Are the way that substitutions are utilised in soccer optimal?

After some research I stumbled onto a soccer metrics page: http://www.soccermetrics.net/team-performance/substitutions-and-their-impact-at-euro-2012 and http://www.soccermetrics.net/team-performance/sudden-impact-substitution-at-the-2010-africa-cup-of-nations.

The data both confirms and disproves some of my observational comments. In these games there was 1 unused substitution slot every two games, i.e. 1 in 2 games used all 6. Also since these were tournament games the end time was not necessarily always known so that may bias the stats for subs held out until extra time which never came.

The distribution seems fairly natural, but the author questions the optimal time for making substitutions as well, without giving any firm conclusions.

I think it would be interesting to compare this with more data on league games, which unfortunately I can't find.

I agree with all points made by @alamooot except I'm still unsure on the frequency of injury meaning I still question whether saving a sub in case of this is better than bringing on a fresh pair of legs.

Your question is somewhat subjective and the answers (or at least this one) might be found to be opinion based. But for me there are a few factors to consider:

1. You only get 3 substitutions
2. You want to utilize your staring line up
3. The reason behind substitutions

Using these points I'll argue that the value of substitutions are not under utilized the way they happen in most professional soccer games.

1. You only get 3 substitutions

If the coach makes 2 or 3 quick substitutions early in the game, they can end up regretting it later in the match. Coaches want to leave themselves with the leeway to make changes based on their own team's or the opposing team's performance.

Imagine making all your substitutions in the first 60 minutes. If one of your players gets injured, gets a yellow or red card, or your team goes down a goal you can't bring in new players to make adjustments.

So the substitutions are made while keeping it in mind how much time there's left in the game. When you substitute late in the game, you minimize the odds of having to make another substitute for readjusting team strategy. Late substitutions can be linked to replacing tired players and other reasons we'll cover in the third point.

Similarly if coaches leave their substitutions all towards the last few minutes of the game they lose the opportunity to make in game adjustments throughout the game. So doing substitutions too early or too late under utilizes their values.

If teams had more than 3 substitutions, you'd then naturally see more substitutions. But also earlier ones, purely because you'd have more substitutions later on to adjust the game plan.

As argued, more substitutions has leads to earlier substitutions in 2020 post the COVID-19 shutdowns. FIFA allowed for up to 5 substitutions for these matches. These extra 2 line up changes a side have made the half time substitutions far more common.

2. You want to utilize your staring line up

Replacing players in the first half is very uncommon as they still have the energy to play for many more minutes. Most football players are used to playing full match so substituting them before half time is a waste of a substitution in most cases. Players also get a 15 minute break at half time so no need to make substitutions in first half to preserve player energy. Furthermore the coaching staff has days between matches to decide which players give the best chance to win based on their level of fitness, skills, injuries, bookings, and etc. So the starting line ups are perceived to give the maximum utility and reduce the need for substitutions.

The only times you see first half substitution is when players get injured, when the coach needs to desperately make an early adjustment because of coming to the game with the wrong tactics (assuming that the coach is willing to accept having made wrong tactics), or to punish players for not following team tactics. An example of the last point is Mario Balotelli substituted for making a ridiculous back heel shot in a friendly game.

So in general substitutions are all done in the second half to maximize the overall utility of all players.

3. The reason behind substitutions

The time of substitutions are dependent on the reason behind the substitutions.

Substitutions to replace injured players can happen at any minute of the game. On the other end of the spectrum you see the "last minute" substitutions when a team wants to kill time as they're happy with the outcome of the game and want to drag the outcome to the end of the game.

Another common type of substitution is at the half time. This usually signals that the coach is either not happy with the player performance or is in need of a change of tactics.

Lastly, you see substitutions between the 70th to 80th minutes where an older player comes in. One of the reasons behind substitutions are to bring in players with fresh legs. Older players tend to get tired faster and so substituting them earlier means they can't play at a high level till the end of the game. Or if they start the game they're likely to be substituted in the second half for the same reason.

There are also times that there is no need to do all 3 substitutions. For example if your team is up by 2 goals and you have control of the game, have made one substitution and there's 10 minutes left to go, then there's no need to do all 3 substitutions.

So in summary substitutions are made precisely to increase the utility of the players on pitch, hence substitutions are not under utilized.