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# The Hiring Problem

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Twist on classic hiring secretary problem. Hire above a threshold, hire above the

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### The Hiring Problem

1. The Hiring Problem: Going Beyond Secretaries Sergei Vassilvitskii (Yahoo!) Andrei Broder (Yahoo!) Adam Kirsch (Harvard) Ravi Kumar (Yahoo!)Michael Mitzenmacher (Harvard) Eli Upfal (Brown)
2. The Secretary ProblemInterview n candidates for a position one at a time. After eachinterview decide if the candidate is the best.Goal: maximize the probability of choosing the best candidate.
3. The Secretary ProblemInterview n candidates for a position one at a time. After eachinterview decide if the candidate is the best.Goal: maximize the probability of choosing the best candidate.
4. The Secretary ProblemInterview n candidates for a position one at a time. After eachinterview decide if the candidate is the best.Goal: maximize the probability of choosing the best candidate.
5. The Secretary ProblemInterview n candidates for a position one at a time. After eachinterview decide if the candidate is the best.Goal: maximize the probability of choosing the best candidate.
6. The Secretary ProblemInterview n candidates for a position one at a time. After eachinterview decide if the candidate is the best.Goal: maximize the probability of choosing the best candidate.
7. The Secretary ProblemInterview n candidates for a position one at a time. After eachinterview decide if the candidate is the best.Goal: maximize the probability of choosing the best candidate.
8. The Secretary ProblemInterview n candidates for a position one at a time. After eachinterview decide if the candidate is the best.Goal: maximize the probability of choosing the best candidate.
9. The Secretary ProblemInterview n candidates for a position one at a time. After eachinterview decide if the candidate is the best.Goal: maximize the probability of choosing the best candidate.
10. The Secretary ProblemInterview n candidates for a position one at a time. After eachinterview decide if the candidate is the best.Goal: maximize the probability of choosing the best candidate.
11. The Secretary ProblemInterview n candidates for a position one at a time. After eachinterview decide if the candidate is the best.Goal: maximize the probability of choosing the best candidate.This is not about hiring secretaries, but about decisionmaking under uncertainty.
12. The Hiring Problem
13. The Hiring ProblemA startup is growing and is hiring many employees: Want to hire good employees Can’t wait for the perfect candidate
14. The Hiring ProblemA startup is growing and is hiring many employees: Want to hire good employees Can’t wait for the perfect candidateMany potential objectives. Explore the tradeoff between number of interviews & the average quality.
15. The Hiring modelCandidates arrive one at a time.Assume all have iid uniform(0,1) quality scores - Forapplicant i denote it by iq . (Can deal with other distributions, not this talk)
16. The Hiring modelCandidates arrive one at a time.Assume all have iid uniform(0,1) quality scores - Forapplicant i denote it by iq . (Can deal with other distributions, not this talk)During the interview: Observe iq Decide whether to hire or reject
17. Strategies
18. StrategiesHire above a threshold.
19. StrategiesHire above a threshold.Hire above the minimum or maximum.
20. StrategiesHire above a threshold.Hire above the minimum or maximum.Lake Wobegon Strategies: “Lake Wobegon: where all the women are strong, all the men are good looking, and all the children are above average”
21. StrategiesHire above a threshold.Hire above the minimum or maximum.Lake Wobegon Strategies: Hire above the average (mean or median)
22. StrategiesHire above a threshold.Hire above the minimum or maximum.Lake Wobegon Strategies: Hire above the averageSide note: [Google Research Blog - March ‘06]: “... only hire candidates who are above the mean of the current employees...”
