  In the quantum version of prisoners' dilemma, each prisoner is equipped with
a single qubit that the interrogator can entangle. We enlarge the available
Hilbert space by introducing a third qubit that the interrogator can entangle
with the other two. We discuss an enhanced interrogation technique based on
tripartite entanglement and analyze Nash equilibria. We show that for
tripartite entanglement approaching a W-state, there exist Nash equilibria that
coincide with the Pareto optimal choice where both prisoners cooperate. Upon
continuous variation between a W-state and a pure bipartite entangled state,
the game is shown to have a surprisingly rich structure. The role of bipartite
and tripartite entanglement is explored to explain that structure.
