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The gloves game Two sellers go to the market. Each has a left-hand glove. At the same time, three other sellers come to the market.
Let N = {1, 2, 3, 4}. Write down a list of conditions guaranteeing that in the coalitional structure {N} Player 1 has no justified objection against Player 2.
Suppose there are two players, N = {I, II}, and four states of nature, S = {s11, s12, s21, s22}.
here exists a belief space in which the set of states of the world contains K states, and there are 2K - 1 different belief subspaces.
Prove that there exists a Bayesian equilibrium (in behavior strategies) in every game with incomplete information in which the set of players is finite.
Find an example of a belief space in which there exists a state of the world ? ? Y , an event A that is common belief among the players.
In this exercise we show that Theorem does not hold without the assumption that P(?) > 0 for every state of the world ? ? Y .
Describe in words the beliefs of the players about the state of nature and the beliefs of the other players about the state of nature in each of the states.
Recall that when the set of states of the world is a topological space, we require that a belief subspace be a closed set.
Can the beliefs of the players be derived from a common prior? If so, what is that common prior? If not, justify your answer.
Construct a belief space in which the described situation is represented by a state of the world and indicate that state.
Laocoon declares: “I ascribe probability 0 ¨ .6 to the Greeks attacking us from within a wooden horse.”
There are two players, N = {I, II}, and two states of nature S = {s1, s2}. A chance move chooses the state of nature.
John, Bob, and Ted meet at a party in which all the invitees are either novelists or poets (but no one is both a novelist and a poet).
Walter, Karl, and Ferdinand are on the road to Dallas. They arrive at a fork in the road; should they turn right or left?
Find an additional Bayesian equilibrium by identifying a strategy vector in which all the players of all types are indifferent between their two possible action
Are the beliefs of the players consistent? In other words, can they be derived from common prior beliefs? If you answer no, justify your answer.
Prove that the unique Bayesian equilibrium where Player I plays A when s = 1 is for both players to play A under all conditions.
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 ?.