A husband and wife live together in a beautiful little house. Both enjoy having a clean and beautiful house, which is the result of both of their cleaning efforts, but both the husband and the wife dislike putting any actually effort into cleaning. Also, they both value the cleanliness of the house differently.
They each choose a non-negative number, , representing how much effort to devote to cleaning, and each has the utility function
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In order to find the Nash equilibrium, we must find the best response function of both the husband and the wife. I'll find the one for the husband as a function of k; the function for the wife will be essentially the same (except for the disutility term), for k = 1.
The husband's problem is to choose such that his utility is maximized, given his wife's choice for . Therefore, taking as a given, we find the value of that maximizes by taking the derivative of this function with respect to and equating it to zero. We get:
We now ...
This job locates the pure strategy Nash equilibrium.