*P*, and a student,

*S*. The final grade for the class is the result of a test consisting of two questions,

*Q*and

_{1}*Q*, such that

_{2}*G = Q*and 0 ≤

_{1}+ Q_{2}*G*≤ 100. Each question is objective and the student's performance on each is given by

*G*, such that

_{i}= E_{i}+ B_{i}*E*is the effort the student has invested in the question (a variable) and B

_{i}_{i}represents the question-specific baseline score the student will receive regardless of any effort invested (

*B*is thus a constant). In the absence of any studying, the student will thus receive a grade equal to

*B*.

_{1}+ B_{2}

The student is crafty but lazy. He knows his baseline levels of competence with certainty. The student has a fixed budget of time *T* in which to allocate *E* (total effort) and *R* (recreation). The student's utility is maximized by selecting an effort level *E = E _{1} + E_{2}* that minimizes E while guaranteeing that he receives a grade

*G*, the passing grade in the class (0<

^{pass}*G*<100). (In other words, the student's utility is increasing in R, decreasing in E, and increasing in G up to a point.) Consequently, if the student receives a maximum score on one question, he knows he must only invest enough effort into studying for the second one such that

^{pass}*G*.

^{pass}= (G_{1}+G_{2})/2

The professor is altruistic but bad with people; she wants to maximize the student's total learning but is both indifferent to the share of time the student invests in recreation (which she views as wasted anyhow) and does not know which question the student is better at answering. If the professor believes that learning is a nondecreasing function only of effort invested, and if the professor derives utility only from the amount her students learn in the class, how can she design an incentive to maximize the student's time spent studying?

The professor announces that the final exam will continue to consist of two questions, but that the grade will now be based on only one question to be chosen with probability *p*, 0 < p < 1. (The credibility of this threat may be enforced through several ways, including the use of a binding agreement with a colleague or a reputation for whimsical or arbitrary behavior.) In expectation, the student's grade is now *G = p*Q _{1} + (1-p)Q_{2}*.

Let us assume that under the original policy the student at baseline levels would have received a score of 90 on the first question and 60 on the second, for an average of 75. If the passing mark is 70, then the student need not study any further. But if there is a 50-50 shot that each question will be graded, then the expected grade remains 75 but the expected utility payoff is much different. He retains the benefits of his time spent in recreation (R) but he now has a 50 percent chance of suffering disutility from failing the class. Assuming the total utility from R is less than the utility from passing the course, he should instead invest a sufficient amount of studying in his second question.

This scheme has apparently been tried at least once, although the student seemed unappreciative of the professor's altruism. In real life, I suspect it would prove unworkable--it might well lead to too many false negatives. On the other hand, its logic seems no less arbitrary than the act of actually designing an exam in which the questions are unknown to the student until the moment he or she opens the blue book.