A supervisor assigns an A and a B to two employees, Rose and Sara, based on their relative performance in a project. The score depends on both ability and effort on preparing for the project of the employees. Rose has higher ability than Sara. Each employee can choose either high (H) or low (L) effort. Low effort is costless and high effort is costly. For Rose, the cost of high effort is 2 and the values of A and B are 4 and 2 respectively. For Sara, the values of A and B are 6 and 4 respectively. Both employees simultaneously choose their own effort levels. Consider the following cases.
Case 1: The difference in ability between Rose and Sara is large and Sara can get the A only if she chooses high effort and Rose chooses low effort. Suppose that the cost of high effort for Sara is 1.
(a) Write down the game in strategic form.
(b) What is the equilibrium? Show your steps.
(c) Does the high ability employee always get the A?
Case 2: Just as Case 1, Sara can get the A only if she chooses high effort and Rose chooses low effort. Now the cost of high effort for Sara is higher at 4.
(d) Write down the game in strategic form.
(e) What is the equilibrium? Show your steps.
(f) The supervisor wishes the employees putting high effort on preparing the test. Will he be satisfied?
Case 3: Just as Case 2, the cost of high effort for Sara is 4. Now the difference in ability between Rose and Sara is small and Sara can get the A if she chooses high effort.
(g) Write down the game in strategic form.
(h) What is the equilibrium? Show your steps.
(i) Does the change of the difference in ability between the employees increase Sara's chance to get the A?
Case 4: Just as Case 3, the difference in ability between Rose and Sara is small and Sara can get the A if she chooses high effort. Now the cost of high effort for Sara is back to 1.
(j) Write down the game in strategic form.
(k) What is the equilibrium? Show your steps.
(l) Will the test reflect the ability of the employees?