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.

Last time we looked at the sort of winning and losing streaks that can be expected in heads-up sit-and-go play under the assumption that the player’s win rate is consistent. If a player has a tendency to play better or worse depending on emotional state, however, then his actual win rate at any given point in time might be either higher or lower than his long-term average. If we assume that being on a winning streak tends to make a player feel good and play well, and that being on a losing streak tends to make the player tilt and play worse, we would expect that to be reflected in their winning and losing streaks.

We’d like to be able to model tilt mathematically, both to assess a given player’s tendency to tilt and to project his or her future performance on that basis. There are a number of simple models one could try, however, and it’s impossible to guess which most accurately reflects real players’ behavior without first looking at some data.

This week we’re therefore going to head “into the wild,” using Sharkscope to look at a number of high-volume heads-up players on PokerStars in an attempt to find hard evidence for the effects of tilt on real players.

The problem at hand

Unfortunately, despite the level of detail Sharkscope provides in its statistics (especially to a subscribing user), it doesn’t quite suit our purposes because it lumps all kinds of tournaments together. Since we’re interested only in players’ performance at heads-up Sit-and-Gos, that’s a problem.

Now, winning heads-up players tend to specialize in this format, so it’s not hard to find players with over 10,000 games to their name, at least 95% of which are heads-up Sit-and-Gos. Unfortunately, even 95% isn’t really good enough if we want to look at losing streaks.

The problem is that heads-up games obviously pay 50% of seats, whereas other single-table games typically pay 33% and multi-table tournaments somewhere in the range of 10-15%. This means that long losing streaks are not uncommon in the other game types, especially in the multi-table events; even if a player has 9500 heads-up games and only 500 other games to her name, then, it’s still entirely possible that a high proportion of her longer losing streaks came during periods when she was playing something other than heads-up.

The good news is that this cuts the other way as well, and we can be very confident that most of a player’s winning streaks come from heads up play, especially if that’s their primary game. Even if we assume that emotional effects are predominantly negative – that running bad makes you play bad, but the converse is not true – the fact that a player is sometimes on tilt would mean that their average win rate (which includes periods of tilt) is still somewhat lower than their peak, non-tilting win rate. That, in turn, means that we would expect to see most players experiencing longer and more frequent winning streaks than their sample size and overall win rate would suggest.

Longest win streaks vs. expectation

The most obvious thing to start with, then, is an examination of players’ longest win streaks in comparison to the predicted streaks for a consistent win rate, which we calculated last time. If we assume that tilt stems from losing games, then players should be in peak form while on a winning streak, so if their peak win rate exceeds their average, then we’d expect to see a lot of players whose longest streaks exceed what simple statistics predict.

As it turns out, however, this is not the case. Looking at a sample of 25 winning heads-up players with between 4000 and 40,000 games and win rates between 51% and 59% (mostly in the 53-56% range), I found that only two of them had win streaks of 3 games or more above expectation and as a whole their longest win streaks were on average shorter than would be predicted, although only by a fraction of a game.

Here is a graph showing how these players’ longest win streaks stacked up compared to the statistical predictions – a positive number meaning the player had a longer-than-expected best win streak, and a negative meaning the player’s best streak was shorter than expected. As you can see, ignoring a couple of outliers, the overall trend is that players are performing as you would expect based on their overall win rate, with no emotional effects discernible.

This seems to suggest that tilt, if it happens, is a rare enough occurrence not to bring good players’ average win rates down very much – i.e. these players’ peak win rates are not much above their averages.

Whatever the case, it means that we’ll have to look at something other than longest streaks in order to find evidence for tilt affecting people’s play.

Frequency of medium-length streaks

Even if the effects of tilt on a player’s overall win rate are too small for the discrepancy to become apparent when looking at the longest winning streaks, you might expect them to show up in the player’s shorter streaks. Unfortunately, we still can’t consider the losing streaks for players who have non-heads up games mixed into their data: even if only 5% of the player’s games are anything but heads-up, it’s hard to estimate what percentage of, for instance, their 3-game losing streaks would come from those games and correct for that. You would expect it to be at least 5%, but possibly higher, and our uncertainty in that regard may turn out to be higher than the effect we’re looking for.

So, all we can do is continue to look at winning streaks. If losing games puts a player on tilt, we would expect their losses to be more clustered together than if the results were totally random. Since the only two possible results of a heads-up match are wins and losses, then if losses are more clustered together, the wins must be as well. A higher-than-expected number of short win streaks might therefore point to short-term clustering of losses as well.

Of the 25 players I looked at, I picked the ten for which I have the best data (i.e. most samples and fewest non heads-up games mixed in) and looked at their 3-, 5- and 7-game winning streaks in comparison to a simulation.

As you can see, the results for 3-game streaks look promising. Eight of the ten players have more 3-game win streaks than you’d expect for a consistent player and half of them have significantly more. Meanwhile, only one player is significantly below expectation. Since the players have widely varying sample sizes, we’re showing deviation from the predicted number of streaks as a percentage rather than a raw number.

Looking at the 5- and 7-game streaks, however, we see the opposite trend. Although a few players have significantly more of these streaks than expected, overall the trend seems to be that players have far fewer of these streaks than you would see for a stable win rate. It seems unlikely that winning four to six games in a row would put a player on tilt, so there must be something else going on to explain this; perhaps overconfidence is an issue, or maybe players have a tendency to move up and take shots at higher stakes and therefore tougher competition when they’re on a perceived heater.

Continuing the hunt

The results for the 3-game streaks look like there’s some kind of very short-term clustering going on, although this might have more to do with the ability to rematch a given opponent repeatedly, rather than anything happening in the player’s own head. It seems, therefore, that we’re going to have to find a way to look at losing streaks directly in order to continue our hunt for tilt.

Fortunately, among the 25 players I looked at, I did find one who has played nothing but heads up, and one who has only 3 non heads-up games out of 4818. Having only two data sets to work with means that we have to take any trends we see with a grain of salt, but it’s better than nothing. These two players will serve as case studies on which we can try to describe the effects of tilt mathematically.

Next time, we’ll see what stories their results history can tell us.

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