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A numerical method based on a relaxation algorithm and the Nikaido-Isoda function is presented for the calculation of Nash-Cournot equilibria in electricity markets. Nash equilibrium is attained through a relaxation procedure applied to an objective function, the Nikaido-Isoda function, which is derived from the existing profit maximization functions calculated by the generating companies. We also show how to use the relaxation algorithm to compute, and enforce, a coupled constraint equilibrium, which occurs if regulatory, generation, and distribution (and more) restrictions are placed on the companies and entire markets. Moreover, we use the relaxation algorithm to compute players' payoffs under several player configurations. This is needed for the solution of our game under cooperative game theory concepts, such as the bilateral Shapley value and the kernel. We show that the existence of both depends critically on demand price elasticity. The numerical method converges to a unique solution under rather specific but plausible concavity conditions. A case study from the IEEE 30-bus system, and a three-bus bilateral market example with a dc model of the transmission line constraints are presented and discussed.