  We study a generalization of nonlocal games---which we call extended nonlocal
games---in which the players, Alice and Bob, initially share a tripartite
quantum state with the referee. In such games, the winning conditions for Alice
and Bob may depend on outcomes of measurements made by the referee, on its part
of the shared quantum state, in addition to Alice and Bob's answers to randomly
selected questions. Our study of this class of games was inspired by the
monogamy-of-entanglement games introduced by Tomamichel, Fehr, Kaniewski, and
Wehner, which they also generalize. We prove that a natural extension of the
Navascues--Pironio--Acin hierarchy of semidefinite programs converges to the
optimal commuting operator value of extended nonlocal games, and we prove two
extensions of results of Tomamichel et al. concerning monogamy-of-entanglement
games.
