How to resolve the Premier League

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.

Position	Team	Games played	Goal difference	Points	PPG
1	Liverpool	29	45	82	2.828
2	Man City	28	37	57	2.036
3	Leicester	29	30	53	1.828
4	Chelsea	29	12	48	1.655
5	Man Utd	29	14	45	1.552
6	Wolves	29	7	43	1.483
7	Sheffield Utd	28	5	43	1.536
8	Tottenham	29	7	41	1.414
9	Arsenal	28	4	40	1.429
10	Burnley	29	-6	39	1.345
11	Crystal Palace	29	-6	39	1.345
12	Everton	29	-9	37	1.276
13	Newcastle	29	-16	35	1.207
14	Southampton	29	-17	34	1.172
15	Brighton	29	-8	29	1.000
16	West Ham	29	-15	27	0.931
17	Watford	29	-17	27	0.931
18	Bournemouth	29	-18	27	0.931
19	Aston Villa	28	-22	25	0.893
20	Norwich	29	-27	21	0.724

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 PPG_H stands for points per home game and PPG_A for points per away game.

Premier League home and away PPG.

	Home			Away
Team	Played	Points	PPGH	Played	Points	PPGA
Liverpool	15	45	3.000	14	37	2.643
Man City	13	29	2.321	15	28	1.867
Leicester	15	30	2.000	14	23	1.643
Chelsea	15	24	1.600	14	24	1.714
Man Utd	15	29	1.933	14	16	1.143
Wolves	15	22	1.467	14	21	1.500
Sheffield Utd	15	24	1.600	13	19	1.462
Tottenham	14	26	1.857	15	15	1.000
Arsenal	15	26	1.733	13	14	1.077
Burnley	15	23	1.533	14	16	1.143
Crystal Palace	15	22	1.467	14	17	1.214
Everton	14	25	1.786	15	12	0.800
Newcastle	14	21	1.500	15	14	0.933
Southampton	15	14	0.933	14	20	1.429
Brighton	14	18	1.286	15	11	0.733
West Ham	14	15	1.071	15	12	0.800
Watford	14	17	1.214	15	10	0.667
Bournemouth	14	17	1.214	15	10	0.667
Aston Villa	13	17	1.308	15	8	0.533
Norwich	14	15	1.071	15	6	0.400

One way of obtaining a single number for each team would be to take the average of each team's PPG_H and PPG_A. 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 PPG_H and PPG_A for each team to take this difference into account.

Adjusted home values

First let's consider the points per home game (PPG_H). We can calculate an adjusted value (Adj PPG_H) 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 (F_H) 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 PPG_A appears in the numerator and denominator of the expression above.)

We now define the adjusted PPG_H for team T as

$$Adj \;\;PPG_H = PPG_H \times F_H.$$

This adjusted PPG_H 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 $$\mbox{Average $PPG_A$ over opponents $T$ played at home}=\frac{1.643+1.714+1.500+...}{14}=1.069$$ and $$\mbox{Average $PPG_A$ over all $T$'s opponents}=\frac{2.643+1.867+1.643+...}{19}=1.191.$$ Then $$F_H=\frac{1.069}{1.191}=0.898.$$ 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 F_H1), their PPG_H has effectively been scaled down.

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

Adjusted PPG for the home games of the teams at the foot of the Premier League table.

Team	Played	Points	PPGH	FH	Adj PPGH
Brighton	14	18	1.286	0.898	1.154
West Ham	14	15	1.071	1.078	1.155
Watford	14	17	1.214	0.982	1.192
Bournemouth	14	17	1.214	0.985	1.196
Aston Villa	13	17	1.308	0.964	1.261
Norwich	14	15	1.071	1.067	1.143

Adjusted away values

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

$$F_A = \frac{\mbox{Average $PPG_H$ over opponents $T$ played away} }{\mbox{Average $PPG_H$ over all $T$'s opponents}},$$ Then $$Adj \;\;PPG_A = PPG_A \times F_A.$$

Adjusted PPG for the away games of the teams at the foot of the Premier League table.

Team	Played	Points	PPGA	FA	Adj PPGA
Brighton	15	11	0.733	1.037	0.760
West Ham	15	12	0.800	1.004	0.804
Watford	15	10	0.667	1.021	0.681
Bournemouth	15	10	0.667	0.960	0.640
Aston Villa	15	8	0.533	0.961	0.513
Norwich	15	6	0.400	0.987	0.395

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.

Position	Team	Played	Adj PPG	PPG	Points
15	West Ham	29	0.979	0.931	27
16	Brighton	29	0.957	1.000	29
17	Watford	29	0.937	0.931	27
18	Bournemouth	29	0.918	0.931	27
19	Aston Villa	28	0.887	0.893	25
20	Norwich	29	0.769	0.724	21

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.

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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.