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Network games have provided a framework to study strategic decision making processes that are governed by an underlying network of interdependencies among agents. However, existing models do not account for environments in which agents simultaneously interact over multiple networks. In this paper, we propose a model of multiplex network games to capture the different modalities of interactions among strategic agents. We then explore how the properties of the constituent networks of a multiplex network can undermine or support the uniqueness of its Nash equilibria. We first show that in general, even if the constituent networks are guaranteed to have unique Nash equi- libria in isolation, the resulting multiplex need not have a unique equilibrium. We then identify certain subclasses of networks wherein guarantees on the uniqueness of Nash equilibria on the isolated networks lead to the same guarantees on the multiplex network game. We further highlight that both the largest and smallest eigenvalues of the constituent networks (reflecting their connectivity and two-sidedness, respectively) are instrumental in determining the uniqueness of the multiplex network equilibrium. Together, our findings shed light on the reasons for the fragility of the uniqueness of equilibria in multiplex networks, and potential interventions to alleviate them.