Based on the theoretical basis of weighted Nash equilibrium in multi-objective games, the existence results of weighted Nash equilibrium points in multi-objective games under the condition that the vector-valued payoff function was pseudo-continuous were discussed. The game space of pseudo continuous vector-valued payment function was constructed, the definition of weighted Nash equilibrium point was given, and the set-valued mapping of multi-objective game was defined, and the set-valued mapping was proved to be non-empty, convex and USCO mapping. By using Fan-Glicksberg fixed point theorem, Fort theorem and the definition of intrinsic equilibrium point, the generic stability of weighted Nash equilibrium point was discussed under simultaneous perturbation of weight vector, payment function and strategy set. It is concluded that in the sense of Baire's classification, the problem we construct is essential, that is, the weighted Nash equilibrium points in multi-objective games have generic stability.