Existence advertising, price competition, and asymmetric market structure

We examine a two stage duopoly game in which firms advertise their existence to consumers in stage 1 and compete in prices in stage 2. Whenever the advertising technology generates positive overlap in customer bases the equilibrium for the stage 1 game is asymmetric in that one firm chooses to remain small in comparison to its competitor. For a specific random advertising technology we show that one firm will always be half as large as the other. No equilibrium in pure price strategies exists in the stage 2 game and as long as there is some overlap in customer bases the mixed strategy equilibrium is far from the Bertrand equilibrium