xPoint Goalie Pull Model
"I was happy, just getting some answers" - Nas
Introduction
As a fan, watching your team trail by a goal in the waning minutes of a hockey match can be nerve-wracking, and perhaps even frustrating. Why is that goalie still between the pipes?! What are we waiting for?? It's now or never! Coaches have to make a tough judgment call in the thick of battle as to whether to pull the goalie and when, based on a myriad of variables such as expected goals, expected goals against, probability of scoring, line matchups, etc..
When we break down the problem into first principles, we find the objective of any NHL team is to maximize standing points—the points that determine playoff qualification and define season success. When a team is down by a goal with just a few minutes left, scoring to tie the game is worth everything, while conceding an empty-net goal doesn't change much—if a team loses by one goal or five, the number of standing points they gain is unaffected.
This dynamic calls for a different approach to risk and reward. While the traditional focus on goal differential doesn’t fully capture this, the strategy of pulling the goalie does. Think of pulling the goalie as a calculated risk, like a poker player going all-in on a strong hand. You’re not guaranteed to win, but the potential payoff far outweighs the downside. Statistically speaking, we find the act of pulling the goalie actually reduces the risk of losing outright—and should be viewed as a form of "insurance."
By taking on the controlled risk of an empty net, a team gives itself a better chance to score and extend the game, which aligns with the real goal: accumulating those critical standing points. This shift in mindset highlights the need to evaluate in-game decisions through the lens of maximizing points, rather than minimizing goals allowed.
History
The act of pulling the goalie is one that can be credited back to coach Art Ross, who first attempted this move during a 1931 playoff game between the Boston Bruins and the Montreal Canadiens. Down 0-1 with a minute to play, he sent goaltender Tiny Thompson to the bench and inserted a sixth attacker. While the move didn’t work, a tradition was born.
While there have been a few whitepapers on the topic of when to pull the goalie–all of which use different data sets and create different models–all of the authors agree that the statistically-optimal time to pull a goalie when trailing by one goal is earlier than current NHL practice.
Despite this awareness, data from the 2023-2024 NHL season shows that teams down one goal are still pulling their goalies with well under two minutes remaining in the game. This is much later than what xPoint finds to be optimal. While small differences in assumptions or parameters could make the case for somewhat later pulling times than our model recommends, no reasonable estimates can justify the current NHL practice.
The xPoint Model
The xPoint model is dynamic. Using play-by-play data from the 2023- 2024 NHL season, the xPoint model makes goalie pull time recommendations based on Team A vs Team B (e.g. New York Rangers vs Columbus Blue Jackets). We calculate this using six inputs:
- The current goal differential
- The time remaining in the game.
- The probability of your team scoring 5v5
- The probability of your opponent scoring 5v5
- The probability of your team scoring 6v5
- The probability of your opponent scoring 5v6 with an empty net.
The screenshots below illustrate the need for a dynamic model. When the Columbus Blue Jackets play the New York Rangers, our model suggests the optimal time to pull their goalie when down a goal is with 160 seconds remaining:
When the Columbus Blue Jackets play the Philadelphia Flyers, however, our model suggests the optimal time to pull their goalie when down a goal is with 400 seconds remaining:
League Averages
To avoid confusion, we will use an average NHL team versus an average NHL team to run through our calculations, using the same 2023-2024 play-by-play data. Proceeding in this way creates a simple model with attractive intuition. We estimate average values of 0.70% per 10-second interval for inputs 3 and 4, 2.60% for input 5, and 6.40% for input 6.
Using the inputs above, the team that pulls its goalie multiplies its opponent’s chance of scoring by just over nine, while multiplying its own chance of scoring by nearly four. Pulling the goalie is a negative expected value move in terms of goals. But the number of expected goals is not the appropriate criterion. What matters here is the expected number of standings points. A team down a goal with little time remaining in the game will gain a lot by scoring and lose little if the other team scores. Losing by two goals in hockey is no worse than losing by one.
Calculations
To solve for the optimal goalie pull strategy, we use dynamic programming, solving the problem at the end and then working backwards. With 0 seconds left in the game, a win is worth 2 standing points, a tie is worth 1.5 standing points (50% chance of losing in OT/shootout + 50% chance of winning in OT/shootout), and a loss is worth zero. To find the optimal expected standing points based on the pull/don’t pull scenario, we use the formula = max(1.5qa + 0qo + 0qb, 1.5pa + 0p0 + 0pb). If we ignore the tiny probability of multiple goals within a 10-second interval, a team trailing by 1 goal with 10 seconds to go has a 0.70% chance of tying the game with their goalie in place; the chance rises to 2.60% if the team pulls its goalie. That means that its expected standing points are 1.5 × 0.70% = 0.01052 if it doesn’t pull, and 1.5 × 2.60% = 0.03895 if it does, so we should definitely pull the goalie in this situation. The chance of the other team scoring here doesn’t change a thing; you still get zero standing points.
Working backwards in our dynamic program, we use Bellman’s Equation to find the solution to subproblems thereafter:
Results
We find that an average team trailing by one goal should pull their goalie with 4 minutes and 40 seconds remaining in the third period. At 4 minutes and 50 seconds, the expected standings points of pulling is not greater than that achieved by leaving the goalie in the net. When optimizing for an average team trailing by two goals, we find the team should pull its goalie with 9 minutes and 50 seconds remaining in the third period. While these results vary from team-to-team, directionally the data tells us that NHL teams are acting sub-optimally across the board.
Additional Impact
The act of pulling the goalie has proven to have statistical significance in other areas of the game as well. Our research shows that roughly 10% of the time a goalie is pulled, the defending team takes a penalty in favor of the attacking team. We believe this is due to factors such as increased offensive pressure, fatigue in defending players, panic in high-pressure situations, and tactical risks taken by the defending team. During the 2023-2024 NHL season, this figure was 10.7%. If we multiply 10.7% by the success rate of a team’s Power Play unit, we can use this figure to determine the increased probability of additional standing points.
The Big Picture
A team that practices optimal goalie pulling gains an average of 0.07 more points per game–six standing points over an 82-game season–over a team that never pulls the goalie. Putting this into practice, we find that a team like the Columbus Blue Jackets would have earned 7.31 additional standing points during the 2023-2024 season simply from pulling their goalie at an optimal time. CBJ’s expected standing points increases to 7.51 once we factor in the likelihood of drawing a penalty with their 15.1% powerplay success rate. This is a material difference in expected performance that comes with zero cost or work, but rather from simply acting in a statistically-optimal way.
Appendix
Check out the full whitepaper in PDF form here: https://drive.google.com/file/d/1Ffeuyh7YKaKtLCfFraOmTTU1NOq8jqw5/view?usp=sharing