My newest Game Theory post about the value of rotation was inspired by a Gab Marcotti tweet:

God forbid you're in a CL semifinal + you prioritize winning the CL over a top 4 finish…

— Gabriele Marcotti (@Marcotti) May 1, 2016

He was speaking of Manchester City playing a “B Team” in the weekend’s Premier League fixture, prioritizing their mid-week CL semi-final return fixture against Real Madrid instead. The tweet was fairly controversial, especially among City’s fan base, and gave me a lot to think about. So as I like to do, I think about it from a utilitarian perspective and try to game the expected value for each choice.

The idea behind rotating before a big game is that you can increase your chances of winning the big game while diminishing your likelihood of winning the rotation game and diminishing your chances of obtaining a given league position. For Manchester City, they are currently in a fight for fourth place with Manchester United (and to a lesser extent after this weekend’s fixtures, Arsenal).

The first step is to think about which is more important to Manchester City fans: winning the Champions League semi-final (and possibly the entire tournament), or getting Arsene Wenger’s famous “fourth trophy” and ensuring Champions League football next year. I can see arguments for both, and despite the mocking of Wenger’s qualifying record, as a Milan fan I know the pain of missing out of Champions League football after you’ve become accustomed to it.

However, the expected happiness from advancing to the finals vs. securing 4th place is mitigated by a pretty significant factor: the probability of winning the semi-final match with a full strength squad, which leads us to the following equation.

**Expected Utility**_{(Rotation) }=* Pr(Advance to CL Finals _{(Rotation})*(Value of Advancing to CL Finals)-Pr(Miss CL Next Year_{(Rotation)})*(Pain of Missing CL Next Year)*

Manchester City’s expected utility (“good”) from rotating the squad is basically calculated by how much value they get from advancing to the Champions League finals^{1} multiplied by their probability of advancing to the finals. Then you subtract the probability that the rotated squad causes them to miss the CL next year multiplied by the pain of missing out. **In short: the biggest driver of value here is whether Manchester City fans think they can beat Madrid given a 0-0 draw in the home leg. If you don’t think this outcome if pretty likely,** then the first half of the equation approaches zero, meaning that the pain of missing next year hurts more than any potential pleasure gained from rotation. **In this case, it doesn’t make sense to rotate the squad.**

**However, if you assign a high probability to winning the semi-final at the Bernabeu** then the first half of the equation becomes higher, meaning that the potential pain of missing next year is less significant. **In this case, it makes perfect sense to rotate.**

But this isn’t the only factor. There’s a second equation at play here, which I present now:

**Expected Utility**_{(Full Strength}_{) }= Pr(Advance to CL Finals_{(Full Strength)}*(Value of Advancing to CL Finals) -Pr(Miss CL Next Year_{(Full Strength})* (Pain of Missing CL Next Year))

This represents the expected utility gained from playing a full strength squad. The equation is largely the same, but the values change because Manchester City played a full strength squad on the weekend. Presumably their likelihood of winning mid-week decreases because of fatigue (and potential injuries), while their likelihood of securing Champions League football next year increases because they have a greater likelihood of getting what would have been a crucial three points against Southampton.

**If you’re a Manchester City supporter and believe that the odds of beating Madrid are low**, then your values likely don’t change for the first half of the equation while your values for the second half of the equation increase. **In this case, you want a full strength squad during the weekend**.

**If you’re a Manchester City supporter and you believe that a fresh squad will beat Madrid while a fatigued squad will lose**, then your values for the first half of the equation are lower than they were previously. This lowers your expected value in a significant way, **meaning you want a rotated squad over the weekend.**

The final decision is calculated by which equation gives you a higher expected utility: which version makes you happier? **Ultimately the question depends on two major factors: how likely you think Manchester City is to beat Madrid on the road, and how much pain you’ll feel if they fail to qualify next year.** If you don’t have faith that they can pull of an upset mid-week, then you’ll oppose rotation and prioritize the league. If you believe there’s a chance, then you’ll support rotation and going all-in for this year’s Champions League.

**Part 2: Pellegrini’s Lame Duck Status**

Normally we can roughly argue a manager’s incentives are aligned with his team’s and the fans. However, Manchester City have done something strange this year, announcing Pep Guardiola will be the new manager of Manchester City regardless of what Manuel Pellegrini does this year. This introduces a new wrinkle, one that I think fully explains why he did what he did. I want to return to the expected utility equation from earlier, because the logic is the same while the values are different given Pellegrini’s unusual incentives here.

**Expected Utility**_{(Rotation) }=* Pr(Advance to CL Finals _{(Rotation})*(Value of Advancing to CL Finals)-Pr(Miss CL Next Year_{(Rotation)})*(Pain of Missing CL Next Year)*

Because Pellegrini is a lame duck manager with zero interest in what happens to Manchester City next year, he experiences literally zero pain from Manchester City missing out on the Champions League next year. Pep Guardiola gets all the benefits if he qualifies, and Pep gets all the pain from missing out if he doesn’t. The second half of this equation is literally zero, so it becomes completely irrelevant to our calculations. So when we combine the two equations from earlier, we get the following:

**Expected Utility**_{(Pellegrini) }=* Pr(Advance to CL Finals _{(Rotation})*(Value of Advancing to CL Finals)-Pr(Advance to CL Finals_{(Full Strength)}*(Value of Advancing to CL Finals)*

Because the value of advancing to the Champions League Finals is the same for Pellegrini in both cases, we can cancel that term out and we’re left with the following:

**Expected Utility**_{(Pellegrini) }=* Pr(Advance to CL Finals _{(Rotation})*-Pr(Advance to CL Finals_{(Full Strength)}*

Even if the probability is virtually zero in both circumstances, and even if the value of rotation is virtually zero, Pellegrini strongly prefers^{2} rotating the squad over the weekend to maximize his probability of winning the Champions League, something that could presumably bolster his CV and improve the contract at his next job. **Manuel Pellegrini has literally no reason to not rotate the squad, even if he sees virtually no value in it.**

The Manchester City case described here is a relatively unusual one, which is why it’s interesting to me. The conflict between a manager’s incentives, the fans’ incentives, and reasonably different incentives between fans makes this a difficult case to think about and one worth exploring more and provides a lively discussion.