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Does the increase to five substitutes really benefits the big clubs after all?
tl; dr: it's complicated. Long thread ahead (0/25
tl; dr: it's complicated. Long thread ahead (0/25
An interesting article by economist/football scientist Ignacio Palacios Huerta on whether increasing the number of subs in football necessarily favors the big teams: https://www.theguardian.com/football/blog/2020/jul/29/football-five-substitutes-rule-does-not-benefit-the-big-clubs-and-here-is-why. Interesting, but I think some people are misinterpreting (1/.
IPH notes that the increased number of substitutions *could* actually favor the smaller teams, but it could also favor the big guys, as the conventional wisdom says. (2/.
It all depends on the relative quality of "starters" versus "subs" in big versus small teams: (3/.
In the first scenario, there is a dropoff in quality of the subs for the big team, but not the small team --> small team has the advantage. In the second, it's the opposite. (4/.
So, what do we know about the differences in quality between starters and subs in small vs big teams? (5/.
Here is a little exercise, which I'm sure can be improved in lots of ways, but hopefully gets the point across. I have taken data on player values at the beginning of the 2016-2017 season from TransferMarkt (6/.
[Aside: TransferMarkt values sometimes give whacky results for individual players, but on average they are not that bad. a TM-based prediction model was the best at predicting the outcome of the 2018 World Cup] (7/.
Then, for each team, I calculate the TM value of the most valuable player, the 2nd most valuable player, etc., all the way up to the 18th most valuable player on the squad (8/.
Then I calculate the average TM value for each rank of the Top 4 teams and the bottom 4 teams in the league (by market value) (9./
Example: the highest ranked player in the top-4 teams were Aguero (60), Hazard (65), Pogba (70), and Alexis Sanchez (55). The average value of the top ranked player is 62.5 (10/.
In the bottom 4 teams: Jordan Rhodes (Boro, 10), Solomon Rondon (WBA, 17), Andre Gray (Burnley, 10), and Abel Hernandez (Hull City, 8). Average value=11.25 (11/.
Then repeat for each rank and each league, calculate the difference between the top 4 and the bottom 4. What comes out? (12/.
This: (13/.
The difference in player values between the top and bottom 4 clubs is very large for the very top players (Aguero vs Rondon), but much smaller for the 14th or 16th best player on the club (Fernando versus Cristian Gamboa). (14/.
There are differences in magnitudes, but the pattern is the same in all 5 big leagues. (15/.
Based on IPH's arguments, this means that the big teams would see a bigger dropoff in quality with the 5-sub rule, so the rule actually favors the small clubs, right? (16/.
Not so fast. IPH's model talks about player "quality", while I am using players' monetary value according to TransferMarkt. Those are not the same thing (17/.
Take three players, similar roles and similar age: Neymar (EUR 128M today), Heung-Min Son (64) and Thorgan Hazard (32). Linear model assumes that the difference in "quality" between Neymar and Son is twice as large as the difference in quality between Son and T. Hazard (18/.
What if quality is multiplicative instead? Diff between Neymar and Son the same as the difference between Son and T.Hazard? Well, let's use log(value) instead of value, and let' see what happens (19/.
This. (20/.
Now not so clear what the message is. In some leagues there is a negative slope, in some the relationship looks pretty flat, in some the slope is positive. 
(21/.
(21/.
For what it's worth, I have a slight preference for log values and use that in my prediction models. Various measures of team performance are more strongly correlated with average team log values than with average team values. (22/.
Bottom line: maybe the conventional wisdom was not so far off, and the increase to 5 subs does actually favor the big guys. Or at least it doesn't clearly favor the small ones. (23/.
Of course, more and better analysis is welcome, as well as comments and suggestions. Fin (24/24
