Is This the Fairest Way to Split the Rent?

by Lark Turner

Is This the Fairest Way to Split the Rent?: Figure 1

A mathematical theorem called Sperner’s Lemma could hold the key to living in harmony with roommates, at least when it comes to dividing the rent fairly, according to The New York Times.

Rentals can be difficult to split up among roommates, especially in older units where room size isn’t uniform and each may have different access to natural light. In DC, splitting a rowhouse with roommates may mean one person lives in an English basement room while another enjoys a spacious master bedroom, the rent divided up as fairly as possible. Sperner’s Lemma suggests that because each roommate has different priorities when it comes to space and pricing, there’s a happy medium at which each person is content with their chosen room and the price point at which they’ll be renting it. This creates the ideal “envy-free” living situation where no roommate is jealous of another roommate’s perceived superior deal.

The lemma calculates the price points at which every roommate finds their happy place. The method describes a triangle with three points: Point A, Point B, and Point C. The interior of the triangle is further divided to represent the variety of ways that rent could be split. When using the method to calculate rent, each interior triangle corresponds to a different split, where A pays x, B pays y and C pays z. The rent can be more accurately and “fairly” divided the more the triangle is divided. The more mathematically-minded can read about the theorem in depth here.

A New York Times tool used to split the rent incorporates the theorem into its calculations. It works by changing the division of prices from room to room until everyone ends up at a price point and room they’re happy with.

Before using the tool, the creators advise making sure it’s necessary. They note that the tool won’t work if one person doesn’t want any of the rooms, or if one roommate would always take the most expensive room even if rent is free on another room.

UrbanTurf looked at a recent rental listing on Craigslist and used Sperner’s Lemma to calculate who might pay what given a hypothetical set of roommates.

The house in Petworth is offered at $2,800/month.

Here are our hypothetical roommates:

Roommate A, “Wheels”: Has a car and a decently paid job. Can’t pay more than $1,300/month.
Roommate B, “Millennial”: Bikes to work and makes a tidy sum. Can’t pay more than $1,800/month.
Roommate C, “Student”: A grad student who works nights to bring in extra income. Can’t pay more than $1,000/month.

And the rooms:

Master Room: The traditional master bedroom with three bright windows bringing in lots of natural light.
Narrow Room: A narrow bedroom with an interior window, meaning less privacy.
Back Room: A big bedroom with just one window near the back of the house. It’s quieter during the day.

The roommates decide ahead of time that the parking spot is worth about $100 a month, a little below market price since they know just one of them needs it. As a result, Wheels reduces her top price to $1,200/month. Going into negotiations, Wheels wants the master, and so does Millennial, while Student wants the back room because it’s the most quiet in the morning.

Using the tool with our hypothetical roommates in mind, UrbanTurf made decisions we thought made sense for each roommate and ended up with this:

Is This the Fairest Way to Split the Rent?: Figure 2

Wheels pays $816 for the back room, because with the cost of the parking pad she couldn’t compete with Millennial’s ability to pay for the room she most wanted (even without factoring in parking, it still wasn’t in her budget). Millennial got the room she wanted for $1,458 a month because she had the most money to work with. And Student ended up with the cheapest room, which she didn’t really want — but she loves the price at $525. At the end of the day, everybody feels like they got a good deal on their room.

Check out the NYT’s tool here and see whether you can live in an envy-free rent situation in DC.

See other articles related to: splitting the rent, sperners lemma

This article originally published at https://dc.urbanturf.com/articles/blog/a_fair_way_to_split_the_rent_one_formula_may_crack_the_code/8418


  1. Paul said at 8:25 pm on Tuesday April 29, 2014:
    If you're living with your girlfriend/boyfriend/fiance/spouse then I can understand factoring in ability to pay into dividing rent. But when dealing with Craiglist roommates or friends only the merits of the rooms themselves should influence how the rent is divided. The writeup doesn't even say the Master Bedroom has its own bathroom. So why would someone pay nearly 3x the price of the narrow room for it and feel good about that? Frankly I can't see a spread of more than $300 high to low justifiable in this example scenario.
  1. Lark Turner said at 8:50 pm on Tuesday April 29, 2014:
    Hi Paul, If you check out the listing, you'll notice how much better the master is than the much smaller and less private narrow room, which (although you can't quite tell from the listing) seems like it may have been intended as more of a den or study. Of course, we used a hypothetical situation, and these differences could mean much less – or more - to different people. That's the beauty of the tool! Lark
  1. Foster said at 10:14 pm on Tuesday April 29, 2014:
    Even if the master had a private bathroom, no one in their right mind would pay nearly $1500 to live in Petworth. That's crazy! Also, this theorem doesn't seem to account for "common space" rent. my housemates and I took half of our total monthly rent and divided it evenly between all of us to establish a "common space" rent. From there we established a set dollar amount for the two bathrooms since they are different sizes and then split that amount between the # of people that share each bathroom (3 and 2). The remainder of the total monthly rent was divide up based on room size. Monthly rent calculation for each housemate is as follows: share of "common space" rent + bathroom rent + room rent = total individual monthly rent (utilities are additional and are split evenly)
  1. David Bediz said at 10:51 pm on Tuesday April 29, 2014:
    In college we had a house with five roommates. We assigned point values to each room including value for exclusive use of certain bathrooms, all relative to each other. For example, the large master bedroom with en-suite bathroom and a bay window was assigned 50 points, and smaller rooms were assigned less points, all the way down to the closet-like smallest room for 10. The total rent was divided according to each room's ratio of points to the total number of points. Worked for us!
  1. Lance said at 6:21 pm on Wednesday April 30, 2014:
    One thing not factored in is whether the owner would be willing to rent the house out as a group home at the price advertised or if they'd not raise the overall monthly rent given that they may have to deal with negatives such as lost rent if one roommate leaves unexpectedly and the other 2 stay ... but won't cover the 'gone' roommates share of the rent ...

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