The problem is relatively well-known. IN its basic form there are two firms competing either on location or on some product characteristic. They can each choose a number in [0;1] and the consumers are uniformly distributed along [0;1]. Each consumer buys from the firm that is closer to his preferences. The Nash equilibrium is not hard to foresee: both firms will end up at 0.5 .
My question is what will happen if there are three firms? Intuitively it seems that there can be no Nash equilibrium.
Can anyone help me to show this algebraically, or at least logically?
EDIT: An important assumption needed (especially) for the 3 player case is the fact that if the consumers are indifferent between two or three firms they choose at random. THis means that if all 3 firms a re in the same location, they all get 1/3 of the profits.