Profiling Opponents in the Absence of Statistical Convergence

There are three basic ways we use statistics in poker:

1. To help profile our opponents in order to make better playing decisions.

2. To analyze our opponents’ play in order to aid in game selection and seat selection, and …

3. To analyze our own play for diagnostic purposes.

While online stat tracking software programs such as PokerTracker or Hold’em Manager gather a wealth of data during game play, making the correct play against a particular opponent in a particular situation is not always as simple as looking at a statistic and having that stat equate directly to an action. More often, we must use a statistic in the context of the player’s profile (Is the player tight-aggressive, loose-aggressive, tight-weak, nit, maniac, etcetera?), in context of other statistics we have on a player, as well in the context of the actions that player has taken in other situations.
This is particularly true in the absence of statistical convergence – that is, in situations when we haven’t had enough trials with a particular opponent for a certain statistic to become statistically significant in and of itself.

The Absence of Statistical Convergence

There are two basic reasons why a given statistic fails to converge: Either you simply haven’t played enough hands with a given opponent for a given stat to become statistically significant, or a situation comes up so infrequently such that a particular statistic will essentially never converge.

Some statistics converge rather quickly. Voluntarily Put Money in Pot Percentage (VP$IP%) and Pre Flop Raise Percentage (PFR%) statistics are recorded nearly every hand , and thus become meaningful in a relatively short period of time. After about 50 hands with an opponent, you should start to get a general sense for what your opponent’s VPIP/PFR profile looks like, and after about 100 hands with an opponent, you should start to have a pretty good idea. After about 200 hands, you should have a reasonable proxy. And after about 1,000 hands, a player is pretty much what his VP$IP/PFR says he is for all practical purposes.

It helps that the two most important profiling statistics are the ones that converge the fastest.

Other statistics take a little bit longer to converge, because they require some specific things to happen. For example, in order for Flop Continuation Bet Percentage to be tracked, a player must have a c-bet opportunity, which means he must first put in a raise before the flop. Consequently, if a player only raises before the flop 15 percent of the time, then the c-bet percent statistic will only be tracked at most on 15 percent of hands. Similarly, in order for Check-Raise Flop Percentage to be tracked, a player must (1) play a hand, (2) be out of position against at least one opponent, and (3) have an opponent acting behind the player in question bet after the player checks.

Likewise, Pre-Flop 3-Bet Percentage and Fold to Pre-Flop 3-Bet Percentage also take some time to converge, while Pre-Flop 4-Bet Percentage and 3-Bet Pre-Flop and Fold to 4-Bet Percentage take even longer.

Still more obscure stats – like Flush Board Flop C-Bet Percentage or Flush Board Flop Check-Raise Percentage (both stats can be found in the online repository for PokerTracker 3) – may never approach convergence against typical opponents. This is because a player might only have the opportunity to make one play or the other 50 times in 10,000 hands – or five times in 1,000.

Let’s say you decide that if a player check-raises on flush board flops more than 10 percent of the time, that you will three-bet bluff him. Over 1,000 hands with this opponent, he has check-raised on a flush board flop once in five opportunities, for a 20 percent check-raise percentage on such flops. How confident are you that this particular opponent actually check-raises more than 10 percent of the time on flush board flops? How confident are you that this opponent doesn’t only check-raise with the nut flush, and that this five-hand sample just happens to capture the one time he flopped the nuts?
The fact is, most players don’t play 10,000 hands with more than a handful of opponents, much less whatever it would take for a flush board statistic to approach convergence.

Moreover, let’s say that you have played 10,000 hands with a given opponent; because our opponents are constantly evolving, it is quite possible – perhaps even probable – that a given opponent’s behavior may be quite different over the most recent 1,000 hands than in the first 1,000 hands, making the 10,000-hand sample less relevant than we would like. Meanwhile, even with the most basic statistics, we are constantly forced to make decisions without the benefit of statistical convergence, if only because we are constantly playing against new opponents (or at least most of us are, particularly at smaller stakes).

Does this mean that you can’t use the statistics when they haven’t converged, or that flush board statistics are useless?

Of course not. At this point, you have three reasonable choices:

1. Default to basic strategy. With little or no information on an opponent, you should default to basic strategy, which involves making the best play – on average – against the universe of opponents.

2. Play the percentage directly. Take the statistic at face value and play the percentage as if it were statistically significant, or – and more to the point…

3. Profile the player. Use the statistic in the context of other statistics, the way the player has played other hands, your history with the opponent, and other profiling factors in order to improve the accuracy of your decision on a case-by-case basis.

We’ll discuss these options in further depth in future issues of Card Player. ♠

Card Player columnist Jeff Hwang is author of Pot-Limit Omaha Poker: The Big Play Strategy and the three-volume Advanced Pot-Limit Omaha series. Jeff is a consultant for the soon to be released PokerTracker 4, public beta available now.