Extremal Perfect Monotone Equilibria in Bayesian Games with Strategic Complementarities

We study sharp bounds on equilibrium behavior in Bayesian games when players may make mistakes. Under strategic complementarities, the set of behavioral perfect equilibria has a largest and a smallest element, both in pure strategies. We ask when these bounds are monotone and how they depend on the way mistakes are made. For monotone supermodular games, we consider limits of equilibria under increasing perturbations, in the sense that conditional tremble distributions have full support and increase stochastically with type. We show that the largest and smallest perfect equilibria exist, which are pure and monotone. Under the general perfection notion, the extremal perfect equilibria continue to be monotone in the setting with finite actions. With infinite actions, the extremal perfect equilibria can be nonmonotone, and the set of perfect monotone equilibria can even lack extrema.