The universal belief space

Abstract

Chapter summary In this chapter we construct the universal belief space , which is a belief space that contains all possible situations of incomplete information of a given set of players over a certain set of states of nature. The construction is carried out in a straightforward way. Starting from a given set of states of nature S and a set of players N we construct, step by step, the space of all possible hierarchies of beliefs of the players in N . The space of all possible hierarchies of beliefs of each player is proved to be a well-defined compact set T , called the universal type space . It is then proved that a type of a player is a joint probability distribution over the set S and the types of the other players. Finally, the universal belief space Ω is defined as the Cartesian product of S with n copies of T ; that is, an element of Ω, called state of the world , consists of a state of nature and a list of types, one for each player. Chapters 9 and 10 focused on models of incomplete information and their properties. A belief space Π with a set of players N on a set of states of nature S , is given by a set of states of the world Y , and, for each state of the world ωϵ Y, a corresponding state of nature s(ω) ϵ S and a belief π i (ω) ϵ Δ( Y ) for each player i ϵ N .