Why splitting rent by room is harder than it looks

Six people, one rental, five bedrooms of wildly different quality. Ask any group how to handle it and you will get as many confident answers, all of which are wrong in their own way. The interesting part is that they are wrong for reasons you can name in advance.

The argument always sounds like it is about money. It is not. It is about two or more factions setting their rules of engagement, and a group that has to choose one.

Attempt one: split it evenly

The total is $4,800 for the week. Six people. Everyone pays $800. Done, and everyone dusts their hands of the decision until they step foot into the rental.

Then someone gets the master bedroom with the ensuite and the balcony, and someone else gets the cot under the stairs that looks suspiciously like a wizard child should be occupying it. Both paid $800. One of them is now subsidising the other's luxury, and they both know it. Nobody says anything on day one. Somebody says something by day four.

The even split is not neutral. It is a strong claim that all the rooms are worth the same. That happens to be false in most houses that were built for people to live there. Its real appeal is that it requires no conversation, which is also precisely why it defers the conversation to the trip itself when people are just trying to enjoy themselves.

Attempt two: everyone ranks the rooms

So the group gets more sophisticated. Everyone lists the rooms in order of preference, and some arrangement is found that makes as many people as possible happy with their placement.

This is a real improvement, and it fails on intensity. Ranking captures the order of your preferences and throws away how strongly you hold them. Four people rank the master first. Three of them mildly prefer it. The fourth would genuinely have paid $200 extra for it, because they are the one who cannot abide without a private bathroom.

The ranking cannot tell those four people apart. So the tiebreak becomes something arbitrary — a coin flip, seniority, who booked the house, who asked first. The person who wanted it most gets nothing for wanting it more, and there was never a way for them to say so.

Attempt three: auction the rooms

Fine, says the group: let people put money behind it. Highest bid gets the master. Now intensity counts, because intensity is exactly what a bid measures.

This solves the previous problem and creates another one. Consider what happens to the person who bid $150 for the master and lost to someone who bid $200. They do not get the master. They also do not get anything instead of the master. They are in a bunk bed, and the $200 went to “the group” in the abstract — a slightly lower bill for everyone, spread so thin that no individual feels compensated.

Their experience is pure loss. They wanted a thing, they said so with money, they were outbid, and they received nothing in return for losing.

Attempt four: split the difference

The natural next move — and the one this project made first — is to build something in between. Take the rankings. Take the bids. Take some baseline notion of what each room is worth. Weight them together into a score, and assign rooms in score order.

It feels balanced. It is defensible to a degree. Every one of those weights is a number somebody chose, and when a person asks why is the ranking worth 40% and the bid worth 35%, there is no answer that is not, ultimately, “it seemed about right.” A blend of two mechanisms does not inherit the good properties of either. It guarantees nothing, and it now has constants to argue about on top of the outcome. On the right track, but not entirely there yet...

We shipped a version of that, ran a real trip through it, and then checked what it had actually done. It had charged six people out of eighteen more than they had said their own room was worth to them. That is not a tuning problem. The full numbers are here.

The dilemma is false

Here is the part that is genuinely surprising, and the reason this problem is worth writing about rather than just arguing about.

You do not have to choose between “preferences count” and “nobody ends up feeling robbed.” There is a way to get both at once, it has been known since the 1980s, and it comes with a proof.

The idea is called envy-freeness. An allocation is envy-free when no person prefers anybody else’s room-and-price combination to their own, judged by the numbers they submitted themselves.

Notice what that does to the auction problem. Somebody still pays a premium for the master. But that premium has to clear a test before the result is allowed to stand: everyone who did not get the master has to prefer the room they did get at its price — not accept it, prefer it, measured against the numbers they submitted themselves. You were not overruled and you were not outbid. Your own figures said you would rather have your room and the difference in cash. That is a categorically different experience, and the difference is structural rather than a matter of presentation.

Two things can be meant by that. The everyday sense of prefer is about how you feel: the master has the balcony, the bunk room has a ladder, and no arithmetic changes which one you would rather wake up in. The sense used here is narrower and it is arithmetic. You can want the balcony and still be $340 ahead in a room you said you would happily take, and both of those are true at once. Envy-freeness is a claim about the second one. It does not promise you will stop noticing the balcony; it promises that if you do the subtraction, there is no trade in the house you would rather have made — and that we actually ran that subtraction instead of assuring you of it.

It is also not a compromise between the three attempts above. It is a different object, with guarantees none of them have — including the fact that a solution always exists, no matter how strange the group’s preferences are.