This post originally appeared on Hockey Graphs.
The NHL is in the middle of a goalie pulling frenzy. While the year is still young, coaches of teams who are losing by a goal have been pulling their goalie roughly around the 1:40 mark of the 3rd period the last two years, about 40 seconds earlier than they were in previous years. This development, of course, is a long time coming – analysts have been arguing for years that teams should be more aggressive in removing their netminders.
The main reason that teams have been pulling their goalies earlier is the difference in the value of a goal for and against when teams are losing. If you’re down by 2 in the third period, allowing another goal hardly impacts the amount of points you expect to earn, while scoring a goal can be a major boost to your expected points.
This is something that Micah McCurdy has written about extensively, and his idea of leverage shows how the incentives to play offense or defense change as you approach the end of the game depending on the score. Leverage is simply the change in expected standings points due to a goal for (Offensive Leverage) or against (Defensive Leverage) at a given point in time for a given score differential. The plot below shows how leverage changes for both the home and away teams depending on the score and time left in the game.
This change in approach is exciting from an analytics perspective, as it represents a tangible way that numbers can be used to actively advocate for demonstrably better strategies. And the adoption of this more aggressive but technically sound approach leads almost naturally to a new question: should teams pull their goalie when they’re trailing late on the power play?
Answering this question obviously requires us to make a lot of assumptions – after all, no one is really doing this, so we’re going to have to guess how it’ll work in a real game to do the math on when it’s worthwhile and when it’s not.
The first assumption that we’ll make is that teams will play cautiously with this strategy, and that they’ll replace their goalie with 30 seconds left in the power play if they haven’t scored by that point in time. This may be difficult to do in practice, but in our simulation we’ll assume that teams are able to replace their sixth skater with a goalie instantaneously at the 1:30 mark and we’ll update the scoring rates we’re using accordingly at that point.
We’ll also assume that scoring rates for and against are consistent with what we’ve observed in the past. We’ll use 6v4 goals for and against per 60 for play in the last 2 minutes of the game as our baseline rate since 6v4 play earlier in the game is almost entirely delayed penalty calls which will obviously decrease the rate of goals against per 60. And we’ll use overall 5v4 scoring rates as our baseline scenario, and as the expected scoring rate for the last 30 seconds of play. We’ll assume that scoring rates are constant across the whole power play which is obviously false, but probably defensible for the purposes of this analysis.
With these assumptions we can start by digging into the scoring rates to see how much of an impact playing at 6v4 has on goal scoring.
Goals Per 60 | 5v4 | 6v4 | 1:30 6v4 + 0:30 5v4 |
For | 6.4 | 12.6 | 11.0 |
Against | 0.8 | 8.7 | 6.7 |
Looking at the numbers it’s clear that (as we’d expect) scoring both for and against goes way up when you pull your goalie. Goals For per 60 at 6v4 are nearly twice that of 5v4, while Goals Against per 60 are more than eight times higher when teams play with an extra attacker. Clearly this is a high risk strategy, but does the payoff justify the cost?
To get that answer we’ll have to look at the offensive and defensive leverage charts and apply the scoring rates above to calculate the impact on expected points we’d expect to see from using each strategy for a power play starting in a given minute. The formula for this is simple: for a strategy S (either 5v4 or 6v4 + 5v4) and a power play starting at minute M, take the offensive and defensive leverage at M + 1 (we’ll assume that goals will occur on average in the middle of the power play) and multiply those numbers by the probability of scoring or allowing a goal on a 2 minute power play using that strategy.
xPts Value (S, M) = P(GF | S) * Off Leverage(M+1) – P(GA | S) * Def Leverage(M+1)
When you do this for both 5v4 and the hybrid 6v4/6v4 strategy above, we can plot the expected power play value (in standings points) at any given point in time to see where the hybrid goalie pull strategy might be effective.
As you can see, the value of the hybrid strategy is highly dependent on the score, venue, and time remaining. When teams are down by 2, there’s a good argument for pulling the goalie on a power play at any point in the 3^{rd} period, with some small positive value appearing as early as the 15 minute mark in the 2^{nd} period (5 minutes remaining). On the other hand, when teams are down by only 1 goal, the model suggests more caution, with visiting teams seeing positive value in pulling the goalie on a power play when there are 15 minutes left in the 3^{rd}, while home teams see the break-even point closer to the half-way point of the 3^{rd}.
We can also look at the problem by plotting the difference between the 5v4 and hybrid-6v4 strategy to look at the net benefit of pulling the goalie on the power play. For teams down 2, that peaks out around 0.025 standings points for an away team (fairly consistent throughout the 3^{rd} period), and 0.0375 for a home team around the half-way point of the third. When teams are down by 1, obviously the value continues to increase as the game goes on, as there’s a strong argument to keep the goalie out even when playing at 5v5.
While an extra 0.025 standings points may not seem like a big deal, it does help to view that number it in context: a team down 2 at home with 15 minutes to play is expected to earn roughly 0.19 points anyways, so a boost of 0.025 is effectively increasing your expected points by ~13%. With 10 minutes left, that same team should expect to earn just 0.11 points, so the 0.039 extra points you earn are increasing your expected points by 35%.
You could also look at the value of this strategy it in terms of dollar value.- while assigning a dollar value to a standing point is tough, Patrick D of Fear the Fin put it at roughly $1.25M/point in 2013-14 which works out to ~$1.5M/point in 2018-19 dollars. Josh and Luke Younggren also put a rough estimate of the cost of a win last year at ~$4.8M/win, or $2.4M/point. If we split the middle on these two estimates and call it $2M/point we get the following estimates of the dollar value of executing this strategy at any given time.
These aren’t huge numbers, but even trying this strategy four-to-five times per year is worth more than the cost of a well paid NHL analyst nearly half the cost of a player on a league minimum deal. It’s not the same as adding a superstar to your team, but every marginal benefit can help in today’s NHL.
Of course, there are many reasons why this strategy might not work. The data we used at 6-on-4 was entirely from end-of-game scenarios where the defenders may be wary of pushing hard for a goal and may be more content to just get the puck out of their zone than they would be with more than 5 minutes left to play.
It’s also possible that the specific act of replacing the goalie on the fly may be more difficult than it sounds on paper. Teams will have to both designate a player to be responsible for leaving the zone around the 30 second mark, and will need to make sure they don’t errantly pass the puck to that player around the time they’re supposed to be leaving the ice (not an easy task in the middle of the flow of a power play).
Nevertheless, the math seems to show that this might be a strategy that could be worth trying out for teams. Do I expect teams to jump on board and take this up overnight? Not at all. But with the trend towards more aggressive coaching when teams are down late, it’s only a matter of time before someone gets crazy enough to let me try my weird experiments on their AHL team decides that the extra edge may be worth the odds looks they’ll get when they first try it.
