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Quantifying Variance is a biweekly column in which we’ll take a look at some of the math underlying poker, with the goal of understanding just how probable or improbable various occurrences actually are, and how to tell the difference between what is random and what is not.

Since this series started, my goal has been to find clear statistical evidence for what I’m calling long-term agential variance, which is basically a fancy way of saying “tilt,” although it also potentially includes “reverse tilt,” or confidence effects.

Last time around, we took three high-volume heads-up players as case studies, and found that their win rates did fluctuate to a statistically significant extent over the course of a winning or losing streak. The trouble was that these fluctuations appeared quite individual; for each of them, the effects cropped up to different extents, after streaks of varying length, and sometimes even affected them in opposite ways.

My next question was whether, if you looked at a group of players together, there would be some tendency displayed in the aggregate. That is, I wondered whether it’s fair to say that people in general tilt at a certain point, even if an individual player might actually be clutching just then, instead. It turns out that the answer is yes.

The relevant metric

In doing the case studies, I was looking at the players actual win rates after streaks of a given length, since that’s ultimately what we mean by agential variance: the component of variance that comes from variations in how the person is playing. In order to calculate the win rate, I had to convert from exact streaks (as given by Sharkscope) to cumulative streaks.

To be clear, the difference is that an exact streak of say 10 wins means that we have a loss, followed by exactly 10 wins, followed by another loss. A cumulative streak of 10 wins includes all the times that the player won at least 10 games in a row. The reason I call this cumulative is because it is equal to the player’s exact 10-game streaks, plus their exact 11-game streaks, plus their exact 12-game streaks, etc.

In collecting and analyzing the data, I noticed one interesting fact, which is that most winning players have significantly more (exact) 2-game losing streaks than 2-game winning streaks. This is initially surprising, because with more wins than losses overall, you’d intuitively expect more streaks of all lengths. The reason this is not the case is fairly obvious when you think about it, however: a winning player is more likely to extend a winning streak than a losing streak, so a greater proportion of their wins than their losses will be come in the course of streaks longer than two games.

Once I’d thought about it like that, I realized that one of the reasons that it has seemed surprisingly hard to get a clear look at tilt is that the exact streaks presented by Sharkscope are inherently deceiving. When a player has fewer streaks of a given length than you expect, there are two possible and contradictory reasons for this: the first is that they might be reaching that point less often than you’d expect, but the other is that they might be extending the streak at that point more often than you’d expect, rather than terminating it there. Without distinguishing one possibility from the other, then even when we can spot a disproportionate number of streaks of a given length, it’s hard to tell whether that means the player is actually running hot, or if they’re running cold.

By looking at cumulative streaks instead, we eliminate this confusion by including those streaks which were extended, and everything becomes much clearer. If the player has more cumulative streaks than you’d expect, then they’re reaching that point more often. If they have fewer, then they’re terminating their streaks earlier.

And when we look at a group of 15 high-volume players through that lens, we find that there is a pretty clear picture indeed.

Tilt, plain as day

Note: For all the graphs we’re going to look at, the vertical axis is the percentile difference between the player’s actual streaks of a given length or longer, and what is predicted by a computer model. So, for instance, if the player was predicted to have 100 streaks of a given length or longer, +10% means they actually had 110, while -10% means they actually had 90.

Quite often when you’re dealing with data, especially data which contains a lot of random noise, spotting a trend comes down to visualizing things the right way. Here is what it looks like when we naively plot each player’s graph of cumulative losing streaks, for length 2 through 8:

We can observe that most of the graphs seem to be above the zero-line most of the time, so it seems likely that people in general go on losing streaks more often than you’d expect if they were playing consistently. Aside from that, however, it’s all just a jumble.

But look what happens when, instead of tracking individual players, we sort the data for each streak length, from furthest below expectation to furthest above, and then plot each one as a separate bar graph. (Note that the colors no longer correspond to individual players.)

Now the trend is much clearer. For all lengths of streak beyond three, most of the players are experiencing more streaks than you’d expect, very few are experiencing significantly fewer, and for the longer streaks, quite a large proportion of the players are streaking at a rate of 15% or more above expectation.

If we take their average, it looks like this:

Given the amount of noise in the individual players’ data, this is a remarkably consistent trend, showing how, as losing streaks get longer, the average player gets more and more likely to keep losing. The falloff in their average win rates is considerably more chaotic, but although the graph bobs up and down, the important thing is that the average impact on winrate remains negative the whole way:

So after losing a few games in a row, the average player is winning about 2% fewer games than usual. This may sound like a small effect, but most winning head-up players only win a few percent more than 50% to begin with, so the players with a greater-than-average propensity to tilt may well be statistical losers while suffering through a bad streak.

After much analysis, then, we’ve found what we pretty much knew all along: that losing games and being in a bad mood makes a lot of people play worse. More interesting is the question of what happens to people on a winning streak. Here is the winning streak data for the same 15 players, sorted and graphed separately for each length of streak, as we did for the losing streaks:

It looks somewhat similar to the losing streaks, but with two important differences. Firstly, note the vertical axis values: whereas many players were exceeding expected losing streaks by 15% or more, here it is only a few outliers that deviate from expectation by even 10%, either positively or negatively. So it seems that any emotional effects from winning are considerably more subtle. Secondly, although most of the players have slightly more of the long streaks than expected, a significant number of them have far fewer. Again, because we’re sorting the data from lowest to highest, the same color doesn’t necessarily correspond to the same player, but in fact most of the individual players do show fairly consistent trends. Overlaying all 15 at once produces a jumble, as we saw with the losing streaks, but here are three players to look at as examples:

All three seem to experience slightly more short winning streaks than expected, up until we get to cumulative streaks of four or more. After that, however, they part ways. Yellow is scarcely (but positively) affected, but blue seems to benefit a lot from confidence, while red suffers from it. This is in keeping with our findings last time, that individual players tend to respond to streaks in general, and winning streaks in particular, in idiosyncratic ways.

What about on the whole, though? If we take their averages, is there a trend?

Barely. It’s probably significant that the graph stays at or above the zero line the whole way, but note that the average deviation is less than 1%. This means that even if, on the whole, players play slightly better when on a winning streak, this collective trend is vastly outweighed by players’ individual personalities and idiosyncratic responses.

In summary, then, what we’ve found is that almost everyone is affected negatively by losing streaks of around five to eight games, many of them dramatically so. By contrast, winning streaks do affect some players one way or another, but in no consistent way, and usually to a lesser extent than losing streaks. Using this information, we can try to profile players’ poker personalities, and maybe even quantify their tendency to tilt. This is what I’ll be trying to do in our next instalment.

Alex Weldon (@benefactumgames) is a freelance writer, game designer and semipro poker player from Montreal, Quebec, Canada.