Advanced Game Theory and Applications problem

4. An instructor is too lazy to grade all of his students' exams. Instead, he uses the following method: after completing the exam, the instructor makes a copy of each exam, then allows students to take their own exam and answer key home to grade themselves. After grading one's own exam, the student emails the instructor her final score (which can range from 0 to 100). This reported score is then official, with one exception: the instructor will randomly audit one out of every five exams. If audited, as long as the student's reported score is the same or below the actual score, the student's reported score stands. However, if the student's reported score is higher than what it should be, the student's score will be adjusted to the correct score, and then penalized 2 additional points for every point overstated. Assume that there is no subjectivity in grading exams (answers are clearly right or wrong), and students are risk-neutral. a) Explain whether this time-saving strategy by the instructor results in students truthfully reporting their scores. Support your answer by writing an IC condition comparing truthful score reporting versus understating one's score, and another IC condition comparing truthful score reporting versus overstating one's score. b) Suppose the penalty is changed to 3 points for every point overstated. Explain whether this affects your answers from part a. c) Calculate the minimum penalty per overstated point that would satisfy the IC condition for reporting one's actual score truthfully. d) Suppose that a standard grading scale is used (90 to 100 = A, 80 - 89 = B, 70 - 79 = C, 60 - 69 = D, and 0 - 59 = F), and this exam determines a student's course grade. Explain whether this grading scale has any effect on your previous results. A descriptive answer would suffice (no additional calculations needed). - 2. to u.s