The whole thing is a worthwhile read, but the one line that stood out to me was this:
The point is that [Roman Polak and Andreas Borgman] got time against the opposition’s top line in circumstances in which that line didn’t really pose an attacking threat.
The overall idea that Tyler presented is pretty intuitive but also incredibly important: if you come on the ice when your opponent is finishing their shift, they’re unlikely to be mounting much of an offensive push, and most of their effort will likely be focused on advancing the puck out of their zone and getting of the ice.
The implication of this is that traditional measures of quality of competition might be understating the actual differences in QoC between players. Ten seconds of ice-time against Connor McDavid when he’s trying to get off the ice isn’t the same as 10 seconds of ice time against Connor McDavid when he’s trying to score a goal.
The key question, however, is whether there is an actual drop in shot rates at the end of a shift, and how long that drop lasts for. We know from past work by Micah McCurdy and Gabe Desjardin how shot rates change depending on how a shift starts, but how they change at the end of a shift is a relatively unexplored area.
The plot below shows how shot rates change relative to the time remaining in a player’s shift (the right side of the chart is the end of a player’s shift). I’ve limited the analysis to shifts that ended with an on-the-fly change, since shot rates at the end of a shift that results in a stoppage are extremely high (these are mostly shots right before the puck is frozen) and face-off starts are obviously a whole different ballgame.
As you can see, the data seems to be fairly in line with what Tyler was arguing – shot rates against go up at the end of a player’s shift while shot rates for drop slightly. We see a slight rise in both rates for and against in the last 1-2 seconds, which are likely just shots that are occurring as a player is leaving the ice and are unlikely to be related to their individual efforts.
When we look at the data broken down by position, however, we see a more nuanced situation which gives us a better view of how different roles impact a player’s results.
While the trend we saw above seems to hold for defencemen, the progression of a shift for forwards is much different. Defencemen tend to outshoot their opponents in the early part of their shift, but mostly get outshot from the last 20 seconds of their shift onward.
Forwards, on the other hand, have a fairly even shot attempt ratio through most of their shift, with the overall pace of play dropping off towards the end of their shift. The key thing to note here is that if you’re coming onto the ice for the last 10 or so seconds of an opposing forward’s shift, you’re probably not going to be seeing the best effort from them.
These trends even hold when we break things down by quality of player. In the chart below I binned (yes, I know) players by average TOI rank to create four lines for forwards and three pairs of defencemen (this is an imperfect method, but should give us a reasonable view of what’s going on).
The data shows pretty much what you’d expect – each line and pairing shows the same trend, with the major difference between the top and bottom 6 being that top 6 players tend to outshoot their opponents early on, while bottom 6 lines get outshot throughout their shifts. The key again though is that even first and second line players see drops in their output towards the end of their shifts.
All of these results have clear implications for many of the regression-based models that are being developed today. These models treat each interval (a consecutive period with the same 10 skaters on the ice) as a row in their input, and have dummy variables for each player on the ice and to account for the impact of how the interval started (e.g. defensive zone face-off or on-the-fly). The regression then attempts to find the coefficients for each player and starting state that best explain the observed results (generally Expected Goals or Corsi, for or against).
But what this analysis suggests is that we might need to know additional information about how long a player has been on the ice in order to adjust for a player’s “intent” over the interval. Players who have just started their shift likely intend to generate chances, while players who have been on the ice for more than 30 seconds may only intend to get the puck out of their zone and get off the ice. If I’m a player who consistently plays against top line players only at the end of their shift I may be getting credit for shutting down these players (who otherwise have good offensive results) when in reality their suppressed offensive output may simply be a structural product of how the flow of the game works.
The big question, of course, is whether the differences in which players face better or worse opposition at the end of their shift are meaningful (spoiler alert: I don’t know). Tyler’s data certainly suggests that they are, but there’s likely more digging to be done to determine the extent that this phenomenon impacts the metrics that we use to evaluate players. Nevertheless, these results clearly show that there’s more about the structure and flow of the game that we can mine from the existing data, and that quality of competition may be more complex than previous attempts to measure it have suggested.
]]>It’s the Olympics again, which means it’s time for everyone’s favorite activity: watching Canada underperform at ice-hockey! And while Hilary Knight breaking the hearts of Canadians is fun for everybody, the only thing that’s more fun is watching Hilary Knight break the hearts of Canadians while you have a statistical model that predicts each team’s likelihood of winning a medal! That’s right, Hockey Graphs is taking on the challenge of predicting the Women’s Olympic Hockey Tournament results.[1]
While the US enters as the betting favorites having won 4 consecutive World Championships and the last three 4 Nations Cups, their archrivals from north of the border haven’t lost a game at the Olympics since the 1998 finals. Although pretty much everyone expects the big two to meet again in the finals, the US’ failure to make the finals in the 2006 winter games offers proof that nothing is guaranteed in a short tournament.
Ideally, we’d use every Olympian’s past statistical performance to estimate each team’s overall strength; however, because of the scarcity of historical women’s hockey data available online, the model that we’ll build won’t be as detailed as we’d like it to be. We won’t use any individual-level data, which means the model has no idea that Alex Carpenter and Megan Bozek won’t be joining the US squad in PyeongChang this year.
Instead, we’ll build our model using past international results from the Women’s World Championships since 2009, Olympic results since 2002, and the International Friendlies results tracked by scoreboard.com since 2012. We’ll take the regulation outcomes of each game and use those to predict the goals for and against each team based on their past offensive and defensive performance, who they’ve played and whether they’re the home (or host) team.
We’ll use a mixed effects model which will shrink the team offensive and defensive factors towards the mean and also help deal with the smaller sample sizes we have for some countries. In order to account for each nation’s changing strength and roster over time, we’ll weigh recent results more heavily and reduce the weight for friendly competitions (which may not have featured the best possible rosters each country could’ve put together). We’ll decrease the weight of past games by 70% for each year since they occurred and weight friendlies as being 50% as important as World Cup/Olympic games.[2]
So who does our model like this year?
With the help of @Cane_Matt, who made a model for me from chicken wire and lint, here are some probabilities for th… twitter.com/i/web/status/9…
—
Micah Blake McCurdy (@IneffectiveMath) February 09, 2018
The Americans enter the tournaments as favorites, with a 52% chance of taking home the Gold, followed closely by Team Canada at 45%. The US are actually favored to beat Canada 56% of the time in a head-to-head matchup on neutral ice, but their odds are taken down slightly by likely having to play Finland (who our model thinks are the third strongest club) in the semi-finals.
While the model has clear favorites in Division A, Division B is more of a toss-up, with each team having at least a 43% chance of advancing on to the quarter finals. The wildcard may be Korea: with little data on how the unified team will play, the model is assuming that each country’s individual results will be a good proxy for how they’ll perform in the future. This may be understating their actual ability, however, as the combined team will likely be stronger than either individual squad.
One neat part of our model is that the team strengths will continue to update as the games are played. This means that as the model gets more details about who is playing well this tournament, our estimates for the win probabilities in the games left to be played will also update.
Want to follow along with the fun? Hockey-Graphs alumnus and “Scotch from a shot glass” enthusiast Micah Blake McCurdy will be tweeting out updated predictions before and after every game.
[1] It should really just be called the Olympic Hockey Tournament, since we all know the Men’s tournament is going to be a snooze.
[2] These weights were determined by training the model using a range of weights on all games before the 2014 Olympics, and finding the weights that produced the lowest logloss on games after the 2014 Olympics.
]]>While the whole panel was endlessly informative and extremely entertaining, one thing Daryl Morey said about the Rockets approach to implementing strategies really stuck with me:
“One thing I think all of us have done is more take the lessons that are sort of obvious that everyone has agreed to and taken them to the logical conclusion. Which is, for example us, it’s better to make 3 than 2 on a shot…genius, right…taking it to it’s logical conclusion, which is shoot 50 of them a night.”
Basically what Morey was saying is that if all the work you’ve done has shown that a strategy is beneficial, don’t hedge your bets and only go half-in on it. It’s this idea that’s been floating around in my head for the last 2 days since I wrote about optimizing contract structure – if GMs could theoretically save cap space by setting up contracts to pay more money to players upfront, how much total cap room could they create simply by structuring their contracts in the most efficient way possible?
The answer, of course, depends on the assumptions you make (since no one has been foolish enough to let me run my crazy experiments on an actual NHL team). First and foremost is the impact of the discount rate – higher discount rates make this a more effective cap optimization strategy, while lower discount rates have less total benefit. But this is easy enough for us to test out – we can simply run our analysis using various discounts and observe the range of impacts the each discount rate gives us.
There’s also the question of what the optimal contract structure is. While the ideal contract from a player’s point of view would be something like 99.9% of the money in the first year with the remaining cash spread out over the last N-1 years of the deal, the CBA has certain rules to prevent this kind of cap circumvention. Specifically, there are 2 major criteria that all contracts have to meet with regards to when payments occur:
We can add 2 other criteria that are necessary to ensure that the NPV is maximized while minimizing the actual dollars spent:
While these rules give us a general sense of what the best structured contract looks like, they don’t give us an exact answer as to what the optimal contract structure is. When the length of the contract is 1 or 2 years, the optimal structure is easy enough to define. For a 1 year deal, the AAV is the total salary in the first year, so we don’t have anything to do – there’s no way to actually optimize it. For a 2-year deal, the optimal structure is to pay P in year 1, and 0.65P in year 2 – that’s the biggest drop you can get, and we want to move as much money forward as possible.
But for contracts that are 3 years or longer, there are actually many ways to structure a contract that meet the rules we established above, but that aren’t necessarily optimal. For example, if we call the salary in the first year of a contract P, we could simply decrease the salary of contract in a straight line until we hit 50% at the end of the contract (and then solve for the value of P that makes the NPV equal to the NPV of the actual contract signed). While this seems like a logical solution in theory, in practice it’s actually not aggressive enough in front-loading contracts, and many current deals are actually better structured as they exist already.
One method of structuring that’s mostly optimal[1] goes like this:
This won’t always give us the most optimal contract structure, but it will generally be a non-trivial improvement over how contracts are currently structured. We can solve for P in the same way we described above: simply find the value of P that makes the NPV of our optimal contract the same as the NPV of the actual contract. We can then find the AAV as:
AAV (Optimal) = 1/N * [P * floor(N/2) + 0.65 * P + (N – floor(N/2) – 1) * 0.5 * P]
As a simple example, let’s look at P.K. Subban’s most recent contract that he signed with the Montreal Canadiens. That deal has an AAV of $9M per season, but is structured in far from an optimal manner: the bulk of the payments occur in the middle of the deal. In theory, the Habs could have offered to pay him more in the first 4 years in order to knock down the total cost (and with it the AAV). But what would the optimal way to structure it be?
If we assume a discount rate of 5%, the NPV of Subban’s deal was $60.9 million dollars when he signed it. Using the structure we described above, he could get the same value if he signed a deal that paid roughly $11.3M in years 1-4, $7.3M in year 5, and $5.65M in years 6-8. The AAV of that deal would have been ~$8.7M, saving the Habs Preds $300k per year – not a bad chunk of change just for rearranging some payments.
If we repeat this exercise for each player who has signed since July 1, 2014 and total up the savings by team, we can get a reasonable estimate of what optimizing each contract’s structure could be worth to a GM.[2]
Discount Rate | Min Cap Hit Saved Per Team | Average Cap Hit Saved Per Team | Max Cap Hit Saved Per Team |
2.5% | $0.06M | $0.38M | $0.83M |
5.0% | $0.12M | $0.75M | $1.62M |
7.5% | $0.18M | $1.11M | $2.37M |
10.0% | $0.24M | $1.45M | $3.07M |
As we noted above, the impact is highly dependent on the assumptions you make about the time value of money, but it seems to me that there could be some value to this strategy, particularly if you’ve got an owner with deep pockets. The average team would save nearly $400k under the most conservative of assumptions, an amount which could be the difference between adding that marquee player at the deadline and sticking with your current roster.
Now there are obviously a few caveats to this analysis that could reduce the potential impact. First, as we mentioned above, you’d need to have a very nice owner to give you the financial flexibility to structure deals like this. This may be a challenge in cash-strapped markets, but this strategy could actually be a good way for teams whose potential spending is being restricted by the salary cap to flex their financial muscle a bit.
Second, as I noted in my last piece, the savings from front-loading need to be weighed against the potential additional costs if you need to buyout a contract or if a player retires. If we exclude players who are 30 or older when they signed, the expected cap savings tends to drop by $50k-200k, depending on the discount rate. That’s not enough to remove all merit from the strategy, but it does knock away some of the benefit that we noted above.
Third, it’s not necessarily clear that players would be willing to accept a lower cap hit, even if it was in their best interest financially. Players may be more concerned with their cap hit than the actual financial details of their contract, and may be reluctant to accept a deal that makes them look worse than one of their peers.
Nevertheless, it does look like there could be some cap benefit to a team with an open-minded owner who’s willing to take a risk. While correctly evaluating a player’s future performance will always be more important than these kind of accounting tricks, finding new ways to squeeze a bit of extra value out of your limited cap space may give team’s just enough room to add that piece that pushes them over the edge.
[1] There are only 3 contracts that are “more optimal” in their current state than this one, so I feel pretty safe saying this is pretty close to being optimal.
[2] Excludes contracts under $1MM and the Vegas Golden Knights.
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For players, there’s a strong incentive to get as much money as soon as you can. Not only does this protect you against a buyout to a certain degree, but because of the time value of money the actual net value of the contract is greater to you the more you front-load your contract. If a player has the choice between a 3-year deal where they’re paid $4M each year, and a 3-year deal where they earn $5M in the first year and $3M in the last year, they should opt for the latter deal, since an extra dollar now is worth more than an extra dollar in the future. The AAV on both deals is the same, but the player comes away with more value if they get their money sooner.
For owners, obviously, the incentive runs the other way – you want to back-load your players’ contracts as much as you can, since you can take any amount you save in year 1, invest it until year N, and then pay the player using the principal while pocketing any interest you’ve earned. But if you insist on pushing salary later in a deal, players may ask for more money in total to compensate, inflating the overall AAV on the contract.
Because of this, it’s not really fair to compare contracts that have the same AAV but different payment structures. If two players have a 6.5M cap hit, but one player is getting more of their money earlier, it likely means that their team believes the player getting paid sooner is more valuable.
While it’s easy enough to compare two deals that have the same cap hit but different payment structures, it gets a bit more difficult when the cap hits, or even the contract lengths, are different. One way we can make a fair comparison, however, is by calculating the net present value (NPV) of a contract, and then using the NPV to figure out what the equivalent cap hit would be, if all the payments were the same amount over the life of the contract.
While this may seem a bit complicated (and perhaps even unnecessary), a simple example shows how this can impact a team’s cap position. In his most recent deal with the Lightning, Steven Stamkos’ signed for an average annual value (cap hit) of $8.5M per year, earn $64M total of the course of the 8 years of the deal. But because of the way it was structured, Stamkos will earn nearly 75% of that over the first 5 years, taking home just $20.5M in the last 3 year seasons. How much money cap hit would he have commanded if the Lightning had insisted on equal payments over the life of the deal?
If we assume a 5% discount rate[1], the total payments on Stamkos’ deal had a net present value of just over $58.5M when he signed the deal. To create an 8-year deal with the same present value and with constant payments each season, you’d need to pay an average of $8.625M per year. Not a huge difference, but that means that the Lightning saved roughly $125K on their cap just by structuring the deal in a manner more friendly to the player. If you’re able to do that for five long term deals, you’ve saved yourself the cost of a whole ELC.
Which players have seen their cap hit reduced the most by structuring? The table below has the 10 deals with the greatest savings from structuring since the July 1st, 2015.
Player | AAV | Equal Payment AAV | Savings From Structuring |
Carey Price | $10,500,000 | $10,804,266 | $304,266 |
Jamie Benn | $9,500,000 | $9,799,430 | $299,430 |
Anze Kopitar | $10,000,000 | $10,287,619 | $287,619 |
Brent Burns | $8,000,000 | $8,219,435 | $219,435 |
Connor McDavid | $12,500,000 | $12,693,446 | $193,446 |
Brent Seabrook | $6,875,000 | $7,066,703 | $191,703 |
Ryan O’Reilly | $7,500,000 | $7,666,430 | $166,430 |
Brad Marchand | $6,125,000 | $6,274,331 | $149,331 |
Andrew Ladd | $5,500,000 | $5,647,751 | $147,751 |
Milan Lucic | $6,000,000 | $6,146,556 | $146,556 |
Unsurprisingly, big value deals for longer terms have saved teams the most cap space, but the list of names here also raises a slight problem with the “pay them early strategy”, and that’s the possible increased buyout cost or cap recapture penalties. While ideally a GM’s player evaluation skills are so good that they never run into a scenario where they no longer want a player that they signed to a long term deal, a quick look at the table above shows quite a few names that may be candidates for buyouts in the future.
As such, this buyout/retirement risk needs to be considered alongside any initial cap benefit that you might get from front-loading a deal, and giving front-loaded deals to players past their prime should almost always be avoided as the savings may not be worth the eventual cost.
We can also look at which deals have theoretically left teams with less cap space than they otherwise would have had on a balanced deal.
Player | AAV | Equal Payment AAV | Cost From Structuring |
Michael Matheson | $4,875,000 | $4,741,956 | -$133,044 |
Adam Larsson | $4,166,667 | $4,086,521 | -$80,146 |
Damon Severson | $4,166,667 | $4,087,939 | -$78,727 |
Brandon Saad | $6,000,000 | $5,933,451 | -$66,549 |
Oscar Klefbom | $4,167,000 | $4,103,884 | -$63,116 |
Aleksander Barkov | $5,900,000 | $5,837,267 | -$62,733 |
Nathan Mackinnon | $6,300,000 | $6,249,147 | -$50,853 |
Alexander Wennberg | $4,900,000 | $4,850,978 | -$49,022 |
Hampus Lindholm | $5,250,000 | $5,206,106 | -$43,894 |
Vincent Trocheck | $4,750,000 | $4,711,394 | -$38,606 |
There are two things to note here: first, the list of names on here are almost exclusively from clubs that rank towards the bottom of the league in attendance, meaning that these structuring decisions may have been made by ownership rather than the front office.
Second, the magnitude of the “costs” here are much smaller than the savings from front-loading deals. This may reflect the fact that while owners would obviously prefer to push back payments, the people negotiating these deals (GMs and players) often see the same benefit from front-loading them, and so there are simply fewer deals that significantly push money back into later years.
The last thing we can look at is which teams have saved or spent the most cap space on structuring over the last few seasons.
We see a bit of an expected trend here, as most teams have seen a net cap benefit from pushing payments forward, while some of those with weaker finances have actually theoretically paid more to delay some of the cash payments until later. For most teams it doesn’t amount to a significant savings, but some clubs (Dallas, Montreal) have seen more than half-a-million in theoretical savings (although whether they’re really savings is debatable when you consider that they’re locked in with Ben Bishop and Carey Price in net for the 2022-23 season).
While there clearly may be benefits to teams to ensuring they structure their deals properly, how big those benefits are is heavily dependent on the assumptions you make about the discount rate. We’ve used a 5% rate here, but if it’s higher (i.e. good markets) the benefits jump, while in lower growth environments, the benefits of structuring will be muted.
We’ve also significantly over-simplified our analysis here with our handling of signing bonuses. Since signing bonuses are paid out on July 1st of every year, while salaries are paid over the course of the season, they’re actually even more valuable than regular salary to a player. While owners may be hesitant to give out signing bonuses (there’s a risk you pay a significant cost to a player without having them actually play for your team if you trade them before the season starts) they’re a significant way in which teams can offer more value to a player without seeing a corresponding jump in cap hit.
[1] Entirely and completely arbitrary, but probably somewhat reasonable.
]]>While defensemen are often taught to take the pass and leave the shooter to the goalie, Gardiner’s execution left a lot to be desired. As Justin Bourne noted, when you give the shooter that much space you’re basically turning a 2-on-1 into a 1-on-0.
