The setup is again as in Exercise 9.31, but this time Peter chooses the integer k randomly according to a geometric distribution with parameter 1/2, that is, P(k = n) = 1/2n for each n ∈ N. How does this affect your answers to the questions in given Exercise?
Exercise :-
Peter has two envelopes. He puts 10k euros in one and 10k+1 euros in the other, where k is the outcome of the toss of a fair die. Peter gives one of the envelopes to Mark and one to Luke (neither Mark nor Luke knows the outcome of the toss). Mark and Luke both go to their respective rooms, open the envelopes they have received, and observe the amounts in them.
(a) Depict the situation as a model with incomplete information, where the state of nature is the amounts in Mark and Luke's envelopes.
(b) Mark finds 1,000 euros in his envelope, and Luke finds 10,000 euros in his envelope. What is the true state of the world in your model?
(c) According to the information Mark has, what is the expected amount of money in Luke's envelope?
(d) According to the information Luke has, what is the expected amount of money in Mark's envelope?
(e) Peter enters Mark's room and asks him whether he would like to switch envelopes with Luke. If the answer is positive, he goes to Luke's room and informs him: "Mark wants to switch envelopes with you. Would you like to switch envelopes with him?" If the answer is positive, he goes to Mark's room and tells him: "Luke wants to switch envelopes with you.
Would you like to switch envelopes with him?" This process repeats itself as long as the answer received by Peter from Mark and Luke is positive. Use your model of incomplete information to show that the answers of Mark and Luke will be positive at first, and then one of them will refuse to switch envelopes. Who will be the first to refuse?
Assume that each of the two would like to change envelopes if the conditional expectation of the amount of money in the other's envelope is higher than the amount in his envelope.