PBE means perfect bayesian equilibrium

Advertising. In this question we will work through an advertising model. Suppose a seller is selling a good of quality 6 6 {63,195}, each equally likely, where 6;,- denotes high quality and 9;, denotes low quality. Assume SH > 6;, > U. The seller knows the quality of the good but the prospective buyer does not. The seller can advertise: after observing 9, she chooses a level of advertising, a E [0, 00), which is observed by the buyer. The buyer interacts with the seller for two periods. The buyer rst observes the level of advertising chosen by the seller, (1. Then in the rst period, the buyer chooses whether or not to purchase the good at a xed price p, :51 E {0, 1}, where 1 corresponds to purchase. If the buyer purchases the good, she observes 19. In the second period, the buyer again chooses whether or not to purchase the good at the same xed price p, 372 e {0, 1}. Quality is drawn once at the beginning of the game (Le. it remains the same in both periods). The seller's payoff is her revenue minus the cost of advertising, 30(331 + 562) a. The buyer's payoff is her value for the good minus the price if she purchases, (6 p)(m1 + m2), and zero otherwise. Suppose p E (3:09}; + 61,), 193). Advertising is wasteful, in that it is a cost for the seller but it does not add value to the product for the buyer. (a) Show that the unique outcome is :31 = x2 = 0 when advertising is not possible. ( b) Consider the following strategy prole for the rm: type :93 chooses a level of advertising as\" > 0 and type 6;, chooses a = 0. Using Bayes rule, derive the buyer's beliefs after observing (i) advertising level (1* and (ii) zero advertising. (0) Suppose that after a level of advertising a. E {0, {1\"}, the buyer believes p(6H|a.) = 0 if a a\". Combined with the beliefs you derived in part (b), derive the buyer's optimal strategy given these beliefs. (d) Show that for any level of advertising as\" E [1), 2p], the seller's strategy outlined in part (b), and the beliefs and buyer's strategy outlined in part (c) constitute a separating PBE. (e) 1Why is there no PBE with {1* 2p