Beyond the poor execution though, there’s a more general question about the wisdom of playing the pass and not the shot. Taking the pass and trusting your goalie to stop the shot is an idea that’s drilled into defencemen’s minds from a young age, and on its face it makes a lot of sense – we know that shots of passes go in more often than those off shots, so if you’re looking to minimize goals against (a pretty good idea for a defencemen) your best bet is to take away the higher percentage play.
The problem with this thinking though is that if you play the pass every time it will start to affect your opponent’s behaviour. Shooters will become more aggressive and work towards better shooting locations, which will start to eat away at some of the advantage from defending the pass. There’s a balance to be struck between preventing the more dangerous play and becoming too predictable. Playing the pass more often will probably always be the right play, the question is how often defenders should change things up and play the shot.
On the flip side of things, attackers face a similar decision when they choose whether to shoot or pass. A player with the puck on a 2-on-1 will obviously prefer to setup the higher percentage opportunity (or should prefer to a pass at least, I’ve certainly played with players who would shoot 100% of the time), but if they choose to pass every time they become easy to defend. We know that how each player acts will impact the other’s optimal decision, the question is whether we can predict how often each player will (or should) choose each action based on their incentives (expected goals).
This kind of problem makes it the perfect time for some game theory game theory. The decision of how often to pass/shoot or defend the pass/defend the shot is just a matter of finding the Nash equilibrium in a mixed strategy game. For each player, we want to find the percentage of the time they should pass (or defend the pass) so that their opponent is indifferent between defending the pass (passing) and defending the shot (shooting). In other words, for an attacker we want to find P such that:
P * xG(Pass | Defend Pass) + (1 – P) * xG(Shot | Defend Pass) =
P * xG(Pass | Defend Shot) + (1-P) * xG(Shot | Defend Shot)
To run the math though, we need a few numbers that will help us calculate the cost-benefit for both the attacker and the defender. First, we need to know how often teams score when they shoot or pass on a 2-on-1. While the NHL doesn’t track this kind of stat, we can use data from Ryan Stimson’s passing project to estimate the shooting percentage when a team passes or shoots on a 2-on-1. Within Stimson’s data, teams that had their last pass before a shot in the offensive zone scored on 26.8% of their shots, while teams that had their last pass before a shot on a 2-on-1 outside of the offensive zone scored on just 14.2% of those shots. While these are broad estimates (there were only 374 2-on-1 shots recorded in his dataset), they should be good enough for our purposes.
Next, we need to know how often teams are successful when they chose to pass on a 2-on-1. Unfortunately, no one has tracked this data so we’ll have to pick numbers that seem reasonable. We also need to provide estimates by the defenders actions, so we’ll assume that the attacker is successful in making a pass 50% of the time when the defender chooses to play the pass, and 80% of the time when the defender chooses to play the shot.
Lastly, we need to know how often a defender will block a shot when they choose to play the shot rather than the pass. Once again, we don’t have the actual data to calculate this number, but in this case we know that defenders block around 25% of the shots that are taken during all situations in a game. We can assume that the block rate on a 2-on-1 is much lower, so let’s put it at 12.5% – again it’s a guess, but it’s a fairly low number and it should be good enough to give us a general estimate.
With all of our assumptions out of the way, we can look at the expected goals for each scenario (attacker shoot or pass, and defender play the shot or pass).
Attacker Shoot[3] | Attacker Pass | |
Defender Play Shot | (1 – 0.125) * 0.142 = 0.124 | 0.8 * 0.268 = 0.214 |
Defender Play Pass | 0.142 | 0.5 * 0.268 = 0.134 |
Obviously if you’re a defender you want to minimize your expected goals against, so your best case scenario is that the attacker shoots and you play the shot. But if you’re the attacker and you know that the defender is likely to play the shot to get their best case, you’ll probably pass, since your expected goals are higher if you pass when the defender plays the shot. But if you’re a defender and you know the attacker is going to pass, you’ll play the pass. And then if you’re the attacker…well you see how you could go on for a while, right?
But, if we use the equation we had above, we can figure out the equilibrium for this problem, that is how often the attacker should shoot so that the defender doesn’t care whether they play the shot or the pass (and similarly, how often the defender should play the pass so the attacker doesn’t care whether they shoot or pass).
And as it turns out the conventional wisdom of always play the pass is *almost* right – the equilibrium for this game (based on our assumptions that we noted above) is that defenders should play the pass roughly 92% of the time and defend the shooter just 8% of the time. On the other hand, shooters should take the shot 82% of the time, while trying the pass just 18% of the time. And while these numbers are really heavily dependent on our assumptions, they do make a lot of sense – often the conventional wisdom exists because it’s right, and if you’re a forward who knows the defender is probably playing the pass, you’re likely going to opt for the shot most of the time, while occasionally taking a risk for the higher percentage tap in.
But even though the results make sense, what if our assumptions are actually wrong? The blocked shot number might not matter that much and is probably within a reasonable range given that we actually had some data to base it off of, but the pass completion rates were kind of drawn out of thin air. If these numbers are different in reality, our view of the optimal strategy for both forwards and defenders would change as well.
We can see the impact of our assumptions by looking at how the equilibrium for the attacker and defender change as we vary the pass success rates depending on defender choice. We’ll assume that the probability of a pass being successful is always higher if the defender chooses to play the shot than the pass, which is why you won’t see any data in the top left half of the graphs below.
First, let’s look at defenders – each point on the graph below represents how often the defender should play the shot, depending on the pass completion percentage when playing the shot (x-axis) and the pass completion percentage when playing the pass (y-axis).
There are 3 things that stand out in this graph:
We can also look at what happens to the attacking player’s decision making when we change our assumptions.
Attackers don’t show the same discrete decision regions as defenders – there’s no rates that would cause an attacker to always shoot or always pass – but we do see the same wide range of results that we saw for defencemen. While we had originally estimated that forwards should shoot 82% of the time, if we had instead assumed that attackers were successful completing a pass just 70% of the time that the defender played the shot, that equilibrium number would drop to 75%.
While it’s difficult to know where the true equilibrium lies without access to better data on player positioning and pass success rates, it’s likely that there’s no “one-size-fits-all” approach for defending an odd-man rush[4]. Even when one strategy seems to be clearly preferable to another, becoming too predictable can give your opponent an advantage and will certainly make their decision making simpler. Not having strict rules but rather broad guidelines about how to play in a given situation will ultimately lead to better results, and at the very least will help defenders avoid looking like Jake Gardiner did last night.
[1] It’s unlikely that Gardiner knew that Daley is one of the league’s most lethal defencemen on the penalty kill, sitting third amongst blueliners in shorthanded goals since 2014-15 with 2.
[2] Again, not really defying the odds since Trevor Daley may be the last player a goalie wants to face one-on-one on the power play, but how could Jake Gardiner possibly have known that.