23. Threshold HiringSet a threshold t , hire if iq ≥ t .
24. Threshold HiringSet a threshold t , hire if iq ≥ t .
25. Threshold HiringSet a threshold t , hire if iq ≥ t . t
26. Threshold HiringSet a threshold t , hire if iq ≥ t . t
27. Threshold HiringSet a threshold t , hire if iq ≥ t . t
28. Threshold HiringSet a threshold t , hire if iq ≥ t . t t
29. Threshold HiringSet a threshold t , hire if iq ≥ t . t t
30. Threshold HiringSet a threshold t , hire if iq ≥ t . t t
31. Threshold HiringSet a threshold t , hire if iq ≥ t . t t t
32. Threshold HiringSet a threshold t , hire if iq ≥ t . t t t
33. Threshold HiringSet a threshold t , hire if iq ≥ t . t t t
34. Threshold AnalysisSet a threshold t , hire if iq ≥ t . 1+tEasy to see that average quality approaches 1 2Hiring rate . 1−t
35. Threshold AnalysisSet a threshold t , hire if iq ≥ t . 1+tEasy to see that average quality approaches 1 2Hiring rate . 1−tQuality stagnates and does not increase with time.
36. Maximum HiringHire only if better than everyone already hired.
37. Maximum HiringHire only if better than everyone already hired.
38. Maximum HiringHire only if better than everyone already hired.
39. Maximum HiringHire only if better than everyone already hired.
40. Maximum HiringHire only if better than everyone already hired.
41. Maximum HiringHire only if better than everyone already hired.
42. Maximum HiringHire only if better than everyone already hired.
43. Maximum HiringHire only if better than everyone already hired.
44. Maximum HiringHire only if better than everyone already hired.
45. Maximum HiringHire only if better than everyone already hired.
46. Maximum HiringHire only if better than everyone already hired.
47. Maximum HiringHire only if better than everyone already hired.
48. Maximum HiringHire only if better than everyone already hired.
49. Maximum HiringHire only if better than everyone already hired.
50. Maximum AnalysisStart with employee of quality qLet hi be the i-th candidate hired
51. Maximum AnalysisStart with employee of quality qLet hi be the i-th candidate hiredFocus on the gap: gi = 1 − (hi )q gi
52. Maximum AnalysisStart with employee of quality qLet hi be the i-th candidate hiredFocus on the gap: gi = 1 − (hi )q giConditioned on gn−1 :
53. Maximum AnalysisStart with employee of quality qLet hi be the i-th candidate hiredFocus on the gap: gi = 1 − (hi )q giConditioned on gn−1 : gn ∼ U nif (0, gn−1 )
54. Maximum AnalysisStart with employee of quality qLet hi be the i-th candidate hiredFocus on the gap: gi = 1 − (hi )q giConditioned on gn−1 : gn ∼ U nif (0, gn−1 ) gn−1 E[gn |gn−1 ] = 2
55. Maximum AnalysisStart with employee of quality qLet hi be the i-th candidate hiredFocus on the gap: gi = 1 − (hi )q giConditioned on gn−1 : gn ∼ U nif (0, gn−1 ) gn−1 E[gn |gn−1 ] = 2 1−q E[gn ] = n 2
56. Maximum AnalysisStart with employee of quality qLet hi be the i-th candidate hiredFocus on the gap: gi = 1 − (hi )q giConditioned on gn−1 : gn ∼ U nif (0, gn−1 ) gn−1 E[gn |gn−1 ] = 2 1−q E[gn ] = n Very high quality! 2
57. Maximum AnalysisStart with employee of quality qLet hi be the i-th candidate hiredFocus on the gap: gi = 1 − (hi )q giConditioned on gn−1 : gn ∼ U nif (0, gn−1 ) gn−1 E[gn |gn−1 ] = 2 1−q E[gn ] = n Extremely slow hiring! 2
58. Lake Wobegon Strategies
59. Lake Wobegon Strategies Above the mean: 1 Average quality after n hires: 1 − √ n
60. Lake Wobegon Strategies Above the mean: 1 Average quality after n hires: 1 − √ n Above the median: 1 Median quality after n hires: 1 − n
