George Washington is designing a fundraiser to drain the malaria-infested swampland that will become Washington, DC....

 

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George Washington is designing a fundraiser to drain the malaria-infested swampland that will become Washington, DC. He isn't a player in this game for now but wait until part (b). The players, and potential contributors, are Abigail and John Adams. Abigail chooses a gift x € [0,10], and John chooses his own donation y € [0,10]. As they will be DC residents at some point, and they are also somewhat altruistic patriots, each of them gets a benefit B(x, y) = (x + y) from the total improvements to the city. However, each of them bears a private cost equal to the amount donated. So for now, the payoffs are: John: u₂(x, y) = B(x,y) - y Abigail: u₁(x, y) = B(x, y) - x a. Determine the Nash equilibrium of the game, and explain how much Abigail and John give. Also, suppose we define total welfare as W = U₁+U₂. Work out what the donation levels that maximize welfare would be. b. George decides to spice up the fundraiser by combining it with a lottery. (You could also think of this as a raffle.) There will be a L dollar prize (to be funded out of the donations). Strategies are still x, y € [0,10], but each dollar donated buys the donor one ticket in the lottery. So Abigail wins the prize with probability x/(x+y), and John wins with probability y/(x + y). After funding the lottery, the total improvements to DC are now x + y - L. So the new payoffs are: x Abigail: u₁(x, y) = (x + y −L)-x+ L, John: u₂(x, y) = (x + y − L) − y + x₂ L x+y x+y Treat L as a fixed parameter. Solve for the new Nash equilibrium. Discuss the donation levels and compare to what you got in part (a) and to the donations that would maximize welfare. Do your best to provide some economic intuition for the difference. Finally, if George wants to maximize welfare, how large should the prize L be? George Washington is designing a fundraiser to drain the malaria-infested swampland that will become Washington, DC. He isn't a player in this game for now but wait until part (b). The players, and potential contributors, are Abigail and John Adams. Abigail chooses a gift x € [0,10], and John chooses his own donation y € [0,10]. As they will be DC residents at some point, and they are also somewhat altruistic patriots, each of them gets a benefit B(x, y) = (x + y) from the total improvements to the city. However, each of them bears a private cost equal to the amount donated. So for now, the payoffs are: John: u₂(x, y) = B(x,y) - y Abigail: u₁(x, y) = B(x, y) - x a. Determine the Nash equilibrium of the game, and explain how much Abigail and John give. Also, suppose we define total welfare as W = U₁+U₂. Work out what the donation levels that maximize welfare would be. b. George decides to spice up the fundraiser by combining it with a lottery. (You could also think of this as a raffle.) There will be a L dollar prize (to be funded out of the donations). Strategies are still x, y € [0,10], but each dollar donated buys the donor one ticket in the lottery. So Abigail wins the prize with probability x/(x+y), and John wins with probability y/(x + y). After funding the lottery, the total improvements to DC are now x + y - L. So the new payoffs are: x Abigail: u₁(x, y) = (x + y −L)-x+ L, John: u₂(x, y) = (x + y − L) − y + x₂ L x+y x+y Treat L as a fixed parameter. Solve for the new Nash equilibrium. Discuss the donation levels and compare to what you got in part (a) and to the donations that would maximize welfare. Do your best to provide some economic intuition for the difference. Finally, if George wants to maximize welfare, how large should the prize L be?

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