This exercise is similar to given Exercise, but instead of announcing the probability of a particular event given their private information, the players announce whether or not the expectation of a particular random variable is positive or not, given their private information. This is meant to model trade between two parties to an agreement, as follows. Suppose that Ralph (Player 2) owns an oil field. He expects the profit from the oil field to be negative, and therefore intends to sell it. Jack is of the opinion that the oil field can yield positive profits, and is therefore willing to purchase it (for the price of $0). Jack and Ralph arrive at different determinations regarding the oil field because they have different information. We will show that no trade can occur under these conditions, because of the following exchange between the parties:
- Jack: I am interested in purchasing the oil field; are you interested in selling?
- Ralph: Yes, I am interested in selling; are you interested in purchasing?
- Jack: Yes, I am interested in purchasing; are you still interested in selling?
- And so on, until one of the two parties announces that he has no interest in a deal.
The formal description of this process is as follows. Let (N, Y, F1, F2, s, P) be an Aumann model of incomplete information with beliefs where N = {I, II}, let f : Y → R be a function, and let ω ∈ Y be a state of the world. f (ω) represents the profit yielded by the oil field at the state of the world ω. At each stage, Jack will be interested in the deal only if the conditional expectation of f given his information is positive, and Ralph will be interested in the deal only if the conditional expectation of f given his information is negative. The process therefore looks like this:
- Player I states whether or not E[f | FI](ω) > 0 (implicitly doing so by expressing or not expressing interest in purchasing the oil field). If he says "no" (i.e., his expectation is less than or equal to 0), the process ends here.
- If the process gets to the second stage, Player II states whether his expectation of f , given the information he has received so far, is negative or not. The information he has includes FII(ω) and the affirmative interest of Player I in the first stage. If Player II now says "no" (i.e., his expectation is greater than or equal to 0), the process ends here.
- If the process has not yet ended, Player I states whether his expectation of f , given the information he has received so far, is positive or not. The information he has includes FI(ω) and the affirmative interest of Player II in the second stage. If Player I now says "no" (i.e., his expectation is less than or equal to 0), the process ends here
- And so on. The process ends the first time either Player I's expectation of f , given his information, is not positive, or Player II's expectation of f , given his information, is not negative.
Show that this process ends after a finite number of stages. In fact, show that the number of stages prior to the end of the process is at most max{2|FI| - 1, 2|FII| - 1}.
Exercise
Consider an Aumann model of incomplete information with beliefs in which
N = {I, II},
Y = {1, 2, 3, 4, 5, 6, 7, 8, 9},
FI = {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}},
FII = {{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 describes 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.