George, John, and Thomas are standing first, second, and third in a line, respectively. Each one sees the persons standing in front of him. James announces: "I have three red hats and two white hats. I will place a hat on the head of each one of you."
After James places the hats, he asks Thomas (who can see the hats worn by John and George) if he knows the color of the hat on his own head. Thomas replies "no." He then asks John (who sees only George's hat) whether he knows the color of the hat on his own head, and he also replies "no." Finally, he asks George (who cannot see any of the hats) if he knows the color of the hat on his own head.
(a) Construct an Aumann model of incomplete information that contains 7 states of the world and describes this situation.
(b) What are the partitions of George, John, and Thomas after James's announcement and before he asked Thomas whether he knows the color of his hat?
(c) What are the partitions of George and John after Thomas's response and before John responded to James's question?
(d) What is George's partition after hearing John's response?
(e) What is George's answer to James's question? Does this answer depend on the the state of the world, that is, on the colors of the hats that the three wear?