The Roommate’s Dilemma

Well hello!

Lobby and Fobby

Creepy hallway

Kitchen

Living room

Slave Bedroom

Slave Bedroom

Master Bedroom

All rooms lead to balcony

All rooms lead to balcony

Another view of the living room. Also, an idiot.

Kitchen again

The reason this place is affordable

It will be totally possible to dive from here

The Roommate’s Dilemma is an instantiation of the Cake Cutting Problem. Basically, n=2 (dashing, handsome, etc.) roommates need to be assigned to n rooms, which are not identical. Since they are not identical, the rent should not be split equally. Each room presents different utility to the roommates, so the assignment and rent split should reflect this and try to maximize everyone’s utility. How to do it “fairly”?

The solution to the cake problem is to let one person cut the cake and let the other choose. Thus, the person has an incentive to cut the cake as equally as possible, lest the other take the bigger. Neglecting physicalities, such as the impossibility of splitting something exactly equal, this algorithm ought to leave both parties satisfied that the outcome is fair. If the chooser is not satisfied, it is his own fault for being a dumbass and picking the inferior cut; if the cutter is not satisfied, he has only himself to blame for not cutting it fairly.

The direct analogue of this solution is to let one person split the rent and let the other choose which room to live in. I started thinking about this, though, and I think this is actually not fair. It is not possible for the cutter to live in a room and pay less than what he thinks it is worth, but it is possible for the chooser.

For example, suppose that roommate 1’s utility is at 1200/800, and 2 is at 1500/500, for rooms A and B, respectively. If 1 is cutting, then he has no choice but to place the rent at 1200/800 (assuming he doesn’t know 2’s utility split). For were he to do otherwise–for example, at 1300/700, then 2 may choose room B, leaving 1 with room A, and having to pay higher than what he thinks room A is worth. Only at 1200/800 would he be neutral about the outcome of 2’s choice–it’s an equilibrium of sorts. Then, 2 will choose room A, and pay 300 less than what he thinks it’s worth. Thus, basically, 2 has benefited $300, while 1 has benefited $0 from this mechanism.

Of course, in real life, 1 can try to guess 2’s utility. If he knew what his utility was, he could place the split at 1499/501. 2 will then pick A (assuming he knows the value of an honest dollar), leaving 1 with a $299 discount.

The system we’ve agreed on is to each write down our utilities by secret ballot, then average the utilities, and assign the rooms “down”. For example, in the example above, the averages are 1350/650. Then, 1 would take B and pay 650, and 2 would take A and pay 1350. Then each has mutually benefited $150 (essentially, wealth is created from the utility differential).

Now to decide what my utilities are and see if I will actually be happy with the outcome of this experiment… Today I measured the sizes of the rooms, using my armspan, which ought to be the same as my height. The smaller room is 2×2.4 Siwei’s, and the larger is about 2×3.5. However, the balcony door to the smaller room is bigger, so it’s brighter. Hmmm…