Good News for Smart Players

It would be nice if you could sit down at a poker table, win a few dollars, and be pretty sure you're playing with positive expectation at that table – kind of like arm wrestling, but it doesn't work that way. Variance in poker session outcomes is high, so a single result doesn't tell you that much. What's worse, at the lower limits, the overall proportion of winning players is fairly low. So, if you tell me that you played poker for a few hours, or even a few sessions, and won a bit, I'd still be foolish to guess that you're a winning player.

It sounds a little unfair to tell someone who has played three or four poker sessions and won three or four times that he's probably a loser, but it follows from some relatively uncontroversial premises. First, winning poker players are fairly rare (especially at the lowest limits). Second, losing players often have winning sessions. Given these premises, a player who goes on a short winning streak is still more likely to be a lucky loser than a solid winner.

The process by which these different bits of information are incorporated into conditional probability estimates comes from Bayes' Theorem, an important theorem in statistical reasoning. Although Bayes' Theorem is a precise statistical relationship, I think for now that it's best for us to think about this intuitively. The theorem provides a mathematical basis for updating our beliefs about the world when we get new information. So, for example, suppose we believe that a player is 90 percent likely to be a losing player – that's the prior probability – and then someone tells us that player has won eight sessions in a row. Bayes' Theorem gives us some guidance on how to adjust the prior probability downward, given this new information. If winning nine sessions in a row turns out to be just a bit less likely for losing players than for players in general, we update our estimate of 90 percent for this player downward a bit. If it turns out to be a lot less likely, we adjust our estimate downward a lot.

Bayes' Theorem has relevance to poker in many areas. Every time a player acts, we get information to help update our beliefs about his holdings as well as about how he plays in general. Updating the probability distribution of a player's holdings based on his action is probably the most interesting application of Bayes' Theorem in poker, and well worth some detailed discussion. But I'd like to devote this column to something a little more mundane: the probability that a player is a winning poker player. This is a somewhat different approach to the traditional one (looking at the confidence intervals of your earning rate), although the two approaches can be integrated.

At some level, most of us have an intuitive grasp of how to apply Bayes' Theorem to a problem like this – as long as we pose the right questions. Suppose there are two guys you've known for a while, and who have just taken up poker. Bob is an emotional wreck, and not particularly bright. Doug is a genius, and the coolest customer you've ever known. If Bob tells you he won $100 in a $2-$4 hold'em game, your first thought would probably be that he's a lucky loser. If Doug tells you the same thing, you'd be much more likely to think it's a pretty representative outcome for a winning player. What you're doing, in essence, is incorporating your knowledge of Bob-like and Doug-like people. You may be overcompensating (in truth, most Doug-like people are losers, too, especially if they're just starting out), but at least you're trying.

Unfortunately, people tend not to give the prior probabilities enough weight, especially when the appropriate probability is not so salient. Suppose only 10 percent of poker players are winners. If Fred is an average-looking guy you just met, you might guess that he's neither a winner nor a loser, because his appearance really doesn't tell you anything. But in truth, you should guess that he's a loser, because we already postulated that 90 percent of poker players are. If Fred has played 10 sessions and is dead even, you might be tempted to guess he's basically a break-even player. But since breaking even over 10 sessions is not that unlikely, even for losing players, you would be wrong to upgrade him too much.

A critical element of applying Bayes' Theorem to these kinds of problems is estimating the prior probability appropriately. Suppose we're talking about a player (Doug, again) who has been playing and studying for a few years, is very smart, and is temperamentally well-suited to poker. Maybe our estimate of the prior probability that he's a winner should be drawn just from players like that. Call it 30 percent. Knowing this about Doug, we should be willing to believe he's a winning player after somewhat less evidence than for other players. Of course, it will still take quite a bit of evidence to cross that 50 percent barrier, because even solidly losing players win quite often. So, we shouldn't let a few wins convince us of anything. But we should be much more likely to believe good things about a dedicated, smart, emotionally stable player, based on the same amount of evidence. Similarly for Bob the loser, it will take quite a bit for us to believe the probability that he's a winner is even 10 percent, even though that's the rate in the general population, because the prior probability for people like him is somewhat less.

One of the reasons that Bayes' Theorem is a good way to start an argument (even, I'm told, among academic statisticians) is that estimating the prior probability is slippery. Statistics provides some guidance, but no simple rule, for choosing a reference population. Suppose you're incredibly smart, but have poor emotional control. Do you want the rate for smart people, or the one for emotional wrecks? These rates will probably give you very different answers, yet neither one is clearly more appropriate. Ideally, we'd like one for smart emotional wrecks, but we might not have that information, or it might be drawn from far fewer examples. In general, though, more specific is better. "Smart poker player" is more specific than "poker player," so if those are your two choices, and you figure they're both pretty accurate, take the more specific one. Is it more optimistic? That's because, all else being equal, you'd rather be smart.

There's some good news in there somewhere for people who are convinced they're winners even though their numbers don't quite support it. If you're more like Doug than like Bob, the base rate for your reference population is probably pretty favorable. When you tell people you've been winning a lot, they will be skeptical that you're really a winning player, because the most specific base rate they have for you is the general poker-playing population. They don't know you're a Doug, so you can forgive them. But you know you're diligent, experienced, knowledgeable, emotionally stable, analytical, insightful, and quick-thinking. Deep in your heart, you know you're not like all those other guys. You might not be sure if you have quite enough evidence to overcome the base rate (which due to all those positive qualities is a whopping 40 percent, still not quite enough to convince anyone you're a winner without seeing some results), but at least you know you should be cut a bit of slack if your results aren't statistically significant.

Or do you? The real problem comes when you want to choose your own prior probability – that is, the rate for the population of people most like yourself. Most everyone who plays poker would probably like to believe they're smart, dedicated, well-suited to poker, and so on. More to the point, even if the true rate of poker winners is 10 percent, if we asked every poker player to estimate the rate of winners among people just like them, it would probably come out to more like 60 percent. Nobody likes to think he's a loser, much less a typical loser. So, if we leave it up to you, you'll probably overestimate your prior probabilities of just about everything good. Or, you'll look to a reference population of (say) smart, analytical types, conveniently forgetting that you could just as easily have chosen impatient people with rotten intuition.

What we really need is a highly intelligent, well-informed, statistically savvy, completely neutral observer. That's going to be hard to arrange. So, as a stand-in, I've been working on a little questionnaire that, if you answer it honestly, will serve reasonably well. It's not quite ready, and I'll save it for a time when I'm ready to write in more detail about Bayesian estimation of your earning rate, which can get somewhat more involved than arithmetic with probabilities. But in the meantime, I thought it would be helpful to provide a little comfort to anyone who's convinced he's a winner but just can't muster the hours to prove it.