# Game Theory: Hotelling game with 3 players

• Mar 24th 2011, 03:01 AM
Val92
Game Theory: Hotelling game with 3 players
Hi,
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?

Thank you :)

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.
• Mar 24th 2011, 10:54 AM
SpringFan25
i suggest this approach;

1) show that there is no pooling equlibrium (where they all choose the same number). This is easy enough - if all 3 firms are on the same spot they get 1/3; but if one firm moves one $\displaystyle \epsilon$ towards the centre, it will get at least 0.5. In the special case where all 3 firms are at the center already; any firm can increase profits by moving one $\displaystyle \epsilon$ away from the centre .

2) show that where at least one firm is not "on top of" the other two, it can increase its profits by moving slightly. By the same logic above, the firm on the "outside" of the group can increase profits by moving one $\displaystyle \epsilon$ towards the other two. The same logic would apply if all 3 firms were in different positions; either outside firm can increase profits by moving towards the middle one.

Since there is no equlibrium with all firms in the same spot, or with them separated, there is no equilibrium.
• Mar 26th 2011, 02:08 AM
Val92
Thanks, that's what did!