Who pays the monthly bills in a roommate agreement?
How To Achieve Rental Harmony
In the article Rental Harmony:
The author advances an envy-free partitioning system. Moreover, for dividing the payment of rent among roommates. The rent-division system is such that for a single room, the individual components are divided. And a set of three conditions that include the good condition of the house. Moreover, miserly tenants, and closed preferences become satisfied. The article Rental harmony presents a robust and practical way of partitioning to avoid envy and unfairness. Because each party pays for what they desire and can afford.
Sperner’s lemma for Triangles
Edward begins the article by stating his friend’s dilemma when he was moving out to a household with quarters of various dimensions and structures . Edward claims that his friend and roommates had trouble assigning each other rooms and the total amount of rent that each party would pay every month. The article thus advances a way of partitioning the rent such that every party preferred a different room. The rent division issue was a type of fair-division question that seeks to divide something proportionally among a group of people. The fair-division question helps to attain envy-free divisions where every person feels that they have received their best share .
Edward claims that the fair division approach for portioning the room, predicated on a simple combinatorial lemma. One that is as powerful as it is simple.
To illustrate how the fair division works. Edward considers a triangle T that is split into several minor triangles referred to as basic triangles with apexes labeled by 1s, 2s, and 3s. The kind of labeling that Edward chose obeys two conditions that include all the significant apexes of T having various markers and the marker of an apex along the edge of T matching the marker of one of the significant apexes that span the edge . Edward claims that labeled triangulation of T that fulfills the two conditions is referred to as Sperner labeling.
Edward claims that we require the idea of an n-simplex whereby a 0-simplex is a point while a 1-simplex is a line section, 2-simplex is a triangle, and 3-simpex is a tetrahedron. The splitting of an n-simplex would be the assortment of minor n-simplices whose unification is S. For a more practical application, Edward requires the audience to view the n-simplex S as a household that is split into several rooms that are the basic simplices. A side of the room will be a door if it has the initial n of the n + 1 markers.
Edward further discusses Simmons’ tactic to cake-cutting. Where he visualizes a rectangular piece shared among a number of people. The audience is old to imagine using n-1 knives to slice along the surfaces that are corresponding to the left side of the piece. The comparative sizes of the cake denote the cuts. Assuming that the full size of the piece is 1, the sum of the individual components is given by x1 + x2 + . . . + xn = 1. Edward hypothesizes that for a hungry crowd with closed preference sets, there are envy free cake divisions .
The third method that Edward advances is the ’n-player case’. Where more than 3 people wish to share a given space. The process takes very basic simplex in a triangulation. And then splits it by demarcating the sides’ barycenter. In every length and then linking them to create a new triangulation. The triangulation of the space can be made smaller through further iteration .
After explaining the three procedures theoretically. Edwards proceeds to show how the cake-cutting technique applies in a real-life scenario, such as in dividing the rent. Edward discusses the rental harmony theorem. Whereby for n roommates in an n-bedroom household, the house partitions in a fair manner. To realize the rental harmony theorem, Edward proposes certain conditions upheld.
- The house must be in good condition such that in any division of the rent. Every person finds his or her assigned room as acceptable.
- The second condition denotes miserly tenants. Whereby every person always wishes to have a free room that does not cost any rent.
- The third condition denotes a closed preference set. Whereby an individual who desires a room for a centrolineal series of prices. Would prefer the room at the least price.
Edward issues a caveat that the rent-division issue may be seen as a simplification of the cake-cutting issue.
Where a group of people seeks to divide up a set of goods equally. Since the rooms are indivisible, the cake-cutting solutions that Edward advanced could not be applied to solve the issue. Since the rent is paid according to particular sections. Moreover, it cannot be separated into smaller components and then re-assembled. Disregarding the use of known envy-free partition methods that were discussed earlier.
To ensure a cake a cutting method that is free of envy, Edward advanced an even more complex technique that he refers to as ’rental harmony: cake-cutting with a twist’ . Edward theorized that assuming there are n roommates, with n rooms to allocate from number 1 to n; then xi denotes the i-th room price and the total rent due sums up to 1. It follows that x1 + x2 + . . . + xn and xn = 1 and xi = 0.
The next step is triangulating the simplex by barycentric subdivisions, after which the pricing scheme follows.
The new rent division scheme would satisfy condition 1 that ensures that the house is in good condition, and condition 2 where just some quarters are free, and proprietors would prefer to choose each one depending on how much they can pay for it .
In conclusion, the article presents a robust and practical way of partitioning to avoid envy and unfairness. Because each party pays for what they desire and can afford. Edward first presents Sperner’s lemma for triangles and then Simmons’ Approach to Cake-cutting. Before settling on ’rental harmony: cake-cutting with a twist. Lastly, as the most effective way of portioning rent without envy arising.
References  Francis Edward Su. Rental harmony: Sperner’s lemma in fair division. The American mathematical monthly, 106(10):930–942, 1999.