The Chore Rotation

Live with anyone, anywhere, and you will run into this sort of situation. Things need to get done to help keep wherever you're living in reasonable/clean/respectable/whatever condition, and you want everyone to be a part of it, at least in theory.

I personally had to deal with some of this recently, so this was the first topic that came to mind. I was also discussing this with my girlfriend, so I figured it deserved mentioning.

For ease of analysis, I will look at the case where the chore in question is cleaning, though it extends easily to more general chores. I will also consider the two player case, though, again, the multi-player case is a pretty natural extension.

I guess I need to state my ASSUMPTIONS, and these will likely be the same for any situation I describe: Players are rational(in rough non-technical terms, can do a lot of computations really fast), self interested(will always act to increase their own utility), and are informed(know all the rules, alternatives and outcomes of the game).

Rational is an okay assumption in games where there are not many outcomes, so not many problems there.
Self-interested is a pretty good assumption considering how most people act in real life.
Informed is tricky. For small games with few alternatives it's reasonable, but it's still tricky. I'll try to avoid it when I can.

SETUP: Each player has two choices: clean(A) or don't clean(B). There is a benefit associated with having a clean house, we'll call it H, which is >0. However, there is a cost associated with cleaning. We'll call this C. The benefit to a dirty house is 0.

GAME1: House is cleaned each week. Each player is in charge of cleaning half the house, and the part of the house that is cleaned rotates between each player each week. There is a "gentleman's agreement" that cleaning will get done that both players agree to, ie, they are going against the self-interested assumption. This agreement also includes that when cleaning gets done, the whole house must be cleaned in a given week.(This is a fairly common agreemen among housemates)

The utility for both players in this case is H - C. We will assume that H>C so that there is net positive utility to a clean house.
Things are going along swimingly. However, one week Player 1(p1) is unable to clean for whatever reason. They have only option D, and so have to do it. Player 2(p2) decides that this week, she will do the cleaning.

p1 utility = H, p2 utility = H -2C. H - 2C could be <0, but this is only for one week.

The next week, however, p1 is able to clean, but the agreement before is nulled since she couldn't clean before. She is now self interested once again. Her utility from the last week is H which is > H - C, so she has no incentive to start cleaning again.(perhaps she is lying to the other player, continuing her story from the week prior)
p2 is now in a difficult position. If H - 2C > 0, then she will likely still clean, since if she doesn't clean, her utility is 0. This is stll unfair, as H > H -2C, so p2 is still getting a raw deal.
If H - 2C < 0, then she won't clean either, since her utility will increase from a number less than 0 to 0.
At this point the players are in a Nash equilibrium. There is no incentive to single-handedly start cleaning again until H - 2C > 0.

At that point people get angry and start yelling at each other to clean.

GAME2: Similar to above, each player is in charge of half the house, but it does not rotate. The benefit to half a clean house is obviously (1/2)H. The agreement this time is that each person cleans only the half of the house they are responsible for.(Another common agreement)

The utility for both players adhering to the agreement is once again H - C which is > 0.

Let's say a similar thing happens here, that p1 can't clean for one week for whatever reason. p2 then cleans her half, and gets a utility of (1/2)H - C. This could be less than, equal to, or greater than 0, as before. If we multiply these expressions by 2, we get H - 2C is greater than, less than, or equal to 0, depending. We are in the same situation as before.

Thus, we get into situations where once the agreement/system breaks, the players end up in a Nash Equilibrium where no one is willing to start the system back up.

The philosphy I follow with regards to cleaning is thus: Clean up after yourself. Not only does this keep your own personal space clean, it helps keep public spaces clean, and shifts blame of dirtiness to the other players. It puts you on what I call the "high horse", where a player is in a position to incentivize(read: guilt) other players into action.

Any other cleaning strategies out there?