How could additional substitutions affect the title race?

Like their German counterparts, the clubs of the highest Austrian league have decided to take usage of FIFA's step to allow two additional substitutions for the remainder of the season. While it has been argued that this rule change will especially benefit the bigger teams due to their stronger squads, we can use historic data (i.e. the season before lockdown) to deduce which effects the new norm can have, especially on the most important decisions (league winner, European starters and relegation). I also propose a new measurement of squad rotation, which will be affected by the rule change.

A first look at the data shows that most of the teams in the league are quite happy to use the substitutions they are allowed to make (see Graph 1); three of the twelve clubs of Austria's highest league used all of their 66 (22 league games so far). Another seven teams missed very few possibilities to make a change during their games. Only one team (Altach) is more reluctant to make changes within games, rejecting one substitution almost every other game.

Graph 1 (click to enlarge)

There is, however, no relationship between the use of substitutions and teams' position in the table. Among the three clubs at the top of the substitution usage table, there are both highly (LASK) and lowly (Wattens, St. Pölten) classified clubs.

Pure usage figures alone don't tell us a lot about how the rule changes might influence clubs' behaviour, although we can suppose that teams which are very reluctant to even use all three substitutions will also use the additional ones more cautiously.

Likewise, the patterns of substitution timing might be affected. Right now, clubs tend to make their first substitution between minutes 50 and 60 (see Graph 2), although there are some significant differences between them. You would suppose that the first, second and third change will come earlier, so the lines on Graph 2 will be lower on the y-axis. Knowing that they have two more opportunities, coaches might even take advantage of tactical changes in the first half more often.

Graph 2 (click to enlarge)

You can also expect the number of substitutions at half-time to go up, due to the restrictions to three slots in which teams can take off players during the second half. Only seven percent of substitutions so far were done at the break, with newly promoted Wattens the most active team at this time (nine). On the end of the scale, substitution resistant Altach and Sturm have only used this opportunity twice, with all the other teams recurring three, four or five times to it.

On a club level, teams which tend to take their first changes later also tend to take their second and third later. My hypothesis would be that these patterns won't change all that much, rather we will see more double or even triple subs later on in the game by teams such as Sturm, Austria and Wolfsberg (Admira is a bit more difficult to assess given that they have their third head coach this season already). 

So far for the historic data. Beyond these, we are more interested in the way the new rule might affect teams and their playing performance during the rest of the season. We need therefore a way to assess the quality of those players which might benefit from more substitutions, i.e. the players who have not been regular starters or are not regularly subbed on when not in the starting eleven.

One possibility to assess their quality is to simply look at the number of minutes they have played (league only), assuming that higher quality players tend to get more game time. We therefore rank the players of each team by the number of minutes played and compare afterwards those players with the 12th to 14th most minutes (regular subs) to those with the 15th and 16th most. The latter are those who should benefit the most from the upcoming rule change.

Graph 3 (click to enlarge)

Higher levels of minutes played by either subgroup indicate in theory higher levels of quality, hence the club should benefit more from the new rule. As we can see in Graph 3, there is a mostly linear relationship between both indicators, with the notable exceptions of probably-not title candidate LASK and Hartberg. Both of them rely on a strong core of 14 more or less regular players, but lack depth afterwards. The latter are an interesting case, as they have reached the upper play-off group, which means that they cannot be relegated anymore but might reach the Europa League, which would be a huge success given their stature.

There is no surprise that Wolfsberg were the only club decidedly against the rule change, given their location in Graph 3. They neither had nor will have strong impact coming from the bench, which is the more interesting as they also competed at the European level before Christmas and are still in contention for another year of international football.

Leaders Salzburg should definitely be able to exploit the new rules given the large resources at their disposal, giving them an edge in the title race (which also includes third-placed Rapid). At the other end of the table, Mattersburg might have a little advantage and Admira a small drawback in a relegation battle which will be fierce, with the six teams in the lower play-off separated by four points only.

