On the existence of Pareto undominated mixed-strategy Nash equilibrium in normal-form games with infinite actions

We show that each mixed strategy of a normal-form game can be reformulated as a pure strategy of an induced Bayesian game. Moreover, a normal-form game has a Pareto undominated mixed-strategy Nash equilibrium if and only if its induced Bayesian game has a pure-strategy Bayesian Nash equilibrium. By relying on the existence result of Pareto undominated pure-strategy Nash equilibrium in Bayesian games in Fu and Yu (2015), we also show that every normal-form game has a Pareto undominated mixed-strategy Nash equilibrium.