  Network games are an important class of games that model agent interactions
in networked systems, where players are situated at the nodes of a graph and
their payoffs depend on the actions taken by their neighbors. We extend the
classical framework by considering a game model where the strategies are
positive semidefinite matrices having trace one. These (continuous) games can
serve as a simple model of quantum strategic interactions. We focus on the
zero-sum case, where the sum of all players' payoffs is equal to zero. We
establish that in this class of games, Nash equilibria can be characterized as
the projection of a spectrahedron, that is, the feasible region of a
semidefinite program. Furthermore, we demonstrate that determining whether a
game is a semidefinite network game is equivalent to deciding if the value of a
semidefinite program is zero. Beyond the zero-sum case, we characterize Nash
equilibria as the solutions of a semidefinite linear complementarity problem.
