The Roommate’s Dilemma
Figuring out how to coexist with a roommate can be a tricky process, especially if you’ve never had to live with one before. Luckily for me, my roommate here in Ithaca is my brother; we’re both students up here, very close, and far from home, so having each other is really nice and comforting.
There is only one key difference between us, which causes our living together to not run smoothly- we have very different standards of cleanliness. My brother doesn’t care if the apartment is clean or dirty, he just hates to clean, and I don’t mind cleaning, but I hate when the apartment is dirty. It usually ends up being that I clean the apartment, and he does nothing, and I’m going to use game theory to see if it makes sense that this tends to be the most frequent outcome. (Note: this is going to be similar to the prisoners dilemma https://en.wikipedia.org/wiki/Prisoner%27s_dilemma)
In this graph, I will be the left side (M) and my brother will be the top (B). 
The two possible strategies for each of us are to clean and to not clean. As for the values, I’ve chosen our emotions on the matter, as how we feel about the apartment determines if we’ve won or lost. The values are being upset (denoted with a -1), neutral (denoted with a 0), and happy (denoted with a +1).
I’ll go through each possible outcome and why I’ve selected the values I’ve chosen:
- Outcome one: We both clean. If we both clean, I’ll be ecstatic with how clean the apartment is and be very happy. My brother however, will be quite upset that he had to clean. (+1,-1)
- Outcome two: I don’t clean and he cleans. Since the cleaning was done by just one person, it wasn’t done as thoroughly, so I’m neutral about the cleanliness of the apartment. He however, will still be upset about having to clean the apartment. (0,-1)
- Outcome three: I clean and he doesn’t clean. I don’t mind cleaning, I just want to have a clean apartment, but I still can’t do as good of a job by myself as I can with the both of us, so the apartment is only somewhat clean. I am neutral about the cleanliness of the apartment. My brother though, since he doesn’t care about the state of the apartment, as long as he doesn’t have to clean, is happy he doesn’t have to clean. (0,+1)
- Outcome four: Or as I like to call it, the worst case scenario. We both don’t clean. The apartment turns into a pigsty, and I’m very upset. My brother is slightly bothered by the extreme mess, but still hates cleaning, so he’s neutral. This usually only happens when I go on a cleaning strike out of frustration, or during finals week. (-1,0)
The best outcome for the cleanliness of the apartment, as well as for me, is outcome 1. The best possible outcome for my brother is outcome 2. It doesn’t look like there is a pure strategy Nash Equilibrium, so let’s find find the best mutual response.
If my brother cleans, then I should also clean. If my brother doesn’t clean, then I should also clean. As for my brother, if I clean, then he shouldn’t; if I don’t, then he still shouldn’t clean. Here is where the unfortunate mixed NE comes in; regardless of whether or not my brother cleans, I should clean, and regardless of if I clean or not, my brother shouldn’t. Outcome three is the outcome the always occurs, because it our best strategies towards each other.
Which is why I’m always stuck doing the dishes, and vacuuming, and taking out the trash, and everything in between. I love you bro, but I hate your mess.