Soccer Transfers and Game Theory

Soccer is a fascinating sport. Yes its fun to play, and it is definitely the most popular sport in the world, but at the professional level its a whole different game. And that’s because the manager plays a much more important role at the professional level. A good manager can compensate for a lack of talent in the team. Soccer strategy can be the reason why the underdogs come out on top (shout out to Jose Mourinho for winning the Champions League with Porto in 2004).

But the reason why I’m super fascinated with this aspect of professional soccer is that I, along with most people, don’t quite understand what good strategy looks like. How exactly do you create a synergistic formation for your team given your players? How do you render an oppositions tactics useless by planning in advance? How exactly do you know which player to play and when to play them for maximum performance in your match? If you can answer these questions on a macro-level, then you’re a legend in the making.

I have yet to write up a number of soccer theories and commentaries that I may have, but I thought I’d start with some basic game theory that I recently read about. While reading my daily dose of SkySports Soccer News, I came across this link where Man City apparently ‘bullied’ Totenham into pulling out their bid for Criag Bellamy.

Now here is a game theoretic analysis of the situation, since Man City has an unlimited supply of money (those bastards) they were able to use their financial power and control this ‘game’. They essentially played an ultimatum game with Tottenham by threatening to purchase Wilson Palacios, another target for the Spurs (for those not acquainted with soccer, Spurs is the nickname for Tottenham), unless Tottenham dropped out of the race to sign Bellamy.

Now there are a few things to note. At the time, City had bought De Jong, another midfielder and hence weren’t really looking for a midfielder so this threat may not be completely credible, but since City do have all the resources in the world I’m sure they would have welcomed a good midfielder and the opportunity to send a message to all potential competitors that they should not try and go up against City when trying to sign a player.

I’ve made a Decision Tree diagram below, though this is not completely accurate because I’ve assumed that by bidding for Bellamy, Tottenham will actually manage to sign him. To make this diagram more accurate we would need probabilities of Bellamy wanting to join Tottenham and add another decision node, but I’m not willing to speculate on these probabilities and I’m going to assume there was a very high chance of this happening which is why City had to threaten Spurs.

So according to this decision tree, what is the optimal choice for Tottenham? Lets use backward induction to deduce the subgame perfect Nash Equilibrium:

If Tottenham bid for Bellamy, Man City’s best option is  to sign Palacios (assuming that in the long run they aren’t making a loss through this transaction since Palacios isn’t too similar to De Jong and could really add to the City side). But if Spurs don’t bid, then City’s best option is to not sign Palacios since they don’t really want him. So what should Tottenham do? In both scenarios they get one player, so they should pursue the option that gives them the player they want the most. By the result, we can assume that player was Palacios (though in reality I think we should have added another decision node to consider the possibility of alternative players that would have come to Spurs had they not signed Bellamy, and in reality we know Defoe was one of these alternatives).

So the sub-game Nash Equlibrium was for Spurs to not bid for Bellamy, and for City to sign Palacios if they Spurs did bid, and to not sign Palacios if Spurs did not bid. The expected outcome is for Spurs to not bid and for City to not sign.

By this logic the threat “You know, if you don’t drop out of Bellamy, we’ll sign Palacios too and you won’t get either of them” is not credible. But that is under the assumption that Bellamy was more likely to sign for Spurs (if he wasn’t then why would Man City threaten Spurs?).

We also should evaluate the assumption that City would benefit from signing Palacios. He would have probably cost them around £ 14,000,000 which is what Spurs paid for him. Would that amount be worth how much Palacios would add to the team, and the message for other teams to not mess with City? That depends on the City management team, if yes, then the sub-game Nash Equilibrium we found was valid. If this assumption is not true, then Spurs should have just gone ahead and bid for Bellamy and Palacios and were likely to sign both.

Another assumption I made is that Palacios would definitely be signed by City if they wanted. I’m not going to question this one since I don’t know too much about the actual situation.

Anyways, given how complicated this apparently simple situation was, I can only imagine how hard it must be for coaches to design the perfect formation to compete in the Champions League Final.

And as an afterthought I’d like to apologize to all who I may have offended by referring to football as soccer in this post.