Weak order polytopes

The weak order polytope P n WO is the polytope in R n(n?1) whose ver-tices correspond to the members of the family of reeexive weak orders on f1; 2 : : : ; ng when each coordinate of R n(n?1) is associated with a diierent ordered pair (i; j) of distinct points in f1; 2; : : : ; ng. The vertex w that corresponds to weak order W has w (i;j) = 1 if i-j in W and has w (i;j) = 0 otherwise, with P n WO the convex hull of all such w. The paper focuses on facet-deening inequalities, vertex adjacency and symmetries. We relate P n WO to the theories of probabilistic choice and preference aggregation, prove a basic lifting lemma that carries facet deening inequalities for P n WO into P n+1 WO , identify complete sets of facet-deening inequalities for n 4, give a complete account of vertex adjacency and extended transitive orientations, determine all symmetries of P n WO , and note how the polytope relates to the linear ordering and partition polytopes on f1; 2; : : : ; ng.