Five people get together for five nights at the beach. Dinner duty is distributed so that each person is assigned a night to buy and cook dinner and provide the wine. At the end of their stay, the receipts from the five nights will be summed up and each person will pay 1/5th of the total. These five people are all gourmands and each is trying to impress the other. Suppose that five different amounts can be spent on preparing a meal: 50, 80, 150, 250, and 400. As reported in the table below associated with each amount of expense is the benefit received by the person who made the meal and the benefit received by all of those who ate it (which includes the person who made it). For example, if 250 was spent on the meal, then the person who made it receives a total benefit of 280 (230 from cooking it and 50 from eating it) and each of the other four people
receive a benefit of 50 from eating it. A player’s payoff equals the benefit
from making his or her meal plus the benefit from eating the five meals
minus his or her share of the expense. For example, suppose person 1
makes a meal that costs 50, person 2 makes a meal that costs 400, and
the other three people each makes a meal that costs 150. Person 1’s payoff
is 120, which equals 100 (which comes from the meal she produced)
plus 200 (which comes from consuming a meal valued at 20, a meal
valued at 60, and three meals each valued at 40) minus 180 (which
equals total expense of 900 divided by five). Assume the five friends
simultaneously decide how much each will spend to prepare the meal
for the night he or she has been assigned.
a. Find all Nash equilibria.
b. If someone was cooking only for herself and thus only attached value
to eating the meal, how much would she spend? (Note: In answering
this question, divide the cost of a meal in the table below by 5 since
that person is now only cooking for one.)
c. If all five friends could collectively commit themselves to spending
the same amount per meal, what level of expense would they
jointly choose?
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