Poker Digest Vol. 3, No. 16, July 28 - August 10, 2000
In my last article I touched upon some issues involved in ranking preflop hold'em hands and I had no intention of saying more at this time on the topic, but since writing that article, I have observed considerable Internet discussion and feel compelled to revisit the topic. I wish to introduce some mathematical structure into the discussion. I find this structure useful and hope readers will find that to be the case as well. So let's plunge into the mathematics first and then apply it to the poker setting.
In colloquial English we have words to describe ways of ordering objects -- ``bigger than'' -- ``heavier than'' -- ``better than'' and so on. Typically, we take a finite set of objects and put them in a first-to-last order with respect to some property. The mathematical prototype that is familiar to everyone is the set of positive integers for which one is the smallest element, two is next, followed by three, and so on, ad infinitum. Mathematicians have abstracted the idea of order in a way that captures such examples and allows us to discuss them easily in the abstract setting.
One common abstraction is the notion of a partially ordered set henceforth known as a poset. Instead of using a particular property like ``heavier than'' or ``longer than'', we shall use the general term ``related'' and think of the ordering as depending on the ``relation''. Then a poset is a set equipped with some relation satisfying three properties: No element is related to itself; if x is related to y, then y cannot be related to x; and if x is related to y and y is related to z, then x is related to z.
Let's see that the relation ``heavier than'' gives us a poset. Something cannot be heavier than itself so that the first property holds. Clearly, if x is heavier than y, then y cannot be heavier than x and the second property is satisfied. Finally, if x is heavier than y and y is heavier than z, then x certainly is heavier than z so the third property holds. This means the relation ``heavier than'' defines a partial order.
The reason we call such an order relation a partial order is because we do not insist that any two objects are related to each other, that is, the order relation may only be partial. Let's look at an example of such a relation.
Let the set be the positive integers between 1 and 50 inclusive. Let's say that x is related to y if x and y are not the same integer and x divides y. Under this relation an integer is not related to itself; if x and y are different and x divides y, then y does not divide x; if x divides y and y divides z, then x divides z. So this relation defines a poset, but there are many pairs which are not related such as 5 and 7, or 9 and 28, or 16 and 17, and so on.
If a poset has the additional property that for any two distinct elements x and y, either x is related to y or y is related to x, then the poset is called a linearly ordered set. Linearly ordered sets are extremely important because all the elements can be lined up in order from worst to best or vice-versa. (Convince yourself this is the case simply by considering the three properties of a poset and the fact any two distinct elements are related one way or the other.) In colloquial English when people are thinking about an ordered set, they normally are thinking in terms of a linearly ordered set.
Let's now move to the arena of ranking preflop hold'em hands. As soon as we start to try to apply the theory of posets to this ranking problem, we encounter difficulties arising from the problem of translating imprecise real-world conditions to precise mathematical structures. Nevertheless, let's see what we can say.
Much of the acrimonious discussion on the ranking of pre-flop hold'em hands arises from the fact some people are talking as if the 169 possible hands form a linearly ordered set. I am going to argue it is a mistake to think this way. One big problem (discussed in the last article) is that the relation defining the ordering is terribly fuzzy. That is, in certain cases the notion of one hand being better than another hand is not clear-cut. In my opinion, the best thing to do for problem comparisons is simply to agree there is no order between them. In other words, treat the 169 hands as a poset.
How do we treat the 169 hands as a poset and what do we lose if we do so? In order to treat the 169 hands as a poset, I think we should introduce some threshold a pair of hands must pass before they are considered to be related. Since the notion of ``better than'' is fuzzy, I do not object to using something like ``essentially everyone agrees this hand definitely is better than that hand.''
Now suppose we have imposed a poset structure on the 169 hands. An important feature of any poset, including the practical one under discussion, is that they contain linearly ordered subsets. It is these linearly ordered subsets we want to find and work with. Here are my suggestions for important linearly ordered subsets:
In fact, working with these lists separately is the intuitive approach most people take, and is the way experts discuss preflop hands in their articles and books. For example, if you find yourself holding a pocket pair, you make your decision on what to do pre-flop based on the situation when the bet reaches you and how you feel pairs play in such a situation. How pairs compare to suited connectors does not enter your mind at this point. Similarly, if you find yourself holding a hand from L(2), you base your decision on what to do in terms of the situation at your table.
The way to use the above linearly ordered subsets is to decide whether or not the game you are in warrants playing any hand from a given list, and if you decide you will consider playing some hands from a given list, your decision is then how far down the list you will go under appropriate conditions.