Game Theory of Cleaning

It’s a new semester and a lot of students are hoping that this time they will be able to

overcome the roommate problems they had encountered last semester. But even

though the number on the calendar might have changed, it’s the same old story with

the same old problems.

Have you ever had a roommate or flat-mate? What was your biggest problem? In my

case it is room-cleaning.

Let’s imagine a following situation - I and Ming-Ming are roommates. Each of us

prefers a clean room to a dirty room, but neither is thrilled about cleaning the place,

and we are not rich enough to hire a service. What is worse, there is no common

language in which we can discuss the problem. So, what does game theory predict

will happen?

First, we assume that Magda and Ming Ming are rational. What implies that each of

us is motivated by maximizing her own payoff, utility. Here is how we can model this

situation as a game. Magda and Ming Ming - each of us can choose either “clean” or

“not clean”.

We can assign numerical payoffs to identify the most desired outcome.

1) The best scenario is when I don’t clean and our roommate does it instead,

then I get payoff 6. I did nothing and it’s clean!

2) If both of us clean then I get payoff 5.

3) If both of us don’t clean then I get payoff 3.

4) And the worst scenario when I spent a lot of time and cleaned the room and

my roommate didn’t even lift a finger, then I get payoff 2.

Clean Don’t Clean

Clean 5, 5 2, 6

Don’t Clean 6, 2 3, 3

As I and Ming Ming cannot communicate we have to guess what the other person is

doing. To analyze this sort of game, we need to decide on a course of

action given what you think your roommate will do.

1) If Ming Ming decides to clean the apartment,

if I clean 5 utile and I don’t 6 utile.

2) If Ming Ming chooses not to clean then,

if I clean 2 utile and if I don’t 3 utile.

In other words, no matter what Ming Ming does, I always get higher payoff if I

choose not to clean!

The reasoning is precisely the same for Ming Ming: No matter what I do, Ming Ming

always gets a higher payoff if she chooses not to clean.

Therefore, the outcome where both roommates choose “Don’t clean” is a dominant

strategy equilibrium. This is true despite the fact both of us would be better off if we

chose to clean the room. There is a reason to choose not to clean. The roommate’s

incentives reinforce my own inclination not to drain, and vice versa, ad infinitum. So,

to be exact it’s not our laziness but the construction of the game imposes such

conditions that we are stuck in our cleaning dilemma. If we could cooperate to clean

and both change our behavior, then it would be possible to have the best outcome

where the room is clean.

So, what is the lesson here? Rational choices can lead to bad outcomes and… I’m a professional in making excuses!

And now… it’s time to get real! Next week the dormitory supervisor will check the

state of our room! So as you can see, there are alternative mechanisms inducing

cooperation and making us to clean the room.