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This paper analyses the robustness of networked systems from a game-theoretical perspective. Networked systems often consist of several subsystems sharing resources interdependently based on local preferences. These systems can be modelled by a dependence game, which is a generalisation of stable paths problem. A unique pure Nash equilibrium in a dependence game can characterise the robustness of the represented networked system, precluding oscillations and nondeterminism. We show that the absence of a structure termed a generalised dispute wheel is useful to ensure the existence of a unique pure Nash equilibrium. Furthermore, we consider more sophisticated settings: tie-breaking over non-strict preferences and asynchronous communications among subsystems. We also obtain stronger results that the absence of a generalised dispute wheel can be useful to ensure the consistency of tie-breaking and asynchronous convergence to a pure Nash equilibrium.