So Chelsea drew 1-1 against Wigan Athletic after blowing a late lead thanks to some dodgy play by certain players who will remain nameless, and that's sparked off a debate about Andre Villas-Boas tactics in the match. Ignoring the introduction of Salomon Kalou at halftime, which was pretty undeniably sensible unless irrationally hating everything Kalou is your bag, the major talking point was a pair of defence-minded substitutions which saw Juan Mata and Daniel Sturridge withdrawn to protect a 1-0 lead.
The debate hasn't really managed to progress beyond the 'You smell' 'NO YOU SMELL' state as it is, and while that's partially because the Fernando-Torres-needs-to-play-more-crowd is pissed off that he didn't see the pitch it's mostly to do with the fact that as currently framed, there's no single fact strong enough to convince anyone on either side to change their position.
And that's where maths(!) comes in.
Those who've been around this blog for a while are pretty familiar with the framework of win probability (WPA), which basically gives us the likelihood of one average team beating another average team in any given situation thanks to the magic of statistics. With a few tweaks, we can adjust it to account for strength disparity between teams, so let's do the maths that will get us there.
Chelsea score 47 percent more goals than the average Premier League team and concede percent fewer. Wigan score 35 percent fewer goals than the average Premier League team and concede 30 percent more. With home field advantage counting for about a half-goal swing, some basic arithmetic gets us to an average score of about 2.5-1 in Chelsea's favour. And now we can plug that into our WPA framework at the time the first defensive substitution was made.
In the 66th minute with the score at 1-0, Chelsea had an 86% chance of winning, a 12% chance of drawing and a 2% chance of losing. A second goal at that point would have been worth a grand total of 0.26 points. A Wigan goal in the 67th minute, on the other hand, would have cost -0.94. What can we conclude about the tactical state of the game at that point? It's easy: Preventing a goal was more than three times as important as scoring another one.
Basically, in order to argue that an attacking substitution was required, you'd have to imagine that the last striker left on the bench in Fernando Torres would have made it 3.6 times the difference that John Obi Mikel would have on the defence. Considering that Torres has scored five goals in his Chelsea career, that strikes me as implausible.
So, if you're on the 'that substitution was crazy' train (and similar rules actually apply to the Malouda switch, although by that point we're looking at an impact differential of almost six to one) you basically have to assert that the attacking team you'd have wanted to run out is at least 3.5 times better at doing its job than the defensive unit that was actually deployed. Have fun with that.