Nash equilibrium

Two firms compete by choosing a price between 0 and 1 and a message for their product. T1 firms produce an identical product, but the messages chosen by the firms influences the ability of consumers to compare the products. There are two possible messages: a and b. There is a unkown mass of consumers(i.e. the size of the market is 1). If a consumer is able to compare the two products, she chooses the cheapest one (and buys each with probability 1/2 if they have the same price). Otherwise, she buys each with probability 1/2. If at least one of the firms chooses message,all consumers can compare the products. If both firms choose message b, a fraction a= l (i.e. all consumers)are unable to compare the products Firms have constant marginal costs of c=0 and no fixed costs
(a) Suppose the timing of the game is as follows. In stage l, Firms I and 2 simultaneously choose a message. In stage 2, after observing the messages, both firms simultaneously choose a price
Suppose both firms chose message a in stage 1. Explain the Nash equilibrium or equilibrium to the pricing game. What profits do the firms earn?
ii. Suppose both firms chose message b in stage 1. Explain the Nash equilibrium or equilibrium to the pricing game. What profits do the firms earn?
iii. Identify the pure messaging strategies of both players in all subgame perfect Nash equilibrium to the game
iv. Is there a subgame perfect Nash equilibrium to the game in which players use mixed
message strategies? Explain
(b)Suppose instead that both firms simultaneously choose both prices and messages in a single period. Identify any pure strategy Nash equilibria to the game, if an equilibrium exist
Explain your answer.

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