2. (4 points) There are two ice cream vendors, Ben and Jerry, on a crowded strip

Question

2. (4 points) There are two ice cream vendors, Ben and Jerry, on a crowded strip of each. There are five permissible locations for ice cream vendors to locate, cleverly named positions 1, 2, 3, 4, and 5. Position 1 is at the far north end of the beach, Position 5 at the far south end, and Position 3 in the middle, while Positions 2 and 4 are midway between their respective neighbors. Each location may accommodate more than one vendor. There are 1,000 beachgoers scattered uniformly across the beach (200 at each position) who will buy from the closest vendor. i. Fill in the payoff matrix below, with payouts reflecting the number of customers each vendor receives. (Example: If Ben is at Position 1 and Jerry is at Position 4, Ben will get 400 beachgoers at locations 1 and 2, and Jerry will get 600 of the beachgoers at Positions 3, 4, and 5. If customers are equidistant from Ben and Jerry's position, half will go to one and half to the other.) Jerr Position 1 Position 2 Position 3 Position 4 Position 5 Position Positiorn Position Ben Position 4 Position ii. Find the Nash Equilibrium or equilibria in this game. Where are Ben and Jerry likely to end up locating?

Explanation / Answer

i)

ii) If Ben is at P1, Jerry will be at P2 as it offers him the maximum pay-out.

If Ben is at P2, Jerry will be at P3 as it offers him the maximum pay-out.

If Ben is at P3 Jerry will be at P3 as it offers him the maximum pay-out.

If Ben is at P4, Jerry will be at P3 as it offers him the maximum pay-out.

If Ben is at P5, Jerry will be at P4 as it offers him the maximum pay-out.

If Jerry is at P1, Ben will be at P2 as it offers him the maximum pay-out.

If Jerry is at P2, Ben will be at P3 as it offers him the maximum pay-out.

If Jerry is at P3, Ben will be at P3 as it offers him the maximum pay-out.

If Jerry is at P4, Ben will be at P3 as it offers him the maximum pay-out.

If Jerry is at P5, Ben will be at P4 as it offers him the maximum pay-out.

This gives the Nash equilibrium (P3,P3). Both will end up at Position 3.

J E R R Y B P1 P2 P3 P4 P5 E P1 500,500 200,800 300,700 400,600 500,500 N P2 800, 200 500,500 400,600 500,500 600,400 P3 700,300 600,400 500,500 600,400 700,300 P4 600,400 500,500 400,600 500,500 800,200 P5 500,500 400,600 300,700 200,800 500,500

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