[3] These numbers are definitely wrong, since what we observe (the 14.2%) is a blend of player’s shooting when the defender is playing the pass and when the defender is playing the shot, but without knowing how often they’re actually doing each we can’t really break it up any better.
[4] Unless Trevor Daley has the puck while shorthanded, in which case you always cover Trevor Daley.
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This post originally appeared on Hockey Graphs.
Last night Dave Hakstol and the Flyers were the first team to get burned by the NHL’s new offside challenge rule. With a one-goal lead over Nashville and just 2:41 left in the 3rd period, Philadelphia was dinged for not one but two minor penalties at the same time. And on the ensuing 5-on-3 power play, Scott Hartnell banged in a loose puck to tie the game up.
Philly, however, decided there was something not quite right about Hartnell’s goal. They thought that Filip Forsberg may have snuck into the offensive zone just slightly ahead of the puck on the zone entry that preceded the tying marker. The Flyers decided to challenge, hoping that video review would negate the Preds’ goal and put them back on top with just under two minutes to play.
When news first came out of the league’s proposal to change the rules, there was a lot of skepticism that it would act as much of a deterrent to frivolous challenges. While no coach wants to see their team go on the penalty kill after conceding a goal, the odds were still stacked pretty heavily in favour of challenging even in low probability scenarios. In a normal even-strength situation, your probability of success doesn’t need to be all that high in order to make a challenge worthwhile, in fact you’re safe challenging a lot of the time with less than a 25% certainty of success.
But the Flyers situation was anything but normal – they were already guaranteed to be down at least one man after the goal regardless of whether they challenged or not. Even if they were successful, they’d still need to kill off another 64 seconds at 6-on-3 and then survive for an additional 41 seconds at 6-on-5.
While the play was definitely tight (Forsberg even turned back immediately, indicating he may have even been worried he was ahead of the puck), it was far from an easy call, and the risks to the Flyers were definitely high. It’s impossible to know how frequently plays like this one end up getting overturned in the long run, but we can figure out how certain the Flyers should have been that they’d be right in order to make the challenge worthwhile.
To do this, we have to consider the 3 possible scenarios for Philly:
To run the math on each of these scenarios we need to know how often each team is likely to score in each strength-state, ignoring the possibility of a non-empty-net shorthanded goal.
Strength State | Goals Per 60 |
6-on-3 | 23.6 |
6-on-4 | 10.7 |
6-on-5 | 6.9 |
5-on-3 | 20.0 |
5-on-4 | 6.2 |
3-on-6 | 2.6 |
4-on-6 | 9.3 |
5-on-6 | 15.4 |
With these numbers we can calculate Philadelphia’s expected points in each of the three scenarios above, and then figure out how often Hartnell’s goal would need to be overturned to make the challenge worthwhile.
The simplest scenario to look at is if Dave Hakstol leaves the bench, fumes for a minute or two, and doesn’t call for the linesmen to review the play. In this case we have 2 possible outcomes to consider:
What’s the probability of the Flyers getting through those 36 seconds? If teams score 6.2 goals per 60 at 5-on-4 that translates to a 6.2% chance of the Predators scoring during their 36 seconds of 5-on-4 time.
Philly’s expected points are then:
xPoints (No Challenge): 6.2% * 0 Points + (100% – 6.2%) * 1.5 Points = 1.41 Points
The next simplest scenario to look at is the one that actually happened. The refs reviewed the play, determined there was no conclusive evidence that Nashville was offside, and the Preds took advantage of their extra-man advantage to score the game winning goal. Once again there are 2 basic outcomes of this scenario:
This time our expected points calculation is a bit more complicated. At 5-on-3 teams score 20.0 goals per 60, which means there’s a 20% chance Philly will concede on the 5-on-3, while at 5-on-4 there’s a 7.1% chance a team will score in 41 seconds. We have to reduce that 7.1% chance though to account for the fact that a second goal won’t have the same impact if Nashville scores on the 5-on-3.
Therefore the probability that they’ll score at least one goal is:
Probability >1 Nashville Goal = 20% + (100% – 20%) * 7.1% = 25.7%
And their expected points when we lose the challenge are:
xPoints (Lose Challenge): 25.7% * 0 points + (100% – 25.7%) * 1.5 Points = 1.12 Points
We can see right away that this is a decent drop from the No Challenge scenario, as the odds are reasonably high that with over 30 seconds on a 2-man advantage you’ll score. The question then is how much of a boost the Flyers get from erasing a goal.
Lastly, we need to consider the case where Dave Hakstol is proven right and looks like a hero (at least temporarily). This is the most complicated scenario since we have to take into account that the clock would reset back to 1:45 if the play was indeed offside, giving Nashville an additional 28 seconds of 2-man advantage to play with.
We also need to assume the Preds would keep their goalie pulled (at least until they tie the game) which means we have to consider the possibility of a shorthanded goal by Philly.
To simplify our calculations, we’ll use Nashville’s net goal differential in each empty-net scenario to model our problem. There are 3 cases we need to consider for this scenario:
What’s the probability of each scenario?
The net goal differential per 60 of a team at 6-on-3 is 21.0, which over 64 seconds makes a 37.3% chance of scoring. We have to multiply that number by the probability they’ll also score at 5-on-4 after that. Assuming Nashville gets 32 seconds at 5-on-4 (half of the 64 seconds from the first penalty), 32 seconds is worth about 0.055 goals at 6.2 goals per 60. That makes the total probability of Philly giving up two in the last 36 seconds of their penalty kill equal to:
Probability of 2 Nashville Goals = 37.3% * 5.5% = 2.1%
In the 1-goal scenario, Nashville has 64 seconds to score at 6-on-3, or, if they fail to do that, 41 seconds to score at 6-on-5. The probability of one goal in 64 seconds of 6-on-3 is again 37.3%, while the probability that they score at 6-on-5 is actually negative at -6.1%, as teams who have pulled their goalie actually get outscored in the long run.[3]
Probability of 1 Nashville Goal = 37.3% + (100% – 37.3%) * (-6.1%) = 31.3%
This makes the total expected points for the Flyers if they win the challenge equal to:
xPoints (Win Challenge): 2.1% * 0 points + (31.3%) * 1.5 Points + (100% – 2.1% – 31.3%) * 2 = 1.80 Points
Here’s where you see the real benefit to challenging – if they wind up being right, they’ve earned nearly half a point more than if they just let the play go on. So while there’s risk to the decision, there’s also a decent reward if they’re proven correct.
Knowing Philadelphia’s expected points in each possible scenario, we can calculate the Flyers’ Break Even Certainty for the challenge. The Break Even Certainty is simply the probability of overturning the call on the ice which makes the expected points in the challenge scenarios equal to the expected points if they don’t challenge.
