Hotelling Model with Three Firms [20 marks]

Consider again Hotelling's linear model with exogenous prices and endogenous locations. As before, let the product space be the unit interval, [0,1]. Let there now beTHREEfirms, unimaginatively known as Firm 1, Firm 2, and Firm 3. All three firms are required to set the same price,p, for their product, and they all have zero production costs.

The three firms compete by simultaneously choosing 'location'. That is, each of the firmsichooses a locationfi[0,1]. The consumers observe the profile of locations,f= (f1, f2, f3), and then each consumer non-strategically purchases from the firm whose location closest to her. That is, if a consumer is located at pointx[0,1], she will purchase from firmiarg mini|fix|. Assume that if two or more firms minimise this distance then a consumer is equally to purchase from each.

(a) 20 marks Prove that there is no pure strategy Nash equilibrium for this game.