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Extensive-Form Perfect Equilibrium Computation in Two-Player Games

We study the problem of computing an Extensive-Form Perfect Equilibrium (EFPE) in 2-player games. This equilibrium concept refines the Nash equilibrium requiring resilience w.r.t. a specific vanishing perturbation (representing mistakes of the players at each decision node). The scientific challenge is intrinsic to the EFPE definition: it requires a perturbation over the agent form, but the agent form is computationally inefficient, due to the presence of highly nonlinear constraints. We show that the sequence form can be exploited in a non-trivial way and that, for general-sum games, finding an EFPE is equivalent to solving a suitably perturbed linear complementarity problem. We prove that Lemke's algorithm can be applied, showing that computing an EFPE is $\textsf{PPAD}$-complete. In the notable case of zero-sum games, the problem is in $\textsf{FP}$ and can be solved by linear programming. Our algorithms also allow one to find a Nash equilibrium when players cannot perfectly control their moves, being subject to a given execution uncertainty, as is the case in most realistic physical settings.