61. Lake Wobegon Strategies Above the mean: 1 Average quality after n hires: 1 − √ n Above the median: 1 Median quality after n hires: 1 − n Surprising: Tight concentration is not possible Hiring above mean converges to a log-normal distribution
62. Hiring Above MeanStart with employee of quality qLet hi be the i-th candidate hiredFocus on the gap: gi = 1 − (hi )q gi
63. Hiring Above MeanStart with employee of quality qLet hi be the i-th candidate hiredFocus on the gap: gi = 1 − (hi )q giConditioned on gn : (in+1 )q ∼ U nif (1 − gn , 1)
64. Hiring Above MeanStart with employee of quality qLet hi be the i-th candidate hiredFocus on the gap: gi = 1 − (hi )q giConditioned on gn : (in+1 )q ∼ U nif (1 − gn , 1) n+1 1Therefore: gn+1 ∼ gn + U nif (0, gn ) n+2 n+2
65. Hiring Above MeanStart with employee of quality qLet hi be the i-th candidate hiredFocus on the gap: gi = 1 − (hi )q giConditioned on gn : (in+1 )q ∼ U nif (1 − gn , 1) n+1 1Therefore: gn+1 ∼ gn + U nif (0, gn ) n+2 n+2 U nif (0, 1) = gn 1− n+2
66. Hiring Above Mean U nif (0, 1)gn+1 = gn 1− n+2
67. Hiring Above Mean U nif (0, 1)gn+1 = gn 1− n+2 t U nif (0, 1)Expand: gn+t = gn 1− i=1 n+i+1
68. Hiring Above Mean U nif (0, 1)gn+1 = gn 1− n+2 t U nif (0, 1)Expand: gn+t = gn 1− i=1 n+i+1Therefore: n U nif (0, 1) gn ∼ (1 − q) 1− i=1 i+1
69. Hiring Above Mean U nif (0, 1)gn+1 = gn 1− n+2 t U nif (0, 1)Expand: gn+t = gn 1− i=1 n+i+1Therefore: n U nif (0, 1) gn ∼ (1 − q) 1− i=1 i+1 n 1 1 E[gn ] = (1 − q) 1− = Θ( √ ) i=1 2(i + 1) n
70. Hiring Above MeanConclusion: 1 Average quality after n hires: 1 − Θ( √ ) n Time to hire n employees: Θ(n3/2 ) Very weak concentration results.
71. Hiring Above MedianStart with employee of quality qWhen we have 2k + 1 employees. Compute median mk of the scores Hire next 2 applicants with scores above mk
72. Hiring Above Median mk
73. Hiring Above Median mkNew hires: with quality above mk
74. Hiring Above Median mk
75. Hiring Above Median mk+1
76. Hiring Above Median mk
77. Hiring Above Median mkInductive hypothesis: U nif (mk , 1)
78. Hiring Above Median mk Inductive hypothesis: U nif (mk , 1)New hires: are U nif (mk , 1)
79. Hiring Above Median mk U nif (mk , 1)
80. Hiring Above Median mk+1 U nif (mk , 1)
81. Hiring Above Median mk+1 U nif (mk+1 , 1)
82. Hiring Above Median mk+1 U nif (mk+1 , 1) Induction holds.
83. Hiring Above Median mk+1 U nif (mk , 1)
84. Hiring Above Median mk+1 U nif (mk , 1)The median:
85. Hiring Above Median mk+1 U nif (mk , 1)The median: smallest of k + 1 uniform r.v., each U nif (mk , 1)
86. Hiring Above Median mk+1 U nif (mk , 1)The median: smallest of k + 1 uniform r.v., each U nif (0, gk )
87. Hiring Above Median mk+1 U nif (mk , 1)The median: smallest of k + 1 uniform r.v., each U nif (0, gk ) gk+1 |gk ∼ gk Beta(k + 2, 1)
88. Hiring Above Median kExpand (like the means): gk ∼ g Beta(i + 1, 1) i=1
89. Hiring Above Median kExpand (like the means): gk ∼ g Beta(i + 1, 1) i=1 1 E[gn ] = Θ( ) n
90. Hiring Above Median kExpand (like the means): gk ∼ g Beta(i + 1, 1) i=1 1 E[gn ] = Θ( ) n log nCaveat: this is the median gap! The mean gap is: Θ( ) n
91. Hiring Above MedianConclusion: log n Average quality after n hires: 1 − Θ( ) n Time to hire n employees: Θ(n2 ) Again, very weak concentration results.
92. ExtensionsInterview preprocessing: Self selection: quality may increase over timeErrors: Noisy estimates on iq scores.Firing: Periodically fire bottom 10% (Jack Welch Strategy) Similar analysis to the median.Many more...
93. Thank You