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We investigate the complexity of bounding the uncertainty of graphical games,and we provide new insight into the intrinsic difficulty of computing Nashequilibria. In particular, we show that, if one adds very simple and naturaladditional requirements to a graphical game, the existence of Nash equilibriais no longer guaranteed, and computing an equilibrium is an intractableproblem. Moreover, if stronger equilibrium conditions are required for thegame, we get hardness results for the second level of the polynomial hierarchy.Our results offer a clear picture of the complexity of mixed Nash equilibria ingraphical games, and answer some open research questions posed by Conitzer andSandholm (2003).

Bounding the Uncertainty of Graphical Games: The Complexity of Simple Requirements, Pareto and Strong Nash Equilibria