Guide · Splitting rent by room
Why splitting the extra rent evenly isn’t fair
Three roommates, three different rooms, and the obvious-sounding method: price each room, then split whatever’s left over the rent equally. It feels fair. A published proof, and one worked example, show it usually isn’t.
The short answer
Mathematician Francis Su’s 1999 paper on rent division proves an envy-free split always exists — prices where nobody would rather have someone else’s room at that person’s price. sourcedThe obvious method most households actually use — assign rooms to maximize total value, then split the leftover evenly — is not that split. It can leave one roommate quietly wishing they had someone else’s room, at that price, every single month.
The naive method, and where it breaks
Say what each room is worth to you, assign rooms to maximize the total value everyone gets, then divide the rent so everyone ends up the same amount ahead of their own number. It’s the natural first idea, and it’s exactly right for two people. From three roommates up, it can fail — and it doesn’t announce when it does.
Three roommates, a $2,400 rent, three rooms: house judgment— the method and the proof it fails are the cited papers’; this specific set of numbers is a worked example, not a claim about any particular household.
- Alex values the big bedroom at $1,100.
- Sam values the small bedroom at $620.
- Jordan values the middle bedroom at $900.
That assignment maximizes the total value the house gets from its own rooms. Splitting the $220 above rent evenly — about $73.33 each — prices the rooms at $1,026.67, $546.67 and $826.67. Every roommate is exactly $73.33 ahead of their own number for their own room. It looks fair from every seat.
Except Jordan’s number for Sam’s room was $700 — and at Sam’s price of $546.67, that room would leave Jordan $153.33 ahead, not $73.33. Jordan would rather have Sam’s room at Sam’s price, by $80 a month.The prices sum to the rent exactly, and someone still envies someone else’s deal.
What actually fixes it
Su’s paper doesn’t just prove an envy-free split exists — later work by Gal, Mash, Procaccia and Zick identifies which envy-free split is fairest: the one that maximizes the smallest amount anyone ends up ahead by, rather than splitting the total evenly. sourcedThat method — the “maximin” point — doesn’t hand every room to whoever values it most in isolation; it finds the one set of prices where no comparison between any two people’s room-and-price ever runs the wrong way, checked over every pair in the house.
It isn’t a hypothetical fix. A rent-division tool implementing this exact method ran publicly from 2014 to 2022 and settled more than 30,000 real households’ rent splits. sourcedThe math isn’t new — what’s new is that it can run instantly on ordinary numbers a household types in, rather than requiring anyone in the room to understand the proof.
What this doesn’t mean
- It doesn’t mean splitting evenly always fails. With exactly two roommates, splitting the surplus evenly and the envy-free maximin split are the same answer — the gap only opens up from three people on.
- It doesn’t mean there’s one universally “correct” price. Many envy-free price sets can exist for the same house; the maximin point is the fairest single one among them by this specific standard, not the only envy-free answer mathematically possible.
- It only works on numbers everyone actually believes.The proof guarantees fairness relative to the values typed in — it can’t rescue a split where someone lowballed a room they secretly wanted, or padded one they didn’t.
So what should your household actually pay?
The rent split calculator runs this exact envy-free maximin method on your own rooms and your own numbers — no solver, no spreadsheet, just everyone’s honest value for their own room in, and a room and a price out that nobody in the house would trade away.
Split the rent →Sources
- Francis Edward Su, “Rental Harmony: Sperner’s Lemma in Fair Division,” American Mathematical Monthly 106 (1999) — the proof that an envy-free rent division always exists.
- Gal, Mash, Procaccia & Zick, “Which Is the Fairest (Rent Division) of Them All?” Journal of the ACM 64 (2017) — identifying the maximin point as the fairest envy-free price set.
- “Multi-Apartment Rent Division,” AAAI 2024 — the record of Spliddit’s rent-division app implementing this method for over 30,000 real households, 2014–2022.
Informational only. This guide reports the cited mathematics as published and labels our own worked example as ours. It isn’t a tenancy agreement — whoever signed the lease still owes the landlord the full rent regardless of how the household splits it internally. Last reviewed: August 2026.
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