(Price Competition with Brand Loyalty) There are two firms, Firm A and Firm B, which produce cell phones at no cost. There is a continuity of consumers in the segment line [0, 3], where each point is...

(Price Competition with Brand Loyalty) There are two firms, Firm A and Firm B, which produce cell phones at no cost. There is a continuity of consumers in the segment line [0, 3], where each point is a consumer. Each consumer will buy (at most) a phone if the price is less than r, where r> 0 is a fixed real number. Consumers in the interval [0, 1] are loyal to Firm A and will buy Firm A's phone if the price pA is not more than r. They will not buy anything if Firm A asks for a price pA above r. Consumers in the interval [2, 3] are loyal to Firm B and will buy Firm B's phone if the price pB is not more than r. They will not buy anything if Firm B asks for a price pB above r. The rest of the consumers (in the interval [1, 2]) are not loyal to any firm and will buy the phone only when the minimum price in the market is less than or equal to r and of the firm with the lowest price. If the two firms have the same price, half the unloyal consumers will buy from A and the other half of B.

i need to answer : Show that there is no equilibrium in pure strategies and Find the symmetric equilibrium in mixed strategies.