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 how much to donate.

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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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