Question

Assume that one firm will always locate at the top of the circle, and call that 0°. Then we know, for example, that with three firms, one Nash equilibrium has the firms locating at 0°, 120°, and 240°; and that the “most extreme” Nash equilibrium has the firms locating at (for example) 0°, 90°, and 180°. Here, “most extreme” means “as far as possible from perfect rotational symmetry”; you could also interpret it as “having the largest possible gap between some two firms”. In this problem, each firm earns $1 for every customer (degree) closest to it.

1. Describe the Nash equilibrium positions for 5 firms. (Both the “symmetric” one, and the most extreme equilibrium you can find.)

Answer 1

we know that for symmetric one will be at 0, 90, 180 and 270 degrees. The same for extreme equilibrium will be 0, 45, 90, 135 degrees.

EXPLANATION;

First I will explain the symmetric case. The top position is marked as 0 degree. So this firm will get a sale from 0-45 degrees and also from 315-0 degrees. So its total sale would be $90 since these degrees are closer to it. Then for the firm at 90 degree its closest sale will be from 45-90 degrees and from 90-135 degrees. So again its sale would be $90. And so will be for all other firms. Now in order to check whether its a Nash equilibrium all you need to do is say firm at 0 degree relocates its firm at 10 degree others staying same. Then it would get a sale from 10-50 degrees that is a sale of $40. And also it would get sale from 320-10 degrees that is of 50 degrees or of $50. So total sale is again $90. So there is no unilateral incentive for the firm to deviate. So it is a Nash equilibrium.

Similarly the most extreme will also have a Nash equilibrium at 0, 45, 90, 135. Here again one can verify that deviating unilaterally from these positions would not make them strictly better off. So this is a Nash equilibrium.