In the Champions League tie between Manchester United and Real Madrid which finished last night, for roughly 145 minutes the two sides played at even strength, and United outscored Real 2-1. For roughly 35 minutes, Real were a man up, and outscored United 2-0. Real went through.

The shape and flow of the game changed instantly after Nani’s controversial sending off. Whether or not his particular red card was justified, it seems to me that the whole idea of the red card itself is not, and it would make more sense if teams were able to replace a sent-off player, using one of their substitutions.

There are three reasons to punish players for committing fouls: disincentive, retribution and compensation for the other side. I don’t think the red card rule, as it stands now, can be justified on any of them. The threat of being sent off is a strong personal disincentive, regardless of the effect on the team. And the league can always add suspension and fines to the punishment after the game, when they’ve had a chance to review the foul.

As for compensation, the red card gives the injured team a disproportionate advantage. We saw that last night. It’s a very crude device. In an article published in the Journal of the American Statistical Association in 1994, G. Ridder, J.S. Cramer and P. Hopstaken found that after a red card ‘the scoring intensity increases by 88 per cent for the team with 11 players’ (a.k.a. ‘team 1’) while remaining unchanged for the penalised team, and that ‘a red card early in the match increases team 1’s probability of victory substantially’. Andrew Titman and colleagues at Lancaster University made similar findings last year, while observing ‘no direct effect of yellow cards on goal scoring rates’. What makes the crudity of the punishment particularly obvious is that two yellow cards add up to a red. So you get this anomaly: a team with two players who commit a bad foul each is punished much more lightly than a team with one player who commits two fouls.

This leaves retribution. It seems a little odd to talk about that in the context of football. It’s a game. But the league itself would be better placed to decide appropriate punishment after the heat of the match, with the benefit of video replays. Players often get a bad rap for protesting, but on the whole I find that less offensive than the posturing of the referee, officiousness brandishing authority in the face of brilliance.

There’s another kind of argument fans sometimes make. That injustice is part of the game. That it heightens the drama (Ridder, Cramer and Hopstaken observed that ‘a motivation for the more frequent use of the red card is to increase the number of goals scored in a match... it has the desired effect'). Sometimes that’s right, though last night the red card killed off the match pretty quickly.

Besides, most athletes are competitive people – they want to win because they’re better, and they want the outcome of the game to prove it. There’s not much joy to be had from beating a guy with his hands tied.