Achieving rental harmony with a secretive roommate

Abstract

Given the subjective preferences of n roommates in an n-bedroom apartment, one can use Sperner's lemma to find a division of the rent such that each roommate is content with a distinct room. At the given price distribution, no roommate has a strictly stronger preference for a different room. We give a new elementary proof that the subjective preferences of only n-1 of the roommates actually suffice to achieve this envy-free rent division. Our proof, in particular, yields an algorithm to find such a fair division of rent. The techniques also give generalizations of Sperner's lemma including a new proof of a conjecture of the third author.

v2: additional coauthor, significantly expanded version with new results. 11 pages, 2 figures. v1: Supplemental material for the PBS Infinite Series episode "Splitting Rent with Triangles", available at https://www.youtube.com/watch?v=48oBEvpdYSE . 4 pages, 2 figures