[Break Even Certainty] * xPoints (Win Challenge) + (1 – [Break Even Certainty]) * xPoints (Lose Challenge) = xPoints (No Challenge)
Re-arranging, we get:
[Break Even Certainty] = (xPoints (No Challenge) – xPoints (Lose Challenge)) / (xPoints (Win Challenge) – xPoints (Lose Challenge))
which in our case works out to 42.4%. That is to say, as long as Hakstol and his staff thought they had better than a 42.4% chance of being right, challenging the call was the correct move.
While that level of required certainty doesn’t make it an easy decision, it’s also not a super high probability. The play was definitely close, and you can’t blame Hakstol for taking the chance on potentially icing the game at that point. Our calculations may even overstate the true risk to the Flyers: the numbers we’ve used assume average goal scoring rates, but the Predators would be starting in the neutral zone if the call did get overturned, and would need to re-enter the Flyers end before they get any offensive opportunities.
Though Dave Hakstol is certainly going to live with that failure in the back of his mind for a long time, the reality is that he may have made the right call and just got a bit unlucky. Even in a worst-case scenario like this one, the odds still lean towards challenging, as the benefits of erasing one of your opponent’s goals will almost always outweigh the risk of going on (or staying on) the penalty kill.
[1] We’ll ignore the case where they give up a goal at 5-on-4 and then score after that for simplicity’s sake.
[2] Again, we’ll ignore the fact that Nashville gets about 43 seconds of 4-on-3 time in OT to make the math easier.
[3] This is a conservative estimate, as it includes a lot of delayed penalty time from the middle of the game where it’s (nearly) impossible for the defending team to score.
]]>Those spurious challenges are one reason why the NHL modified the rules around coach’s challenges yesterday. Starting next season, instead of a failed challenge simply resulting in the loss of a team’s timeout, clubs will now face a 2 minute penalty for losing an offside challenge. Upon hearing of this change many fans were apoplectic, complaining that this rule change could bury teams who were already reeling from giving up a goal against, and would severely limit the willingness of coaches to challenge even legitimate missed offside calls.
Fan reaction notwithstanding, however, the question coaches should be asking is whether they should be changing their approach in response to the new rules. The threat of killing off a penalty for a failed challenge may seem like a big deal, but it’s important to note that teams only score on roughly 20% of their power play opportunities. Fans will surely remember when a failed challenge leads to a power play goal against, but there will certainly be occasions when the potential gain from overturning your opponent’s goal outweighs the risk.
The question for any coach then is how sure you have to be of your video coach’s recommendation in order to call for a video review. If you think there’s only a 5% chance of success it’s unlikely that the 1 in 20 odds of taking a goal away outweigh the cost of the power play you’re going to give up 95% of the time. But that decision won’t always be so easy: what if you’re 33% sure you’ll be successful, but you’re already down 1 with just 10 minutes left to play? Is it worth the risk of going down two goals with half a period left for a 1 in 3 shot of being tied?
To answer this question we can look to the always excellent insight of Micah Blake McCurdy of Hockeyviz.com. Last year, Micah wrote about the concept of leverage, which he defined as the cost (benefit) in expected standings points for a team allowing (scoring) a goal given the current score state and time left in the game.
Offensive leverage is the increase in expected points from scoring a goal, while defensive leverage is the cost in expected points from conceding a goal. Leverage allows us to estimate how big the cost of a failed challenge will be by looking at how a team’s expected points will change based on the success or failure of their challenge.
Because a team that’s deciding whether or not to challenge has already conceded a goal, a coach who decides to challenge faces two possible outcomes:
We can use this information to model how a team’s expected points will change based on the probability of success of a challenge.[i]
Change in Expected Points = P(Success) * Offensive Leverage + P(Fail) * 0.2 * Defensive Leverage
As long as a team’s Change in Expected Points is positive, a coach should feel confident that challenging a goal is the right choice. We can then calculate the Break Even Certainty, which is the confidence that a coach needs to have in order to ensure that the Change in Expected Points is positive.
Break Even Certainty = (0.2 * Defensive Pressure)/(0.2 * Defensive Pressure + Offensive Pressure)
With this formula and Micah’s leverage data, we can plot the Break Even Certainty for a team playing at home given the current score (after the goal that’s being challenged) and time remaining. So, when should teams challenge under the NHL’s new rule?
For a home team that’s tied before the challenge (i.e. they were up 1 and gave up a goal that they think may have been offside, the green line in the plot above), the break-even rate is pretty consistent around the 18% mark until roughly the 35^{th} minute of the game, at which point the break-even rate starts to rise steadily throughout the rest of the game. This makes a lot of sense – when you get closer to the end of the game, the cost of giving up a goal on the penalty kill rises, as you have less time to score a tying goal afterwards (and you lose the OT point). As such, you need to be more sure you’re going to get it right before you call for a replay review.
The other curves appear mostly where we’d expect[ii] – a team that’s leading needs to have a much higher certainty than a team that’s trailing, which isn’t exactly surprising. What’s interesting to note is how low a team’s certainty needs to be in order to make challenging worthwhile. A team that’s trailing at basically any point of the game needs less than a 15% chance of success to see a positive expected value,. And even a team that’s ahead by 2 goals can still challenge with less than 50% certainty for most of the entire game.
All of which is to say that while these rule changes may get rid of a few longshot challenges, it will still be worthwhile for a team to challenge in many instances where their odds are far below even. While killing off a 2 minute penalty may seem like a substantial deterrent, the benefits of potentially saving a goal are simply too large to ignore in many instances. As long as the offside challenge remains an option, finding ways to discourage its abuse will be difficult for the league.
The flipside of this, however, is that teams should continue to challenge even if they’re less than certain on their likelihood of success. The value of a timeout is significantly reduced now that teams can’t call one after an icing, and as we’ve seen above, the required success rates are generally so low that coaches can afford to take a flyer every now and then with the hopes of saving a goal.
[i] We’ll ignore the lost 5-on-5 time for simplicity here. Practically speaking, this would be a further net negative for a trailing team, but it should be small enough that it doesn’t significantly impact our conclusions.
[ii] There are a few weird areas where small samples and/or smoothing may make things look odd, but in general the modelled results are pretty intuitive.
This post originally appeared on Hockey Graphs.
One of the weird things about sports that I find fascinating is how often coaches and players seem to go out of their way to avoid having a negative impact on the game, even at the expense of potential positive impacts. People often seem to prefer to “not lose” rather than to win, which can result in sub-optimal decision making, even in the presence of evidence to show that the correct decision is not being made.
There are many examples of this across sports, but the biggest two in hockey are pulling the goalie and playing with 3 forwards on the power play. Analysts have been arguing for many years now about why teams should pull their goalies earlier, but it’s only been in recent seasons that teams have become more aggressive in getting their netminders out earlier.
