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How to resolve the Premier League

Football and virus

Due to COVID-19 football leagues have been suspended.

Due to the ongoing COVID-19 pandemic, football leagues around the world have been suspended and for many there is the threat that the season will not be completed. This has raised the spectre of deciding the positions of teams in the league by points per game (PPG). Indeed in Scotland this appears to be a strong possibility. However, in its conventional form, using PPG could be considered an unfair method, as some teams have played "easier" opponents than others and some teams have played more home games than others. This article suggests an adjustment to PPG to account for these factors.

Looking at the current English Premier League table below, PPG has been determined using the standard calculation

  \[ PPG=\frac{\mbox{Number of points}}{\mbox{Number of games played}}. \]    
Premier League table at suspension on March 9, 2020.
PositionTeamGames playedGoal differencePointsPPG
2Man City2837572.036
5Man Utd2914451.552
7Sheffield Utd285431.536
11Crystal Palace29-6391.345
16West Ham29-15270.931
19Aston Villa28-22250.893

We can see that using PPG would change the positions very little from using points. The only difference would be that Sheffield Utd would swap places with Wolves, and Arsenal would swap with Tottenham.

Nevertheless, if we look at the relegation scrap specifically, it is very close so we need to be confident that PPG accurately represents the teams' relative performances so far. This is debatable. There are 2 major concerns:

  1. Some teams will have played easier opponents than others.
    For example, Brighton were due to play Liverpool, Man City and Man Utd (the 1st, 2nd, and 5th highest teams) within their nine remaining games. This highlights that their PPG score so far was probably gained playing against relatively easier opponents than those their relegation rivals faced, and hence it could be argued that their PPG should be scaled down to reflect this.
  2. Some teams have played fewer home games than others.
    For example, Aston Villa have only played 13 home games whereas their relegation rivals have all played 14. As teams tend to get more points on average at home than away, the PPG calculation should be adjusted to allow for this.

Home and away

To accommodate these factors it is important to look at home and away results separately. In the table below PPGH stands for points per home game and PPGA for points per away game.

Premier League home and away PPG.
Man City13292.32115281.867
Man Utd15291.93314161.143
Sheffield Utd15241.60013191.462
Crystal Palace15221.46714171.214
West Ham14151.07115120.800
Aston Villa13171.3081580.533

One way of obtaining a single number for each team would be to take the average of each team's PPGH and PPGA. However, that would not take account of the fact that some teams have had an easier ride so far than others. So let's see how we can adjust the PPGH and PPGA for each team to take this difference into account.

Adjusted home values

First let's consider the points per home game (PPGH). We can calculate an adjusted value (Adj PPGH) for a particular team, call it team T, by considering the strength of the opponents which team T has played at home. First we define the home opponents factor (FH) for team T as the relative strength of the opponents team T played at home:

  \[ F_ H = \frac{\mbox{Average $PPG_ A$ over opponents $T$ played at home} }{\mbox{Average $PPG_ A$ over all $T$'s opponents}}. \]    

(Note that if team T played an opponent at home, then that opponent played away — this is why PPGA appears in the numerator and denominator of the expression above.)

We now define the adjusted PPGH for team T as

  \[ Adj \; \; PPG_ H = PPG_ H \times F_ H. \]    

This adjusted PPGH should better represent a team's performance than PPG does.

Let’s illustrate this for Brighton, who have played all teams at home except for Liverpool, Man City, Man Utd, Arsenal, and Newcastle. In this case we have

  \[ \mbox{Average $PPG_ A$ over opponents $T$ played at home}=\frac{1.643+1.714+1.500+...}{14}=1.069 \]    


  \[ \mbox{Average $PPG_ A$ over all $T$'s opponents}=\frac{2.643+1.867+1.643+...}{19}=1.191. \]    


  \[ F_ H=\frac{1.069}{1.191}=0.898. \]    

This gives

  \[ Adj \; \; PPG_ H =1.286\times 0.898 = 1.154. \]    

Because Brighton have played easier opponents at home on average than the other teams (i.e. their FH<1), their PPGH has effectively been scaled down.

Doing similar calculations for the other five teams in the relegation battle, we can obtain each team's adjusted PPGH for its home games:

Adjusted PPG for the home games of the teams at the foot of the Premier League table.
TeamPlayedPointsPPGHFHAdj PPGH
West Ham14151.0711.0781.155
Aston Villa13171.3080.9641.261

Adjusted away values

Repeating this process for the away results, the away opponents factor (FA) of a team T is the relative strength of the opponents team T played away:

  \[ F_ A = \frac{\mbox{Average $PPG_ H$ over opponents $T$ played away} }{\mbox{Average $PPG_ H$ over all $T$'s opponents}}, \]    


  \[ Adj \; \; PPG_ A = PPG_ A \times F_ A. \]    
Adjusted PPG for the away games of the teams at the foot of the Premier League table.
TeamPlayedPointsPPGAFAAdj PPGA
West Ham15120.8001.0040.804
Aston Villa1580.5330.9610.513

Overall adjusted PPG

We can then obtain the overall adjusted PPG for each team by finding the average of their home value and away value:

  \[ Adj\; \; PPG=\frac{Adj\; \; PPG_ H+Adj\; \; PPG_ A}{2}. \]    

By utilising averages, we are allowing for teams to have played different numbers of home or away games from each other, which the standard PPG doesn't do.

Using the Adj PPG values, the bottom of the final Premier League table would be as follows:

Final Premier League positions using the adjusted PPG.
PositionTeamPlayedAdj PPGPPGPoints
15West Ham290.9790.93127
19Aston Villa280.8870.89325

As it turns out, in this scenario the same 3 three teams would be relegated using Adj PPG as the original PPG. Aston Villa's increase due to their extra home game is wiped out by their difficult run-in. However, Brighton's PPG has reduced considerably due to their easier past opponents, resulting in dropping a place. This would have serious implications had they been 17th beforehand.

In conclusion, I suggest that the adjusted PPG would be a fairer way than the standard PPG for assessing the performance of teams in a curtailed season. If the Scottish FA or any other body decide to implement a PPG system, they and the clubs should carefully look at its shortcomings and consider the option of using an adjusted PPG. However, I think the very best solution would be to complete the fixtures when it becomes possible to play again, whenever that may be.

Slim Khan

About the author

Slim Khan is a Teaching Fellow at Warwick University, and is particularly interested in mathematics and statistics related to games and sports.

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