Game Theory: The Logic of Not Caring About the Champions League

John Burn-Murdoch’s thought-provoking piece about English soccer teams prioritizing qualifying for the Champions League than winning the Champions League has inspired a fair amount of discussion on Twitter, so I wanted to walk through the logic of such a strategy and the implications for English soccer.

You all should read the piece, but the general point he makes is that you can tell English soccer teams don’t prioritize the Champions League because they are more likely to play their full-strength squads in the league than they are in the European games. This, rather than a lack of quality in English teams, can explain at least some of why England’s results have been so abysmal this year. But is this the right choice?

My disclaimer here is that I only took one game theory class in graduate school, but this seems to be a fairly simple probability exercise. Are you better off playing a full-strength squad in the Champions League and the Premier League, or playing a rotated squad in the Champions League and a full-strength squad in the Premier League?1

Champions League

The figure represents the reduced form game tree here. “Nature” begins with a Champions League fixture, and the manager is given two choices: play a full-strength squad or rotate players out of the Starting XI? Playing a full-strength squad increases the probability of winning (Pr(win))/decreases the probability of a loss (Pr(Loss)). The payoff for winning remains the same, but the “pain” of losing decreases because the media and fans don’t second guess you the next day.

So what are the benefits of winning each fixture? The benefits of winning a Champions League fixture include the obvious: points for the fixture, representing an improved probability of qualifying for the knockout stages. That has monetary benefits, including presumably at least one more home match where you can charge fairly high ticket prices, and some added money from the total pool (although not nearly as much as you get just for qualifying). It also has non-tangible benefits such as prestige for the club (and maybe even “big club” status), which could help you sign higher quality players later, and improving morale. These are not insignificant benefits.

However, because we’re looking at an issue of probability here, it’s not as simple as “If we play a full-strength squad, we win, but if we rotate, we lose.” English clubs should still be considered favorites over many of their competitors even with a rotated squad, although the win probability decreases by a certain %.2. So rather than realizing the whole benefit, we have to multiply the expected benefits/pain by that percentage, and then have to multiply that number by how much this increased/decreased the likelihood of qualifying for the next round. It’s impossible to assign actual numbers to the benefits/pain, but we can see that it conceivably becomes a very small number when we look at the changes in expected value. You can see how this number would become virtually 0 for the Europa League, both in terms of money and the non-tangible benefits, and the pain would decrease because English fans, until recently, haven’t cared about the Europa League.

On the other side, if a team plays a full-strength side, that decreases their likelihood of winning their next EPL fixture. Losing that fixture comes with potentially tangible pain: particularly losing points that could cause a team to miss out on next year’s Champions League. This comes with a very significant financial loss, as well as a much bigger loss of face (and “big sidedness”) which will hurt recruitment of players next season. There are also pain issues of angering the fans and media, which are exaggerated if the next match is some sort of rivalry or important match against another top side, and maybe pain issues associated with losing against a relegation-level side because you played the wrong players. From a managerial standpoint, one can also see the potential for pain in terms of players not getting enough playing time who begin to complain, and the potential of over-working starters who are playing two games a week instead of having time to let little injuries heal. Once again, these are multiplied by the change in probability of winning the individual game, and then multiplied by the change in probability of the result affecting either Champions League qualification next year or winning the title. 3 These probabilities are also very small, but the benefits of qualifying for the Champions League seem to exponentially outweigh the benefits of qualifying for the knockout stages, so you can see how this number would still be higher.

It’s a simple expected value calculation: how much do you trust your rotated squad and what value do you get from the league vs. Europe? There are next level concerns as well: if you’re going to lose in the knockout stage to Barcelona or PSG, do you get any benefits from playing a full-strength squad and then playing two extra games? Do any of the English teams consider themselves likely to make it past the quarterfinals ahead of Barcelona, Madrid, Bayern, Juventus, or even PSG? The economics and “big club” nature of soccer lend me to believe expected value calculations favor not playing a full-strength squad in the Champions League unless you think you have a chance to make the semi-finals or beyond, and these teams likely have a strong enough second XI where they could rotate with little consequence regardless. Insert your own probabilities and values for winning/pain for losing, but I think you’d have to make some pretty strong assumptions about the Champions League to make it worth playing a full-strength squad.

  1. Two disclaimers here: I don’t present the third option: a rotated squad in England and full-strength in Europe for a lack of space. The results would be the same as full-strength in both, although presumably more exaggerated because Pr(win) in the next EPL fixture would be even lower. Second, I present a reduced form of the game here – I don’t look at the added effects of how an extra game could affect players 10+ games down the road. Again, this would exaggerate Pr(win) in subsequent EPL games and would argue for rotation.
  2. As an example, Manchester City without Aguero, Silva, and Kompany a few weeks back lost about 10% winning probability against a lower side
  3. I don’t figure there’s a big difference to supporters between 2nd and 3rd, but there is a massive difference between 1st and 2nd.

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