Gravity: Part 1. A Metric for Event Importance

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Gravity: Part 1. A Metric for Event Importance

Some goal attempts are more important than others. 2-0 up, ten minutes to go, and a chance to score. Whether the goal goes in or whether it doesn’t is unlikely to affect the outcome of the game. Not a lot of pressure on the striker really. How about a chance to score at 1-1 with five minutes to go? That’s a different story. Scoring will almost certainly win the game. But 1-1 with 70 minutes to go? Not so important to hit the back of the net, and presumably so not so much mental pressure on the striker.

We can see intuitively that the importance of a goal – and in fact the importance of any event in a match – depends both on the scoreline and the time remaining in the match. In this post I’m going to develop a metric for this, and later I’ll look at what it means for scoring goals, winning matches, player pressure and of course xG.

Morris (1977) defined the ‘importance’ of a point in a game of tennis as the probability player A wins the game if he wins the point, minus the probability he wins the game if he loses the point.  This makes intuitive sense; the importance of a point depends on how much difference it makes to winning and losing.  A similar principle applies to football of course.  Events that change the probability of winning a lot are more important than events that only change the probability of winning a little.  So for example, when there is a large difference in the scoreline, additional goals by either team have a relatively small effect on the match outcome and such goals are therefore not important; on the other hand when the scores are close and the match is almost over, additional goals are valuable because they have a big effect on the probable outcome of the match.

Gravity

One difference between tennis and football is that football matches can be drawn, while tennis matches cannot; most tennis is played in a knockout format, where losing the match means elimination, and the only currency worth talking about is winning. In football, winning is not the only currency; draws also have value, both in a league format where a club’s league position is determined by the total points accrued, and in a competitions like the FA cup where drawn matches are replayed.   So to apply Morris’s insight to football, we need to broaden the definition of importance.  I propose a metric called Gravity, (denoted Γ, gamma) which reflects the importance of an event in a football match as follows:

The importance, or “gravity ” of an event is the expected value of the match outcome if the event occurs minus the expected value of the match outcome if the event does not occur.

(Those unfamiliar with the idea of expected values can find a good video tutorial by PatrickJMT here.)

The outcome values can be defined in any way we choose.  For example, a draw would be a highly valuable outcome to a team in the group stages of a tournament and just needing one point to proceed to the next stage, but would not be at all valuable to a team needing to win. And the events can be any events we choose. Most events in football like passes and tackles probably have very low Γ, but saves, goals, red cards spring to mind as high Γ events.

Calculating Goal Gravity

In what follows I will limit the discussion to goal events. Also, because most football is played in a league context I will measure the value of the match outcome in league points.

The Gravity (Γ) of an attempt on goal is therefore:

Expected Points if the goal is scored minus Expected Points if the goal is not scored.”

Because a win is worth three points and a draw is worth one point, the expected points at any moment in a match is:

E\left( p \right) =3\ast P\left( win \right) \quad +1\ast P\left( draw \right)

where E(p) is the expected points and P(win) and P(draw) are the probabilities of winning and drawing at that moment,

To calculate expected points we need to calculate the probabilities of each possible outcome (win, lose or draw).  To do so, we assume that goals in football are produced by a Poisson process. We can then calculate the probabilities we need if we know the current scoreline, the time remaining in the match, and the scoring rate (i.e. goals per 90) of each team.  The principle of the calculations is quite straightforward, although the calculations themselves are a bit tiresome:

Suppose team A has scored X goals and team B has scored Y goals, and team A is leading.  If p(A,X) denotes the probability that team A scores X goals in the time remaining, we can say that:

The probability of a draw is:

the probability that ….
p(A,0) & p(B, X-Y) +                         A scores no goals and B scores X-Y goals   OR
p(A,1) & p(B, X-Y + 1) +                   A scores one goal and B scores X-Y + 1 goals OR
p(A,2) & p(B, X-Y+2)                        A scores two goals and B scores X-Y + 2 goals OR
… etc

