Secretary Bound

Secretary Bound
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From Wikipedia, the free encyclopedia
Mathematical problem involving optimal stopping theory

^ Jump up to: a b c Ferguson, Thomas S. (August 1989). "Who Solved the Secretary Problem?" . Statistical Science . 4 (3): 282–289. doi : 10.1214/ss/1177012493 .

^ Hill, Theodore P. (2009). "Knowing When to Stop". American Scientist . 97 (2): 126–133. doi : 10.1511/2009.77.126 . ISSN 1545-2786 . S2CID 124798270 . For French translation, see cover story in the July issue of Pour la Science (2009).

^ Gnedin 2021 .

^ Gardner 1966 .

^ Gnedin 1994 .

^ Bearden 2006 .

^ Palley, Asa B.; Kremer, Mirko (8 July 2014). "Sequential Search and Learning from Rank Feedback: Theory and Experimental Evidence" . Management Science . 60 (10): 2525–2542. doi : 10.1287/mnsc.2014.1902 . ISSN 0025-1909 .

^ Moser, Leo (1956). "On a problem of Cayley". Scripta Math . 22 : 289–292.

^ Sakaguchi, Minoru (1 June 1961). "Dynamic programming of some sequential sampling design" . Journal of Mathematical Analysis and Applications . 2 (3): 446–466. doi : 10.1016/0022-247X(61)90023-3 . ISSN 0022-247X .

^ Rose, John S. (1982). "Selection of nonextremal candidates from a random sequence". J. Optim. Theory Appl . 38 (2): 207–219. doi : 10.1007/BF00934083 . ISSN 0022-3239 . S2CID 121339045 .

^ Szajowski, Krzysztof (1982). "Optimal choice of an object with ath rank". Matematyka Stosowana . Annales Societatis Mathematicae Polonae, Series III. 10 (19): 51–65. doi : 10.14708/ma.v10i19.1533 . ISSN 0137-2890 .

^ Vanderbei, Robert J. (21 June 2021). "The postdoc variant of the secretary problem" . Mathematica Applicanda . Annales Societatis Mathematicae Polonae, Series III. 49 (1): 3–13. doi : 10.14708/ma.v49i1.7076 . ISSN 2299-4009 . {{ cite journal }} : CS1 maint: date and year ( link )

^ Ghirdar 2009 . sfn error: no target: CITEREFGhirdar2009 ( help )

^ Gilbert & Mosteller 1966 .

^ Bearden, Murphy, and Rapoport, 2006; Bearden, Rapoport, and Murphy, 2006; Seale and Rapoport, 1997; Palley and Kremer, 2014

^ Shadlen, M. N.; Newsome, W. T. (23 January 1996). "Motion perception: seeing and deciding" . Proceedings of the National Academy of Sciences . 93 (2): 628–633. Bibcode : 1996PNAS...93..628S . doi : 10.1073/pnas.93.2.628 . PMC 40102 . PMID 8570606 .

^ Roitman, Jamie D.; Shadlen, Michael N. (1 November 2002). "Response of Neurons in the Lateral Intraparietal Area during a Combined Visual Discrimination Reaction Time Task" . The Journal of Neuroscience . 22 (21): 9475–9489. doi : 10.1523/JNEUROSCI.22-21-09475.2002 . PMC 6758024 . PMID 12417672 .

^ Heekeren, Hauke R.; Marrett, Sean; Ungerleider, Leslie G. (9 May 2008). "The neural systems that mediate human perceptual decision making". Nature Reviews Neuroscience . 9 (6): 467–479. doi : 10.1038/nrn2374 . PMID 18464792 . S2CID 7416645 .

^ Costa, V. D.; Averbeck, B. B. (18 October 2013). "Frontal-Parietal and Limbic-Striatal Activity Underlies Information Sampling in the Best Choice Problem" . Cerebral Cortex . 25 (4): 972–982. doi : 10.1093/cercor/bht286 . PMC 4366612 . PMID 24142842 .

^ Flood 1958 .

^ Gardner 1966 , Problem 3.

^ Bruss 1984 .

^ Ferguson 1989 .

^ Kesselheim, Thomas; Radke, Klaus; Tönnis, Andreas; Vöcking, Berthold (2013). "An Optimal Online Algorithm for Weighted Bipartite Matching and Extensions to Combinatorial Auctions". Algorithms – ESA 2013 . Lecture Notes in Computer Science. Vol. 8125. pp. 589–600. doi : 10.1007/978-3-642-40450-4_50 . ISBN 978-3-642-40449-8 .

The secretary problem demonstrates a scenario involving optimal stopping theory [1] [2] that is studied extensively in the fields of applied probability , statistics , and decision theory . It is also known as the marriage problem , the sultan's dowry problem , the fussy suitor problem , the googol game , and the best choice problem .

