  Bell showed 50 years ago that quantum theory is nonlocal via his celebrated
inequalities, turning the issue of quantum nonlocality from a matter of taste
into a matter of test. Years later, Hardy proposed a test for nonlocality
without inequality, which is a kind of "something-versus-nothing" argument.
Hardy's test for $n$ particles induces an $n$-partite Bell's inequality with
two dichotomic local measurements for each observer, which has been shown to be
violated by all entangled pure states. Our first result is to show that the
Bell-Hardy inequality arising form Hardy's nonlocality test is tight for an
arbitrary number of parties, i.e., it defines a facet of the Bell polytope in
the given scenario. On the other hand quantum theory is not that nonlocal since
it forbids signaling and even not as nonlocal as allowed by non-signaling
conditions, i.e., quantum mechanical predictions form a strict subset of the so
called non-signaling polytope. In the scenario of each observer measuring two
dichotomic observables, Fritz established a duality between the Bell polytope
and the non-signaling polytope: tight Bell's inequalities, the facets of the
Bell polytope, are in a one-to-one correspondence with extremal non-signaling
boxes, the vertices of the non-signaling polytope. Our second result is to
provide an alternative and more direct formula for this duality. As an example,
the tight Bell-Hardy inequality gives rise to an extremal non-signaling box
that serves as a natural multipartite generalization of Popescu-Rohrlich box.
