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## Settling the Rent Argument

In the New York Times this past week, Albert Sun's article "To Divide the Rent, Start With a Triangle" has created some interest.

The basic problem: how to decide on fair rents when people move into apartments that have unequal ameneties, such as different bedroom footages or views...assuming equal shares are not appropriate. And to complicate the problem, some individuals weigh room factors differently...light access, fire escape access, floor level, room shape, proximity to a bathroom, the one parking spot, etc.

To the rescue in this rent dilemma: Something called Sperner’s lemma involving a decision triangle, as published in Dr. Su's paper “Rental Harmony: Sperner’s Lemma in Fair Division” (1999). It is related to the "fair" division of a cake, knowing some desire the frosted flowers or edges with more frosting. Su's claim: “The trick is to design a procedure to have everyone act in their own self-interest and have an outcome that’s fair.”

The key: A small, fully labeled triangle represents the rooms and prices in a hypothetical apartment. Then potential renters use their "needs" to label/choose the triangles at each interior corner, using an algorithm (built by Su with E. Peterson and P. Vingorad), meandering through multiple "smaller" triangles until an acceptable "point" is reached where all potential renters are satisfied.

At each stage, each renter must pick the room they like best, based on the prices currently assigned to each room in the traingular simplex. As the algorithm continues, the prices get closer and the decisions become harder...but the process remains fair and acceptable based on your previous choices. Now, you can play with the decision process (and Sperner's Lemma) via an on-line visual calculator. By playing with it, you should start seeing why it does promote fair division.

Finally, this same procedure has been used to divide Germany after World War II, allocate deep-sea mining rights, determine the order order of names of co-authors on a scientific paper, and split valuable property after a divorce or death. Again, mathematics to the rescue in the real world.

Finally, if interested in the mathematics more than the rent-aspect, consider Wikipedia's elaboration or NRICH's game version.

Note: One NYT reader (i.e. John Smith) provided some interesting commentary relative to the rent-decision method: "I suspect Sperner lived alone. God alone knows what an ordeal it must been to deal with him when the chore wheel came up. "Now, I only used 11 spoons this week, compared to B's 25 spoons and six forks and C's 4 knives and 18 sporks. I lost track of who used how many ladles, but I've been able to generate a randomization protocol for calculating ...." Be grateful the biggest problem you have is which of three clean, safe rooms in an apartment with running water, heat, food, etc. you're going to have to 'settle' for. Sheesh."

Source: http://www.nytimes.com/2014/04/29/science/to-divide-the-rent-start-with-a-triangle.html