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This letter proposes a new approach to the computation of extremal-Nash equilibria in a wholesale power market with transmission constraints. The approach uses linearization techniques to formulate the extremal-Nash equilibrium problem as a single-stage mixed-integer linear programming problem which can be solved with standard software. Through the introduced concept of extremal-Nash equilibria, the derived structure can efficiently locate all Nash equilibria of the game. We show that this approach offers significant performance improvements over existing approaches to computing Nash equilibria.