The Annals
Annals of
ofProbability
Probability
The
1991, Vol.
Vol. 19,
19, No.
No.1,
342-353
1, 342-353
1991,
MINIMAX-OPTIMAL STOP
STOP RULES
RULES AND
AND DISTRIBUTIONS
DISTRIBUTIONS
MINIMAX-OPTIMAL
IN SECRETARY
SECRETARY PROBLEMS
PROBLEMS
IN
By THEODORE
THEODORE P.
P. HILL1
HILLl
BY
AND ULRICH
ULRICH KRENGEL
KRENGEL
AND
Institute of
of Technology
Technology and
and University
University of
of Gdttingen
Gottingen
Georgia Institute
For the
the secretary
secretary (or
(or best-choice)
best-choice) problem
problem with
with an
an unknown
unknown number
number N
N
For
of objects,
objects, minimax-optimal
minimax-optimal stop
stop rules
rules and
and (worst-case)
(worst-case) distributions
distributions are
are
of
derived, under
under the
the assumption
assumption that
that N
N is
is aa random
random variable
variable with
with unknown
unknown
derived,
distribution, but
but known
known upper
upper bound
bound n.
n. Asymptotically,
Asymptotically, the
the probability
probability of
of
distribution,
selecting the
the best
best object
object in
in this
this situation
situation is
is of
of order
order of
of (log
(log n)-Y.
n) - 1. For
For
selecting
example, even
even if
if the
the only
only information
information available
available is
is that
that there
there are
are somewhere
somewhere
example,
between 1
1 and
and 100
100 objects,
objects, there
there is
is still
still aa strategy
strategy which
which will
will select
select the
the best
best
between
item about
about one
one time
time in
in five.
five.
item
1. Introduction.
Introduction. In
In the
the classical
classical secretary
secretary problem,
problem, aa known
known number
number of
of
1.
rankable objects
objects is
is presented
presented one
one by
by one
one in
in random
random order
order (all
(all n!
n! possible
possible
rankable
orderings being
being equally
equally likely).
likely). As
As each
each object
object is
is presented,
presented, the
the observer
observer must
must
orderings
either select
select it
it and
and stop
stop observing
observing or
or reject
reject it
it and
and continue
continue observing.
observing. He
He may
may
either
stop must
never return
return to
to aa previously
previously rejected
rejected object,
object, and
and his
his decision
decision to
to stop
must be
be
never
based solely
solely on
on the
the relative
relative ranks
ranks of
of the
the objects
objects he
he has
has observed
observed so
so far.
far. The
The
based
This
selected.This
is selected.
goal
to maximize
maximize the
the probability
probability that
that the
best object
object is
the best
is to
goal is
well
is
or best-choice
problem, also
known as
the marriage
marriage problem
problem or
best-choice problem,
problem, is well
as the
also known
problem,
for
(1989)
Ferguson
and Ferguson (1989) for
known, and
the reader
reader is
referred to
to Freeman
Freeman (1983)
(1983) and
is referred
known,
and the
aa history
literature.
ofthe
the literature.
reviewof
and review
historyand
random
is aa random
butis
is not
notknown,
known,but
Suppose
ofobjects
numberof
objectsis
totalnumber
thetotal
nowthat
thatthe
Supposenow
positive
fixed
n
is
a
known
variable
N
taking
values
in
{I,
2,
...
,n},
where
n
is
a
known
fixed
positive
where
..
variable N takingvalues in {1,2, , n},
highest
the
in
to
guarantee
the
observer
play
in
order
to
guarantee
the
highest
integer.
How
should
order
play
observer
integer.How should the
the
is the
what
and
this
probability
of
selecting
the
best
object,
what
is
this
probability
and
what
is
what
is
probability
object,
best
the
of
selecting
probability
these
is
to
determine
this
paper
worst
distribution
for
N?
