I found the following two Theorems when studying games with incomplete information.
"Consider a finite incomplete information (Bayesian) game. Then a mixed strategy Bayesian Nash equilibrium exists."
"Consider a Bayesian game with continuous strategy spaces and continuous types. If strategy sets and type sets are compact, payoff functions are continuous and concave in own strategies, then a pure strategy Bayesian Nash equilibrium exists. "
The lecture slides I found did not have any proofs for the above statements. Does anyone know any good reference/books to look up the said proofs?
It can be found on p. 34 of Fudenberg and Tirole (1991), which is referencing a result obtained separately by Debreu (1952), Glicksberg (1952), and Fan (1952).
The above quotes are likely from the lecture notes given in an online MIT course by Asu Ozdaglar in 2010. Some explanation of the proof and be found in an earlier lecture.