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A Note on Regression, The Gambler’s Fallacy and The Devils’ Kyle Palmieri

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Expect Kyle Palmieri to be something like his normal self the rest of the way, but don’t expect his season totals to look that way.

Pittsburgh Penguins v New Jersey Devils Photo by Bruce Bennett/Getty Images

Kyle Palmieri started off this season with a 9-game goal drought. That’s an unusual stretch for him — he’s endured a goalless streak that long only twice in the last 3 seasons. And he’s never exceeded a 10-game drought. It’s difficult to say whether it’s his age, the new system, or just plain bad luck that caused this; but I’m sure even Kyle would agree that it couldn’t come at a worse time. He’s in the final year of his contract and looking to finally cash in on being one of the most reliable goal-scorers in the NHL over the past half-decade and then COVID straps the team’s salary caps and he hits a dry spell.

You have to feel for the guy. But, it’s not as though he hasn’t shown significant signs of life since the tough start. You might not know it by looking at Twitter, but since his 9-game goalless start, Palmieri actually leads the Devils with 7 goals and trails only Pavel Zacha in primary points with 10. During that 23-game stretch, he’s shooting 12% — his career average is 12.6%. (Editors Note: He scored again against Boston so he’s got 8 goals now, but I’m not re-doing the calculations in the whole piece)

Because he still isn’t where we’d like him to be in the NHL leaderboard, and likely won’t be even by the end of the year, I thought it’d be a good opportunity to talk about what it means when we say Kyle Palmieri is “due for regression”.

First of all, while most readers of this blog likely needn’t be told, I still think it’s important to point out that, despite the connotation, “regression” can be either positive or negative. It simply means that we will expect numbers to trend towards previous expectations. But how that happens, (more precisely, how quickly that happens) is sometimes less intuitive.

Let’s imagine a fake player with a “true” shooting talent of 10%. Let’s say he misses his first 90 shots. How many of the next 10 do you expect him to make? You might be inclined to say “10” because if his true shooting talent is 10% and he missed the first 90, he needs to make the next 10 to get up to 10%. In reality, the past doesn’t matter. 10 shots are 10 shots and if he’s a 10% shooter, we expect him to score, on average, 1 of those 10 shots.

The 10-in-a-row mistake is actually a very common one — so common that it’s been given it’s own name: The Gambler’s Fallacy. It is essentially the mistake of believing that, if something happened less than expected in the past, it is more likely to happen in the future (and vice-versa). You’ve probably heard your friends say something like “he’s due” when describing a playing in a goal drought. If, by that, they mean that he’s more likely than normal to score because it’s been a while, they are guilty of the fallacy.

Now how does this apply to Kyle Palmieri?

Let’s start with the TL;DR version of this piece before I show you some graphs. Palmieri has never stopped shooting the puck and since the first 9 games, has been scoring on a typical percent of his shots. As such, we should expect him to produce at career-average goal rates the rest of the way, but since he started off quite a bit behind, we’d expect him to finish the season a few goals short of what we’d previously projected.

To model why it is that this is what we’d expect, let’s first define a few of the necessary stats. In his time as a Devil, Kyle Palmieri has averaged 2.67 shots a game with a shooting percentage of 12.6%. In a 56-game season, that’d correlate to about 19 goals. In the first 9 games, he took 2.33 shots per game and went 0-for-21. In the last 23, he took 59 shots (2.57/game) and scored on 7 (11.9%) of them. So his shot rate has not particularly changed and his shooting percentage over the past 23 games is along his career average.

Let’s simulate a hypothetical 56-game Palmieri season under the following assumption: he had 9 goalless games and then produced a career-average amount of shots and converted a career-average percent of them.

Technical Notes: I simulated 1000 seasons assuming shots were Poisson-distributed and goals were Bernoulli trials of those shots.

I gave the fake Kyles all 9 goalless games and then you can see that, on average, they creep towards Palmieri’s career average. They approach it asymptotically which means they’ll never quite reach it. This is especially pronounced in a shortened season where there is less time for regression. In total, of the 1000 simulations, the graph below shows where his year-end shooting percentage fell vs where his career average is. By virtue of having started off with 9 goalless games, being “typical” for the rest of the season means finishing the season with slightly worse than typical numbers (see below).

If you actually counted all those dots, you’d see that only Palmieri only eclipsed his career-average shooting percentage 17% of the time when he started without a goal through 9.

The shooting percentages are the real mechanism of the regression, but for a player looking for a new contract, the counting stats probably matter even more. And with shooting regression comes goal regression.

Below is a graph of how many goals Fake Palmieri had scored by each game of the season — the bigger the circle the more simulations that goal count occurred in. As you can see, all simulations started out with 9 goalless games and then he scored anywhere from 4 to 28 goals the rest of the way, but the average was right around 15 goals.

You’ll recall from the beginning of the piece that he was expected to score around 19 goals in the season. Just like in the shooting percentage example, it wouldn’t be bizarre for him to get to his career goal rate and put up 19-20 goals and nab himself the big contract (~17% chance). But it’s more likely is that he finishes with 14-15 goals given his slow start.

What I’m trying to say using all these graphs is that even IF Palmieri has not gotten any worse at shooting or scoring, he may still end up behind where we thought he would coming into the season, because that’s what happens when you start the season off with 0 goals in 9 games. You might wonder if the distinction matters. After all, 15 goals is 15 goals regardless of when the were scored, no? Well if you want to know if it matters, ask yourself if there’s a difference between these players.

Player A: A 30-year-old forward primarily known for scoring underachieves expectations by about 5 goals and 2 shooting percentage points.

Player B: One of the most consistent scoring wingers in the NHL over the past 5 years started off a season with a new GM, new coach, new teammates, a shortened camp, penalty-killing duties and 9 goalless games. He performed at a career-average level for the remaining 47 games.

Obviously, both of those players are Palmieri. And those players would likely get 2 very different contracts if the GMs granting them were unfamiliar with the Gambler’s Fallacy. Kyle Palmieri was due for regression and it’s coming ... but it might just be coming a bit slower than you think.