Secretary Markets with Local Information Ning Chen1 Martin Hoefer2 Marvin Künnemann2,3 Chengyu Lin4 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

Motivation: The Voice

Motivation: The Voice

Motivation: The Voice

Motivation: The Voice

Motivation: The Voice

Motivation: The Voice

Motivation: The Voice

Question: What is the best strategy for a coach?

Classic Secretary Problem

Classic Secretary Problem

Classic Secretary Problem

Classic Secretary Problem

Classic Secretary Problem

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

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

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.

Classic Secretary Problem: The Algorithm threshold 1 𝑒 fraction

Classic Secretary Problem: The Algorithm

Classic Secretary Problem: The Algorithm

Classic Secretary Problem: The Algorithm

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

Generalized Secretary Problem

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

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

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

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.

Generalized Secretary Problem First Attempt: Bad Example

Generalized Secretary Problem First Attempt: Bad Example

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

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 )

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 )

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 )

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 )

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?

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

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.

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?

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

Thank you!