Three married couples in the state of Maryland—Bob and Carol, Ted and Alice, and Ross and Mike (remember, same-sex marriage is legal in the state of Maryland)—are thinking about renting a boat to go sailing on the Chesapeake Bay. The cost of a boat rental is $600. Each of the three couples puts some amount of money in an envelope. Thus, each player in this game is a couple. If the total amount collected is at least $600, then the boat is rented. If the amount collected is more than $600, then the money left over after renting the boat is spent on wine. If the total amount collected is less than $600, then they do not rent the boat, and the money is spent on a dinner. Assume the benefit to a couple from the boat trip is 400, the benefit from each dollar spent on wine is 50 cents, the benefit from each dollar spent on dinner is 40 cents, and the personal cost of the contribution to a couple equals the amount of contribution. For example, if the boat is rented, $50 of wine is purchased (so a total of 650 was contributed), and Bob and Carol contributed $100 then their payoff is 400 + 0.5 x 50 - 100 = 325. If $400 was collected (so the boat was not rented) and spent on dinner, and Ross and Mike contributed $200 then their payoff is 400 x 0.4 - 200 = -40. Let x denote the contribution of Bob and Carol, y the contribution of Ted and Alice, and z the contribution of Ross and Mike. a. Is it a Nash equilibrium if x = 0, y = 0, and z = 0?b. Consider a strategy profile such that 0 < x + y + z < 600. Derive valuesfor x, y, and z such that it is a Nash equilibrium.c. Consider a strategy profile such that x + y + z > 600. Derive valuesfor x, y, and z such that it is a Nash equilibrium.d. Find all Nash equilibria.
Three married couples in the state of Maryland—Bob and Carol, Ted and Alice, and Ross and Mike (remember, same-sex marriage is legal in the state of Maryland)—are thinking about renting a boat to go sailing on the Chesapeake Bay. The cost of a boat rental is $600. Each of the three couples puts some amount of money in an envelope. Thus, each player in this game is a couple. If the total amount collected is at least $600, then the boat is rented. If the amount collected is more than $600, then the money left over after renting the boat is spent on wine. If the total amount collected is less than $600, then they do not rent the boat, and the money is spent on a dinner. Assume the benefit to a couple from the boat trip is 400, the benefit from each dollar spent on wine is 50 cents, the benefit from each dollar spent on dinner is 40 cents, and the personal cost of the contribution to a couple equals the amount of contribution. For example, if the boat is rented, $50 of wine is purchased (so a total of 650 was contributed), and Bob and Carol contributed $100 then their payoff is 400 + 0.5 x 50 - 100 = 325. If $400 was collected (so the boat was not rented) and spent on dinner, and Ross and Mike contributed $200 then their payoff is 400 x 0.4 - 200 = -40. Let x denote the contribution of Bob and Carol, y the contribution of Ted and Alice, and z the contribution of Ross and Mike.
a. Is it a Nash equilibrium if x = 0, y = 0, and z = 0?
b. Consider a strategy profile such that 0 < x + y + z < 600. Derive values
for x, y, and z such that it is a Nash equilibrium.
c. Consider a strategy profile such that x + y + z > 600. Derive values
for x, y, and z such that it is a Nash equilibrium.
d. Find all Nash equilibria.
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