Consider a Hotelling duopoly model between firm 1 and 2 located at the
extreme points. The firms have zero marginal cost, and the consumers are uniformly distributed on [0, 1] with transportation cost au . The firms have imperfect
information about consumer demands: they know whether a consumer belongs
to group A having x in [0, 1/4] or group B having x in [1/4, 1]. Assuming that
the firms are allowed to choose different prices for different groups (i.e., thirddegree price discrimination or group pricing), compute the equilibrium prices
and profits.