# How to set up the payoffs properly for a division of labor game

I'm admittedly a novice when it comes to game theory (currently a few lectures into Yale's intro course lectures), so hopefully people will indulge me what may be a dumb question.

I was trying to understand how Nash Equilibrium affects division of labor (which was one of the examples used in the class). In particular, I was trying to represent a simple marital dispute: who cleans the house. I'm assuming that both partners would prefer equally that the house be clean, and that they have similar cleanliness standards. Furthermore, I'm assuming that if they both clean, they contribute to the work equally (or, at least, in a way that both of them are equally satisfied with). John Gottman's books and personal experience both assure me that these are rather naive assumption, but bear with me.

The two Nash Equilibria are a) where they share the work equally or b) where neither of them clean (the "bad" equilibrium). This is because neither of them wants to be the one that gets stuck doing all of the work. My reasoning behind that is that neither one would like to be the one who has to do all of the cleaning; however, they would both prefer it if the other person did all of the cleaning.

I tried to formulate that as follows:

Husband Cleans Husband Doesn't Clean
Wife Cleans 1,1 2,0
Wife Doesn't Clean 0,2 0, 0

On the plus side, them sharing the cleaning is effectively a compromise - they both share the cost equally and they both get the benefit equally. On the other hand, the following facts bother me:

• Does the way that I've structured the payoffs imply that there's no payoff at all for the spouse that does all the work? Yes, they have the added "cost" of doing all the work themselves, but this would say that the cost of the added work would entirely offset the benefit of having a clean house - it's just as "bad" for them as if the work hadn't gotten done at all (which seems strange to me, because some spouses might prefer to do all of the work themselves than to have it not done at all).
• Similarly, there's no "penalty" for having the house not cleaned - they just don't get a payoff. I'm a little bit confused if that's ok, though; if neither people get what they want, should the payoff be negative because they're unhappy about that fact, or should it just be 0 because they didn't get what they wanted (and therefore got no payoff)?
• To my previous point: if I adjust the payoffs, I would like to make sure that I do so in such a way that both of them cleaning or neither of them clean are still equilibrium solutions.

Are my concerns well-founded here, or did I do the table correctly after all?

If you want "both don't clean" to be a Nash equilibrium (NE), then your payoff matrix doesn't work. With those payoffs, cleaning is a strictly dominant strategy and "both clean" is the only NE.

A more systematic approach would be to assume that a clean house has utility $$b>0$$ for both partners, but cleaning alone generates a disutility of $$c>0$$. Cleaning is likely to have increasing marginal disutility, such that sharing the cleaning work costs only, say, $$c/4$$. Then the payoff matrix is

Husband Cleans Husband Doesn't Clean
Wife Cleans $$b-c/4$$,$$b-c/4$$ $$b-c$$,$$b$$
Wife Doesn't Clean $$b$$,$$b-c$$ 0, 0

If $$b>c$$, then "only wife cleans" and "only husband cleans" are two strict NE (and there's also a symmetric mixed one).

If $$b, then "wife doesn't clean" and "husband doesn't clean" are strictly dominant strategies and "no one cleans" is the unique NE. If $$b but $$4b>c$$, then "both clean" is Pareto better than "no one cleans" (and uniquely maximizes total payoffs), which implies that your game is a Prisoner's Dilemma.

With this approach, of course, "both clean" is never a NE, since saving the costs of cleaning while enjoying the benefits of a clean house is always preferred to participating in the cleaning. But this is what you seem to assume ("they would both prefer it if the other person did all of the cleaning"). This assumption contradicts your wish for having a NE where both clean.

However, you could go on to argue that the game they actually play is the infinitely repeated version of this 2x2 game. This opens up the possibility that "both always clean" is a NE-outcome of the repeated game, e.g. based on Trigger strategies, even for the Prisoner's Dilemma case (with small enough discount rate).