Secretary markets with local information

  • Ning Chen
  • Martin Hoefer
  • Marvin Künnemann
  • Chengyu Lin
  • Peihan Miao


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


  1. 1.
    Alaei, S., Hajiaghayi, M.T., Liaghat, V.: Online prophet-inequality matching with applications to ad allocation. In: Proceedings of 13th Conference Electronic Commerce (EC), pp. 18–35 (2012)Google Scholar
  2. 2.
    Babaioff, M., Dinitz, M., Gupta, A., Immorlica, N., Talwar, K.: Secretary problems: weights and discounts. In: Proceedings of 20th Symposium Discrete Algorithms (SODA), pp. 1245–1254 (2009)Google Scholar
  3. 3.
    Babaioff, M., Immorlica, Nicole, K., David, Kleinberg, R.: Online auctions and generalized secretary problems. SIGecom Exchanges, 7(2) (2008)Google Scholar
  4. 4.
    Babaioff, M., Immorlica, N., Kleinberg, R.: Matroids, secretary problems, and online mechanisms. In: Proceedings 18th Symposium Discrete Algorithms (SODA), pp. 434–443 (2007)Google Scholar
  5. 5.
    Babichenko, Y., Emek, Y., Feldman, M., Patt-Shamir, B., Peretz, R., Smorodinsky, R.: Stable secretaries. In: Proceedings of 18th Conference Economics and Computation (EC), pp. 243–244 (2017)Google Scholar
  6. 6.
    Bateni, M.H., Hajiaghayi, M.T., Zadimoghaddam, M.: Submodular secretary problem and extensions. ACM Trans. Algorithms 9(4), 32 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Buchbinder, N., Jain, K., Singh, M.: Secretary problems via linear programming. In: Proceedingsof 14th International Conference Integer Programming and Combinatorial Optimization (IPCO), pp. 163–176 (2010)Google Scholar
  8. 8.
    Chen, N., Hoefer, M., Künnemann, M., Lin, Chengyu, M., Peihan: Secretary markets with local information. In: Proceedings of 42nd Intl. Coll. Automata, Languages and Programming (ICALP), vol. 2, pp. 552–563 (2015)Google Scholar
  9. 9.
    Cownden, D., Steinsaltz, D.: Effects of competition in a secretary problem. Oper. Res. 62(1), 104–113 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Devanur, N., Jain, K., Sivan, B., Wilkens, C.: Near optimal online algorithms and fast approximation algorithms for resource allocation problems. In: Proceedings of 12th Conference Electronic Commerce (EC), pp. 29–38 (2011)Google Scholar
  11. 11.
    Dimitrov, N., Plaxton, G.: Competitive weighted matching in transversal matroids. Algorithmica 62(1–2), 333–348 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dinitz, M., Kortsarz, G.: Matroid secretary for regular and decomposable matroids. SIAM J. Comput. 43(5), 1807–1830 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dütting, P., Feldman, M., Kesselheim, T., Lucier, B.: Prophet inequalities made easy: stochastic optimization by pricing non-stochastic inputs. In: Proceedings of 27th Symposium Foundations of Computer Science (FOCS), pp. 540–551 (2017)Google Scholar
  14. 14.
    Dütting, P., Kleinberg, R.: Polymatroid prophet inequalities. In: Proceedings of 23rd European Symposium Algorithms (ESA), pp. 437–449 (2015)Google Scholar
  15. 15.
    Dynkin, E.: The optimum choice of the instant for stopping a Markov process. In: Sov. Math. Dokl, vol. 4, pp. 627–629 (1963)Google Scholar
  16. 16.
    Feldman, M., Svensson, O., Zenklusen, R.: A simple O(log log(rank))-competitive algorithm for the matroid secretary problem. In: Proceedings of 26th Symposium Discrete Algorithms (SODA), pp. 1189–1201 (2015)Google Scholar
  17. 17.
    Feldman, M., Tennenholtz, M.: Interviewing secretaries in parallel. In: Proceedings of 13th Conference Electronic Commerce (EC), pp. 550–567 (2012)Google Scholar
  18. 18.
    Feldman, M., Zenklusen, R.: The submodular secretary problem goes linear. In: Proceedings of 56th Symposium Foundations of Computer Science (FOCS), pp. 486–505 (2015)Google Scholar
  19. 19.
