Ordered Kripke Model, Permissibility, and Convergence of Probabilistic Kripke Model

We define a modification of the standard Kripke model, called the ordered Kripke model, by introducing a linear order on the set of accessible states of each state. We first show this model can be used to describe the lexicographic belief hierarchy in epistemic game theory, and perfect rationalizability can be characterized within this model. Then we show that each ordered Kripke model is the limit of a sequence of standard probabilistic Kripke models with a modified (common) belief operator, in the senses of structure and the (epsilon-)permissibilities characterized within them.