Similarly, a lot has been written about how much better 4 forward units are on the power play, but adoption of this approach is still far from universal, likely due to coaches fearing being blamed when their teams allow a shorthanded goal.
With that in mind, one of the ideas I’ve been pushing for a while now is that teams should be playing with 4 forwards at 5-on-5 when they’re down late in the game. The thinking behind this tactic is pretty simple: when a team is down late they have little to lose from allowing a goal, and a lot to gain from scoring one.
Micah McCurdy has written previously about this imbalance, presenting the idea of the “leverage” of a given situation. Leverage encompasses two things – first, the change in expected standings points of a goal scored at a given point in time with a given score (or the offensive leverage), and second, the change in expected standings point of a goal allowed at a given time with a given score (or the defensive leverage).
The chart below has the average leverage values for a team down 1 in the 3^{rd} period (I’ve folded together different score states and home/away for simplicity). We can see that as you get closer to the end of the game, the value of scoring a goal increases exponentially (since there’s little time left to score after that, or to be scored on).
Similarly, as we approach the end of the game, the cost of allowing a goal also goes down towards zero – it doesn’t matter if you lose 1-0 or 2-0, so giving up an additional goal has little effect near the end of a game.
All of which is to say that the incentives for a team down late in a game aren’t balanced – the benefit to scoring vastly outweighs the cost of allowing a goal. Given this imbalance why would teams not try playing with 4 forwards when they’re trailing late in the game?
Coaches certainly seem to realize that there’s a point where it’s worth it to throw caution to the wind, as they’ve been pulling the goalie for an extra attacker as far back as 1931. Switching to 4 forwards could be used as an intermediate step, to ratchet up pressure without completely giving up on playing defence.
Unfortunately, this strategy hasn’t exactly caught on with teams, and so we don’t really have a good dataset to work with[1]. While teams have played with 4 forwards at 5-on-5 for over 700 minutes since 2009-2010, most of that time was immediately following the end of a power play, which doesn’t exactly make for an unbiased sample.
Given our limited data then, how can we evaluate whether this strategy might work, without having to convince a handful of coaches to try it long enough to get a sufficient sample?
One way to do it would be to guess the impact of using 4 forwards on scoring rates, and then evaluate the impact on a team’s expected points due to the change in scoring rates. If we can come up with a reasonable estimate of how much scoring would increase for a team playing with 4 forwards (and for the team playing against only 1 defenceman), we can use Micah’s leverage numbers to find the optimal time (if there is one) to switch to 4 forwards.
The question then becomes what a reasonable scoring impact would be. A fair starting assumption would be that using 4 forwards will increase both the rate of goals scored and allowed at 5-on-5 – you’ll get a bonus offensively, but at a cost of weaker defence. But we still need to know how big that increase is for each team to run the number
We could take an initial estimate from the data we have on the power play, where a team’s Goals For Per 60 is roughly 1.26 times higher with 4 forwards than with 3 forwards. Similarly, teams using only 1 defencemen on the man advantage allow goals 1.48 times as often as those with 2. If we assume a constant GF60 and GA60 across the third period, we can use the following scoring rates:
Scoring Rates When Down 1 in 3rd | 3F-2D | 4F-1D Estimate |
Goals For Per 60 | 2.33 | 2.94 |
Goals Allowed Per 60 | 2.08 | 3.08 |
One thing that’s interesting to see is that you will likely end up being a negative goal differential team if you use 4 forwards. While this seem to kill off the idea of using 4 forwards (generally you don’t want to be outscored in hockey, or so I’ve been told), because the value of a goal scored is much greater than the cost of a goal allowed when you’re down 1, it makes sense to increase the overall rate of scoring for both teams for a short period to try to tie the game.
Now that we have an initial estimate for scoring rates we can take a look at the net estimated benefit of adopting a 4 forward approach when down 1. To simplify the situation, I’ve made two important assumptions about how teams will approach this strategy:
First, once a team switches to using 4 forwards, they’ll continue to use 4 forwards until they score to tie the game, or are scored on to go down 2 (the latter point is to simplify the calculation). Second, under both strategies, teams will pull their goalie with 1.5 minutes to play in the 3^{rd} (and playing with 4 forwards prior to pulling the goalie will have no impact on how teams perform with an empty-net).
With all the pesky details ironed out, we can (finally) look at whether this strategy might work. The graph below shows the predicted change in expected points versus the amount of time remaining when the team switches over to 4 forwards.
There’s three things to note here: first, for a good portion of the 3^{rd} period, switching to 4 forwards would be a net negative. This makes sense given the net negative long term goal differential for a 4F team.
Second, the 4 forward approach is likely to result in a positive change in expected points if adopted any time after the 10 minute mark in the 3^{rd} period.
Finally, the ideal time to swap a defencemen for an additional forward, assuming our goal scoring estimates are right, is around the 5.5 minute mark in the third period. Any time before that and you’re likely allowing your opponents too many opportunities to score; any time after that and you’re probably not giving yourself enough of an opportunity to score.
Although this initial approach gives us some indication that a 4 forward approach holds promise, we don’t have enough data to know whether our assumption of a 1.26x increase in goal scoring is valid.
To get around that, we can test out various other scoring impacts and see how they change our estimate of when (or if) we should switch to 4 forwards, assuming the increase in goals against stays constant. For the purposes of this exercise, we’ll leave the goals against impact constant at 1.48, as that seems like a safe upper bound to me.
While the size of the impact might vary, there’s still some indication that switching to 4F-1D late makes sense. In fact, even if you only see a 5% increase in your goals for rate, it’s still a net positive to switch after the 2:30 mark.
Given that the evidence seems to point towards an offense-oriented 4F approach providing benefits when down late, how large would those benefits be for an average team? If we go back to the original graph and assume that teams change over at the optimal point (around 5.5 minutes for a 1.26X offensive impact), every time a team uses that approach it would be worth about 0.022 points.
Since 2009-10, 35% of games have had one team leading by a goal with 5.5 minutes left, meaning an average team would play in roughly 28.7 of those games over the course of a season. If we assume that they’re leading in half of those 28.7 and trailing in the other half, that works out to a total benefit of 0.32 points over the course of a season. It’s not enormous, but at a rough cost of $1.5 million per standings point that’s a pretty valuable tweak for an NHL team[2].
Even though switching to 4 forwards late likely won’t be the difference between making or missing the playoffs, it is one area where teams can give themselves a marginal boost in the standings without having to go out and change their roster at all. Given how difficult it can be to improve even slightly in today’s NHL, it does as if it may be a worthwhile tactic for a team looking to get an additional edge in the season to come.
[1] I remember the Sens using this strategy once, but I can’t think of any other times a team has done it intentionally.
[2] NHL teams, please make cheques payable to “Matt Cane”. Cash is also accepted.
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