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For each e ? [0, 1] find Bayesian equilibria in threshold strategies, where a has uniform distribution over the interval [1/4 , 1/2].
In each of the two strategic-form games whose matrices appear below, find all the equilibria.
Describe this situation as a game with incomplete information, and find the set of Bayesian equilibria of this game.
Minerva ascribes probability 0.7 to Hercules being able to lift a massive rock, and she believes that Hercules believes that he can lift the rock.
Eric believes that it is common belief among him and Jack that the New York Mets won the baseball World Series in 1969.
Using two states of the world describe the following situation, specifying how each state differs from the other: “Roger ascribes probability.
Let ? be a belief space equivalent to an Aumann model of incomplete information and let Bi be player i’s belief operator in ?.
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