How to set up the payoffs properly for a division of labor game
If you want &quot;both don't clean&quot; to be a Nash equilibrium (NE), then your payoff matrix doesn't work. With those payoffs, cleaning is a strictly dominant strategy and &quot;both clean&quot; is the only NE.
A more systematic approach would be to assume that a clean house has utility $b&gt;0$ for both partners, but cleaning alone generates a disutility of $c&gt;0$. 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 $b&gt;c$, then &quot;only wife cleans&quot; and &quot;only husband cleans&quot; are two strict NE (and there's also a symmetric mixed one).
If $b&lt;c$, then &quot;wife doesn't clean&quot; and &quot;husband doesn't clean&quot; are strictly dominant strategies and &quot;no one cleans&quot; is the unique NE. If $b&lt;c$ but $4b&gt;c$, then &quot;both clean&quot; is Pareto better than &quot;no one cleans&quot; (and uniquely maximizes total payoffs), which implies that your game is a Prisoner's Dilemma.
With this approach, of course, &quot;both clean&quot; 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 (&quot;they would both prefer it if the other person did all of the cleaning&quot;). 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 &quot;both always clean&quot; 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).