A new chapter begins for most people with the decision to venture out on their own. They declare their independence by moving out into their own apartment away from their hometown where they can start on a clean slate, meet new people, explore a new city… and face monthly rent bills. People seek after roommates who are tolerable to live with, clean after themselves, are not socially awkward, and most importantly, can help with the rent bill. How do roommates choose to divide up the rent bill when rooms in the apartment are of different size with different amenities (e.g. bigger window, larger in size, balcony, facing the street)?

This question came up in one of my classes this past semester as a discussion question. One student lived in a fraternity where rooms and rent were chosen by a ranking system. The rent was agreed upon and assigned to the rooms and in some way, either by seniority or points earned through chores, the fraternity brothers would be ranked in order. Then each would pick a room with a pre-assigned rent by rank. There was no incentive to lie about which room to choose because all agreed upon the procedure. Another student proposed an auction system in which each person bids on every room and the highest bidder for each room “wins” the room and pays that bid. Another way is to bid percentages for the rooms and the highest bidder of each room pays that percentage of the total rent. There exists advantages and disadvantages for all mechanisms.

Suppose the following scenario. There are three apartment-mates, two rooms, and the rent is $1650 per month. How should the three people decide how to split the rent and who goes into which room? In this type of situation, it would work best when there are differences in preferences (e.g. one person wants the single, one person wants a double, and the other does not care). However, this is not always the case, so how should they solve the room and rent division problem given incompatible interests?

It depends on what the goal is. Some prefer efficiency while others prefer to maximize utility.

But maybe the roomies want fairness. In game theory, the fair division mathematical theory considers the real life problem of dividing a good between people and consists of “n” players (i.e. the people involved in the decision) who have entitlements and different opinions about the value of the good. The goal is to derive the best algorithm that divides the good most fairly. Since each player may have a different definition of fairness such as one may define it to be a fair proportion of the good while the other wants the division to be envy-free in which each recipient believes that no other recipient has received more than they have or that no player wishes to swap their portion with any other player. Therefore, the first rule in fair division is that “fair” division must be defined and agreed upon. Then the players must define and agree upon a procedure that lists certain actions to be followed by the players and all players must follow that procedure.

A commonly used example of fair division is the splitting of a cake between two people who must decide how to divide the cake so that each player thinks they got a fair share of the cake. One mechanism is that player one cuts the cake and player two chooses which one he wants; that way, there is no incentive to lie or cheat because player one will want to cut the cake as fair as possible and player two will obviously choose the one he wants the most. By the rules, player one cannot complain about player two’s choice, and player two cannot complain about how player one cuts it because that is the procedure that they both agreed on. The two players cannot both cut the cake and choose their piece too.

Out of all the methods, I side with the fairness algorithm. It is the most practical that does not deal with number calculations and deep understanding of incomplete information and auction theory. Francis Su, professor of mathematics at Harvey Mudd, created a very easy-to-follow fair division calculator that takes inputs and solves real life problems. Since you can’t cut your cake and eat it too – you can’t have the best of both worlds, then you might as well go for the equilibrium and be satisfied with social fairness. There is no room for greediness!