Consistencies of Beliefs

Type spaces are analysed, and we identify three types of consistency of the players' beliefs in a type space: consistency, universal consistency, and strong consistency. Furthermore, we propose a new interpretation of a type space, defining it as a collection of the players' beliefs. This new interpretation introduces a radically different perspective on key theoretic notions, such as the prior, common prior, and posterior. In this framework: the terms of prior, common prior, and posterior are replaced by and interpreted as adequate aggregation of beliefs, consistency of the players' beliefs, and beliefs respectively. Through examples, we demonstrate that, under the proposed approach to type spaces, the ex-ante stage is not only meaningless but nonsensical. This is because, strong consistency of the players' beliefs cannot be witnessed by any probability distribution over the state space (i.e., a concrete common prior).