Game Theory

QUESTION 3 27 points Save Answer Consider the game of imperfect information where the newlyweds decide simultaneously whether to go to a football game or an opera, and the new wife (W) is not sure about the type of her husband (H). The husband is of type Insensitive or ty sensitive or type Sensitive. Nature moves first to determine the type of the husband. It is common knowledge that the probability of the type-Insensitive husband is 2/3. The husband can observe his type, but the wife cannot. The payoffs associated with either type are given by the following matrices. Husband type is Insensitive Husband type is Sensitive W Football Opera W Football Opera Football 10, 2 10, -2 Football 2, 2 -2, -2 Opera -4, -2 2,2 Opera -2, -2 4, 4 Determine whether the proposed strategy profile ((football, opera), opera) is a Bayes-Nash equilibrium. (A) Given the wife's strategy, consider whether the husband's strategy is optimal here. Given the wife's strategy, the insensitive husband chooses to go to Given the wife's strategy, the sensitive husband chooses to go to Thus, the proposed strategy, of the husband (is/is not) optimal._ (B) Given the husband's strategy, consider whether the wife's strategy is optimal here. If the wife chooses football, her payoff is If the wife chooses opera, her payoff is Thus, the proposed strategy of the wife (is/is not) optimal. () Therefore, ((football, opera), opera) (is/is not) a Bayes-Nash equilibrium. Excluding ((football, opera), opera), name a Bayes-Nash equilibrium strategy profile. QUESTION 4 21 points Save Answer A potential SEPARATING perfect Bayes-Nash equilibrium for the following signaling game is listed below the diagram, with the sender's strategy as proposed. Complete the relevant blanks to establish a potential separating perfect Bayes-Nash equilibrium, then confirm or disprove that this is or is not a separating perfect Bayes-Nash equilibrium by completing all blanks. Nature Type S Prob = .75 Prob .25 Receiver Receiver Sender WNO Receiver 2 Assume Sender's strategy to be. If her type is s, then choose action Y. If her type is t, then choose action X. Then Receiver's consistent beliefs must be: f the action is X, then the sender is of type s with probability If the action is Y, then the sender is of type t with probability Then Receiver's optimal strategy must be. f the action is X, then Rec chooses action If the action is Y, then Rec chooses action Now check whether the Senders' actions are optimal: When her type is s, action Y (is/is not) optimal. When her type is t, action X (is/is not) optimal. Thus, this (is/is not) a separating perfect Bayes-Nash equilibrium