a social outcome $x$ is the chamber of the subarrangement ${\mathcal{A}}_I$ that contains the chamber corresponding to $x$.

An [*objects scheme*]{} is a set of objects $A=\{{\mathcal{A}}_{I_1},\dotsc, {\mathcal{A}}_{I_k}\}$ such that $\bigcup_{j=1}^k I_j=\{1,\dotsc,n\}$. Note that the sets $I_j$ may have non-empty intersection. The [*size*]{} of an objects scheme is the size of its largest object, $$| A | = \mbox{max}\{| {\mathcal{A}}_{I_1} | ,\dotsc,| {\mathcal{A}}_{I_k} | \}.$$

#### Neighbors of a social outcome

Let be given an objects scheme $A=\{{\mathcal{A}}_{I_1},\dotsc, {\mathcal{A}}_{I_k}\}$. A social outcome $y$ is said to be a [*preferred neighbor*]{} of a social outcome $x$ with respect to an object ${\mathcal{A}}_{I_h} \in A$ if the following conditions hold:

1)  $y \succ  x$,

2)  $y({\mathcal{A}}_{I_h^c})=x({\mathcal{A}}_{I_h^c})$, i.e. $x$ and $y$ belong to the same chamber of the arrangement ${\mathcal{A}}_{I_h^c}$,

3)  \[cond:def\_pref\_neigh\] $y({\mathcal{A}}_{I_h}) \neq x({\mathcal{A}}_{I_h})$, i.e. $x$ and $y$ belong to different chambers of the arrangement ${\mathcal{A}}_{I_h}$.

Note that Condition \[cond:def\_pref\_neigh\] is a direct consequence of the first two, but we have left it for the sake of consistency with the non-strict case. The set of all preferred neighbors of the social outcome $x$ with respect to ${\mathcal{A}}_{I_h} \in A$ is denoted by $\Phi(x,{\mathcal{A}}_{I_h})$. The set of all preferred neighbors of the social 