Consider a game between two friends, Amy and Brenda. Amy wants Brenda to give her a ride to the mall. Brenda has no interest in going to the mall unless her favorite shoes are on sale at the large department store there. Amy likes these shoes as well, but she wants to go to the mall even if the shoes are not on sale. Only Amy subscribes to the newspaper, which carries a daily advertisement of the department store. The advertisement lists all items that are on sale, so Amy learns whether or not the shoes are on sale by showing the newspaper to Brenda. But this is costly for Amy, because she will have to take the newspaper away from her sister, who will yell at her later for doing so. In this game, nature first decides whether or not the shoes are on sale, and this information is made known to Amy (Amy observes whether nature chose S or N). Nature chooses S with probability p and N with probability 1-p. Then Amy decides whether or not to take the newspaper to Brenda (T or D). If she takes the newspaper to Brenda, then it reveals to Brenda whether the shoes are on sale. In any case, Brenda must then decide whether to take Amy to the mall (Y) or to forget it (F). If the shoes are on sale, then going to the mall is worth 1 unit of utility to Brenda and 3 to Amy. If the shoes are not on sale, then traveling to the mall is worth 1 to Amy and -1 to Brenda. Both players obtain 0 utility when they do not go to the mall. Amy’s personal cost of taking the newspaper to Brenda is 2 units of utility, which is subtracted from her other utility amounts. a) Draw the extensive form of this game. b) Does this game have a separating perfect Bayesian equilibrium? If so, fully describe it. c) Does this game have a pooling perfect Bayesian equilibrium? If so, fully describeit.
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Consider a game between two friends, Amy and Brenda. Amy wants Brenda to give her a ride to the mall. Brenda has no interest in going to the mall unless her favorite shoes are on sale at the large department store there. Amy likes these shoes as well, but she wants to go to the mall even if the shoes are not on sale. Only Amy subscribes to the newspaper, which carries a daily advertisement of the department store. The advertisement lists all items that are on sale, so Amy learns whether or not the shoes are on sale by showing the newspaper to Brenda. But this is costly for Amy, because she will have to take the newspaper away from her sister, who will yell at her later for doing so. In this game, nature first decides whether or not the shoes are on sale, and this information is made known to Amy (Amy observes whether nature chose S or N). Nature chooses S with probability p and N with probability 1-p. Then Amy decides whether or not to take the newspaper to Brenda (T or D). If she takes the newspaper to Brenda, then it reveals to Brenda whether the shoes are on sale. In any case, Brenda must then decide whether to take Amy to the mall (Y) or to forget it (F). If the shoes are on sale, then going to the mall is worth 1 unit of utility to Brenda and 3 to Amy. If the shoes are not on sale, then traveling to the mall is worth 1 to Amy and -1 to Brenda. Both players obtain 0 utility when they do not go to the mall. Amy’s personal cost of taking the newspaper to Brenda is 2 units of utility, which is subtracted from her other utility amounts.
a) Draw the extensive form of this game.
b) Does this game have a separating perfect Bayesian equilibrium? If so, fully describe it.
c) Does this game have a pooling perfect Bayesian equilibrium? If so, fully describeit.
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