# Stable Household Formation

I want to model the problem of household formation by a finite number of individuals, each of whom has preferences over sets of housemates. A collection of households is unstable if there is a set of individuals who can all be made happier by leaving their current households and forming a new household together.

More precisely, let ${\cal A}$ be a finite set of individuals and ${\cal P}$ the power set of ${\cal A}$.

For each $A\in{\cal A}$, let $>_A$ be a total ordering on $\lbrace X| A\in X\in {\cal P}\rbrace$.

Given a partition ${\cal X}$ of ${\cal A}$ and an individual $A\in{\cal A}$ write ${\cal X}(A)$ for the (unique) element of ${\cal X}$ that contains $A$.

Then $\cal{X}$ is unstable if there exists $Y\in{\cal P}-{\cal X}$ such that for all $A\in Y$, $Y>_A{\cal X}(A)$. A partition is stable if it is not unstable.

Here is an example with three individuals and no stable partition:

$$\lbrace A,B\rbrace>_A\lbrace A,C\rbrace >_A \lbrace A\rbrace >_A \lbrace A,B,C\rbrace$$ $$\lbrace B,C\rbrace>_B\lbrace A,B\rbrace >_B \lbrace B\rbrace >_B \lbrace A,B,C\rbrace$$ $$\lbrace A,C\rbrace>_C\lbrace B,C\rbrace >_C \lbrace C\rbrace >_C \lbrace A,B,C\rbrace$$

Question 1: Does this question have a name? Obviously it is related to, but not identical with, the stable marriage problem.

Question 2: Are there any general conditions under which it is true but not obvious that a stable partition exists?