---
abstract: 'Marengo and the second author have developed in the last years a geometric model of social choice when this takes place among bundles of interdependent elements, showing that by bundling and unbundling the same set of constituent elements an authority has the power of determining the social outcome. In this paper we will tie the model above to tournament theory, solving some of the mathematical problems arising in their work and opening new questions which are interesting not only from a mathematical and a social choice point of view, but also from an economic and a genetic one. In particular, we will introduce the notion of u-local optima and we will study it from both a theoretical and a numerical/probabilistic point of view; we will also describe an algorithm that computes the universal basin of attraction of a social outcome in $O(M^3\log M)$ time (where $M$ is the number of social outcomes).'
author:
- 'Gennaro [Amendola]{}[^1]'
- 'Simona [Settepanella]{}[^2]'
title: Modularity and Optimality in Social Choice
---

[[**Keywords**]{}:\
Social rule, modularity, object, optimum,\
hyperplane arrangement, tournament, algorithm.]{}

[[**MSC (2010)**]{}: 05C20, 05C85, 52C35.]{}

[[**JEL Classification**]{}: D03, D71, D72.]{}

[[**ACM CCS (1998)**]{}: G.2.2.]{}

Introduction {#introduction .unnumbered}
============

In [@Arrow] Arrow 