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Understanding Variance - with No Limit Holdem and Jose Mourinho
During my time playing online poker, I have learnt a lot about variance. Short term results in poker are effectively meaningless, and have almost no correlation to the strength of your process, so it’s important for a professional poker player to mostly ignore them. I’ve realised that this applies not just to poker, but to sport, business, and life in general. In an extremely results orientated world, there are huge edges to be found by focusing on the process, and choosing how to act based on underlying metrics rather than overvaluing short term results. In this article, I’ve focused on poker and a recent football match to illustrate the dangers of results orientated thinking.
To begin, consider the two charts below. It’s important to realise that when a poker player sits down to play a session, he has an expected, or average, outcome, but his actual results follow a random walk around that outcome. This is akin to flipping a coin. If we flipped ten coins, we could expect five heads, but in reality our results are likely to differ from that in a sequence of ten flips. If we repeated the experiment one thousand times, it would likely average very close to the expected outcome, but there would be some times where we flipped seven heads, and even some where we flipped ten! Each graph below is a representation of 20 such random walks for a certain poker player, with the x axis representing hands played, and the y axis representing units won.
Firstly, a typical month for me during my time as a professional. 60,000 online hands, over approximately 120 hours, at a win rate of 5 units every 100 hands;
Secondly, a typical losing player, playing a 2 hour session in a casino. They can expect their loss rate to be around 50 units per 100 hands, and they have a slightly higher standard deviation than a winning player;
These graphs illustrate the danger of relying on results from small samples. 11% of the time, or more than one month a year, I can expect to lose money after playing for 120 hours. On the flip side, 42% of the time, the losing player can expect to win in his session at the casino. From a financial perspective, the best thing for the losing player to do is to not sit at the table. If he does sit at the table, and wins 283 units (the best run in these simulations for him), the best decision for him is still to never sit at the table again.
This is a difficult concept to grasp, and the primary reason that people lose money gambling. Whilst 40 hands is obviously not a big sample, for the player who plays for two hours and wins 283 units (or in a typical 5/10 game, $2830), it is hard to comprehend they can expect to lose the next time they sit at the table. Many of us will know people who have bragged about such wins. If you responded to them by saying ‘your expectation is highly likely to be negative and you’ve just had a 95th percentile run of positive variance.’, it would probably not be received well! If the same person flipped ten heads in a row, chances are that they would happily accept that they likely wouldn’t repeat the feat at the next attempt. People can easily recognise luck in situations with no perceived skill, but in games like poker, where skill and luck are mixed, they tend to outweigh the contribution of skill.
In general, it is a human tendency to put inappropriate weight on short term results, disregarding the effects of variance. Another example can be drawn from one of the greatest premier league injustices of all time, an Arsenal loss to Manchester United in 2017:
For those who didn’t watch the match, Arsenal had 75% possession and 33 shots to Manchester United’s 8, but managed to lose 3-1. Predictably, Jose Mourinho, the Manchester United manager, was lauded for a great counter attacking display, and Arsenal criticised for their wastefulness.
However, there are obvious problems with the popular narrative;
The chart above (produced by football consultant 11tegen11) illustrates the ‘Expected Goals’ over time in the match. This is a statistic used by clubs to measure how effective their process was. The ‘Expected Score’ is a better predictor of a teams long term results than the real score. For anyone interested in learning more about this metric, Opta have made a great explanatory video. We can see that although the real score was 1-3, the expected score is closer to 4-2. So as the manager, should Mourinho care more about the actual score, or the expected score when he assesses how his team performed?
Obviously, the actual result was positive. But huge premier league databases tell us that with the underlying statistics it is not repeatable. No team, or player, has been repeatably able to score significantly more than their expected score, or concede significantly less. So his team shouldn’t be congratulated for being ‘clinical’ when being clinical isn’t a repeatable skill. If Mourinho recognised this, he could have acted on it and prepared accordingly for the next match. However, if he focused on the actual score, ignored the underlying metrics and concluded his strategy is good, they are unlikely to get the same run of positive variance again.
Clearly, poker and football matches differ slightly from other life situations. But there is variance in all walks of life. Merely understanding that results are the output, the noise, of an underlying process, the signal, can give you a big edge in decision making. For the amateur poker player who had a lucky win, returning to the felt might cost him a few hundred dollars before he realises his mistake. For Jose Mourinho, it was a heavy defeat to Manchester City the next weekend, and ultimately his job in 2019. But as a professional poker player, focusing on the noise and ignoring the signal can mean abandoning a good strategy or continuing a bad one. Sometimes, to deliver results in the long term, it can be best to turn a blind eye to results in the short term.