## How Much do Goals Count?

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Date: Tue, 02 Mar 1999 03:39:21 -0500
From: Marco Daniele Paserman
Newsgroups: rec.sport.soccer
Subject: Serie A value-weighted goals 1997-98: a new approach (LONG)

1. INTRODUCTION
A while ago, I proposed a method to measure more accurately the value of
goals and to use this alternative measure to rank a league's
goalscorers. The basic idea was to weight more heavily decisive goals,
and to downgrade meaningless 90th minute goals that do nothing more than
put the nails in the coffin of a 4-0 win. My initial simple proposal
weighted goals by their ex-post value: so, for example, a goal that put
one team up 1-0 would be weighted very heavily if the scoring team was
able to preserve the lead, but would be worthless if the other team made
a comeback. This led to suggestions by Ken Overton and Paul Mettewie to
create an alternative, ex-ante measure, that would better take into
account the impact that each goal would have on that udefinable aspect
of a game, "momentum".

Indeed, Ken Overton took this criticism seriously, and worked very hard
to construct a data set with all Serie A goals for the 1997/98 season,
and to compute such an ex-ante measure of the value of a goal. Ken's
very interesting results are now posted on Benny's European soccer
web-site (www.soccer-europe.com).
Ken's method weighted each goal based only on the margin of difference
in the score at the time the goal was scored. This resulted in a couple
of flaws: 1) The weighting mechanism was somewhat arbitrary. 2) No value
was give to crunch time goals: a goal that put one team up 1-0 in the
10th minute was valued exactly the same as a goal that put a team up 1-0
in the 85th minute.
Lev Polinsky suggested that one could follow baseball's example and
weight the goals by the change in the expected points gained by the
scoring team as a result of the goal.

Having a little spare time on my hands, I decided to embark in the task
of constructing this new measure. I use Ken's data and I thank him again
for going through the trouble of putting it together. Some interesting
results come out.

2. THE NEW WEIGHTING MECHANISM
[Skip directly to Part 4 if you're allergic to numbers]
As I mentioned above, a goal should be weighted by the additional points
that it is expected to give to the scoring team.
For example, assume that the score is 0-0 in the 89th minute. There's a
very high probability that this match will end in a draw. However, if
one scores a goal in the 89th minute, the probability that the scoring
team wins shoots up, and the probability that the match ends in a draw
drops down to nearly zero. Such a goal is worth approximately 2 net
league points to the scoring team.
Formally, we can define the value of a goal scored in minute t, that
makes the difference in score equal to d goals as:

Value= 3*{P[win | min=t, diff=d] - P[win | min=t, diff=d-1]} +
+ 1*{P[draw | min=t, diff=d] - P[draw| min=t, diff=d-1]}

The first line tells us the change in the probability of winning when
one scores the d-difference goal in the t-th minute; the change is
multiplied by 3, given that a win is worth 3 points. The second line
tells us the change in the probability of a draw, and is multiplied by
one.
This looks like a very neat measure. A nice feature is that it has some
real, not just nominal, meaning. In fact, when we add up the weighted
goal values for each player, we get a measure of the expected number of
points that a player gave to his team as a result of his goals.
Now, all we need is to construct the probability measure for every
possible minute and every possible difference in scores. This can be
done using Ken's data on last year's 306 Serie A matches; however, one
must be a bit cautious, since the sample used to estimate these
probabilities becomes quite small for large margins. On the other hand,
one hopes that these biases are relatively small, and that the overall
measure will still be meaningful.

3. IMPLEMENTATION
In practice, I used two alternative methods to estimate the matrix of
probabilities. The first method estimates different probabilities for
home and away teams: this is actually a more refined and improved method
compared to the one described above, as it also gives extra weight to
away goals. On the other hand, more data is needed to estimate the
probabilities accurately: some of the goals scored in the very first
minutes may receive an inaccurate weight.
The second method ignores the distinction between home and away goals,
but it uses more data to estimate the probabilities.

Here are some examples: the probability that the home team wins a game
given that it is up by one goal in the 15th minute is 62.7%. The
probability that it draws given that it is up by one goal in the 15th
minute is 30.2%. The probability that it wins given that the score is
level in the 15th minute is 45.5%. The probability that it draws given
that the score is level in the 15th minute is 29.5%. A quick application
of the above formula reveals that
the value of a goal that puts the home team up by one goal in the 15th
minute is 0.525. [0.525 = 3*(0.627-0.455) + 1*(0.302-0.295)]

What about a goal by the away team in the 15th minute? The probability
of an away win given that the away team is up by one goal is 57.1%, and
that of a draw is 22.9%. The probability of an away win given that the
score is level in the 15th minute is 25%, and that of a draw is 29.5%.
Hence, the value of a goal that puts the away team up by one in the 15th
minute is 0.898.

The second method (partially) ignores the distinction between home and
away teams. The probability that a team up by 1 at the 15th minute
eventually wins the game is 60.3%, and the probability that it draws is
26.9%. The probability that the home team wins given that the score is
level in the 15th minute is 45.5% and the probability that it draws is
29.5%. The probability that the away team wins given that the score is
level in the 15th minute is 25%, and the probability that it draws is
29.5%. Hence, the value of a goal that puts a team up by one in the 15th
minute is 0.416 if the goal is scored by the home team, and is 1.032 if
the goal is scored by the away team. [Wait a minute: didn't I just say
that in this method the distinction between home and away teams is
immaterial? Well, it is in the "general" case, but one needs to make a
distinction between home and away teams when calculating goals that
either break a tie, or create a tie. The reason for this is that the
probability that a team wins (or loses) given that the score is level is
not well defined].

