P. S.: Post Scriptum

As the price of houses rises, more and more people find that the best solution is to divide a house among friends. Usually each person gets a room. The problem then is: who gets what room and how much should he pay. Usually the total rent is fixed, and usually the rooms are not exactly all the same. Some might be bigger, some smaller. Some might have a better view, more privacy, closeness to the toilet, more silence, and so on. And what’s also important is that different people might value the various elements in different ways.

I present here two ways of splitting the rent and dividing a house. I personally favour (and has designed) the second, but while I was presenting this method to some friends to get some
feedback, I was told the other, it seemed simpler, yet interesting enough to add it. They both assume that:
a) the rent is fixed,
b) there are no favoritism among the will-be-housemate on
who gets to choose first.

The ‘find the objective value first’ method.

Before the rooms are assigned, get together and agree on what are the objective value of each room (i.e. 20% of the rent for this, 50% of the rent for this). The total value must of course be the whole rent. Then randomly select who gets what room (at the agreed price), and as a final action people are allowed to exchange rooms if they want to.
Positive element: it is simple and quite straightforward.
Negative element: it assumes that people can easily agree on the actual relative value of the rooms, and that such value does not change respectively to the persons.

The ‘each person gets the best room’ method.

As I said this is the method that I love most. First of all let each person inspect all the room. Then each person, writes, secretly, the relative value of each room in a piece of paper. The sum of the values must be equal to the requested rent. The idea is to divide the house so that each person gets a room, and pays for that room the value THEY wrote on the piece of paper, while the sum of the valued paid by each person totally covers the requested rent.

Obviously, very often, the collected money would then be higher than the rent. Let’s call the collected money minus the monthly rent, the ‘extra money’.

Often there is more than one solution, that permit to have a some extra money each month. When this happens, the solution that permits to maximize the extra money is chosen. The extra money is then used to pay for the light, any extra expenses, or whatever is needed for the house.

Sometimes there are more than one optimal solution, that is some solutions generate the same extra money, everybody is paying the requested cost for each room, and all other solutions are less optimal. In that case the adopted solution will be one of the optimal one, randomly chosen.

Examples, examples:
Let’s suppose we have a house with 3 rooms (a, b, and c) and 3 persons (A, B, and C). Let’s suppose the total rent being 100.

Person A might find the three rooms equivalent, so he might just write (a: 33.3, b: 33.3, c: 33.3). Person B might instead favour room B, because is more sunny, and she likes to paint, and then she thinks that room ‘a’ is slightly better than room ‘c’, infact she would prefer not to be in room c at all, so she would write: (a: 35, b: 40, c: 25). Person C instead does not care about the sun, but has noticed that room A has more privacy, plus is near the toilet, and since he likes to have his gf as a guest, thinks that having room A would be a better deal. So he votes (a: 40, b: 30, c: 30).

Then the papers are revealed.

Generally when a room has a person that values it more than all the others, and he values that room more than all other ooms, then that room gets taken by that person at the price he has choose.

In our example we have:
A: (a: 33.3, b: 33.3, c: 33.3)
B: (a: 35, b: 40, c: 25)
C: (a: 40, b: 30, c: 30)
which would give us that A would get room ‘c’ paying one third of the rent. B would get room ‘b’ paying 40% of the rent, and C would get room ‘a’ for 40% of the rent… and the collected money each month would be 33.3+40+40=113.3 . The extra money would be 113.3-100=13.3 and would be used to pay for the electricity, water, gas, or whatever.

It is also possible to rinormalise the prices, by lowering them so that the total sum becomes exactly the cost of the rent, while the relative ratio remains the same. In our example
A: (33.3/113.3)*100=29.4
B: (40/113.3)*100=35.3
C: (40/113.3)*100=35.3
and person A would pay 29.4 of the rent (since he took the room nobody wanted)
person B would pay 35.3 of the rent (and took the sunny room)
person C would pays 35.3 of the rent (and took the room with more privacy)

So, what if the situation is not that easy. There isn’t a person that prefers each room? For example you could be in a situation like:
A: (a: 45, b: 45, c: 10)
B: (a: 40, b: 40, c: 20)
C: (a: 40, b: 30, c: 30)
well in this case it is obvious that person A will get either room a or room b. But it is also obvious that room c will go to person C. So C get’s c at 30% of the rent. Both A and B value the room a and b equivalently. But once the room will be assigned person A will pay more than person B, so it seem fair to me that person A chooses a or b and pays 45, and person B gets the remaining room, but pays less (40).

But things can get even more complicated if some people
value some rooms exactly the same:
A: (a: 45, b: 45, c: 10)
B: (a: 45, b: 45, c: 10)
C: (a: 40, b: 40, c: 20)
in which case A and B have obviously to randomly choose who gets what.

Or if the situation is symmethric among the rooms:
A: (a: 40, b: 30, c: 40)
B: (a: 40, b: 40, c: 30)
C: (a: 30, b: 40, c: 40)
In which case you randomly choose if A gets a or c, and then the other follow obviously.

So here we have the first mehtod, where everybody chooses the value together, this is equivalent on the second method if everybody agrees on the relative value:
A: (a: 35, b: 40, c: 25)
B: (a: 35, b: 40, c: 25)
C: (a: 35, b: 40, c: 25)
After which, also in this method, you would randomly pick who gets which room.

Please, let me know if you have tried it and if it was succesful.