Consider a small town that has a population of dedicated pizza eaters but is able to accommodate…

Consider a small town that has a population of dedicated
pizza eaters but is able to accommodate only two pizza shops, Donna’s Deep Dish
and Pierce’s Pizza Pies. Each seller has to choose a price for its pizza, but
for simplicity, assume that only two prices are available: high and low. If a
high price is set, the sellers can achieve a profit margin of \$12 per pie; the
low price yields a profit margin of \$10 per pie. Each store has a loyal captive
customer base that will buy 3,000 pies per week, no matter what price is
charged by either store. There is also a floating demand of 4,000 pies per
week. The people who buy these pies are price conscious and will go to the
store with the lower price; if both stores charge the same price, this demand
will be split equally between them.

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(a) Draw the game table for the pizza-pricing game, using
each store’s profits per week (in thousands of dollars) as payoffs. Find the
Nash equilibrium of this game and explain why it is a prisoners’ dilemma.

(b) Now suppose that Donna’s Deep Dish has a much larger
loyal clientele that guarantees it the sale of 11,000 (rather than 3,000) pies
a week. Profit margins and the size of the floating demand remain the same.
Draw the payoff table for this new version of the game and find the Nash
equilibrium.

(c) How does the existence of the larger loyal clientele
for Donna’s Deep Dish help “solve” the pizza stores’ dilemma?