The basic form of the problem is the following: imagine an administrator who wants to hire the best secretary out of

n

{\displaystyle n}

rankable 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. During the interview, the administrator gains information sufficient to rank the applicant among all applicants interviewed so far, but is unaware of the quality of yet unseen applicants. The question is about the optimal strategy ( stopping rule ) to maximize the probability of selecting the best applicant. If the decision can be deferred to the end, this can be solved by the simple maximum selection algorithm of tracking the running maximum (and who achieved it), and selecting the overall maximum at the end. The difficulty is that the decision must be made immediately.

The shortest rigorous proof known so far is provided by the odds algorithm . It implies that the optimal win probability is always at least

1

/

e

{\displaystyle 1/e}

(where e is the base of the natural logarithm ), and that the latter holds even in a much greater generality. The optimal stopping rule prescribes always rejecting the first

∼
n

/

e

{\displaystyle \sim n/e}

applicants that are interviewed and then stopping at the first applicant who is better than every applicant interviewed so far (or continuing to the last applicant if this never occurs). Sometimes this strategy is called the

1

/

e

{\displaystyle 1/e}

stopping rule, because the probability of stopping at the best applicant with this strategy is about

1

/

e

{\displaystyle 1/e}

already for moderate values of

n

{\displaystyle n}

. One reason why the secretary problem has received so much attention is that the optimal policy for the problem (the stopping rule) is simple and selects the single best candidate about 37% of the time, irrespective of whether there are 100 or 100 million applicants.

Although there are many variations, the basic problem can be stated as follows:

A candidate is defined as an applicant who, when interviewed, is better than all the applicants interviewed previously. Skip is used to mean "reject immediately after the interview". Since the objective in the problem is to select the single best applicant, only candidates will be considered for acceptance. The "candidate" in this context corresponds to the concept of record in permutation.

The optimal policy for the problem is a stopping rule . Under it, 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 . It can be shown that the optimal strategy lies in this class of strategies. [ citation needed ] (Note that we should never choose an applicant who is not the best we have seen so far, since they cannot be the best overall applicant.) For an arbitrary cutoff r , the probability that the best applicant is selected is

The sum is not defined for r = 1, but in this case the only feasible policy is to select the first applicant, and hence P (1) = 1/ n . This sum is obtained by noting that if applicant i is the best applicant, then it is selected if and only if the best applicant among the first i − 1 applicants is among the first r − 1 applicants that were rejected. Letting n tend to infinity, writing

x

{\displaystyle x}

as the limit of (r−1) / n , using t for (i−1) / n and dt for 1/ n , the sum can be approximated by the integral

Taking the derivative of P ( x ) with respect to

x

{\displaystyle x}

, setting it to 0, and solving for x , we find that the optimal x is equal to 1/ e . Thus, the optimal cutoff tends to n / e as n increases, and the best applicant is selected with probability 1/ e .

For small values of n , the optimal r can also be obtained by standard dynamic programming methods. The optimal thresholds r and probability of selecting the best alternative P for several values of n are shown in the following table.

The probability of selecting the best applicant in the classical secretary problem converges toward

1

/

e
≈
0.368

{\displaystyle 1/e\approx 0.368}

.

This problem and several modifications can be solved (including the proof of optimality) in a straightforward manner by the odds algorithm , which also has other applications. Modifications for the secretary problem that can be solved by this algorithm include random availabilities of applicants, more general hypotheses for applicants to be of interest to the decision maker, group interviews for applicants, as well as certain models for a random number of applicants. [ citation needed ]

The solution of the secretary problem is only meaningful if it is justified to assume that the applicants have no knowledge of the decision strategy employed, because early applicants have no chance at all and may not show up otherwise.

One important drawback for applications of the solution of the classical secretary problem is that the number of applicants

n

{\displaystyle n}

must be known in advance, which is rarely the case. One way to overcome this problem is to suppose that the number of applicants is a random variable

N

{\displaystyle N}

with a known distribution of

P
(
N
=
k

)

k
=
1
,
2
,
⋯

{\displaystyle P(N=k)_{k=1,2,\cdots }}

(Presman and Sonin, 1972). For this model, the optimal solution is in general much harder, however. Moreover, the optimal success probability is now no longer around 1/ e but typically lower. This can be understood in the context of having a "price" to pay for not knowing the number of applicants. However, in this model the price is high. Depending on the choice of the distribution of

N

{\displaystyle N}

, the optimal win probability can approach zero. Looking for ways to cope with this new problem led to a new model yielding the so-called 1/e-law of best choice.