The
main
goal
of
this
paper
is
to
determine
these
of
goal
The
main
N?
for
distribution
worst
n. For
For
of n.
as aa function
functionof
minimax-optimal
stop
as
distributions
and distributions
values and
rules,values
stoprules,
minimax-optimal
probability
withprobability
example,
firstobject
objectwith
the first
withthe
= 5,
"stop with
the strategy
strategy"stop
if nn =
5, the
example,if
probability
withprobability
objectwith
secondobject
26/75;
withthe
the second
and stop
stop with
continueand
otherwisecontinue
26/75; otherwise
first
the first
otherwisestop
stopthe
and otherwise
26/49
firstobject;
the first
object;and
thanthe
is better
betterthan
it is
providedit
26/49 provided
object"
observedobject"
time
observed
thanany
is better
betterthan
anypreviously
previously
whichis
is observed
observedwhich
an object
timean
objectis
at
withprobability
probability
bestobject
is
objectwith
This
will
at
thebest
willselect
selectthe
strategy
Thisstrategy
minimax-optimal.
is minimax-optimal.
is best
best
and that
that probability
probability
least
of
is
N (~5),
of N
(< 5), and
distributions
forall
all distributions
least 26/75
26/75 for
= 13/75,
= 1)
13/75,
1) =
possible.
distribution P(N
P(N =
if N
N has
has the
the distribution
Conversely, if
possible. Conversely,
best
the best
= 60/75,
= 5)
will select
select the
= 2) =
= 2/75, P(N
no strategy
strategywill
P(N
then no
60/75, then
5) =
P(N =
P(N =
miniis also
also minidistribution
thisdistribution
object
greater
is
so this
than 26/75,
26/75, so
greaterthan
withprobability
probability
objectwith
Received
1989.
revisedOctober
October1989.
1989;revised
ReceivedMay
May1989;
DMS-89-01267.
NSF Grant
GrantDMS-89-01267.
1Research
supported
andNSF
by
Research
Grantand
ResearchGrant
Fulbright
byaa Fulbright
supported
'Researchpartially
partially
90D05.
AMS 1980
62C20, 90D05.
secondary62C20,
60G40; secondary
Primary60G40;
classifications.Primary
1980 subject
subjectclassifications.
minimaxKey
problem,
best-choice
problem,
marriage-problem,
minimaxmarriage-problem,
best-choice
problem,
problem,
Secretary
and phrases.
wordsand
phrases.Secretary
Keywords
rule.
optimal
distribution,
randomized
stop
randomized
stoprule.
distribution,
minimax-optimal
rule,minimax-optimal
stoprule,
optimalstop
342
342
MINIMAX
STRATEGIES IN
IN SECRETARY
SECRETARYPROBLEMS
MINIMAX STRATEGIES
PROBLEMS
343
343
max.
max. (It
N! orderings
are
(It is
is assumed
assumedthat,
that,given
givenN,
all N!
are equally
orderings
equallylikely,
likely,and
and that
that
N, all
ifan
an object
objectis
is rejected
rejectedand
and no
no more
moreobjects
objectsremain,
remain,the
the game
if
game is
is over
overand
and the
the
best object
been selected.)
best
has not
objecthas
notbeen
selected.)
A
ofresults
number of
A number
resultsare
are known
knownfor
forthe
thegeneral
generalsituation
situationwhere
wherethe
the number
number
of
is aa random
objectsN
of objects
N is
random variable.
variable.Presman
Presmanand
and Sonin
Sonin (1972)
(1972) derive
deriveoptimal
optimal
when N
has aa known
stop
N has
prior distribution
and
ruleswhen
knownprior
stoprules
distribution
and mention
mentionthe
thenecessarily
necessarily
prior distributions.
complex
form("islands")
ofoptimal
complexform
rulesfor
("islands") of
optimalstop
stoprules
forcertain
certainprior
distributions.
Irle (1980)
(1980) gives
givesaa concrete
concreteexample
Irle
prior for
exampleof
ofsuch
such aa prior
forwhich
whichthe
the optimal
optimalstop
stop
rule has
these islands
conditions
rule
has these
islandsand
and sufficient
sufficient
forexistence
conditionsfor
existenceof
of simple
simple"non"nonisland"
stoprules.
rules.Abdel-Hamid,
island" stop
Abdel-Hamid, Bather
Batherand
and Trustrum
Trustrum(1982)
(1982) derive
derivenecessary
necessary
and
conditions
of
and sufficient
sufficient
conditionsfor
foradmissibility
ofrandomized
admissibility
randomizedstop
stoprules.
rules.
Extensionsto
to the
the situation
situationwhere
wherethe
the interarrival
interarrival
timesof
of the
the objects
objectsare
Extensions
times
are
continuousrandom
continuous
have
been studied
randomvariables
variableswith
withknown
knowndistributions
distributions
have been
studiedby
by
Presman
Presmanand
and Sonin
Sonin (1972),
(1972), Gianini
Gianiniand
and Samuels
Samuels (1976)
(1976) and
and Stewart
Stewart(1981).
