Setting an exam as an information design problem

We take a teacher's exam-setting task as an information design problem. Specifically, the teacher chooses a conditional distribution of grades given students' types. After observing their exam results, each student updates her belief regarding her type via Bayes' rule and chooses an action. Students' reactions to the same exam result could be different, depending on their heterogeneous prior beliefs. The teacher's objective is to persuade students to take a certain action (e.g., applying to universities), which some may not choose without an exam. The teacher adopts different grade distributions, depending on the teacher's and the students' heterogeneous prior beliefs.