## Abstract

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 distributed fashion. We cope with this problem and design algorithms for secretary markets with limited information. In our basic model, there are *m* firms and each has a job to offer. *n* applicants arrive sequentially 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 actions or values of other firms. Applicants accept their best offer. We consider the social welfare of the matching and design a decentralized randomized thresholding-based algorithm with a competitive ratio of \(O(\log n)\) 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 a secretary algorithm with a constant competitive ratio when the values of applicants for different firms are stochastically independent. In this case, we show a constant ratio even when we compare to the firm’s individual optimal assignment. Moreover, the constant ratio continues to hold in the case when each firm offers several different jobs.

## Supplementary material

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