Ok, so long post ahead.
Goal-scoring and allowing goals in hockey is generally modeled to be a Poisson Process. Here's a primer on the Poisson Distribution and how it can model goal-scoring in the NHL:
http://www.hockeyanalytics.com/Research_files/Poisson_Toolbox.pdf
It's not a perfect model, but it's probably close enough for practical purposes.
So assuming goal-scoring and conceding each follow a Poisson distribution, and that they are independent of each other (this is not true, but again it's likely close enough for our purposes), we want to know what the likely goal differential will be when a certain player plays.
To model this, we use the Skellam Distribution, which is the difference in two Poisson random variables.
Skellam distribution - Wikipedia
Using players' current expected goals for and against/60 while leading, combined with the Skellam Distribution, we can calculate the probabilities of scoring or conceding in a 1 minute period of time on the ice for each of the 4 centers on the team. I chose 1 minute mainly because the math was easiest and also because it's "close enough" to the average shift length of an NHLer, which is about 45-48 seconds. Here are the results:
Player | xGF/60 | xGA/60 | P(-1 goals) | P(0 goals) | P(+1 goals) |
Nicklas Backstrom | 1.27 | 1.99 | 3.14% | 94.78% | 2.01% |
Evgeny Kuznetsov | 2.09 | 2.18 | 3.39% | 93.25% | 3.25% |
Lars Eller | 1.84 | 2.05 | 3.20% | 93.82% | 2.88% |
Nic Dowd | 1.66 | 2.06 | 3.23% | 94.08% | 2.60% |
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(Note that I ignored the probability of scoring or conceding 2+ goals because the probabilities are negligible. I doubt including them will have much of an impact on what follows.)
The probabilities in the table seem to make sense. On a vast majority of 1 minute increments no goals are scored. Backstrom is the lowest event center on the team in terms of xG, so he's more likely to have a 1 minute period of scoreless hockey than any of the other players. And on the other end, Kuznetsov is the highest event center in terms of xG, so he is least likely to have a 1 minute period of scoreless hockey. But since Kuzy has the best xGF/60 it makes sense that he has the best chance of having a 1 minute period where the Capitals net +1 goals. And since Backstrom has the worst xGF/60 while leading, it also makes sense that he is the least likely to have a 1 minute period of play where the Capitals have a net +1 goals. Similar results for xGA, where Kuznetsov fares worst and Backstrom fares best since they are the worst and best, respectively, in terms of xGA/60.
Given these probabilities above, it's also important to consider what effect scoring a goal, conceding a goal, or doing neither have on win probability given the current game state. I did a little Googling and found the following paper on modeling win probability based on time remaining in the game and score-state (ahead by 1, behind by 2, tied, etc.):
https://dlib.bc.edu/islandora/object/bc-ir:108029/datastream/PDF/view
Based on Christophe Bernier's model of win probability, and based on the the probabilities of scoring or conceding calculated for the top 4 centers on the team in the above table, here are the results for each of the players above in terms of expected win probability added for a 1 minute period of play in the third period, assuming the Capitals are ahead by 1:
So until about 6 or 7 minutes into the 3rd period it looks like Kuznetsov would be the best option to play under this model, which makes some level of sense since the game still isn't close to ending. Preferring the player with the best xG differential seems sensical enough when the end isn't near since getting a 2 goal lead still has a lot of value at this point in the game.
However, as the game gets closer to ending Kuznetsov becomes the least preferable player, and Backstrom, who has the best xGA despite having the worst xGF, becomes the best option with about 2 or 3 minutes left in the game. Again, this makes a lot of sense because conceding a goal will hurt win probability much more than scoring a goal will help win probability toward the end of a game when leading by 1.
But despite this, Kuznetsov still appears to be one of the top 2 center options until about 7 or 8 minutes left in the third period, and is still a top 3 option until about 6 minutes left.
Here are the results when up 2 and up 3:
These results are much less interesting IMO. All 4 centers seem to have a similar impact on win probability no matter how much time is left with a multigoal lead (note the scales are slightly different between these graphs and the Up 1 graph above).
So what do I make of this all in terms of Kuznetsov and third period deployment? If holding a multigoal lead it's probably just best to play all of the centers equally and that there's no need to bench anyone. This will leave everyone the freshest, and probably more able to defend a lead than if a few of them are double-shifting which will likely impact their performance due to exhaustion. If holding a 1-goal lead, then I would play Kuznetsov normally until about 6 minutes are remaining in the game, at which point he might become a riskier option to play and I could see taking shifts away from him being the sensical thing to do.
But based on the model above I would not be restricting his shifts as early as Laviolette has been doing, and especially since he seems to do this with multigoal leads where it doesn't seem sensical at all. His slightly worse defensive impact IMO doesn't make up for how it will hurt the other centers by having them double shift.
I plan on looking at defensemen next.