Incentivizing Hidden Types in Secretary Problem

In the academic year 2021, Longjian (then undergraduate student at Peking University) visited UCSD for an exchange program. He asked me if he could take the first year PhD microeconomics. Normally I don’t allow undergraduate students to take PhD classes, because almost all precedents failed to keep up with the coursework. However, looking at Longjian’s CV and academic transcript, I realized that he had taken many advanced math courses and excelled in all of them, so I took the chance and allowed him to take the course. He ended up in the top quarter of the PhD cohort, so his performance was impressive.

During the course, Longjian regularly came to my office hours and discussed his research interest. He explained to me the secretary problem, that early applicants have no incentive to show up for interviews because they get rejected for sure, and so on. He worked on an example with two or three agents in which the administrator needs to incentivize them to show up for interviews. I told him to use dynamic programming to solve the general case with \(N\) agents. He did, and he also proved the optimality of the full learning equilibrium. Later, I added a result on the asymptotic behavior of success probability, and we wrote this paper together.

Although it took us two years to get the paper published (we got rejected from Games and Economic Behavior, Operations Research, and Economic Theory), I am happy that we eventually published it in a decent journal.