Without going into all the details, let's look at some more goal values
for other types of goals:

Method 1            Method 2
Home        Away       Home     Away
Puts team 1 goal up
in 15th minute           0.525      0.898       0.416    1.032

Puts team 1 goal up
in 75th minute           1.266      1.480       1.236    1.530

Puts team 2 goals up
in 62nd minute           0.397      0.694       0.520    0.520

Equalizes  in 88th
minute                   1.003      0.879       1.011    0.874

These are just some examples, but they give the main idea: goals scored
later in the game are valued more; goals that put the team 1 goal up are
more valuable than equalizers and goals that put the team 2 goals up.
Note also that the home equalizer in the 88th minute is more valuable
than the away equalizer in the 88th minute: this is because home teams
are more likely to score a winning goal in the final minutes (including
injury time).

4. RESULTS
So what does all this give us? Here is a table that ranks Serie A's
leading scorers in 1997/98 according to different methods:

(1)    (2)      (3)         (4)        (5)        (6)

Raw    PAS1     PAS2      Expost1    Expost2      KOV

Bierhoff,O.        27    19.98    20.09       30.50      45.00      19.17
Ronaldo,L.         25    16.95    16.25       27.00      32.05      16.45
Baggio,R.          22    14.44    14.32       16.30      26.25      14.25
Batistuta,G.       21    15.81    15.23       14.35      20.05      14.23
Del Piero,A.       21    11.78    10.71       22.40      27.85      14.50
Montella,V.        20    15.05    15.44       18.77      21.00      14.83
Inzaghi,F.         18    15.28    15.20       19.10      25.25      12.67
Hubner,D.          16    11.41    10.64        8.75       8.25      11.50
Oliveira,L.        15    11.27    12.19       14.35      10.55      10.42
Balbo,A.           14     9.11     8.09       10.25      18.75       8.12
Esposito,C.        14     8.49     6.43       10.05      10.60       7.92
Totti,F.           13    10.64    11.44       10.68      11.60       9.58
Andersson,K.       12     7.67     7.47        8.70      15.50       7.42
Crespo,H.          12     9.77     9.11       11.75      11.25       9.08
Paulo Sergio,S.    12     7.89     7.71        8.88       8.30       7.17
Nedved,P.          11     8.24     8.01       12.25      16.75       8.33
Bellucci,C.        10     7.61     7.03        3.00       3.00       6.53
Boksic,A.          10     3.79     4.35       12.25      13.25       5.67
Chiesa,E.          10     7.30     7.49        9.50      10.50       7.33
Palmieri,F.        10     6.25     6.48        7.25       9.00       5.42
Poggi,P.           10     8.05     7.74        8.25      13.75       6.95
Weah,G.            10     7.40     8.14        8.50       6.50       6.83

The first column shows the raw goal total. Columns (2) and (3) show the
two new measures described above. Columns (4) and (5) show my initial
ex-post measures (you can look at the detailed description in my article
dated May 27 1998), and column (6) presents Ken Overton's measure.
[Formulas:
(4) weight = 3*(1/finalA) if finalA>finalB
1*(1/finalA) if finalA=finalB
(5) weight = 3*(1/(finalA-finalB)) if finalA>finalB
1*(1/finalA) if finalA=finalB
(6) weight = 1/diff if diff>=1
1/abs(diff-2) if diff<=0  ;
where finalA is the final number of goals of the scoring team, finalB is
the final number of goals of the conceding team, and diff is the current
difference in score generated by a goal. The formulas for columns (2)
and (3) are given above.]

This table presents some interesting results. The first two places in
the ranking are unchanged, no matter what measure we use. However, as we
go down the ranking, some interesting things happen. For example,
Batistuta, who ranked very low based on the ex-post measures, redeems
himself when we use the new measures. This is because Bati tended to
score often in big Fiorentina wins last year. These goals received
little value in the ex-post measures, but the ex-post measure ignored
the fact that Bati was often the one who opened the flood gates for
Fiorentina's success. On the other hand, Del Piero does pretty badly in
the new ranking. My guess is that many Del Piero goals were at home and
several came late in the game. On the other hand, Inzaghi scores high
with the new measure: his goals are the heaviest, in the sense that they
have the highest value weighted to actual goal ratio.
Boksic stands out for his very low value-weighted total. Four of his ten
goals were scored in the final minutes, and received a weight of zero.
Compare that to his ex-post total, that places him high in the ranking,
mostly because his goals coincided with Lazio's period of grace.
Finally, Totti and Crespo scored relatively high-powered goals, and
Hubner and Bellucci's goals, which received very little value in the
ex-post measures, are revalued using the new method, even if in the end
these goals didn't save their teams from relegation.

5. CONCLUSION
If you've read this far, you must be a very brave person, or a total
football and statistics junkie like myself. Well, I hope you found these
stats interesting, and I'm looking forward to hear comments and
suggestions.

Daniele
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