Post on 15-Apr-2017
Stopping Rule for
Secretary Problem
Researcher: Haoyang Tian, Wesam Alshami,Dong Wang
Haoyang/ Wesam/ Dong CS 6212/ Arora/ Fall 2015 1
Definition of Secretary Problem:
Imagine an administrator willing to hire the best secretary out of n
applicants for a position. The applicants are interviewed one by one in
random order. A decision about each particular applicant is to be made
immediately after the interview. Once rejected, an applicant cannot be
recalled.
Socrates asked Plato, “Plato, take a walk through the wheat field nearby. Without turning back, walk forward, and pick the most magnificent stalk of wheat you can find. However, you are allowed to pick only one.”
Haoyang/ Wesam/ Dong CS 6212/ Arora/ Fall 2015 2
Definition of Secretary Problem:
Imagine an administrator willing to hire the best secretary out of n
applicants for a position. The applicants are interviewed one by one in
random order. A decision about each particular applicant is to be made
immediately after the interview. Once rejected, an applicant cannot be
recalled.
Obviously, if we hire one randomly, the possibility of the one we picked
is the best qualified is1
𝑛. To increase the possibility, let us introduce
“stopping rule”.
Haoyang/ Wesam/ Dong CS 6212/ Arora/ Fall 2015 3
Definition of Stopping Rule:
The interviewer rejects the first r − 1 applicants (let applicant M be the
best applicant among these r − 1 applicants), and then selects the first
subsequent applicant that is better than applicant M.
For example:
Haoyang/ Wesam/ Dong CS 6212/ Arora/ Fall 2015 4
Optimal Stopping Number r:
Apparently, r ∈ (1, n). But what is the optimal number r that can lead
to highest possibility to hire the best qualified applicant.
We denote P(r) as the possibility of being able to hire the best applicant
when choose r as the stopping number. And denote n as the total number
of applicants.Haoyang/ Wesam/ Dong CS 6212/ Arora/ Fall 2015 5
Optimal Stopping Number r:
Attention: This is
Conditional probability
Haoyang/ Wesam/ Dong CS 6212/ Arora/ Fall 2015 6
Optimal Stopping Number r:
Attention: This is
Conditional probability
Haoyang/ Wesam/ Dong CS 6212/ Arora/ Fall 2015 7
Optimal Stopping Number r:
Attention: This is
Conditional probability
Haoyang/ Wesam/ Dong CS 6212/ Arora/ Fall 2015 8
Optimal Stopping Number r:
Attention: This is
Conditional probability
Haoyang/ Wesam/ Dong CS 6212/ Arora/ Fall 2015 9
Optimal Stopping Number r:
Attention: This is
Conditional probability
Haoyang/ Wesam/ Dong CS 6212/ Arora/ Fall 2015 10
Optimal Stopping Number r:
This is what we get:
≈𝑟
𝑛 𝑖=𝑟𝑛 1
𝑖
When 𝑛 → ∞,
𝑃 𝑟 =𝑟
𝑛 𝑟𝑛 1
𝑖𝑑𝑖
To make it simple, denote𝑟
𝑛as x, and denote
𝑖
𝑛as t.
So, dt = d(i
n) dt =
1
ndi di = ndt.
And the bound change from (r, n) to (x, 1).
Therefore, 𝑃 𝑥 =𝑥𝑛
𝑛 𝑥1 𝑛
𝑛𝑡𝑑𝑡 = 𝑥 𝑥
1 1
𝑡𝑑𝑡 = 𝑥 [ln 𝑡]𝑥
1 = −𝑥 ln 𝑥
Haoyang/ Wesam/ Dong CS 6212/ Arora/ Fall 2015 11
Optimal Stopping Number r:
Therefore, 𝑃 𝑥 = −𝑥 ln 𝑥
𝑃′ 𝑥 = − ln 𝑥 − 𝑥 ∗1
𝑥= − ln 𝑥 − 1
When 𝑃′ 𝑥 = 0 , 𝑥 = 𝑒−1.
So, 𝑃′ 𝑥 is a monotonically increasing function on 0, 𝑒−1 and
a monotonically decreasing function on 𝑒−1, +∞ .
Haoyang/ Wesam/ Dong CS 6212/ Arora/ Fall 2015 12
Optimal Stopping Number r:
After all, When 𝑥 =1
𝑒, 𝑃 𝑥 will reach its maximum
1
𝑒. Because of
𝑥 =𝑟
𝑛, in other words, when r =
𝑛
𝑒, 𝑃 𝑟 will reach its maximum
1
𝑒.
In conclusion, if the interviewer rejects the first𝒏
𝒆applicants after the
interview and then stops at the first applicant who is better than every
applicant interviewed so far, the probability of selecting the best
applicant converges toward1
𝑒≈ 0.368 .
Haoyang/ Wesam/ Dong CS 6212/ Arora/ Fall 2015 13
Verification by programming:
TotalApplicants(n)
Total TestCount
1000 100000
Haoyang/ Wesam/ Dong CS 6212/ Arora/ Fall 2015 14
END
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Haoyang/ Wesam/ Dong CS 6212/ Arora/ Fall 201515