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There was a surprise awaiting me.
Brian Alspach
Poker Digest, Vol. 5, No. 8, April 5 - 18, 2002
Note: Several paragraphs appearing below were edited out of the article actually appearing in Poker Digest. I have reinserted them in order to restore, thereby improving, the expository flow.
This article is a result of recent correspondence with Joel Cresap. He explained that there is a dispute in his home poker group regarding the rankings for five-of-a-kind (subsequently called ``quints'') and straight flushes in games with wild cards. I responded, quickly telling him that quints should be considered the better hand.
As is frequently the case, I later started thinking about the question and realized there is more to it than I initially thought.
My quick answer was an intuitive response based on a distinction between addition and multiplication. If we add a joker to a standard 52-card deck and let the joker be a wild card, there are 13 quints and 204 straight flushes among the 2,869,685 five-card hands. These two numbers indicate how the hands compare with one wild card.
Concentrate on the relative sizes of the numbers of the two kinds of hands after making a minimal change which first allows quints to exist. One number is 13 and the other number is 204. Return to a 52-card deck and allow more wild cards. There may be two wild cards, such as one-eyed jacks; there may be four wild cards, such as deuces wild; there may be six wild cards, or eight, or even other possibilities.
As the number of wild cards is increased, the number of quints and straight flushes is going to increase as well. Here is where the difference between multiplication and addition arises. I'm using 13 and 204 as the basis for an intuitive argument -- even though we are not allowing a joker in the deck anymore -- because these two numbers indicate the relative strengths of quints and straight flushes when quints first become possible.
The additive relationship between 13 and 204 is 191; that is, the difference between the two numbers is 191. As we increase the number of wild cards, if the number of quints and straight flushes increase in an additive fashion, then it does not take much additional increase in the number of quints compared to the number of straight flushes for the number of quints to surpass the number of straight flushes.
On the other hand, the multiplicative relationship between 13 and 204 is that 204 is approximately 16 times bigger than 13. Thus, if increasing the number of wild cards tends to multiply the number of quints and straight flushes, then it is going to be much harder for the number of quints to surpass the number of straight flushes.
My quick response to the question was based on an intuitive belief that increasing the number of wild cards would increase the number of quints and straight flushes multiplicatively. Now don't try to pin me down on this because as soon as you do that, we are passing out of the realm of intuition. I realize that a precise definition of additive growth versus multiplicative growth is completely missing. It shall remain a fuzzy notion in this context. Nevertheless, it points out an interesting difference between additive growth and multiplicative growth.
As I thought about the question, another interesting issue is the use of extreme cases. Mathematicians frequently look at what happens to some phenomenon upon making easy evaluations for two extreme situations. This sometimes yields information on what happens in the more difficult, and relevant, intermediate zones. Let's see what happens here.
We already have seen that the number of straight flushes dominates the number of quints when there is only one wild card. The other extreme is to let all 52 cards be wild. Then every hand could be taken as either quints or a straight flush. In other words, at the other extreme the number of quints and the number of straight flushes is equal.
My first reaction upon considering these extremes was to think that the number of quints starts out much smaller than the number of straight flushes, and then stays smaller than the number of straight flushes until finally catching up when all 52 cards are wild. This would provide an intuitive proof that the number of quints is smaller than the number of straight flushes for reasonable numbers of wild cards. (I don't consider all 52 cards wild reasonable or interesting.)
However, just to be careful, I decided to check what happens when there
are 48 wild cards, where the four non-wild cards have the same rank .
There was a surprise awaiting me.
It is easy to see that every five-card hand is quints. However, if we have
a five-card hand containing two or more cards of rank , then the hand
cannot be a straight flush. Therefore, there are fewer straight flushes
than quints with 48 wild cards.
This meant my reaction that the number of quints stays less than the number of straight flushes for any number of wild cards and finally catches up when essentially all cards are wild was invalid. Clearly, this makes the question of ranking these two types of hands more interesting. There are going to be situations for which straight flushes are better hands.
The final issue I wish to raise in this article is that of paradoxical rankings. What do I mean by this? Let me illustrate with a hypothetical situation first.
Suppose you have a set of 160 objects which come equipped with two possible attributes. Let's call the attributes ``A'' and ``B''. Let's suppose 70 of the objects have exactly attribute A, 60 of the objects have exactly attribute B, and 30 objects have both attributes A and B. Further, let's suppose that there is no separate category for the 30 objects having both attributes.
If we rank attribute A first and attribute B second, then there are 100 objects ranked first and 60 objects ranked second. If, on the other hand, we rank attribute B first and attribute A second, then there are 90 objects ranked first and 70 objects ranked second. In either case, there are more objects ranked first no matter which order we use for the ranking of attributes.
If your ranking scheme is supposed to rank rarer objects higher, then no ranking is possible for the preceding collection of objects. This is what I mean by a paradoxical ranking. Of course, there is a solution for the preceding example. Simply treat objects having both attributes as a separate category and rank them first.
There is an analogue of the preceding example in poker. Namely, there are five-card hands which are both straights and flushes. We now call them straight flushes and treat them as a separate category, but there was a time when they weren't considered as a separate category. Did that cause a problem?
There are 40 straight flushes, 5,108 flushes and 10,200 straights. So if straight flushes were not considered as a separate category, there still would be no problem in the ranking system. They would be treated as flushes and the number of flushes would increase to 5,148 which still is considerably less than the number of straights.
The reason hands having two attributes -- that is, straight flushes -- cause no problem is because the number of them is so small in comparison to the numbers of the hands having only one attribute.
As a matter of amusement, I have been looking for real-life examples of paradoxical rankings. In other words, I'm looking for games people actually play whose rules bring about a paradox in the ranking scheme.
What does this have to do with many wild cards? Note that any five-card hand with four wild cards and, say, a queen, could be treated as five queens or a straight flush (in fact, a royal flush). At the present time there is no separate category for such a hand. We must declare it either quints or a straight flush. Therefore, if many wild cards create enough of these hands with two attributes, it is possible a paradoxical ranking could arise.