The Math Of Player Profiling

In my last article, “Sizing Up the Opposition,” I discussed how many players are too quick to give unknown opponents credit for having rare traits — aggressive postflop play, good hand reading, and so forth. This concept is critical, so I wanted to devote another couple of articles to it.

The process of profiling opponents can be broken into two steps: formulate a hypothesis, then test the hypothesis through observation. “That guy across the table with the long beard is a calling station,” is a hypothesis. Though you may not have used the word “hypothesis” to describe it, in your time you’ve no doubt formulated many hypotheses
like this one.

After you have a hypothesis, you test the hypothesis with further observation. “He called a big river bet and showed bottom pair.” This observation supports the hypothesis. “He called a big river bet and mucked.” This observation also supports the hypothesis, though not as strongly as the first one. This is where the math comes in. How much stronger is the first observation than the second? And after either observation, how sure are we that our hypothesis is correct?

To answer these questions, we use a branch of math called Bayesian inference. Though it’s an extremely powerful tool, the concept behind Bayesian inference is simple: You formulate a hypothesis and estimate a probability that it’s true. Then you make observations, and with each observation you adjust your probability. You adjust the probability up if the observation supports the hypothesis, and you adjust it down if the observation contradicts the hypothesis. If you’ve started with a reasonable estimate, then as you accumulate observations, you become more certain about the likelihood of your hypothesis.

Here’s the thing, while this process may seem reasonable, chances are that the math of it will be counter intuitive at first. Here’s a simple non-poker example to help get your pump primed.

Say there’s a scary new virus out there called Virus X. Virus X incubates asymptomatically for many years. Eventually, it turns your skin bright blue.

Population studies determine that about one asymptomatic person in 100 is a carrier of Virus X. Recently, a company has developed a test for Virus X. The test always comes back positive if you have Virus X, but it returns a false positive 5 percent of the time if you don’t have Virus X. That is, if you don’t have Virus X, you have a 95 percent chance of having a negative test, and a 5 percent chance of having a positive one.

You take a Virus X test, and your test comes back positive. Oh no! What is the probability that you will live out your golden years in Smurf blue? 95 percent? That’s what most people say, but it’s wrong — very wrong, in fact. The correct answer is about 17 percent. How do I get that?

Say we were to test 100 people at random. We would expect 1 out of those 100 to have Virus X. That person would test positive. The other 99 would be Virus X negative, but 5 percent of those would test false positive, which averages to just a hair under 5 people.

Therefore, for every 100 people we test, we would expect about 6 positive tests, 5 of which are false positives, and only 1 of which indicates a real infection. Therefore, if you test positive for Virus X, your chance of actually having the infection is about 1 in 6, or about 17 percent.

Why does it work out this way? You take a test which is 95 percent accurate, and after a positive result you’re still a big favorite to be negative. It’s because Virus X is rare; most people don’t have it. When something is rare, you often can’t identify it accurately with just a single observation, you have to check a few times.

Say you tested everyone again who tested positive. Out of the 6 people who originally tested positive, you would expect the person who is infected to test positive again,
and you would expect a second person to test positive about 25 percent of the time (five people each with a 5 percent chance to test positive). Thus, an average of about 1.25 people will test positive, 1 of whom has the disease. This means that anyone who tests positive twice in a row has about an 80 percent chance to have Virus X. If you tested a third time, a third positive would make you nearly certain of an infection.

This is Bayesian inference in action. The hypothesis is, “I have Virus X.” Before any tests, the probability we assign to that hypothesis is 1 percent, the rate of infection in the general population. After a single positive test, we adjust the probability of the hypothesis up from 1 percent to 17 percent. After a second positive test, we adjust the probability from 17 percent to 80 percent.

Back to poker. Humans are great at developing hypotheses. “I think that guy is bluffing a lot.” Humans are bad at evaluating these hypotheses correctly using Bayesian inference. “See, he made a huge turn raise. That’s his third one tonight. He’s a bluffer, I tell you.” We make two mistakes. First, we tend to ascribe too much certainty to our observations. Second, we don’t correctly account for rarity within the population.

Most observations in poker have uncertainty attached to them. When someone raises the turn and there’s no showdown, we can’t be sure what happened. Maybe the player was bluffing. Maybe he had the nuts. Maybe he had a big draw. This uncertainty in how to interpret the observation is akin to the 5 percent false positive rate in the Virus X test. Many of our poker observations, however, have equivalent false positive rates closer to 50 percent.

Some traits are much more common in the poker population than others. Calling stations are much more common in live small-stakes no-limit games than players who like to five-bet bluff preflop. More observations are required to confirm a rare trait than a common one.

Next issue I’ll offer some specific examples of how I use the principles of Bayesian inference to make accurate reads on my opponents.

Ed has authored six poker books and sold more than a quarter million copies. Ed’s newest book, Reading Hands At No-Limit Hold’em, will soon be available for purchase at notedpokerauthority.com. Find him on Facebook at facebook.com/edmillerauthor.