Competitive secretary problem

We consider a competitive variant of the secretary problem. We study a two-stage game in which, in the first stage, the employer uses a very simple but natural hiring rule: interview a fixed number of applicants that are all rejected and accept the first applicant that is better than any of the previously seen applicants and, in the second stage, a strategic applicant chooses the position in which to be interviewed. The goal of the employer is to maximise the probability of selecting the best applicant and the goal of the applicant is to maximise the probability of being hired. We study two different scenarios. In the first scenario, the strategic applicant is known to the employer and in the second scenario, the strategic applicant is unknown and chosen uniformly at random. For both scenarios, we compute the set of subgame perfect equilibria.