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Prove that in an Aumann model of incomplete information with a common prior P, if in a state of the world ? Player 1 knows that Player 2.
Depict the situation as a model with incomplete information, where the state of nature is the amounts in Mark and Luke’s envelopes.
The setup is just as in the previous exercise, but now Peter tells Mark and Luke that they can switch the envelopes.
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.
Two divisions of Napoleon’s army are camped on opposite hillsides, both overlooking the valley in which enemy forces have massed.
Every (Harsanyi) game with incomplete information can be described as an extensive-form game (with moves of chance and information sets).
This exercise illustrates that a college education serves as a form of signaling to potential employers, in addition to expanding the knowledge of students.
Nicolas would like to sell a company that he owns to Marc. The company’s true value is an integer between 10 and 12 (including 10 and 12).
Emily, Marc, and Thomas meet at a party to which novelists and poets have been invited. Every attendee at the party.
I love Juliet, and I know that Juliet loves me, but I do not know if Juliet knows that I love her.
Construct an Aumann model of incomplete information for each of the following situations.
Romeo composes a letter to Juliet, and gives it to Tybalt to deliver to Juliet. While on the way, Tybalt peeks at the letter’s contents.
Construct an Aumann model of incomplete information that contains 7 states of the world and describes this situation.
Every situation of incomplete information (N, Y, (Fi)i?N , s, ?*) over a set of states of nature S uniquely determines a knowledge hierarchy.
Which events are common knowledge in state of the world ? = 1? Which events are common knowledge in state of the world ? = 9?
Consider an Aumann model of incomplete information 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}}.
Prove that if in an Aumann model of incomplete information the events A and B are common knowledge among the players in state of the world ?.
Given an Aumann model of incomplete information, prove that event A is common knowledge in every state of the world
Give an example of an Aumann model of incomplete information with a set of players N = {1, 2, 3}
In state of the world ?, Andrew knows that Sally knows the state of nature. Does this imply that Andrew knows the state of nature in ??
Prove that (N,W, (Fi n W)i?N , P(· | W)) is also an Aumann model of incomplete information with beliefs, where for each player i ? N
This exercise generalizes Aumann’s Agreement Theorem to a set of players of arbitrary finite size.
Three individuals are seated in a room. Each one of them is wearing a hat, which may be either red or white.
What is the set of possible payoffs of the following game (the Battle of the Sexes game) if the players are permitted to decide.
Prove Theorem : if s * is a Nash equilibrium in behavior strategies, then the pair (s *, µs * ) is sequentially rational in every information set U satisfying P