1. Secretary Markets with Local Information
Ning Chen Martin Hoefer Marvin Künnemann2,3
Chengyu Lin Peihan Miao5
1 Nanyang Technological University, Singapore
2 MPI für Informatik, Saarbrücken, Germany
3 Saarbrücken Graduate School of Computer Science, Germany
4 Chinese University of Hong Kong, Hong Kong
5 University of California, Berkeley, USA

2. Motivation: The Voice

3. Motivation: The Voice

4. Motivation: The Voice

5. Motivation: The Voice

6. Motivation: The Voice

7. Motivation: The Voice

8. Motivation: The Voice

9. Question:
What is the best strategy for a coach?

10. Classic Secretary Problem

11. Classic Secretary Problem

12. Classic Secretary Problem

13. Classic Secretary Problem

14. Classic Secretary Problem

15. Classic Secretary Problem
How to maximize the (expected) value of the hired secretary?

16. Classic Secretary Problem: Worst Case
Arbitrary value
Arbitrary order
No guaranteed competitive ratio for any algorithm!

17. Classic Secretary Problem: Uniform Random Order
Theorem [Dynkin 1963] There is an online algorithm that achieves 𝑒-competitive ratio. This is the best that one can possibly achieve.

18. Classic Secretary Problem: The Algorithm
threshold
1 𝑒 fraction

19. Classic Secretary Problem: The Algorithm

20. Classic Secretary Problem: The Algorithm

21. Classic Secretary Problem: The Algorithm

22. Outline Classic secretary problem Generalized secretary problem
Hardness results
Classic algorithm
Generalized secretary problem
First attempt
General Preferences
Our algorithm: 𝑂( log 𝑛 ) competitive ratio
Lower bound Ω log 𝑛 log log 𝑛 for threshold-based algorithms
Independent Preferences
Correlated Preferences

23. Generalized Secretary Problem

24. Generalized Secretary Problem
Value function 𝑣: Candidates×Companies→ ℝ +

25. Generalized Secretary Problem: Key Changes
Independent companies, competing with each other
No global information for each company
No centralized authority

26. Generalized Secretary Problem: Objectives
Algorithm for each company that maximizes
Social welfare
Competitive ratio w.r.t. optimal social welfare
Outcome for each individual company
Competitive ratio w.r.t. best outcome for individual

27. Generalized Secretary Problem: First Attempt
Traditional algorithm for every company
Reject first 𝑟 applicants
Set a threshold: max value so far
Propose to everyone that exceeds that threshold as long as the company is still available
Proposition 1: This algorithm achieves a competitive ratio Ω 𝑛 log 𝑛 of social welfare.

28. Generalized Secretary Problem
First Attempt: Bad Example

29. Generalized Secretary Problem
First Attempt: Bad Example

30. Generalized Secretary Problem: Our Algorithm
Avoid extensive competition/conflicts for a small amount of candidates
Randomized threshold strategy
Randomized sampling: rejects 𝑟 applicants where 𝑟 ~𝐵𝑖𝑛 𝑛, 1 𝑒
Let 𝑀 be the max value in sampling
Randomized threshold: 𝑇≔ 𝑀 2 𝑥 where 𝑥~𝑈𝑛𝑖𝑓(−1, 0, 1, 2, ⋯, log 𝑛 )
Theorem 1: 𝑂( log 𝑛 ) competitive ratio of social welfare

31. Generalized Secretary Problem: Our Algorithm
Randomized threshold strategy
Randomized sampling: rejects 𝑟 applicants where 𝑟 ~𝐵𝑖𝑛 𝑛, 1 𝑒
Let 𝑀 be the max value in sampling
Randomized threshold: 𝑇≔ 𝑀 2 𝑥 where 𝑥~𝑈𝑛𝑖𝑓(−1, 0, 1, 2, ⋯, log 𝑛 )

32. Generalized Secretary Problem: Our Algorithm
Randomized threshold strategy
Randomized sampling: rejects 𝑟 applicants where 𝑟 ~𝐵𝑖𝑛 𝑛, 1 𝑒
Let 𝑀 be the max value in sampling
Randomized threshold: 𝑇≔ 𝑀 2 𝑥 where 𝑥~𝑈𝑛𝑖𝑓(−1, 0, 1, 2, ⋯, log 𝑛 )

33. Generalized Secretary Problem: Our Algorithm
Randomized threshold strategy
Randomized sampling: rejects 𝑟 applicants where 𝑟 ~𝐵𝑖𝑛 𝑛, 1 𝑒
Let 𝑀 be the max value in sampling
Randomized threshold: 𝑇≔ 𝑀 2 𝑥 where 𝑥~𝑈𝑛𝑖𝑓(−1, 0, 1, 2, ⋯, log 𝑛 )

34. Generalized Secretary Problem: Our Algorithm
Randomized threshold strategy
Randomized sampling: rejects 𝑟 applicants where 𝑟 ~𝐵𝑖𝑛 𝑛, 1 𝑒
Let 𝑀 be the max value in sampling
Randomized threshold: 𝑇≔ 𝑀 2 𝑥 where 𝑥~𝑈𝑛𝑖𝑓(−1, 0, 1, 2, ⋯, log 𝑛 )

35. Generalized Secretary Problem: Lower Bound
Thresholding-based algorithms
Sampling phase
Set a threshold 𝑇
Acceptance phase: give an offer to anyone exceeding 𝑇
Theorem 2: Can’t get better competitive ratio than Ω log 𝑛 log log 𝑛
Every company hires one secretary
Each secretary has identical value to all firms
Centralized control might be necessary?

36. Outline Classic secretary problem Generalized secretary problem
Hardness results
Classic algorithm
Generalized secretary problem
First attempt
General Preferences
Our algorithm: 𝑂( log 𝑛 ) competitive ratio
Lower bound Ω log 𝑛 log log 𝑛 for threshold-based algorithms
Independent Preferences
Correlated Preferences

37. Independent Preferences
All the values are sampled i.i.d.
Theorem 3: Constant competitive ratio with the classic algorithm, both for social welfare and for individuals.

38. Correlated Preferences
Each candidate has a quality 𝑞
Values generated independently from a normal distribution with mean 𝑞
Large variance: single threshold
Small variance: 𝑚 thresholds
In between?

39. Summary Generalized secretary problem General Preferences
Our algorithm: 𝑂( log 𝑛 ) competitive ratio
Lower bound: Ω log 𝑛 log log 𝑛 for threshold-based algorithms
Independent Preferences
Constant competitive ratio
Correlated Preferences
Constant competitive ratio when variance is extremely large or extremely small

40. Thank you!