The Real Computational Complexity of Minmax Value and Equilibrium Refinements in Multi-player Games

We show that for several solution concepts for finite n-player games, where \(n \ge 3\), the task of simply verifying its conditions is computationally equivalent to the decision problem of the existential theory of the reals. This holds for trembling hand perfect equilibrium, proper equilibrium, and CURB sets in strategic form games and for (the strategy part of) sequential equilibrium, trembling hand perfect equilibrium, and quasi-perfect equilibrium in extensive form games. For obtaining these results we first show that the decision problem for the minmax value in n-player games, where \(n\ge 3\), is also equivalent to the decision problem for the existential theory of the reals.

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