Rent split calculator

Methodology

This tool runs the Kindhow Scale, version 1.0, and it is the odd one out. Everywhere else on this site the honest position is that no publication has an opinion about your situation, so the ledger tags what is sourced and admits the rest is ours. Here the literature does have an opinion, it is decades old, and it is not a range — it is a theorem. What is ours on this page is one optional button, and the ledger says so the moment you press it.

The formula

everybody's number for every room       yours — the tool is exact
                                       about these and nothing else

  the pairing                          sourced — an envy-free price
    the assignment maximising the      exists ONLY for the pairing
    total of what people said          that maximises the total
                                       (Svensson; Alkan–Demange–Gale)

  compensation, per person             sourced — what you must be paid
    from shortest paths through the    so you stop wanting to swap;
    difference constraints envy-       zero for at least one person,
    freeness imposes on the house      always

  what you are left with, u            sourced — the maximin point,
    u = (surplus − Σ compensation)/n   the fairest of the envy-free
        + your own compensation        prices (Gal, Mash, Procaccia
                                       & Zick, JACM 2017)

  your rent = your number for your room − u

  ─────────────
  = to the cent, and the house's rents add to the rent exactly

  then, checked rather than claimed:
    every person × every room they      the tightest margin, printed
    did not get, at these prices        on the ledger as a $0 line

Why an even split is the wrong answer

Splitting the rent by the number of people is right in exactly one case: the bedrooms are interchangeable. They never are. Somebody gets the ensuite and somebody gets the room over the boiler, and an even split quietly hands the second person’s money to the first every month for a year.

The worked example above is three friends and $2,400.00. An even split charges every one of them $800.00. Here is what each of them actually said the rooms were worth, and what falls out of it:

Alex pays $253.33 more than the even split and does not mind, because the ensuite is worth more than that to them and they said so. Sam pays $226.66 less. The rents still add to $2,400.00 exactly — to the cent, not to a cent either side — because the remainder is handed out largest-fraction-first rather than dropped on whoever is first in the list.

Who gets which room is not a choice — sourced

The tool never asks who wants which room, and that is not a UI decision. An envy-free set of prices exists only for the pairing that maximises the total of what people said the rooms are worth to them. For any other pairing there is no set of prices that works — not a worse one, none at all, at any numbers. That result goes back to Svensson in 1983 and Alkan, Demange and Gale in 1991, and it is restated in the form used here by Gal, Mash, Procaccia and Zick: an envy-free solution “must include a welfare-maximizing assignment”.

So the assignment is read off everybody’s numbers rather than negotiated. On the worked example the winning pairing is worth $2,620.00 across the three of them, and it puts Jordan in the middle bedroom even though Jordan put more money on the small bedroom than the person who ends up there. That is the theorem overruling the obvious answer, and the next section is about what it costs.

The rule everyone reaches for is wrong — and this is what it costs

Ask anyone who has thought about this for five minutes and you get the same method: pair people to rooms to maximise the total, work out how much the house values the flat above the rent, and split that surplus equally. It is clean, it is obvious, and it is wrong from three people upwards. It is also what the note that specced this tool said to build, which is why it is worth spelling out rather than quietly not doing.

Splitting the surplus equally leaves every person exactly the same amount ahead of their own valuation. That is only envy-free when every room happens to go to the person who values it most — and a welfare-maximising assignment does not guarantee that, because it maximises a sum rather than winning every room individually. On the worked example above the surplus is $220.00, so the equal split hands each of them $73.33 and produces these prices:

Under the equal-surplus prices, Jordan is paying $826.67 for the middle bedroom while Sam pays $546.67 for the small bedroom — and Jordan would rather have that room at that price, by $80.00 a month. That is envy, it is the exact thing the whole exercise was supposed to eliminate, and it survives in a method that looks like it has solved the problem. Every rent splitter that does anything cleverer than dividing by the number of bedrooms is, in our experience, doing this one.

What this tool does instead — sourced

Envy-freeness does not pin down a single set of prices. It pins down a range of them, and some points in that range are meaner than others. Gal, Mash, Procaccia and Zick’s contribution is choosing between them: they single out the maximin point, which pushes the worst-off person in the house as high as they will go subject to nobody envying anybody, as the fairest of them. That is the one computed here, and the one Spliddit deployed to solve more than 30,000 real rent splits before it closed in 2022.

