Every hierarchy of beliefs is a type

Any model of incomplete information situations has to consider the players' hierarchies of beliefs, which can make the modeling very cumbersome. Harsányi [12] suggested that the hierarchies of beliefs can be replaced by types, i.e., a type space can substitute for the hierarchies of beliefs (henceforth Harsányi program). In the purely measurable framework Heifetz and Samet [15] formalized the concept of type space, and proved that there is universal type space, i.e., the most general type space exists. Later Meier [17] showed that the universal type space is complete, in other words, the universal type space is a consistent object. After these results, only one step is missing to prove that the Harsányi program works, that every hierarchy of beliefs is in the complete universal type space, put it differently, every hierarchy of beliefs can be replaced by type. In this paper we also work in the purely measurable framework, and show that the types can substitute for all hierarchies of beliefs, i.e., the Harsányi program works.