Monday 12 September 2011

Did Poor Decisions Really Deprive Arsenal of the Title ?

Last Saturday,Liverpool's Kenny Dalglish suggested that his side had been on the wrong end of four poor decisions already this season when they were denied two penalties by the referee and three points by Stoke.Never mind that most impartial judges thought Mark Clattenburg acted correctly in rejecting Liverpool's claims and even acted leniently in not dismissing Jamie Carragher after the one match winning penalty he did award to Stoke.

It's been a continuing theme in recent months and came only days after the claim that Arsenal were denied finishing level on points with Champions Man Utd by a series of ungenerous calls by the officials.

It's easy to see how managers and fans can start to feel victimised by perceived poor officiating as the commonly voiced opinion that "luck evens itself out over the season" is almost never true.Fortune,be it good or bad has no memory,so if a team genuinely has had a run of random,poor decisions go against them,there is absolutely no reason why they are due a run of good decisions in the immediate future.

So what of the claim made on Arsenal's behalf.If Arsenal had a "significant" decision go against them,then they were credited with extra goals and the final score and points tally was recalculated.However,change an event and you change what happens subsequently.

Therefore an approach based on Expected Points is a much more suitable way forward.

Firstly a brief recap of the methodology.

The win probability for each team is tracked on a minute by minute basis throughout each EPL game.At any point in a game a team will have a probability of winning the game outright and an associated probability of drawing the game.By multiplying the probability of winning the game by 3 and the probability of drawing by 1 and adding the results together you will get an expected longterm average for the number of points that team will achieve from that position or Expected Points (EP) for short.

I'll start with Arshavin's disallowed 88th minute goal in Arsenal's 0-0 draw at home to Sunderland.Arsenal just got a point from the game,but the author of the claims they should have gained all 3.At first glance this seems reasonable.Arsenal would have been long odds on to take all 3 points,but their EP when leading Sunderland in the 88th minute wouldn't have been 3,it would be only 2.869 expected points.Close to three admittedly,but you must allow for the slim possibility that Sunderland could equalise and the even slimmer one that they could come from behind to actually win.

The next quibble is more serious.Arsenal were credited with three points when the referee failed to award them a 89th minute penalty at Wigan with the scores level.Those two extra points are far from assured.After the award of the penalty there is a 75% chance that the kick will be made and the EP will rise to 2.87,but there's also a 25% chance that the kick will be missed and Arsenal's EP will stay at around 1.284.Overall,if the ref had given the penalty Arsenal's EP would have risen to 2.47 points as their kicker ran up to take the shot,well short of the claimed 3 points.We can acknowledge that a penalty should have been given,but that's were the certainty ends.We don't know if it was scored,but we do know that in the longterm,Arsenal taking a penalty at Wigan with the scores tied in the 89th minute will accumulate on average 2.47 points.So that's what we must work with.

Similarly,the penalty that wasn't awarded to Blackpool in the 54th minute when Arsenal led 2-1 was considered irrelevant because the final score was 3-1 to Arsenal.The "extra" goal that Blackpool may have scored would still have only made the score 3-2 to Arsenal.However,If Blackpool had been awarded a penalty,that award would have reduced Arsenal's EP to just over 2 at that point.Again we can go to the point were the new "correct" decision is made,but after that we have to work with probabilities to account for the new course the game has taken.Arsenal are no longer certain to take three points in the alternative reality that saw Blackpool awarded a 54th minute penalty.

Even more problems occur when you consider the case of the red card Fabregas didn't get in the 88th minute against Wolves.The score was 1-0 to Arsenal at the time and the dismissal would have reduced the Londoners EP to 2.78,but just as crucially Fabregas could have been suspended for the next game against Everton in which he scored the winner.Once you start playing the "what if" game,you never know when to stop.

In short,if you introduce a significant new event into game then the only way you can deal with what could happen next is by using expected points.

