Shared autonomy requires principled mechanisms for allocating and transferring control between a human and an autonomous agent. Existing approaches often rely on blending control inputs or heuristic switching rules, which lack theoretical guarantees and fail to account for the dynamics of authority transfer. This paper develops a cooperative game-theoretic framework for authority switching in shared autonomy. We formulate the control switching problem as an identical-interest dynamic game in which authority transitions are embedded into the system dynamics, yielding optimal switching policies rather than ad hoc rules. We establish the existence and characterization of team-optimal policies in pure strategies under stochastic human override, accounting for asymmetric authority where humans retain override capability. For linear-quadratic systems, we derive closed-form recursions for the optimal switching policies and value functions, enabling efficient computation independent of the continuous state. We validate the framework on scalar and multi-dimensional linear systems, demonstrating how optimal switching adapts to varying system dynamics, cost structures, and override probabilities. The results reveal fundamental trade-offs between human adaptability and autonomous efficiency, illustrating the practical benefits of grounding shared autonomy in cooperative game theory.

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