[Solution]Nash Equilibrium

Question 1: Two philanthropists; Jeff and Bill; are thinking of donating some money to a school. Assume that they will simultaneously make their decisions regarding…

Question 1:
Two philanthropists; Jeff and Bill; are thinking of donating some money to a school. Assume that they will simultaneously make their decisions regarding how much to donate.
Let S1 and S2 denote that the amount of money donated by Jeff and Bill, respectively. Also, U1 and U2 denote the utilities received by Jeff and Bill, respectively.
 
 
U1 = (1/2 ) (S1 + S2) – S1
U2 = (1/3 ) (S1 + S2) – S2

Plot the best response functions of Jeff and Bill. Please plot S1 along the x – axis and S2 along the y – axis. Clearly label the graph. Please use different colors to plot the best response functions of Jeff and Bill.
Using the graph above, find out all the pure strategy Nash equilibria.
What is the total amount donated by Jeff and Bill?
Who donates more at a pure strategy Nash equilibrium? Justify your answer.

Question 2:
Please note that in part (2), the utilities are:
If S1 + S2 < 2 , then
U1 = (1/2 ) (S1 + S2) – S1
U2 = (1/3 ) (S1 + S2) – S2
If S1 + S2 ≫ 2 , then
U1 = (1/2 ) (S1 + S2 + 2) – S1
U2 = (1/3 ) (S1 + S2 + 2) – S2
Also, please use the scale 1 = 1 million throughout.
Suppose Emma, a highly successful alumna of the school, announces that she will donate $2 million if the total amount donated by Jeff and Bill is at least $2 million. Jeff and Bill learn about Emma’s commitment, and then simultaneously decide how much to donate.

On a separate graph, plot the best response functions of Jeff and Bill. Please plot S1 along the x – axis and S2 along the y – axis. Clearly label the graph. Please use different colors to plot the best response functions of Jeff and Bill.
Using the graph above, find out all the pure strategy Nash equilibria.
What is the total amount donated by Jeff and Bill?
Who donates more at a pure strategy Nash equilibrium? Justify your answer.

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