To illustrate how Arsenal's Expected Points relate to their actual points,I have calculated their EP at significant points that did occur during the 2010/11 season.I've chosen the 1st goal scored in the game and the first table shows the EP immediately after the goal was scored,the actual points gained at the end of the game and a cumulative total for both columns over the season.

As you can see Arsenal under perform the EP cumulative total by about three points and in 20% of the games they gain considerably less points than the Expected Points. A large amount of the "shortfall" comes from the Liverpool game at the Emirates,where Arsenal took the lead in injury time,elevating their EP to 2.868,but a Liverpool equaliser with the game's last kick left the game tied at fulltime.Liverpool's equaliser was unlikely,but not impossible and it was that possibility that made Arsenal's EP after the first goal only 2.868.

Unlikely events happen and EP accounts for that possibility much better than trying to insert goals additively into games.

Expected Points compared to Actual Points where Arsenal score the game's opening goal.

Home Team. Away Team. Time of 1st
Arsenal's EP
after 1st Goal.
Points Gained by Arsenal.
Newcastle. ARSENAL. 1 2.570 1
Wolves. ARSENAL. 1 2.573 3
Tottenham. ARSENAL. 5 2.244 1
ARSENAL. Stoke. 8 2.793 3
ARSENAL. Tottenham. 9 2.636 0
ARSENAL. Blackpool. 12 2.884 3
Birmingham. ARSENAL. 13 2.571 3
Sunderland. ARSENAL. 13 2.572 1
West Ham. ARSENAL. 13 2.671 3
ARSENAL. Fulham. 14 2.786 3
ARSENAL. Wolves. 16 2.876 3
Blackpool. ARSENAL. 18 2.768 3
Blackburn. ARSENAL. 20 2.670 3
Man City. ARSENAL. 20 2.774 3
ARSENAL. Wigan. 21 2.872 3
ARSENAL. Bolton. 24 2.785 3
Everton. ARSENAL. 36 2.375 3
Aston Villa. ARSENAL. 39 2.381 3
ARSENAL. Chelsea. 44 2.315 3
ARSENAL. Man Utd. 56 2.386 3
ARSENAL. West Ham. 88 2.916 3
ARSENAL. Liverpool. 90 2.868 1
58.286 55

A similar pattern is seen when Arsenal's opponents take the lead.A team of Arsenal's calibre would expect to gain on average 13.8 points from games in 2010/11 where they fell behind to the first goal and instead they only gained 10.It's tempting to explain the two tables by suggesting that Arsenal lacked the defensive capabilities to maintain the leads they held or the attacking edge to retrieve the deficits they faced.However,it's more likely that they were just unlucky,the sample size was too small or that they weren't quite as good as the ratings I used to calculate their "in play" EP implied they were.

Expected Points compared to Actual Points where Arsenal concede the game's opening goal.

Home Team. Away Team. Time of 1st Goal. Arsenal's
EP after
1st Goal.
Actual Points
Gained by Arsenal.
West Brom. ARSENAL. 3 1.670 0
ARSENAL. Aston Villa. 11 1.232 1
Wigan. ARSENAL. 18 1.340 1
ARSENAL. Everton. 24 1.421 3
Fulham. ARSENAL. 26 0.893 1
Stoke. ARSENAL. 28 1.041 0
ARSENAL. Birmingham. 33 1.432 3
Bolton. ARSENAL. 38 0.904 0
Chelsea. ARSENAL. 39 0.299 0
Man Utd. ARSENAL 41 0.401 0
ARSENAL. Newcastle. 45 1.325 0
Liverpool. ARSENAL. 46 0.462 1
ARSENAL. West Brom. 50 1.356 0
Total Points 13.775 10

The next step in testing the claim that poor officiating cost Arsenal the title is to take every game used by the original article and calculate the EP at the time of the first significant decision once it has been replaced by the correct decision.Any subsequent events in that game should be discounted as clearly they almost certainly would not occur.The cumulative EP for these "corrected" games can then be compared to the actual number of points gained to see if the claim is justified.

This will be the subject of the next post.

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