In the classic Secretary problem, a company has to select an employee among n applicants, that are interviewed in random order. By rejecting systemically about 36% of the first applicants, and then picking anyone better than these rejected candidates, the company ensures that the selected candidate is with at least 36% chance the best one. This seminal result has given rise to a fruitful line of works on variations of the model. The main question is to compare the situation where the company knows the levels of the applicants before the interviews, and the situation where it does not. This comparison is quantified by a prophet inequality.
After recalling the basics of prophet inequalities theory and its connections to posted price mechanisms, we will present new prophet inequalities in two settings: one where the company knows the level distributions of the applicant, and one where it does not. These results are based on several collaborations with J. Correa, P. Dütting, Felix Fischer and K. Schewior.