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…

OH MY BLOG

I found out something AMAZING today. Did you know, that it is actually more likely to have 2 cards of the same suit on the flop than to have 3 different suits? I know, I didn’t believe it when I read it either, but numbers don’t lie:

The more you know!

Pi

I bought a pumpkin pie from Ralph’s earlier in the week in anticipation of Pi day. Today I ate a pot pie for dinner and then had some of this pumpkin pie. I suppose if I were really hardcore I would have made pie myself, but… I’m Asian; baking isn’t in my genes. Also, I’m lazy.

The great thing is the sell date on the pie is today, 3.14! Lol!

Proposal for the Assymmetrization of the Alphabet

(Assymmetrization is not actually a word. Desymmetrization is, but it doesn’t quite mean what I mean. Also it doesn’t start with “ass”. So, I’m going to coin a new word.)

So I was sitting in class today. We were doing some proof about G(n,1/2), so the complement of a graph came up, and I was thinking of what notation to use. ? ? Well, then I started thinking, it sure would nice to have the notation automatically reflect the fact that the complement of the complement of a graph is itself. For example, you’d have to stipulate that . This isn’t something that’s obvious from the notation itself; it’s additional information you have to write down.

I started thinking about other such operations, like transposes, conjugates, adjoints, inverses. Let’s talk about multiplicative inverses to illustrate my point. Consider trying to simply an expression such as . When you unravel it, it becomes . You then have to search through the expression, and everytime you see an instance of something being inverted twice, remove the double inversion. If this is automatically reflected in the notation, then you don’t need to do this search procedure.

After a little bit of thinking, the solution is obvious. You reflect the letter, like this

Reflection, like inverse, is a square root of identity, in that applying it twice does nothing. But now here is the masterstroke. If you reflect letters to denote inverse, then it automatically takes care of the fact that the order in a product is reversed!

Clearly, this is the superior way of denoting things, just like applying functions on the right hand side. But of course, you can’t get something good for free. What’s the problem? The problem is that some letters, like A, are reflexively symmetric. So under this scheme, you can only use them to denote square roots of 1! That’s fine, because there are plenty of square roots of 1, but sometimes you like to use certain letters for generic things. For example, one often uses A and B to denote arbitrary matrices.

Now it begins to dawn on the reader what is meant by the title of this post. We must assymmetrize the alphabet. This is quite easy to do. For example:

More generally, we need to make the alphabet reflexively unique. For example, if you’re not careful, then reflecting a “b” will net you a “d”, and sometimes you’ve already committed “d” to something that isn’t necessarily the inverse of “b”. But, as you can see in the examples above, this can be skirted by adding a simple flair.