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Narratives and Random Number Generators

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Ordering and it’s impact on how we view our team

NHL: Colorado Avalanche at Toronto Maple Leafs Tom Szczerbowski-USA TODAY Sports

I’ve always been very interested in the narratives we construct about sports. It’s a bit glib, but there’s an xkcd comic that I’ve always found both funny and true:

Randall Munroe / xkcd.com/904

Of course, it’s quite robotic to view the (very real) players, coaches, and staff who play these sports as a ‘weighted random number generator’, but it would not be unreasonable from a mathematical perspective to model their efforts as some sort of random variable, with parameters determined by the abilities of said players, coaches, and staff. In this sense, the comic is exactly right, as dehumanizing as it may be.

That said, sports would be much less fun if we rooted for random number generators... we root for humans, or at least, what the humans represent. This is what makes sports enjoyable — it’s what gives us some sort of skin in the game. And it’s what make sports so ripe for narrative.

Due to the fact that we care about the human element of sports, we tend to interpret changing results as being caused by the underlying human behaviour changing, rather than a list of outputs spit from a random number generator. This leads to narrative.

We can apply this to the Leafs. I’ve made a line graph indicating the Leafs’ 5v5 shot attempts for and against for every game this year. These quantities are smoothed using the Hanning Smoothing Method. If you’re interested in seeing an implementation of this smoothing method, I would recommend this workbook.

These are essentially the same as one of the charts that are produced by Micah McCurdy at HockeyViz.com. You can see his version of the same graph for the Leafs here. I was obviously heavily influenced by his design, and I would recommend checking out his site and supporting it via Patreon if you can. McCurdy is invaluable to the hockey analytics community, and supporting him helps keep his work up and public.

Anyways, lets take a look at this graph. We can see that the Leafs exploded out of the gate; in their first seven games or so, they were destroying their opposition. Since then, it’s gotten much less rosy. The Leafs have had stretches where they were getting heavily outshot by their opponents, and haven’t again reached the heights of shot differential experienced in their first handful of games.

This has led to the pervasive narrative that the Leafs, as of right now, are underperforming*. In particular, it’s been said that they lean too heavily on their depth players, that they no longer play with speed, that they’re becoming a dump and chase team, that they’re not as fun offensively. Babcock tried to make them a ‘grind-it-out’ team and took them away from what made them great.

* It’s worth pointing out that if a team is ‘underperforming’ for three months relative to a two-week stretch, we should probably start treating the three-month stretch as their average and the two-week stretch as a best case scenario.

Except.... look at that chart again. The Leafs offense (measured by shot generation, at least) has hit the heights of the first seven games numerous times this year. Look at the black bar around games 25-40. In that stretch, the Leafs hit their high in terms of shot generation this year, and were never once below their lowest point of shot generation in the first seven games. Even outside this stretch, there are plenty of times where the Leafs have been in the neighbourhood of that early season burst of offense. It’s not their mean, but when they’re running hot, they get to that range, and they’ve done it at various points throughout the season.

If your argument is that the Leafs should always be there, and that the start of the year should represent their average and not their high-water mark... well, that’s a bit presumptuous. The best offenses in the league this year generate in the low-60s in terms of shot attempts per hour. The Leafs are at about 59; that’s not a huge difference, especially since the Leafs take relatively few point shots. Even for the best offenses in the league, stretches of shot attempts in the mid-60s and beyond are rare.

Where the Leafs have actually struggled is on the shot suppression side (for ease of reference, I’ll just call this defence throughout). They started the season in holding teams to shot attempt rates in the low 50s. Since then, they’ve only flirted with that mark for a game or two at a time, if at all. The narrative that the Leafs have tried to trade off their offense for their defense is therefore a weird one. If that IS what Babcock is trying to do, he’s failing miserably. Because the defense has gotten worse since then, and is the main culprit for the team’s mediocre shot metrics over the last few months.

What actually happened to the Leafs then? Simple. Their defense got worse, while their shooting percentage and save percentage began to normalize slightly. All of which is explicable without the narratives mentioned above. The Leafs were never going to shoot above 12% at 5v5, as they did in their first few games. Andersen was never going to maintain his Toskala-level numbers from the first month of the year. And it’s not terribly surprising that a defence corps playing Ron Hainsey on their top pair isn’t that good. The Leafs defense has a lot of guys playing one spot or two above where they should be.

In my opinion, a lot of the reason for this narrative has to do with the order in which these games occurred. As I said earlier, we tend to feel changing results are caused by a corresponding change in the process (in this case, the players or coaches), rather than being the result of variance. So when the Leafs start by destroying teams and then fade back to mediocrity, it’s in our nature to think of why that might be. And conveniently, Babcock gave many quotes in that time about making the Leafs better defensively, not cheating for offense, and about being responsible. It helps that buying into this narrative conveniently excuses the Leafs of three months of uninspired play, chalking it up largely to coaching decisions and lineup optimization rather than the roster (this is the same reason every underperforming team’s fanbase wants to fire the coach, regardless of how culpable they are).

Consequently, I think it’s fair to say the order of the Leafs’ results this season has had a large impact on the narratives that have resulted. This would be true of any ordering, really. You could construct a wildly different narrative from the same team, simply by switching the order of the games. That’s exactly what I did here:

I simply took a random permutation of the Leafs games this season, and made the same chart as before for this reordered set. In this bizarro world instead of a hot start, we’d see the team scuffling until game 25 or so, largely due to horrid defensive results. But at game 25... BAM! Babcock figures something out. The team sees improved defense, while maintaining, no, improving their offense. If they keep up this pace, they could be a contender!

Except, it’s the exact same team, the exact same set of games, and the exact same results. The only thing that changed is the order. Yet it’s hard not to think that the prevailing thoughts behind the team, the coach, and their potential would be notably different. And this is a random order! If I cherry picked this, I could find a sequence of games to construct just about any narrative you want.

Of course, teams AREN’T as simple to model as random variables on a game to game basis. The players do change. Some get injured (Nikita Zaitsev), some get better during the season (Tyler Bozak), some get worse (Andreas Borgman). There are real, underlying changes to the human side of the team, which impact the on-ice results we observe. In that sense, it makes sense to look at trends and see how a team changes over a season.

However, if you look too deeply, you’ll end up chasing trends that are just ghosts. As shown, even a random ordering of games can result in data that looks like it is trending in a given direction. Sometimes, a team just has a good stretch or a bad one, whether offensively or defensively. The Leafs started off the season on a good stretch for both. They’ve since replicated the offense, while failing to do the same on defense. That should be the narrative of the Leafs season, so far.