The matching game is played in a bipartite graph G = (V1, V2, E) in which edges are connect only vertices Vị to vertices in V2. The players are the vertices in the graph that is Vị U V2. Each player has to select one of its neighbors. Player i gets utility 1 when the selection is mutual (player i selects j and player j selects i) otherwise he gets 0. Provide a formal characterization of the strategy profiles that are pure Nash equilibrium of the matching game. Analyze the complexity of the problems related to pure Nash equilibria for this family of games.

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