The essence of the model is based on the idea that life is sequential and that real-world problems pose themselves in real time. Also, it is easier to estimate times in which specific events (arrivals of applicants) should occur more frequently (if they do) than to estimate the distribution of the number of specific events which will occur. This idea led to the following approach, the so-called unified approach (1984):

The model is defined as follows: An applicant must be selected on some time interval

[
0
,
T
]

{\displaystyle [0,T]}

from an unknown number

N

{\displaystyle N}

of rankable applicants. The goal is to maximize the probability of selecting only the best under the hypothesis that all arrival orders of different ranks are equally likely. Suppose that all applicants have the same, but independent to each other, arrival time density

f

{\displaystyle f}

on

[
0
,
T
]

{\displaystyle [0,T]}

and let

F

{\displaystyle F}

denote the corresponding arrival time distribution function, that is

Let

τ

{\displaystyle \tau }

be such that

F
(
τ
)
=
1

/

e
.

{\displaystyle F(\tau )=1/e.}

Consider the strategy to wait and observe all applicants up to time

τ

{\displaystyle \tau }

and then to select, if possible, the first candidate after time

τ

{\displaystyle \tau }

which is better than all preceding ones. Then this strategy, called 1/e-strategy , has the following properties:

The 1/e-law, proved in 1984 by F. Thomas Bruss , came as a surprise. The reason was that a value of about 1/e had been considered before as being out of reach in a model for unknown

N

{\displaystyle N}

, whereas this value 1/e was now achieved as a lower bound for the success probability, and this in a model with arguably much weaker hypotheses (see e.g. Math. Reviews 85:m).

However, there are many other strategies that achieve (i) and (ii) and, moreover, perform strictly better than the 1/e-strategy
simultaneously for all

N

{\displaystyle N}

>2. A simple example is the strategy which selects (if possible) the first relatively best candidate after time

τ

{\displaystyle \tau }

provided that at least one applicant arrived before this time, and otherwise selects (if possible) the second relatively best candidate after time

τ

{\displaystyle \tau }

. [3]

The 1/e-law is sometimes confused with the solution for the classical secretary problem described above because of the similar role of the number 1/e. However, in the 1/e-law, this role is more general. The result is also stronger, since it holds for an unknown number of applicants and since the model based on an arrival time distribution F is more tractable for applications.

According to Thomas S. Ferguson , the secretary problem appeared for the first time in print when it was featured by Martin Gardner in his February 1960 Mathematical Games column in Scientific American . [1] Here is how Gardner formulated it: "Ask someone to take as many slips of paper as he pleases, and on each slip write a different positive number. The numbers may range from small fractions of 1 to a number the size of a googol (1 followed by a hundred zeroes) or even larger. These slips are turned face down and shuffled over the top of a table. One at a time you turn the slips face up. The aim is to stop turning when you come to the number that you guess to be the largest of the series. You cannot go back and pick a previously turned slip. If you turn over all the slips, then of course you must pick the last one turned." [4]

In the article "Who solved the Secretary problem?" Ferguson pointed out that the secretary problem remained unsolved as it was stated by M. Gardner, that is as a two-person zero-sum game with two antagonistic players. [1] In this game Alice, the informed player, writes secretly distinct numbers on

n

{\displaystyle n}

cards. Bob, the stopping player, observes the actual values and can stop turning cards whenever he wants, winning if the last card turned has the overall maximal number. The difference with the basic secretary problem is that Bob observes the actual values written on the cards, which he can use in his decision procedures. The numbers on cards are analogous to the numerical qualities of applicants in some versions of the secretary problem. The joint probability distribution of the numbers is under the control of Alice.

Bob wants to guess the maximal number with the highest possible probability, while Alice's goal is to keep this probability as low as possible. It is not optimal for Alice to sample the numbers independently from some fixed distribution, and she can play better by choosing random numbers in some dependent way. For

n
=
2

{\displaystyle n=2}

Alice has no minimax strategy, which is closely related to a paradox of T. Cover . But for

n
>
2

{\displaystyle n>2}

the game has a solution: Alice can choose random numbers (which are dependent random variables) in such a way that Bob cannot play better than using the classical stopping strategy based on the relative ranks. [5]

The remainder of the article deals again with the secretary problem for a known number of applicants.

Stein, Seale & Rapoport 2003 derived the expected success probabilities for several psychologically plausible heuristics that might be employed in the secretary problem. The heuristics they examined were:

Each heuristic has a single parameter y . The figure (shown on right) displays the expected success probabilities for each heuristic as a function of y for problems with n = 80.

Finding the single best applicant might seem like a rather strict objective. One can imagine that the interviewer would rather hire a hi
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