The amazing case of the extra car.

The sponsor of the Football World Championship 2015 in Canada is going to give away some gifts to the best scorers. But a last minute change in the number of gifts creates a curious change in their apportionment.

FIRST HALF

The Football World Championship 2015 in Canada has finished.

It's the closing ceremony and some awards are going to be delivered between the winners of the various categories.

In the section of best scorers, first 3 qualified players are:

1.- Yuki Ogimi, Japanese footballer, with 12 goals.

2.- Lotta Schelin, Swedish player, also with 12 goals.

3.- Christine Sinclair, forward and captain of the Canadian team, with 4 goals.

There's a draw to 12 goals in the first place. Tha Japanese Ogimi is the winner because she has made more goal assists.

Second place, also with 12 goals, but less assits than Ogimi, is for Swedish player Schelin.

And the third scorer is Sinclair, the Canadian player, with 4 goals.

The Championship sponsor, Yukon Cars has decided to give away 40 cars.

 

They'll give a car to the president of the Organizing Committee, another one to President of the Referees’ Committee, and the 38 remaining cars will be apportioned among the 3 top scorers in proportion to the goals scored by each one.

At the last minute, President of the Organizing Committee decides to reject his gift because he has just been appointed CEO of Manitoba Motors, a direct competitor of Yukon Cars.

Thus, his car will join the other 38 to be distributed among the 3 best scorers.

These news are greeted with joy by all players. Well, all except for Christine Sinclair, who is, in addition to good footballer, a fond of Mathematics.

Why do you think Christine didn't like this late change?

  

In a few days we will post the solution. Meanwhile, you can think about what's happening. We encourage you to give us your solution by writing below a comment, or just entering its event page on facebook or on google+.

Thank you very much.

SECOND HALF

Apparently it's no sense she's sad because there's one more car to apportion, right?

First of all, it seems that if the sponsor is giving away one more car, this shouldn't harm anyone, on the contrary, someone will benefit because of it.

Moreover, as that the top 2 scorers are tied in goals, and the extra car can't be distributed between both players in a fair way, it seems logical that Christine will be who gets the extra car.

Nevertheless, Christine Sinclair is right to be worried.

Why? Hasn't she got enough parking space for so many cars?

No, that's not the problem. Actually, she's going to offer all the cars she gets among her teammates, who have helped her to score these goals.

Perhaps, like the president of the Organizing Committee, she has either a family meber or a friend who works in Manitoba Motors, the rival brand?

No, not really. In addition, she loves cars manufactured by Yukon Cars.

Then I don't understand...

Let's see the first gifts allocation to the three players, when the sponsor was going to apportion 38 cars.

We see that Ogimi and Schelin should have received 16.2857 cars (mathematicians would refer to a 'quota' of 16.2857) and Sinclair should have got 5.4286 cars. As far as it's not possible to divide a car, we determine that the Japanese and the Swedish will get 16 cars, and Christine will get 5 cars (which corresponds to the 'integer part' of their quotas). So far we've distributed 16+16+5 = 37 cars. And the remainig car belongs to Christine, because she's the player whose fractional part is the largest and closest to next unit.

I didn't understand very well this last deal...

Well, Ogimi and Schelin, in addition to their 16 cars, should have got 0.2857 additional cars, while Christine should have received 0.4286 more cars. So Christine is more entitled to take the remaining car than the other two players. This way of apportionment is called in mathematical terms as the ‘largest remainder method’ or ‘Hamilton's method’.

Now I see it more clearly.

Let's see what happens when the sponsor decides to give away one more car.

Now we see that the first 2 players are given 16 cars, and Christine receives 5 cars. With the two remaing cars, we'll use the same method as before.

Ogimi and Schelin should get 0.7143 more cars, while Sinclair should get only 0.5714 more cars. Therefore, the 2 remaining cars are now assigned to each of the first 2 scorers, because their fractional parts are larger.

So now we give 17 cars to Ogimi, another 17 to Schelin, and only 5 cars to Christine, don't we?

That's right. It turns out that increasing one gift, not only Christine doesn't get one more car, but she loses one of hers.

But this question will be very unusual, right?

You shouldn't think so. Actually, this issue has received special attention from mathematicians, who call it by the name of Alabama Paradox.

Why this name?

Map from the Nations Online Project

This mathematical paradox was detected for the first time in the United States House of Representatives. In it, the number of each state's seats are redistributed every 10 years, regarding on their population growths.

After the 1880 census, a study was made on a possible extension of the number of seats in the House. In this study it was found that the state of Alabama would have got 8 seats in a chamber of 299 representatives, but only 7 with a House size of 300.

And is there any method of proportional apportionment in which these paradoxes do not occur?

Lots of mathematicians have tried to give a solution as fair as possible to the issue of allocation of seats after a vote, but they haven't achieved a perfect method.

Thus, numerous ways have been created to assign seats based on the votes cast, like the Hamilton's method we've seen here.

We've got some divisor methods, such al Jefferson's method (also known as d’Hondt method), Webster's method (method of odd numbers or Sainte-Laguë's method), Huntington-Hill's method, Dean's method, Adams' method, or the Danish method, among others.

In these systems we can find several paradoxes: Alabama paradox, the population paradox, or the new states paradox. That means that every method generates some bias or favoritism to some population vs. other.

And we also have preferential voting systems, among which we have the simple or relative majority, the second round, the Borda count, the Condorcet method, the single transferable vote, or the 'approval voting'.

But they are also imperfect. We can remember our story about the Arrow paradox, for example.

But some methods are better than others, right?

Yes, Webster's method (also called Sainte-Laguë`s method or method of odd divisors) seems to be the one which produces minor injustices, but it's not the most used. That's because there uses to be a political will to prioritize the good governance instead of the strictly proportional representativeness of the Parliaments, and to benefit the major parties, or such sort of things.

Now that we've seen all the problems of the distribution, as in our case, Christine did have reason to worry about when she heard they would deliver a car more...

No doubt, she would have previously read any of these great links: 'The Constitution and Paradoxes', 'Apportionment: The Alabama Paradox' o 'Apportionment and rounding schemes'. 

And regarding on these issues of fair divisions, voting and paradoxes, surely Christine have also read some of our stories: And now, who should kick the penalty?, The problem of the fair division, or The golden goalkeeper.

So, should Christine ask for an apportionment based on Sainte-Laguë method?

Well, despite being the method that produces less distortions, in this particular case, its application would cause an additional problem, as Tom (C.M.) Thomson discovered (many thanks for the comment). While it's also true that Christine may benefit from the use of that method.

Why?

Well, this method is used as follows :

We add up the goals scored by each player, and we calculate the quotients resulting from dividing the number of goals by the successive odd numbers. And we'll give the cars according to the highest resulting quotients.

Let's see how the 38th and 39th cars would be assigned, according to the following table:

The cars would be assigned to the players depending on the highest quotients: 12 - 12 - 4 - 4 - 4 - 2.4 - 2.4 - 1.714 - 1.714 - 1.333 - 1.333 - 1.333 - 1.091 - 1.091 - etc...

As you can see, the first 37 are easily given out (yellow cells). But with cars 38th and 39th we have a serious problem, since the three players are equally entitled to the cars, as they've got the same next quotient: 0.364.

Obviously, there should be some kind of rule to break ties. But if there's no fixed rule, the remaining 2 cars should be delivered by lot, so Christine would have a 2/3 chance of getting the 6th car for her.

Then, no doubt, Christine should ask for this method to be used!