Minutes given to regular substitution players are in any case not the only way to measure squad involvement and depth. One could also analyse the total number of players used and compare teams by the amount of footballers they field during the course of a season, either total or above a certain threshold of minutes. These measurements are fine yet can be improved to get a more detailed picture. I propose therefore a methodology, borrowed from Political Science, which counts not the actual but the effective number of units (in this case, players) in a given system.

Consider the case of two fictious parliaments. In the first one, Party A got 42% of the seats, Party B has 38% and Party C the remaining 20%. In the second one, there is a majority Party D with 60% of the seats, meanwhile the rest of the mandates are distributed among parties E (25%), F (10%) and G (5%). Which one of them has more parties? 

The answer is that it depends. Surely, you could simply count parties in each chamber without further consideration of their numerical strength, which would show that there are more parties in the second example (four) than in the first one (three). This is simple, but also misleading. Our hypothetical Party D would in reality be able to govern quite easily without bothering much about the other parties, meanwhile in the first example, no party alone has enough seats to reach a majority. It is therefore more accurate to count the parties not equally, but rather weighting their seats (or vote share, if you are interested more in the electorate than in the parliament) according to their own strength and relative to the strength of the other represented parties.

That is where the idea of the effective (rather than the actual) number of parties comes into place. The concept is satisfyingly described on Wikipedia, from where I also copied the formula, although I use a slightly different one. 

Formula to calculate the effective number of players

It looks more complicated than it actually is. You basically take 1 (if you work with percentages) and divide it by the sum of the squares of the shares. In the case of our two exemplary parliaments, the number of effective parties in the first one would be 2.8 and in the second one 2.3, so the difference is actually reversed in comparison to the simple count. This difference is however much closer to reality than following the naive assumption and counting all the units the same way.

We can apply this concept to football easily, we simply have to adjust the formula a bit and of course use minutes or starts instead of seats or vote shares. Given that minutes played by all the players in a team are variable (due to injuries and sending-offs, a team's players not always play the sum of 11*90 minutes), we have to make the values comparable. We do this by not dividing 1 by sum of squares but using the square of the sum as the dividend. 

To make this a little clearer, we'll apply this to a very simple example, a team which in a single match makes three substitutions at half time and compare it to a team which plays with the same eleven players during the whole game. The first team would have 

(90+90+90+90+90+90+90+90+45+45+45+45+45+45)^2 

divided by 

(8100+8100+8100+8100+8100+8100+8100+8100+2025+2025+2025+2025+2025+2025)

effective players, i.e. 12.74. The team without substitutions accordingly would have 11 effective players.

By applying this formula to the sum of minutes played and starts throughout the season so far, we can calculate the effective number of players and starters for each team. 

Graph 4 (click to enlarge)

By comparing the actual and effective numbers of players used, we can see how many players teams give minutes and also how evenly they distribute minutes among them. A quick look at graph 4 shows that teams cluster to a certain point. St. Pölten stands out in absolute terms, a strategy not really backed up by results so far (they will start the remaining from the last position). Rapid have given a lot of young players a limited amount of minutes without conceding them starts, meanwhile the rest of teams has used between 24 and 28 players out of which 21 to 27 have been starters. 

The picture clearly changes when we look at the effective numbers of players used and started. St. Pölten still leads the trail in terms of the amount of players used, but Salzburg have more effective starters, indicating a high level of rotation in the league. This might be a decisive advantage over title rivals LASK, who used the same number of players and just one starter less in absolute terms, but whose substitution players received far less minutes among them.

In the middle, there are a lot of teams in a quite congested area of the graph. Although I put the team names with an angle, there was no way to prevent this overplotting, because teams are quite close to each other, hence there are no big differences to expect for the rest of the season. Interestingly, the closely packed teams are all from the lower play-offs, with Austria Vienna having a slightly deeper and more balanced squad than their direct rivals. This might favour them to finish first in the lower play-off, a position that would give them the chance to qualify for Europe in spite of a rather underwhelming campaign so far.

At the end of the scale, there are those three teams which also used their players 15 and 16 the least (Graph 3). Wolfsberg, Hartberg and Sturm will battle for European qualification, but based on their squad depth and quality will not be able to join the title race, even with the point difference halved.