    Ferguson, T.: Who solved the secretary problem? Stat. Sci. 4(3), 282–289 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Göbel, O., Hoefer, M., Kesselheim, T., Schleiden, T., Vöcking, B.: Online independent set beyond the worst-case: secretaries, prophets and periods. In: Proceedings of 41st Intl. Coll. Automata, Languages and Programming (ICALP), vol. 2, pp. 508–519 (2014)Google Scholar
  21. 21.
    Gupta, A., Roth, A., Schoenebeck, G., Talwar, K.: Constrained non-monotone submodular maximization: Offline and secretary algorithms. In: Proceedings of 6th Workshop Internet & Network Economics (WINE), pp. 246–257 (2010)Google Scholar
  22. 22.
    Hoefer, M., Kodric, B.: Combinatorial secretary problems with ordinal information. In: Proceedings of 44th Intl. Coll. Automata, Languages and Programming (ICALP), pp. 133:1–133:14 (2017)Google Scholar
  23. 23.
    Im, S., Wang, Y.: Secretary problems: laminar matroid and interval scheduling. In: Proceedings of 22nd Symposium Discrete Algorithms (SODA), pp. 1265–1274 (2011)Google Scholar
  24. 24.
    Immorlica, N., Kalai, A., Lucier, B., Moitra, A., Postlewaite, A., Tennenholtz, M.: Dueling algorithms. In: Proceedings of 43rd Symposium Theory of Computing (STOC), pp. 215–224 (2011)Google Scholar
  25. 25.
    Immorlica, N., Kleinberg, R., Mahdian, M.: Secretary problems with competing employers. In: Proceedings 2nd Workshop Internet & Network Economics (WINE), pp. 389–400 (2006)Google Scholar
  26. 26.
    Jaillet, P., Soto, J., Zenklusen, R.: Advances on matroid secretary problems: free order model and laminar case. In: Proceedings of 16th International Conference Integer Programming and Combinatorial Optimization (IPCO), pp. 254–265 (2013)Google Scholar
  27. 27.
    Karlin, A., Lei, E.: On a competitive secretary problem. In: Proceedings of 29th Conference Artificial Intelligence (AAAI), pp. 944–950 (2015)Google Scholar
  28. 28.
    Kesselheim, T., Radke, K., Andreas, V.B.: An optimal online algorithm for weighted bipartite matching and extensions to combinatorial auctions. In: Proceedings of 21st European Symposium Algorithms (ESA), pp. 589–600 (2013)Google Scholar
  29. 29.
    Kesselheim, T., Radke, K., Andreas, V.B.: Primal beats dual on online packing LPs in the random-order model. In: Proceedings of 46th Symposium Theory of Computing (STOC), pp. 303–312 (2014)Google Scholar
  30. 30.
    Kleinberg, R.: A multiple-choice secretary algorithm with applications to online auctions. In: Proceedings of 16th Symposium Discrete Algorithms (SODA), pp. 630–631 (2005)Google Scholar
  31. 31.
    Kleinberg, R., Weinberg, M.: Matroid prophet inequalities. In: Proceedings of 44th Symposium Theory of Computing (STOC), pp. 123–136 (2012)Google Scholar
  32. 32.
    Korula, N.M.: Algorithms for secretary problems on graphs and hypergraphs. In: Proceedings of 36th International Coll. Automata, Languages and Programming (ICALP), pp. 508–520 (2009)Google Scholar
  33. 33.
    Krengel, U., Sucheston, L.: Semiamarts and finite values. Bull. Am. Math. Soc 83, 745–747 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Krengel, U., Sucheston, L.: On semiamarts, amarts and processes with finite value. Adv. Prob. 4, 197–266 (1978)MathSciNetGoogle Scholar
  35. 35.
    Lachish, O.: O(log log rank) competitive ratio for the matroid secretary problem. In: Proceedings of 55th Symposium Foundations of Computer Science (FOCS), pp. 326–335 (2014)Google Scholar
  36. 36.
    Lindley, D.: Dynamic programming and decision theory. Appl. Stat. 10, 39–51 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Molinaro, M., Ravi, R.: Geometry of online packing linear programs. Math. Oper. Res., 39(1), 46–59 (2014)Google Scholar
  38. 38.
    Soto, J.: Matroid secretary problem in the random-assignment model. SIAM J. Comput. 42(1), 178–211 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Nanyang Technological UniversitySingaporeSingapore
  2. 2.Goethe UniversityFrankfurt am MainGermany
  3. 3.Max Planck Institute for InformaticsSaarbrückenGermany
  4. 4.Chinese University of Hong KongHong KongChina
  5. 5.University of CaliforniaBerkeleyUSA

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