I think the idea is that the bigger room is more desirable, no? I mean, if you were all paying exactly one third of the rent, then ideally you would like to have the big room? So, you need to find a way to determine who gets the large room.
Say the rent (for convenience’s sake) is 300 a week. This means that, all other things being equal, you’d each pay 100 a week. Now, we need to throw into the equation the fact that there is something in the house (the bigger room) that everyone would like to have. The question is, how much more than 100 a week are you willing to pay for the privilege of having the big room?
If no-one is willing (or can afford) to pay more than 100 a week, then you’re left with having to allocate the room some other way (e.g., draw straws, come to a mutual agreement, etc). But if someone is willing to pay more than 100 a week for the bigger room, the question becomes how much more?
If person A says he’s willing to pay 110, and no-one one is willing to pay more than that, then you’ve solved the problem. Person A gets the room and pays 110, while persons B and C pay 95 each.
But if person B says that she’s willing to pay 120, then you have to ask whether anyone is willing to go over that. If no-one can beat 120, then you have your solution: person B gets the room and pays 120, while persons A and C pay 90 each.
And so on.
In economic terms, this essentially allows each member of the house to ask himself or herself what the marginal value of the large room is to him or her. And also to ask what is the opportunity cost of paying more for the larger room. The person for whom the room holds the highest marginal value will bid that value and pay that amount for the room.
Now, i tend to agree with you that a system like this doesn’t necessarily avoid hard feelings. The people who can’t afford the large room (for whatever reason) might feel envious and annoyed. That’s always a possibility. But, the economic argument goes, using a system like this at least means that the room is allocated using a rational system in which each person has equal opportunity to decide how much the room is worth to him or her, and to bid accordingly.
Now, while i belive that the economic logic here is sound, i’m also of the belief that a shared household is more than just a cave for homo economicus, and that people in such a situation usually base their relationships on more than the idea of rational maximization. If you feel that the economic model is likely to undermine the social bonds of the house, then don’t use it. But just remember that, in the absence of such a model, there is always going to be a certain arbitrariness to determining who gets the room and how much they pay for it.