Repeat given Exercise, but this time suppose the beliefs of each player of each type are:
Parts (b)-(e) of Exercise 10.37 relate to a situation in which the true state of nature is s12. Can each player calculate the minimal belief subspace of the other player at each state of the world? Justify your answer.
Exercise
In this exercise, suppose there are four states of nature, S = {s11, s12, s21, s22}. The information that Player I receives is the first coordinate of the state of nature chosen, while the information that Player II receives is the second coordinate. The conditional probabilities of the players, given their respective informations, are given by the following table (the conditional probability of Player I appears in the left column, while the conditional probability of Player II appears in the top row of the table):
The table is to be read as stating, e.g., that if Player I receives information indicating that the state of nature is contained in {s11, s12}, he believes with probability 1 that the state of nature is s11.
(a) Construct a belief space in which the described situation is represented by a state of the world and indicate that state. Suppose that the state of nature is s12, and that ω is the corresponding state of the world. Answer the following questions:
(b) What are the minimal belief subspaces Y˜I(ω) and Y˜II(ω) of the players?
(c) Is Y˜I(ω) = Y˜II(ω)?
(d) Is there a common prior p over S such that the players agree that the state of the world has been chosen according to p?
(e) Is the state of the world ω ascribed positive probability by p?