The theorem that ends rent fights

Every shared apartment has the room. The one with the ensuite, or the big window that doesn't face a parking lot, while somebody else sleeps in what the listing insisted was a second bedroom. Someone has to take it, and someone has to price it.

In 1999 a mathematician proved that this fight always has an exit. There is a way to assign the rooms and divide the rent so that nobody would rather have anyone else's deal. Always. It's a theorem.

The theorem

Francis Su published “Rental Harmony: Sperner's Lemma in Fair Division” in The American Mathematical Monthly in 1999. The setup is your apartment: some number of housemates, the same number of bedrooms, one total rent. Su proved that a partition of the rent always exists in which each person prefers a different room. The master costs more, the small room costs less, and at those exact prices no one strictly prefers anyone else's room to their own. The technical name for that is envy-free.

The assumptions are mild to the point of being funny. Everyone has to find some room acceptable no matter how the rent is spread around, and everyone has to prefer a free room to one that costs money. Su calls that second condition “miserly tenants,” and he shows late in the paper that even it can be relaxed. The proof itself runs on Sperner's lemma, a 1928 result about labeling the corners of subdivided triangles that sat around for seven decades before anyone pointed it at bedrooms. For three roommates, every possible rent split is a point inside a triangle. The lemma forces a tiny patch of that triangle to exist where three different people each want a different room, and inside that patch is your deal.

People actually use this

This escaped the journals a while ago. In April 2014 the New York Times ran an interactive by Albert Sun called “To Divide the Rent, Start With a Triangle,” with a working rent calculator built on Su's proof, and Spliddit, a fair-division site, deploys the same ideas. The theorem has been extended too: a 2019 paper in the same journal proved you only need the preferences of all but one roommate, so an envy-free deal survives even when one person refuses to fill in the form. The Mathematical Association of America gave Su's paper its Hasse Prize for exposition in 2001, which for a piece of math about arguing over bedrooms feels right.

The utilities-included trap

The theorem prices the rooms. The rest of the household bill has its own evidence, and the sharpest piece of it comes from Sweden. In 2011 and 2012, a private housing company with about 1,800 apartments switched roughly 800 of them from unlimited electricity included in the rent to individual meters billed at market rates. The other thousand or so kept the old deal, which handed researchers a natural experiment they published in PNAS in 2017.

In-apartment electricity use fell by about 36% in the metered apartments, around 25% for the housing area overall, and the drop was permanent. Consumption started falling as soon as tenants were told the change was coming. The reduction was also wildly uneven. It came almost entirely from the heaviest users, and in Uppsala University's own summary, a fifth of the tenants accounted for about two-thirds of it. While electricity was “included,” those households were running up a bill everyone else was quietly splitting.

Same lesson as the theorem, from the other direction. When the price of a thing is attached to the person using it, behavior changes and resentment loses its hiding place. “Included” is where free riding lives.

People cheat, and the math mostly holds

A fair objection: theorems assume honest inputs, and your roommate is not an honest input. A 2022 study in Games and Economic Behavior tested this by running the classic fair-division procedures with real people in a lab. Subjects misreported their preferences, in the paper's own words, “very often,” which wipes out the theoretical envy-freeness guarantees.

And yet the envy-free procedures still came out ahead. The allocations they produced were fairer than the ones from simpler proportional methods, and participants perceived them as fairer too. With repeated play, people learned each other's preferences and told the truth less, yet envy went down anyway. The authors write that the results “support the practical use of the celebrated Selfridge-Conway procedure.” It's lab evidence about cake-cutting protocols rather than a field study of rent, so hold it loosely. But the direction matters: the fairness machinery degrades gracefully when people game it, instead of collapsing.

Where dvup comes in

The part of Su's result I keep coming back to is that the fair split is almost never the equal split. The envy-free prices are lopsided on purpose. The big room costs more because it's worth more, and everyone signs off precisely because the asymmetry matches reality. dvup is built to hold that kind of agreement: set your household up as a group, give each person the rent share you actually struck, and let the running balance keep the deal visible month after month.

Splitting expenses and settling balances in dvup is free, with no cap on how much you track, so the household ledger costs nothing to run. The theorem says a deal everyone prefers exists. The Swedish meters say the deal falls apart when costs pool invisibly. An app's job is the part in between: keeping the numbers you agreed on where everyone can see them, every month, without anyone having to ask.

Questions