Variability in Poker Outcomes

When we think about our poker outcomes, two of the most important concepts are expectation and variance. Loosely, expectation reflects our true earning or loss rate, while variance reflects how consistent that rate is. Although expectation and variance are statistical concepts, even a nontechnical understanding provides a common language for discussing poker outcomes.

Poker players tend to think about expectation first and consider variance mostly unpredictable "noise." But in reality, there are many factors that cause fluctuations that are both systematic and predictable. Perhaps you tend to do better when you're feeling better, when you play hold'em versus Omaha, when you play particular limits, or when you play on weekends as opposed to weekdays. Anything that varies between your poker sessions can be a systematic source of variance if it has a systematic effect on your results. We call this kind of variance "intersession variance" (although in poker, it's usually used to estimate interhour variance). Most players eliminate the biggest sources of intersession variance in their data by keeping track of different games and different limits separately. But even within a single game at a single limit, there are likely some remaining systematic sources of variability, including everything from your mood state to the kind of table at which you sit.

While intersession variance is most relevant to your own results, there are other kinds of variance that don't affect your session results but are still worth understanding. For example, if you wanted to argue that poker is a game of skill, it would be good to show that skill is an important contributor to outcome. This means arguing that differences in skill account for some of the interplayer variance. Better players don't always win, and sometimes the worst player at the table ends up with all the chips. But if you had to predict how well a player was going to do in a given session, you would be much closer on average if you knew the player's level of skill.

One difficulty in understanding sources of variance is that they are easily overwhelmed by unpredictable noise. Skill is certainly a strong contributor to results. But for small time periods, the effect would likely be much smaller than the noise. As you collect data from more and more hours, the differences in skill tend to add up, while differences in noise tend to even out. For intersession variance, it may take a great deal of data (and some meticulous record-keeping) to find out you're one-tenth of a big bet worse when a team you follow is playing on a visible television.

Of course, some of the noise in poker results is due to factors that are for all practical purposes truly random. The random shuffling of the cards creates a tremendous amount of variance that we can't eliminate. Some hours, you'll hit everything in sight, while other hours, you'll make very strong losing hands. If you vary your play randomly (for example, for game theoretic bluffing), that's another source. Sometimes you'll bluff at just the right time, while other times it won't work out as well. There may be other sources of noise that are really due to factors you could learn about but probably won't. Perhaps you play better at certain temperatures than others, but you've never thought to keep records. Temperature-induced variation in your ability to read your opponents will just have to be considered random noise. These sources of random noise are the reason you can't just assume you're a winner at a new game when you have a profit after your first 20 sessions.

The factors that explain variance in your poker outcomes are not fixed in stone, and they're not necessarily constant from one person to the next. You may do a lot better at stud than hold'em, while another player may do as well at both games. Sometimes a true source of variance can be masked because you don't actually vary it. You may be a much better player in the evening than in the morning, but you would never find out unless you played relatively often at both times. In this case, it would be correct to say that time of day isn't a significant contributor to your poker outcomes, but it would be wrong to infer from that that you play equally well at all times. In effect, you just don't have the data to estimate the effect of time of day.

Investing your poker bankroll wisely is really an exercise in understanding the explainable sources of variance in your outcomes, identifying the ones you can control, and making use of the information appropriately. Some of these uses are trivial. If you do better at hold'em than stud, play hold'em. If you do better in the morning than in the evening, play more in the morning. If you do better when you're feeling well, avoid the cardroom when you're sick. Play in the cardrooms, and on the days, and against the opponents that tend to be the most profitable for you.

There are, unfortunately, a few wrenches to throw in the works, even if you do have all the records you need. One of the most important is that it's hard to know which effects are statistically significant. Just because you're averaging $2 per hour better at stud than hold'em, you don't really know that your expectation is higher at stud. That's just your best estimate, based on the data. But the effect is hardly consistent; you still have many good hold'em sessions and many poor stud sessions. How do you know it's a real effect and not just random chance? This question can be restated in terms of the noise in your data. If there were no real difference between stud and hold'em for you, how likely would it be that you'd see a difference of this size or larger? In other words, does the "game" variable explain a large amount of variance in relation to the noise in your data? Answering this kind of question is the goal of inferential statistics, and can get deeply involved.

A second complication is that interactions between factors can make the data tricky to understand. A simple example is illustrated in the following table: