In the Hotelling Location Model, if there are 2 vendors, what is the pure strategy nash? what about 3 vendors?

Question

If there is a stretch of beach 1 mile long with potential consumers uniformly distributed along the beach. what is the pure strategy nash equilibrium for 2 vendors? what about 3 and 4 vendors?

Answer 1

Hotelling Location model is a straight line model where vendors can move freely at anytime.  When there are two vendors they would pick the middle as well to reach nash equilibrium.   This is because the two vendors can control both sides.  

There will be no nash equilibrium if there is 3 or more vendor.  This is because who ever is in the between the two vendors will want to move out and go to the edge this is because they  have the least amount of customers.  This will continue to happen forever.  When the two vendors finally reach the ends the third vendor can move anywhere along the strip and will still be worse off at any point compared to the vendors at the edge.   Therefore there is no nash equilibrium if there are 3 more vendors in a hotelling Model.

Answer 2

The two vendors would move to the center of the beach because it would be the location where they would maximise their utility.  The nash equlibium is the center of the beach.