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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
List all the consistent assessments, and all the sequentially rational assessments of the following game.
What is the corresponding belief of Player II at his information sets? Justify your answer.
Find all the sequential equilibria of the following game. Provide complete and step by step solution for the question and show calculations and use formulas.
The following example shows that the set of sequential equilibria is sensitive to the way in which a player makes decisions: it makes a difference.
In an extensive-form game with perfect recall, is every Nash equilibrium part of a sequential equilibrium?
Pre-trial settlement A contractor is being sued for damages by a municipality that hired him to construct a bridge, because the bridge has collapsed.
Describe this situation as an extensive-form game, where the root of the game tree is a chance move that determines whether Caesar is brave.
Henry seeks a loan to form a new company, and submits a request for a loan to Rockefeller.
Prove Theorem for every Nash equilibrium s* in a strategic-form game, the probability distribution ps* that s* induces on the set of action vectors S.
Prove that if a player in an extensive-form game has only one information set, then his set of mixed strategies equals his set of behavior strategies.
Suppose that there is a strategy vector s-i of the other players such that ?(x; si, s-i) > 0 for each leaf x in the game tree.
Let i be a player with perfect recall in an extensive-form game and let bi be a behavior strategy of player i.
In the following two-player zero-sum game, find the optimal behavior strategies of the two players. (Why must such strategies exist?)
Compute the value of the following game, in mixed strategies, and in behavior strategies, if these values exist.
Adding information to one of the players does not increase the maxmin or the minmax value of the other players.
Suppose that a symmetric two-player game, in which each player has two pure strategies and all payoffs are nonnegative.
Explain why the evolutionarily stable strategy is that at which the number of male leopards born equals the number of females born.
Prove that if the payoff matrix of a two-player zero-sum game is antisymmetric, then the value of the game in mixed strategies is 0.