Stopping Rule for Secretory Problem - Presentation by Haoyang Tian, Wesam Alshami and Dong Wang

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.”


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”.


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:


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


Attention: This is

Conditional probability



Attention: This is




Attention: This is




Attention: This is




Attention: This is




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



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, +∞ .



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 .


Verification by programming:

TotalApplicants(n)

Total TestCount

1000 100000


END

PPT模板下载：www.1ppt.com/moban/ 行业PPT模板：www.1ppt.com/hangye/

节日PPT模板：www.1ppt.com/jieri/ PPT素材下载：www.1ppt.com/sucai/

PPT背景图片：www.1ppt.com/beijing/ PPT图表下载：www.1ppt.com/tubiao/

优秀PPT下载：www.1ppt.com/xiazai/ PPT教程： www.1ppt.com/powerpoint/

Word教程： www.1ppt.com/word/ Excel教程：www.1ppt.com/excel/

资料下载：www.1ppt.com/ziliao/ PPT课件下载：www.1ppt.com/kejian/

范文下载：www.1ppt.com/fanwen/ 试卷下载：www.1ppt.com/shiti /

教案下载：www.1ppt.com/jiaoan/

Haoyang/ Wesam/ Dong CS 6212/ Arora/ Fall 201515

Education

Stopping Rule for Secretory Problem - Presentation by Haoyang Tian, Wesam Alshami and Dong Wang