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 (S)

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 (N). 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. Amy can prove whether or not the shoes

are on sale by showing the newspaper to Benda. 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. That is, 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 travelling 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 game tree of this game (10 Marks)

Hint: Check Gibbons page 183 for guidance

b) Does this game have a separating perfect Bayesian Nash equilibrium? If so, fully describe

it. (10 Marks)

c) Does this game have a pooling perfect Bayesian Nash equilibrium? If so, fully describe

it. (10 Marks)

Hint: A separating equilibrium means that Amy takes different strategies in S and N, while she chooses the

same strategy in a pooling equilibrium. Your answer in (c) might depend on the value of p.