Questions: A university is trying to raise funds to build a state of arts auditorium. Total cost to build the auditorium will be $5 million. The university has identified two possible donors for the project. Let S1 and S2 denote the donations of Donor 1 and 2, respectively. Question 1: Donor 1's utility from donating S1 is (1/2) (S1 + S2) - S1. Donor 2's utility from donating S2 is (1/4) (S1 + S2) - S2. The two donors simultaneously decide how much to donate (or equivalently, Donor 1 does not know Donor 2's donation amount, while deciding how much to donate, and vice-versa.) 1. lot the Best Response functions of the Donors. Plot S1 along the x-axis and plot S2 along the y axis. Properly label your diagram. 2. Find out the Nash equilibrium amount of donations for each donor. If there are multiple Nash equilibria, find out all of them. Justify your answer. Question 2: Now, suppose that the university identifies a third donor. Donor 3 will donate $3 million if S1 + S2 at least $2 million. If S1 + S2 is less than $2 million, then Donor 3 will not donate anything. Donor 3 makes her decision after knowing S1 + S2. Therefore, Donor 1's utility from donating S1 is (1/2) (S1 + S2) - S 1 if S1 + S2 < $2 million, and (1/2) (S1 + S2+3) S1 if S1 + S2 $2 million. Similarly, Donor 2's utility from donating S 2 is (1/4) (S1+ S2) - S2 if S1 + S2< $2 million, and (1/4) (S1 + S2 + 3) S2 if S1 + S2 $2 million. 1. Find out the Nash equilibrium amount of donations. If there are multiple Nash equilibria, find out all of them. Justify your answer. 2. How would your answer to part 1 (of Question 2) above change if Donor 3 announces upfront that she will donate $3 million for the project no matter what Donors 1 and 2 do. Donors 1 and 2 make their decisions after learning Donor 3's announcement. Justify your answer.
Questions: A university is trying to raise funds to build a state of arts auditorium. Total cost to build the auditorium will be $5 million. The university has identified two possible donors for the project. Let S1 and S2 denote the donations of Donor 1 and 2, respectively. Question 1: Donor 1's utility from donating S1 is (1/2) (S1 + S2) - S1. Donor 2's utility from donating S2 is (1/4) (S1 + S2) - S2. The two donors simultaneously decide how much to donate (or equivalently, Donor 1 does not know Donor 2's donation amount, while deciding how much to donate, and vice-versa.) 1. lot the Best Response functions of the Donors. Plot S1 along the x-axis and plot S2 along the y axis. Properly label your diagram. 2. Find out the Nash equilibrium amount of donations for each donor. If there are multiple Nash equilibria, find out all of them. Justify your answer. Question 2: Now, suppose that the university identifies a third donor. Donor 3 will donate $3 million if S1 + S2 at least $2 million. If S1 + S2 is less than $2 million, then Donor 3 will not donate anything. Donor 3 makes her decision after knowing S1 + S2. Therefore, Donor 1's utility from donating S1 is (1/2) (S1 + S2) - S 1 if S1 + S2 < $2 million, and (1/2) (S1 + S2+3) S1 if S1 + S2 $2 million. Similarly, Donor 2's utility from donating S 2 is (1/4) (S1+ S2) - S2 if S1 + S2< $2 million, and (1/4) (S1 + S2 + 3) S2 if S1 + S2 $2 million. 1. Find out the Nash equilibrium amount of donations. If there are multiple Nash equilibria, find out all of them. Justify your answer. 2. How would your answer to part 1 (of Question 2) above change if Donor 3 announces upfront that she will donate $3 million for the project no matter what Donors 1 and 2 do. Donors 1 and 2 make their decisions after learning Donor 3's announcement. Justify your answer.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Transcribed Image Text:Questions: A university is trying to raise funds to build a state of arts auditorium. Total cost to build the auditorium will be
$5 million. The university has identified two possible donors for the project. Let S1 and S2 denote the donations of Donor
1 and 2, respectively. Question 1: Donor 1's utility from donating S1 is (1/2) (S1 + S2) - S1. Donor 2's utility from
donating S2 is (1/4) (S1+S2) - S2. The two donors simultaneously decide how much to donate (or equivalently,
Donor 1 does not know Donor 2's donation amount, while deciding how much to donate, and vice-versa.) 1. lot the
Best Response functions of the Donors. Plot S1 along the x - axis and plot S2 along the y - axis. Properly label your
diagram. 2. Find out the Nash equilibrium amount of donations for each donor. If there are multiple Nash equilibria, find
out all of them. Justify your answer. Question 2: Now, suppose that the university identifies a third donor. Donor 3 will
donate $3 million if S1 + S2 at least $2 million. If S1 + S2 is less than $2 million, then Donor 3 will not donate anything.
Donor 3 makes her decision after knowing S1 + S2. Therefore, Donor 1's utility from donating S1 is (1/2) (S1 + S2) - S
1 if S1 + S2 < $2 million, and (1/2) (S1 + S2 + 3) - S1 if S1 + S2 $2 million. Similarly, Donor 2's utility from donating S
2 is (1/4) (S1+S2) S2 if S1 + S2 < $2 million, and (1/4) (S1+S2 + 3) S2 if S1 + S2 $2 million. 1. Find out the
Nash equilibrium amount of donations. If there are multiple Nash equilibria, find out all of them. Justify your answer. 2.
How would your answer to part 1 (of Question 2) above change if Donor 3 announces upfront that she will donate $3
million for the project no matter what Donors 1 and 2 do. Donors 1 and 2 make their decisions after learning Donor 3's
announcement. Justify your answer.
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