them have both property A and property B, and 220 of them have neither property A nor property B. You can see that if you do not have a separate category for objects having both properties, then no matter how you rank objects with properties A or B, you will end up with a rarer object being ranked lower. If you rank A higher, then 240 will have property A and 190 will have property B. If you rank B higher, then 230 will have property B and 200 will have property A.''
``Yeah, I see there is a problem there,'' murmured Bib.
I continued, ``In the preceding example, we could solve the problem by having a separate category for objects with both properties ranking them highest, followed by those with property B, followed by those with property A, and finally those with neither property. Unfortunately, we can concoct examples where this will not work either by having a large number of objects with both properties. Believe me, it can get to be a real mess.
Does this apply to poker? Of course! We have hands (objects) with a property we call straight and those with a property we call flush. Hands with both properties are called straight flushes. In this case no problem arises by ranking straight flushes higher than either straights or flushes. The reason there is no problem is because the number of hands with both properties is so small relative to the number of hands which are either a straight or a flush. Mathematically we say the size of the intersection of the set of flushes and set of straights is small in relation to the sizes of either of the sets.
Sousem possibly is another kettle of fish. You can have hands which contain both a four-card straight and a pair, or a four-card flush and a four-card straight but not a four-card straight flush, and so on. This possibly could create a substantial distortion in the proper ranking of the hands.''
``We didn't consider a hand with a 4-card straight and a 4-card flush, but not a 4-card straight flush as anything in our game,'' Bib interjected.
``I can understand that, but you have to admit it is a type of a hand and we must initially consider it separately because a player holding such a hand can consider it as either a four-card straight or a four-card flush depending on how those two hands are ranked. Right? Perhaps the number of such hands is big enough that no matter how we rank four-card flushes and four-card straights relative to each other, there will be more of the higher ranked hand. So let's first of all list all the types of hands which may occur. Following that we shall count them and determine the proper ranking. Realize I am initially including a category for any type of hand allowing more than one property. When we do the final ranking of hands, most will be merged with other types of hands, but we leave them there for now in order to check and simplify the computation.''
I then wrote down the following list.
In the next issue we shall do the counting for the above hands. Interested readers might want to get a jump start in doing so as a few of them are challenging. Also, the results contain some surprises.