into this subject in a subsequent paper.

In Section \[sec:preliminaries\] we will recall some basic notions on Hyperplane Arrangements, Salvetti’s Complex and Tournaments. In Section \[sect:model\] we will give basic notions on social rules, we will describe Marengo and the second author’s model, and we will also recall their main results. In the last part of the section we will define the notion of [*u-local optimum*]{} and ([*u-*]{})[*deepness*]{} of a social outcome. The former is a compromise between the notion of local optimum and global optimum. In order to obtain a particular local optimum after the voting process it is enough to have the power of deciding the [*status quo*]{} from which the voting process starts. In contrast, u-local optima are characterized by the property of being obtainable after the voting process by means of object construction power only. This is significant, because it may happen that whoever has the object construction power does not have the power of deciding the [*status quo*]{} from which the voting process starts. The deepness and the u-deepness measure the length of voting processes.

In Section \[sec:results\] we will give results that tie the model described in [@MarengoSette] to tournament theory. This link will 