Wednesday, January 25, 2012

Why Do Score Effects Exist?

First off, I must thank and give full credit to 2+2 poster atakdog. Most of this article is just a small model tweak and different presentation of his work in this post. Worlds are colliding!

We talk a lot about score effects. The idea is that when a team is behind they will tend to dominate and teams that are ahead find themselves in their own end more often than when the score is tied. This occurs because teams have different incentives - if you are ahead then preventing the other team from scoring is more important than scoring yourself, so you'll play less aggressively forcing the other team to work for it. Similarly, if you are behind then scoring becomes far more important so you are willing to take chances, pinch with your defensemen, have them jump into rushes and so on.

In this article, I will take a look at score effects by graphing out the incentive to score and prevent goals. In future articles, to come out in the next week, we will use a similar methodology to look at how the points system (2 for any kind of win, 1 for an OT/SO loss, 0 for a regulation loss) affects incentives and whether there might be better systems out there.

A Simple Model

It wouldn't be a JaredL article, or an atakdog derivative for that matter, without introducing a model. To look at the incentives to score and prevent goals, I took the average goals per team per game since the lockout, about 2.88, and divided by 60 to get the average goals per minute, roughly 0.048. I made this number the probability of either team scoring in any given minute. Using backward induction, I determined the probability of winning the game any given minute with any score difference. So this is two average teams facing each other, scoring at the average rate each minute and each with a 50% chance of winning the extra point in overtime/shootout should it get that far. I also assumed that if at any point in the game one team is up 10 goals then they will certainly win.

You may find it a little strange that we're looking at score effects using a model which assumes that they don't exist. A good way to think about this is to ask what happens if the other team plays exactly the same way no matter what the score is. How should we respond?

Score Tied

Let's start with the score tied. If this were baseball, basketball or either North American brand of football, this would be simpler. Unlike those sports, the NHL tiebreaking rules make hockey games non-zero-sum. If two teams tie, then the total number of points both get goes from 2 to 3. Note that in soccer it is exactly the opposite - if a game ends in a draw, the number of points drops from 3 to 2. The NHL rules actually make the incentive to score and prevent goals different when the score is tied, which we'll cover in greater depth in future articles.

Here is a graph showing the marginal benefit, in league points, of scoring and preventing a goal with the score tied. The horizontal axis represents what minute of the game it is and the vertical how many expected points are gained by scoring or preventing a goal. For scoring this would be the difference between starting the next minute up one and starting the next minute tied. For preventing this is the difference between starting the next minute tied and starting it down 1.

This gives us a somewhat strange pattern. Early in the game, scoring and preventing goals are about equally important. When the score is tied you will gain or lose about a third of a league point on average if a goal is scored. Very late in the game, this changes and getting to the 1-point-bonus round becomes the important thing. It's easy to see why we don't like this, but let's move on and look at score effects right now.

The One-Goal Game

Let's now shift to the score not being tied and start with the team that is ahead in a one-goal game. Here is a similar graph:

You can see that every minute of the game preventing a goal is more important than scoring. It's an interesting coincidence that at the start of the third period preventing a goal is almost exactly twice as important as scoring one. I think this is a bit overstated because my model does not take into account a team pulling the goalie, which I think will make it just a bit more important to be up 2 goals. Something worth noting is that the points system somewhat cushions the cost of conceding a goal here - if you are up one then giving up a goal very late isn't all that bad, at worst it costs you just over half a league point. Perhaps paradoxically, it's worse to give up a late goal when the score is tied than when you are up one.

Here is the graph for the team that is losing:

Perhaps the most clear thing from this is the already obvious justification for pulling the goalie - the last couple minutes giving up a goal almost doesn't matter at all while scoring is worth close to a point and a half. I've wanted to take a closer look at the optimal time to pull the goalie for a while, and hopefully will get to it, but just eyeballing this graph it seems earlier than usual might be better.

Another thing about this graph is that it provides some justification for the definition of close game, I believe first proposed by Eric T. over at Broadstreet and now used by many, including Gabe Desjardins for his power rankings, which he posts far more regularly than I (coming soon, I promise!). Under that definition, a game is close if it's within a goal in the first two periods and tied in the third. While such a definition is always somewhat arbitrary, we can see some justification for it by noting that the incentive to score for the team that is behind stays relatively flat earlier but really moves upward in the third period.

Something worth noting is that it is far more important for the team that is behind. Scoring a goal right at the end of regulation is three times as beneficial for the team that is behind than it is costly for the team that is up. Again, this is due to extra point being given in tie games. To better see the size of these effects, here's a graph with all four together:

Based on these incentives, it's not surprising that so much more of the play is in the leading team's end of the ice. The key for both teams is putting the puck in or keeping it out of that goal. The later in the game, the stronger this effect is and that's mostly because of how big a goal would be for the team that is behind.

Two-Goal Games

I'll be very quick with two-goal games and just show the combined graph. Here it's interesting because right near the end of the game it basically doesn't matter what happens - with a minute or two to go the team up two goals is almost certainly going to win whether they give up a goal or not. The key time is 10-15 minutes out when the trailing team has a decent chance to get another goal and equalize. Again, this slightly overstates the case because it assumes teams leave their goalie in there at the end, but the overall shape of the graphs would be mostly the same:

Score/Prevent Benefit Ratio

Finally, here's a graph of the ratio of the marginal benefit of scoring to preventing the other team from doing the same based on the score and time. The larger this is, the more important scoring a goal is relative to preventing one. If the ratio is greater than one that means scoring is more beneficial than giving up a goal is bad, the opposite if it is less than one.

I cut it off at 5 because it shoots way up for the team that is behind. With two minutes to go, scoring becomes 40 times as important as preventing the other team from putting the puck in your net if you are down 2. It is very clear that while teams that are ahead have some incentive to play more defensively, most score effects are driven by the team that is trailing.

In the next installment, I will look at how score effects, and play with the score tied, would be expected to change under alternative points systems such as 3-2-1-0, the simple 2-1-0 with ties at the end of regulation and the soccer system which is 3-1-0.


  1. "I believe first proposed by Eric T. over at Broadstreet"

    That existed long before I joined the community, I'm afraid.

  2. Ok. I guess you wrote the first thing I remember reading that used it. Thanks for the correction.

  3. Thanks Jared, great analysis. Looking forward to the follow up articles.

  4. Great analysis. Do you know if there is a simple way of including score effects in win probability models? The poisson tends to underestimate regulation ties by about 6%.

    Also, I came across this paper regarding pulling the goalie. Basically they conclude what you expect - that pulling the goalie earlier is probably a good move.