The arithmetic is exact rather than searched for. Write u for what a person is left with — their own number for their own room, minus what they pay. Envy-freeness turns into a system of difference constraints between the us, which has no negative cycle precisely because the assignment maximises the total, so shortest paths between every pair of housemates exist. From those paths falls out how much each person must be compensated to stop them wanting to swap, and the rest of the surplus divides evenly. On the worked example that compensation is $80.00, all of it to Jordan, and the $26.67 each that pays for it is a line on every person’s ledger — including Jordan’s.

The consequence worth naming: nobody is talked out of wanting the better room. They are paid the difference until wanting it stops being worth acting on. That is the whole mechanism, and it is why the answer holds up in a conversation in a way “we agreed it was fair” does not.

The guarantee is checked, not claimed

Every other rent splitter asserts that its answer is fair. This one runs the check and puts the result on the ledger as a $0 line: all 3 × 2 comparisons, every person against every room they did not get, at the prices on the page. The line reports the tightest margin in the house, so you can see how close it came rather than being told it passed.

A margin of exactly zero, which is what the worked example produces, is the normal healthy case rather than a near miss: the fairest prices are the ones that push the worst-off person as high as they go, and the thing stopping them going higher is some comparison sitting exactly flush. A tie is not envy — it means that person is precisely as well off either way. The same check runs in this tool’s own test suite against three thousand randomly generated houses, which is the only reason we are willing to print the word proof on the page.

The one thing here that is ours

The suggest button. “I don’t know what the room is worth to me” is the real reason people never do this properly, so the tool will guess a column for you: the rent split across the rooms by floor space, with a private bathroom counted as worth about 15% more space. Nobody publishes what an ensuite is worth as a fraction of a bedroom and we are not going to pretend otherwise — that number is ours, and it is a place to start an argument rather than the end of one.

It carries a cost the ledger states out loud. Use it, and that person’s baseline line retags itself from sourced to house judgment, because everything below it is then exactly right about a number we invented. The same computed-provenance discipline runs the teacher gift calculator and the babysitter rate calculator. Room size and the bathroom checkbox feed nothing except that button — the answer itself never reads them.

What this tool will not do

• It will not work on numbers you made up for other people. This is the honest limitation and it is the whole shape of the tool: an envy-free division needs everybody’s real numbers. Fill it in with guesses about your roommates and you get a result that is mathematically exact and socially worthless. It is a thing you do together at the kitchen table.
• It cannot stop someone playing it. No envy-free rent-division method is strategyproof; that is a known limit of the problem rather than of this page. Bidding low on the room you want makes it cheaper and also makes it likelier somebody else is given it, which is the natural brake. Everyone’s numbers being visible to everyone is the other.
• It assumes one person per room and one rent. Couples sharing a room, a lodger on a different lease, someone paying for parking, bills split on a different basis — none of that is modelled. Settle the rent here and handle the rest separately.
• It is not a tenancy agreement. Whoever signed the lease owes the landlord the whole rent, whatever the four of you decided between yourselves. This is arithmetic between housemates.
• It stops at 6 people. Not a limit of the method — a limit of a grid of numbers on a phone.

Sources

• Francis Edward Su — Rental Harmony: Sperner’s Lemma in Fair Division, American Mathematical Monthly 106 (1999) 930–942 — that an envy-free rent division always exists, proved with Sperner’s lemma. This is the result the whole page rests on.
• Gal, Mash, Procaccia & Zick — Which Is the Fairest (Rent Division) of Them All?, Journal of the ACM 64 (2017) — that an envy-free solution must include a welfare-maximising assignment (restating Svensson 1983 and Alkan, Demange & Gale 1991), and that the maximin point is the fairest of the envy-free prices. This is the method the tool computes.
• Multi-Apartment Rent Division (AAAI 2024) — on Spliddit’s rent-division application, which ran 2014–2022 and solved more than 30,000 instances — for what Spliddit’s rent-division application was, that it ran the method above, and that it closed in 2022 after solving more than 30,000 instances.

Informational only, not professional advice. Nothing here is legal or tenancy advice — how you divide the rent between yourselves does not change what the lease says anybody owes the landlord. The fairness guarantee is exact, and it is exact about the numbers you type: it holds for the valuations entered and says nothing about whether they are honest ones. The only line on this page marked house judgment is the optional suggest button, and it labels itself the moment you use it. Last reviewed: August 2026. Scale version 1.0. This tool depends on published mathematics rather than on figures that go stale, so its review cycle is about the method holding rather than about a number moving.