The facets and the symmetries of the approval-voting polytope

Characterizing the size-independent model for approval voting of Falmagne and Regenwetter (J. Math. Psychol. 40 (1996) 152) was shown by Doignon and Regenwetter (J. Math. Psychol. 41 (1997) 171) to be equivalent to determining all facets of the approval-voting polytope. Here, we prove that the facets of this polytope correspond in a natural way to certain antichains in a power set. Several results on the approval-voting polytope are then derived. For instance, all facet-defining inequalities are characterized, and the group of automorphisms is completely described. On the other hand, providing an explicit listing of all of the facets is shown to be at least as intricate as listing all connected graphs on a given, finite set.