So far, I always checked if there is a mixed Bayesian Nash Equilibrium in static games with incomplete information if there was at least one mixed Nash Equilibrium in static games with complete information and I am wondering if there is a better way to quickly check if there is a mixed BNE.
So, is there a proposition which states that a mixed Nash Equilibrium in static games with complete information induces a mixed Bayesian Nash Equilibrium in static games with incomplete information or another proposition from which one can quickly see if there exists a BNE?
As Bayes-Nash equilibria are Nash equilibria (Transform a Bayesian game into a extremely large normal-form game), Nash's theorem guarantees their existence. If we have a finite incomplete information Bayesian game then a mixed strategy Bayes-Nash equilibrium exists.
We also have that in a Bayesian game with continuous strategy spaces and types. If the strategy and type sets are compact, and payoff functions are continuous and concave in own strategies, then a pure strategy Bayes-Nash equilibrium exists.