The probability of a win for A is:

the probability that ….
p(A,0) & p(B,  {< X-Y}) +                    A scores no goals and B scores < X-Y goals   OR
p(A,1) & p(B,  {< X-Y + 1}) +              A scores one goal and B scores  < X-Y + 1 goals OR
p(A,2) & p(B,  {< X-Y+2})                   A scores two goals and B scores  < (X-Y + 2 goals OR
… etc

The probability of a win for B is:

the probability that ….
p(A,0) & p(B,  {< X-Y}) +                    A scores no goals and B scores < X-Y +1 goals   OR
p(A,1) & p(B,  {< X-Y + 1}) +              A scores one goal and B scores  < X-Y + 2 goals OR
p(A,2) & p(B,  {< X-Y+2})                   A scores two goals and B scores  < X-Y + 3 goals OR
… etc

So how accurate are outcome probabilities calculated in this way? As a basic check I tried to predict the percentage of home wins, away wins and draws in  a season. In the 2015-16 Premier League season the average scoring rate for home teams was 1.47 goals/90 minutes and the average scoring rate for away teams was 1.23 goals/90 minutes. To predict the full-time results I plugged these values into the equations, with a starting scoreline of 0-0 and 90 minutes remaining.  For the half-time  results I used the same 0-0 starting scoreline, but with 45 minutes remaining. The predicted and actual percentages are shown in Table 1.

Table 1. Predicting Full-time and Half-time Results Using Poisson Scoring Rates

Full timeHalf time
ResultsActualPredictedActualPredicted
Home Wins41.3%43.7%32.9%34.8%
Away Wins30.5%30.7%22.9%26.2%
Draws28.2%25.5%44.2%39.0%

Well considering only two input parameters were needed that’s not too bad.  The Poisson assumption works quite well.

Illustrations of Goal Gravity

Now to get an intuition of goal Γ, let’s see how it evolves over the course of a match in some typical situations.  For simplicity I base all calculations on the average home and away scoring intensities.

Figure 1 shows a match that is goalless throughout, and shows goal Γ for the home and away team as the minutes tick by.  At the start of the match, Γ is low, but the gravity of a goal for the away team is slightly higher because its lower scoring intensity makes a win less probable. A goal becomes more and more decisive as the match progresses, and Γ reaches its maximum value as the time on the referee’s watch approaches 90 minutes. Here a goal would convert an almost certain draw (expected value one point) into an almost certain win (expected value three points). The value of a goal at this juncture is worth two points which is the highest gravity a goal can have.

 

Figure 1. Evolution of Goal Gravity in a Goalless Match

Figure 2 shows what happens when the home team is goalless and the away team scores on the stroke of half time.  The increase in gravity seen in Figure 1 is arrested, and Γ remains almost the same throughout the rest of the match.

 

Figure 2. Evolution of home team Gravity when away team scores at half-time

Figure 3. shows what happens when the home team is goalless and away team scores late in the game.  The high gravity suddenly drops.

Figure 3. Evolution of home team Gravity when away team scores late on

Finally Figure 4 shows the effect of the home team going 2-nil up in the 60th minute.  Gravity drops because now the away team getting a goal back won’t change the probability of a home win very much.

 

Figure 4. Evolution of home team Gravity when home team goes 2-0 up.

 

Those of you who have read this far will probably be wondering why I’m bothering to discuss this rather arcane metric.  But as I hope to show in subsequent posts, it turns out to be a piece of the performance jigsaw we haven’t yet fully appreciated.

References

Morris, C., (1977). “The most important points in tennis.” In Optimal Strategies in Sport, edited by S. P. Ladany and R. E. Machol, 131–140. Amsterdam: North Holland. (Vol 5 in Studies in Management Science and Systems.)

2017-11-20T10:28:07+00:00 September 4th, 2017|Recent Posts|0 Comments

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