The secretary model is a popular framework for the analysis of online admission problems beyond the worst case. In many markets, however, decisions about admission have to be made in a decentralized fashion and under competition. We cope with this problem and design algorithms for secretary markets with limited information. In our basic model, there are
firms and each has a job to offer.
applicants arrive iteratively in random order. Upon arrival of an applicant, a value for each job is revealed. Each firm decides whether or not to offer its job to the current applicant without knowing the strategies, actions, or values of other firms. Applicants decide to accept their most preferred offer.
We consider the social welfare of the matching and design a decentralized randomized thresholding-based algorithm with ratio
that works in a very general sampling model. It can even be used by firms hiring several applicants based on a local matroid. In contrast, even in the basic model we show a lower bound of
$$\Omega (\log n/(\log \log n))$$
for all thresholding-based algorithms. Moreover, we provide secretary algorithms with constant competitive ratios, e.g., when values of applicants for different firms are stochastically independent. In this case, we can show a constant ratio even when each firm offers several different jobs, and even with respect to its individually optimal assignment. We also analyze several variants with stochastic correlation among applicant values.