Hyper-stable social welfare functions

We introduce a new consistency condition for neutral social welfare functions, called hyperstability. A social welfare function α selects a complete weak order from a profile PN of linear orders over any finite set of alternatives, given N individuals. Each linear order P in PN generates a linear order over orders of alternatives,called hyper-preference, by means of a preference extension. Hence each profile PN generates a hyper-profile ṖN . We assume that all preference extensions are separable: the hyper-preference of some order P ranks order Q above order Q′ if the set of alternative pairs P and Q agree on contains the one P and Q′ agree on. A special sub-class of separable extensions contains all Kemeny extensions, which build hyper-preferences by using the Kemeny distance criterion. A social welfare function α is hyper-stable (resp. Kemeny-stable) if at any profile PN , at least one linearization of α(PN) is ranked first by α(ṖN), where ṖN is any separable (resp. Kemeny) hyper-profile induced from PN . We show that no scoring rule is hyper-stable, and that no unanimous scoring rule is Kemeny-stable, while there exists a hyper-stable Condorcet social welfare function.