# Tie-break Criteria and the Manchester Clubs

The 2011/12 Premier League season finished with Manchester United and Manchester City at the top of the table with a joint number of point. Different leagues have different rules on what happens in this situation.

In the past the relevant tie-breaker was ‘goal average’. This is the ratio of goals scored to goals conceded. It has not been used in the English Football League since 1975-76. This seems like an odd measure, as intuitively it is the number of goals you win by which is important, irrespective of offsets. For instance, surely 2-1 is just as good as 1-0?

This is no longer used, presumably for the reason that it makes conceding goals too costly. Goal difference is used instead. This is the difference between the number of goals scored and the number conceded.

In tournaments like La Liga, Serie A, the Champions League group stage and the European Cup group stages, it isn’t immediately used. If two or more teams are equal in points then a new league is considered which contains only these teams and from the games between these teams. This is then ordered first by points, and then by goal difference (and then it will typically continue to goals scored, and so forth). The ordering of this league then orders the joint teams in the original league.

[As an aside, I find it slightly odd that the rules are not applied recursively.]

In other tournaments, like the Bundesliga, the Premier League and the World Cup group stage, then after points goal difference across all games is compared. This is what happened to Manchester United and Manchester City, where Manchester City become the first team to win the league on goal difference. (The only time there was been a draw on points at the top of the table since goal average was abolished was 1988-89, where Arsenal and Liverpool were drawn on both points and goal difference, and Arsenal won it on goals scored)

Both teams won, lost and drew the same number of games, giving them the same score. However, Manchester City had a goal difference of 64 and Manchester United had a goal difference of 54.

Goal difference is better than goal average in that it only cares what the difference in scores is. Nevertheless, it is perhaps too linear. Is winning four games 1-0 and then winning a single game 6-0 better than consistently winning 2-0 over five games? Goal difference treats them as the same. If they aren’t equal, it isn’t particular obvious which is better. The latter team were consistently better, but the former team achieved a large victory. There seems to be an argument either way.

The following is a graph of the distribution of wins of the two Manchester teams. As it is already decided that it is only goal difference in a match which matters, this is all that is graphed. Obviously the total left of ‘0’ denotes the number of losses, and right of it the number of wins.

It should be noted that when the Manchester clubs played each other, United won 1-0 away and lost 6-1 at home. Therefore if a head-to-head system were to be used which went to goal difference, City would still win. Goal average would not help United either – they conceded more goals and scored less, so any sensible combination of these two numbers will see City win. Had the United-City game been 1-0 instead of 6-1, the goal difference delta would have changed by 8, but City still would have won on goals scored.

The graph shows that United did achieve more very large wins than City though. United won of five occasions by five or more goals. City did this ‘only’ twice. Perhaps there is some home for United after all, if we change our values?

Is winning by two goals twice as good as winning by 1? Perhaps not. Winning by one can be luck, but winning by two seems like there might have been some skill involved. Perhaps instead of adding up the games goal differences, one should add up the squares of the games goal differences. Obviously, one would have to subtract instead of add if you lost by that amount, otherwise losing by 2 goals would be just as good as winning by 2 goals!

If this is done, Manchester United get a score of 188, but Manchester City get 212. They will still win.

Okay, let’s say winning by two goals is twice as good as winning by one. But then is winning by three goals only 1.5 times better than winning by 2? Perhaps it is twice as good again. Continuing  5 goals is 16 times better than scoring 1. If you lose, it should be the same as if you win except with the opposite sign, and if it was a draw the value is zero.

If we do this, then Manchester City get 112 points, but Manchester United get 121.

Nevertheless, this method is fairly silly, as it means the game Manchester United won 8-2 at the start of the season screws it all. Perhaps the truth is the opposite: the first goal is very important, but each goal after that becomes less and less important. After all, 5-0 and 6-0 are basically the same score. The last goal doesn’t matter very much at all. Perhaps it shows more like a logarithm than a power.

Using $\log{(1+|d|)} \times \rm{sign}(d)$ gives Manchester United 38.1 points, but Manchester City 32.3.

In summary, there is no easy answer to a tie-breaking criterium that rewards the things you want it to. At least goal difference is simple, and so it seems a good choice.