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? 



Stable Household Formation
This problem has been studied under the name hedonic coalition formation, and your stability notion is called core stability. See this [survey](http://link.springer.com/chapter/10.1007/978-3-642-35843-2_4) by Gerhard Woeginger, [arxiv:1212.2236](http://arxiv.org/abs/1212.2236).


