Question

Two soda vendors (“pop” vendors, if you like) need to set up on the beach in the morning fog.

There are 7 locations along the beach, equally spaced apart, labeled,

A, B, C, D, E, F, and G from north to south. The two vendors simultaneously choose their locations (they can’t see each other until the fog lifts), and then operate from those locations for the entire day. A vendor earns $1 profit for each customer he or she serves. Each vendor wants to maximize profit. Each location along the beach has 10 customers. Each customer will patronize the closest vendor. In case the two vendors are equally close to a given location, each vendor will attract half of the customers in that location.

a) What is Vendor 1’s best response function?

b) Find all Nash equilibria.

Answer 1

ANSWER:

• Nash equilibrium exist at points such that the revenues/ customers are divided equally among the sellers. Hence, either both vendors can sell.
• point D (mid-point of the beach), or they can serve. the extreme ends of the beach. The equation for the Nash equilibrium is: If firm 1 places him . position x, firm 2 would place himself
• position 1-x Hence, the Nash equilibrium shall be points: C a. E, B a. F; A and G.

Best response function Ts: Position x 1 for firm 1 such that revenue is maximized.

• The set of best responses for Firm 1 firm Ts the s. of positions that causes the midpoint (x 1 +x 2)/2 of the line segment. from x 2 to x 1 to be less than where m is the actual midpoint i.e. at point D