game theory

3. (40 points) The following questions relate to the Bayesian extensive- form game given on the next page. Nature moves rst, selecting player 1 to be type L with probability 1/4, type M with probability 1/4, and type H with probability 1/2 Player 1 observes his type, and either chooses D or N If player 1 chooses D, the game ends. If player 1 chooses N, player 2 observes player 1's action but not his type, in which case she must choose either B or NB. [The "story" behind this game is that player 1 is a rm, costlasly producing a. product that is either low, medium, or high quality, and charging a price of $3. The rm observes its quality and decida whether to disclose it to the comumer (D), or not disclose the quality (N) Then the consumer decides Whether to buy or not Low quality provides a use value of 1, medium quality provides a use value of 2, and high quality provides a use value of 4, so if the quality is disclosed, she would only buy a high quality product] (a) (20 points) Com/e11 this game into Bayesian nonnal form. (b) (20 points) Denote player 2's beliefs about the three types of player 1, conditional on her information set, by 41", 41M, and q\" r Find all of the perfect Bayesian equilibria of this game