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Topics in Search Theory
The Secretary Problem
Theme Paper
• Who Solved the Secretary Problem? – Thomas
Ferguson, 1989
Background
• Problem appeared in late 50’s and early 60’s
• Also known as “marriage problem”, “the
dowry problem”
• Highly appealing problem:
– Easy to state
– Striking solution
• Since then the problem has been extended
and generalized in many directions to become
a “field” of study (see [Freeman, 1983])
Problem Statement
• Simplest form:
– There is one secretarial position available
– The number of applicants is known
– The applicants are interviewed sequentially in random order
(each sequence is equally likely)
– You can rank all the applicants from best to worst without ties
– The decision to accept or reject the applicant must be based
only on the relative ranks of those applicants interviewed so
far
– An applicant once rejected cannot later be recalled
– You will be satisfied with nothing but the very best (meaning: your
payoff is 1 if choosing the best of the n applicants and 0 otherwise)
Analysis
• Attention can be restricted to the class of rules
that for some integer r≥1 rejects the first r-1
applicants and then chooses the next
applicant who is best in the relative ranking of
the observed applicants (since we don’t learn
anything new about the population over each interview)
reject first r-1
accept (and stop) if better than max so far
timeline
n
Analysis
• What is the probability n r  of selecting the
best applicant?
– For r=1 or r=n, 1/n
– For r>1:
the probability that this is
the best candidate
 j - th applicant is best  n  1  r  1   r  1  n  1 
    
  

n r    P
 
j r
 and you select it
 j r  n  j  1   n  j r  j  1 
n
max of r-1
>
max of j-r
the probability that the maximum
number in a sample of j-1 is one of
the first r-1 numbers
timeline
n
Analysis
• The optimal r is the one that maximizes this
probability
 r 1  n  1 

n r   
 
 n  j r  j  1 
• For small values of n the optimal r can be
easily computed (see excel file)
• Approximation can be supplied for large n
Kepler’s Problem
• In 1611, the German astronomer Johannes Kepler
lost his wife
• Since his first marriage had been arranged for him,
he decided this time to make his own decision
• He arranged to interview and to choose from among
no fewer than eleven candidates
• The process consumed much of his attention for
nearly 2 years, investigating:
–
–
–
–
–
Virtues and drawbacks
Dowry
Negotiation with the parents
Advice of friends
…
Kepler’s Problem
• Of the eleven candidates interviewed, Kepler
eventually decided on the fifth
• What a surprise! Check the excel sheet for the
optimal strategy here…
• Perhaps if Kepler had been aware of the secretary
problem, he could have saved himself a lot of time
and trouble
• Of course, it is not exactly the same problem:
–
–
–
–
Recall option
Discount factor?
Other utility aspects?
Maybe it’s an optimal stopping rule problem?
Back to Secretary problem:
Analysis
• This is actually a Riemann approximation to an
integral!
• When:
 r 1  n  1 

n r   
 
 n  j r  j  1 
n 
r
x  lim
n
j
t
n
dt 
1
n
1 1
 r  1  n  n  1 
   x   dt   x log x 
n r   
 
x t
 n  j r  j  1  n 
 
Optimal Solution
• First derivative: n r    x log x 
n ' r    log x   1  0
log x   1
x  1 / e  0.367879
• Substituting in n r :
n r   1 / elog 1 / e  1 / e
• Interpretation: wait until 37% of applicants have
been interviewed and then select the relative best
one. The probability of success in this case will be
37%
Differences from One-Sided Search
•
•
•
•
•
Rankings
No search costs
Decision horizon
Recall
Goal function (find the best vs. optimize
expected value)