This exercise generalizes Exercise 9.25. Let (N, Y, F_I, F_II, s, P) be an Aumann model of incomplete information with beliefs in which N = {I, II} and let A ⊆ Y be an event. Consider the following process:

• Player I informs Player II of the conditional probability P(A | FI(ω)).
• Player II informs Player I of the conditional probability that he ascribes to event

A given the partition element F_II(ω) and Player I's announcement.

• Player I informs Player II of the conditional probability that he ascribes to event

A given the partition element F_I(ω) and all the announcements so far.

• Repeat indefinitely.

Answer the following questions:

(a) Prove that the sequence of conditional probabilities that Player I announces converges; that the sequence of conditional probabilities that Player II announces also converges; and that both sequences converge to the same limit.

(b) Prove that after at most 2|Y | announcements the sequence of announcements made by the players becomes constant.

Exercise 9.25

Consider an Aumann model of incomplete information with beliefs in which

N = {I, II},

Y = {1, 2, 3, 4, 5, 6, 7, 8, 9},

F_I = {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}},

F_II = {{1, 2, 3, 4}, {5, 6, 7, 8}, {9}}

P(ω) = 1/9, ∀_ω ∈ Y.

Let A = {1, 5, 9}, and suppose that the true state of the world is ω∗ = 9. Answer the following questions:

(a) What is the probability that Player I (given his information) describes to the event A?

(b) What is the probability that Player II ascribes to the event A?

(c) Suppose that Player I announces the probability you calculated in item (a) above. How will that affect the probability that Player II now ascribes to the event A?

(d) Suppose that Player II announces the probability you calculated in item (c). How will that affect the probability that Player I ascribes to the event A, after hearing Player II's announcement?

(e) Repeat the previous two questions, with each player updating his conditional probability following the announcement of the other player. What is the sequence of conditional probabilities the players calculate? Does the sequence converge, or oscillate periodically (or neither)?

(f) Repeat the above, with ω∗ = 8.

(g) Repeat the above, with ω∗ = 6.

(h) Repeat the above, with ω∗ = 4.

(i) Repeat the above, with ω∗ = 1.