n pre-med students are planning to take the MCAT. Each student must decide whether to take a preparatory course prior to taking the test. Let xi denote the choice of student i, where xi = 0 indicates that she will not take the course and xi = 1 indicates that she will take the course. A student cares about her ranking in terms of her MCAT score and whether or not she took the prep course. Let si denote student i‘s MCAT score and ri denote the ranking of student i among the n students who took the test. Specifically, ri equals 1 plus the number of students who scored strictly higher than student i. To clarify this specification, here are three examples: If (In other words, if nobody’s score is higher than that of student i, then her rank is 1.) If for (In other words, if student i has the lowest score, then her rank is n.) Finally, if t Now, assume that student i’s payoff equals Note that taking the prep course entails a cost to a student equal to c. Note also that a student adds to her payoff by an amount b if her rank increases by 1. Student i’s score is assumed to be determined from the formula is related to the innate ability of the student and is what she would score if she did not take the prep course. If she takes the prep course, she adds to her score by an amount z. Assume that
This means that student 1 is, in a sense, smarter than student 2, student 2 is smarter than student 3, . . . , student n – 2 is smarter than student n – 1 and students n – 1 and n are equally smart. The final assumption is
In this game, there are n students simultaneously deciding whether or not to take the MCAT preparatory course. Derive a Nash equilibrium.