Beware the Hidden Parameters

As the 20th century draws to a close, there have been many lists of the form ``Greatest Whatevers of the 20th Century''. Some of the whatevers have been athletes, novels in the English language, movies and so on. Humans seem to have a certain fascination for this kind of exercise in spite of the fact it makes little sense.

For example, comparing athletes across different sports and separated by a vast chasm of time is impossible -- in my opinion. There's also a huge distortion arising from the source of the list: If a U.S.A. publication produces the list, Australian and West Indies cricketeers, for example, will be ignored. In addition, some of the lists cross personal boundaries. The fact that the most moving novels I remember reading are Janette Turner Hospital's The Last Magician and Robertson Davies' Fifth Business make me not really care where they appear, if at all, on any list.

Let's now look at the connections between lists of the above type and both mathematics and poker. Consider the connection with mathematics first.

In spite of my comments about the silliness of the above lists, a perfectly legitimate mathematical activity is the attempt to associate single numbers with a collection of objects. For example, suppose you are faced with a list of eight horses and are asked to speculate as to the likely order in which they will finish an upcoming race. Most people would try to think of the attributes which are likely to affect the outcome such as recent history of the horse, weight of the jockey, track conditions, length of the race, and so on. They would then carefully consider the importance of the various attributes, how they are related and eventually sift all this information before reaching a conclusion.

The important attributes are what mathematicians call parameters. How important one judges the attributes to be corresponds to attaching some kind of weight to the parameters. Essentially, you are trying to come up with a function which spits out a single number when you plug in values for the various parameters. Of course, for most real life examples there is guesswork involved in deciding what the parameters are, and how important they are in relation to each other. This doesn't stop us from trying.

So what should you do when confronted with a list in order to determine its reliability? The most important feature of the list is a clear statement as to what the list purports to measure. The other important feature of the list is an indication of the parameters involved in producing the list. If you don't know how the list was produced, you must question what it claims. For example, if you see a a list of the top 100 grossing movies of the 20th century, you can see there is a single parameter which is easy to measure. Thus, you can trust such a list. If you see a list of the greatest athletes of the 20th century, you can ask yourself what the list is measuring and how it was determined. Upon doing so, I think you will reach the same conclusion I have reached about that list.

We have important ranked lists in poker. For example, there are many lists ranking preflop hold'em hands. These lists are a source of debate amongst hold'em players. On the other hand, they have played an important role in the evolution of many players. Let's discuss these lists in the context of the criteria mentioned above.

What are these lists measuring? I believe it is a mistake to view any of these lists as a strict first-to-last ordering of the preflop hold'em hands. This viewpoint leads to pointless arguments as to whether hand x,y is better than hand w,z. These arguments are pointless because clear statements as to what better means are few and far between, and the acrimony diverts people from more important issues. The preceding sentence raises a serious issue, namely, why is one particular hand ranked higher than another particular hand? Normally we expect a rational explanation when ranking objects.

There are two possible comparisons which are easy to make: One is to test which hand is better in a head-to-head matchup, and the other is to run each of them through many trials against nine random hands. However, experienced players mount serious objections against each of these methods saying they do not reflect a real poker game. This leads us to another explanation for the rankings in some cases, namely, experienced players rank the two hands based on their extensive experience at the game. This is a touchy-feely approach and difficult to measure.

The parameters used in making the lists typically are not mentioned. That is, they are hidden which leads to the title of this article. People who are using the lists should try to discover the contexts in which the lists make most sense, that is, the values of some of the hidden parameters.

Does the list pertain to no fold'em hold'em or does it apply to tighter games? Does it apply to low-limit games or mid-limit games? Is the list intended for inexperienced players?

Are the lists useful? I believe so. The lists are being used to address the absolutely fundamental question of whether or not a player should see the flop with a particular hand, and under what conditions. This has been helpful to many beginning players and has drawn attention to the importance of considering the value of preflop hands. Some of the lists vary according to parameters like the number of players already in the hand or the player's position. These parameters are being made explicit. There may be other parameters not being made explicit. It is up to the user to think about the potential hidden parameters and what effect they may have on her use of the list.