Pseudocompactness of hyperspaces was studied by J. Ginsburg, who asked whether there is a relationship between the pseudocompactness of X^\omega and the hyperspace exp(X) for a topological space X. For an almost disjoint family \mathcal{A}, maximality is equivalent to pseudocompactness of \Psi(\mathcal{A}) and that of \Psi(\mathcal{A})^\omega. Hence J. Cao and T. Nogura asked whether some/every MAD family has a pseudocompact hyperspace.

Recently, the statement that every MAD family has a pseudocompact hyperspace was proved to be equivalent to the Novak or Baire number \mathfrak{n} being greater than \mathfrak{c}, however, not much more is known about the existence of MAD families with pseudocompact hyperspace. We will address this problem by showing many models and cardinal invariant assumptions that imply the existence of MAD families with pseudocompact hyperspace.