Tuesday 30 September 2014

An impossible transfer

A football agent burdened by failing to make any transfer during the summer, suddenly comes across a unique opportunity. The problem is that the acquiring team and the selling team have seemingly opposing interests. Will he be able to please both sides? The Will Rogers phenomenon and the game theory.

Will Rogers phenomenon
It's been an awful summer for Klaus Händler.

Klaus is a football agent, and during the last months none of the transfers that he has negotiated has come to a successful conclusion.

Some clubs he works with have no money for signing new players because of the economic crisis, and moreover, some of the players he represents are injured.

He's thinking he'll have to sell some of his dearest Lamborghini cars to afford the expenses of his comfortable life style during this winter, while, suddenly, the phone rings.

Selection of wonderful luxury car of the marca Lamborghini

FC Schalke 04 badge
It's Josef, chairman of the FC Schalke 04:

- Klaus, just now we've got 5 forwards in our team, and we want to transfer one of them. Could you take charge of it?

Just after hanging up the phone, it rings again. This time it's Michael, chairman of the Bayer 04 Leverkusen:

Bayer 04 Leverkussen badge
- Klaus, how are you? I call you because today is the transfer deadline day, and we want to sign one additional forward...

It looks like Klaus won't have to sell his luxury cars...

But there's a problem. Both teams are interested in increasing the average goal-scoring ability of their forward line and, at the same time, reducing the average wage of their players.

Increasing the average scoring ability and reducing the average wages at a time

Klaus phones some teams, to see if it's possible any agreement, but all the teams have completed their playing staffs.

It seems clear that it's not possible to transfer a forward from Schalke to Bayer, because of the imposed conditions by both teams. So, what can Klaus do? Could we help him in some way?


Joe Vitruvius
Klaus has decided to call his friend Joe Vitruvius, to see if he can help with this apparently unsolvable problem.

- Joe, I think that if we transfer one player from Schalke 04 to Bayer Leverkussen, the goal-scoring average of Bayer will rise at the same rate as Schalke's average decreases. And, in same way, the average wage of Bayer will go up at the same rate as the Schalke's average goes down.

Everything a team loses, the other wins it

- So, you think that everything a team gets, the other loses it. That's what we know as zero-sum game in mathematical game theory. But it's not always like this. As in real life, we can find situations in which all parts win or lose (non-zero-sum games).

- Yes, but I don't think this is one of the times it works.

- Well, we can try it. You told me that both teams want to increase the average goal-scoring ability of their forward line, and at the same time, reduce their average wage, didn't you?

Increase the average goal-scoring level and reduce the average wage at the same time.

- Yes, that's right.

- if we talk in absolute values, there's no way of doing it. But as we talk about averages, it may be possible. Can you give me the data of goals and salaries of the forwards of the two teams?

- Yes, of course. Here you are.

Salaries and goals of Schalke 04's forwards

Salaries and goals of Bayer 04 Leverkussen's forwards

- Well, let's calculate the means of goals and wages of both forward lines.

Data and averages of wages and goals of the Schalke 04's forwards
Data and averages of wages and goals of the Bayer 04 Leverkussen's forwards
- Now we can verify if there's a solution for your problem. We only have to apply the Will Rogers phenomenon.
Will Rogers, an American humorist
- And... what's that about?

- This paradox is related to a comment that Will Rogers, an American artist, said in the early 20th century: “When the Okies left Oklahoma and moved to California, they raised the average intelligence level in both states".

- He meant that when the citizens from Oklahoma move to California are the less intelligent persons of their state, even so they're more intelligent than California inhabitants, right?

Moving from Oklahoma to California

- That's right. This way, any inhabitant from Oklahoma whose intelligence is under the state average, that moves to California, makes that the average intelligence levels of both states experience an increment.

- And how can we apply it to our case?

- Well. Let's look at the Schalke 04 player, the Peruvian Jefferson Farfán. His goal-scoring ability is smaller than the team's average, but his salary is higher than his teammates average. However, he scores more goals than Bayer Leverkussen's forwards, and earns less money than them.

Comparison of the salary, and goal-scoring capacity of Jefferson Farfán vs. Schalke 04 levels
Comparison of the salary, and goal-scoring capacity of Jefferson Farfán vs. Schalke 04 Bayer Leverkussen levels

- I understand. If we transfer the player from one team to the other, assuming that he will earn the same, and that he will score the same goals, both teams will improve their statistics.

- That's right. Let's see how it works on the table.

Average wages and goal-scorings of Schalke 04's forwards after the transfer of Jefferson Farfán
Average wages and goal-scorings of Bayer 1904 Leverkussen's forwards after the transfer of Jefferson Farfán

- Indeed, both teams have improved their average goal-scoring level, and have reduced their average wages. And does this happen always?
Succulent and tipical German food
- No, not ever. But sometimes it does, like in this case.

- Well, you've just solved my economics this winter. I would have to invite you to a good meal

- I think this will cost you more than a simple meal. I prefer we do the following deal: you'll pay me in function of the goals that Jefferson scores this season. If he scores only one goal, you'll pay me 2 bitcoins. If he scores 2, you'll double the amount, 4 bitcoins. If he scores 3 goals, I will receive the double, that is, 8 bitcoins. And so on.

- That seems a fair deal. I agree.

Jefferson Farfán scoring a goal for Bayer Leverkussen.

The transfer was done. Schalke sold one of its excedent players, Bayer signed an extraordinary forward, and Klaus didn't have to get rid of his precious Lamborghini automobiles. At least during this winter, because Farfán scored 25 goles for his new team…

Can you calculate how many bitcoins Joe Vitruvius received from Klaus Händler at the end of the football season?

Some bitcoins, one of the internet virtual coins

If you're interested in learning more about this topic of the Will Rogers phenomenon and about the game theory, you can visit any of this magnific articles: The "Will Rogers Phenomenon" lets you save lives by doing nothingThe Will Rogers phenomenonGame Theory and the Nash EquilibriumEvolutionary Game TheoryAdvanced Game Theory Overview,

Below this lines  you will find other links, for if you liked this story and you want to share it with your friends.

And don't forget to take a walk by the Carnival of Mathematics. There you'll find lots of excellent math posts that you'll surely like too.