(1981).
More recently,
More
Bruss
Bruss (1984)
recently,
(1984) and
and Bruss
Bruss and
and Samuels
Samuels (1987)
(1987) derive
derivesurprising
surprising
and very
in this
and
strategies
verygeneral
generalminimax-optimal
minimax-optimal
thissame
same context
strategiesin
contextand
and even
for
evenfor
moregeneral
loss functions.
In contrast
generalloss
more
stop
functions.In
to the
contrastto
the minimax-optimal
minimax-optimal
stop rules
rules
in this
derived
paper, which
based on
N, those
derivedin
thispaper,
whichare
are based
on knowledge
ofaa bound
boundfor
forN,
knowledgeof
thoseof
of
Brussand
are based
and Samuels
Bruss
based on
of
Samuelsare
on knowledge
knowledgeof
ofthe
thedistributions
distributions
ofthe
thecontinucontinuous i.i.d.
i.i.d.interarrival
in this
ous
times;
theirs.
interarrival
times;in
thissense
sense our
our results
resultscomplement
complement
theirs.
This
paper is
for
is organized
as follows:
containsnotation,
resultsfor
This paper
follows:Section
Section22 contains
organizedas
notation,results
the classical
problem and
basic results
randomized
and basic
resultsconcerning
randomizedstop
the
classicalsecretary
secretaryproblem
concerning
stop
rule~; Section
of the
the main
main results
resultsand
and examples;
containsthe
the statements
statementsof
Section33 contains
rules;
examples;
Sections
proofs of
stop
of the
rules and
and
and 55 contain
containthe
the proofs
the minimax-optimal
Sections44 and
minimax-optimal
stop rules
distributions,
respectively;
and
containsremarks
remarksand
and asymptotics.
and Section
Section66 contains
distributions,
respectively;
asymptotics.
2. Preliminaries.
A well-known
Preliminaries. A
2.
equivalent
of
well-known
equivalentformulation
formulation
of the
the classical
classical
secretary
problem is
R I , R2,...,
R 2 , ••• , Rn
R n are
is the
the following.
secretaryproblem
are independent
random
following.R1,
independentrandom
c'variables
probability space
positive integer
variableson
on aa probability
wherenn is
is aa fixed
fixedpositive
space en,
Y, P),
P), where
integer
(fQ,!T,
= i)
and
peR) =
for all
{I, 2,
... , n}.
and P(RJ
1for
2, ...
If ~n-deall ii EE{1,
..., , j}j} and
and all
2,...,
all jj EE{1,{I, 2,
dei)=j= j-I
n). If
notes
R I , R2,...,
R 2 , ••• , Rn,
R n , then
notesthe
the stop
stop rules
rulesfor
forR1,
thenthe
the value
of aa stop
value of
stop rule
rule tt EE ~
E
(given
thatthere
thereare
are nn objects)
(giventhat
objects)is
is
= n)
= P{R
= 11 and
> 11 for
> t);
forall
all jj >
and R)
V(tIN =
n) =
V(t/N
t);
P(Rtt =
Ri >
= n)
that
probability of
best object
is the
the probability
that is,
of selecting
the best
the
is, V{t/N
V(tIN =
n) is
selectingthe
objectusing
using the
stop-rule
The
goal
is
to
find
that there
thereare
are nn objects.
The
to
is
findaa tt
stop-rulestrategy
strategyt,
t, given
giventhat
objects.
goal
= n)
making
possible, and
problem is
as large
as possible,
and the
thesolution
solutionto
well
makingV{t/N
V(tlN =
n) as
largeas
to this
thisproblem
is well
known [cf.
known
and Freeman
Freeman(1983)]
is recorded
herefor
[cf.Ferguson
Ferguson(1989)
(1989) and
and is
recordedhere
forease
ease
(1983)] and
= 0,
of
of reference.
reference. Throughout
paper, So
= E{=lithispaper,
and for
forjj ?~ 1,
Throughoutthis
0, and
1, s)
s; =
E
1i' l .
so =
DEFINITION 2.1.
positive integer
integer
DEFINITION
For each
2.1. For
each positive
is the
integern,
the nOIlnegative
n, kkn
nonnegative
integer
n is
satisfying
satisfying
Sn-1
-
Skn-1
2 1 > Sn-1
-
Skn
T.
U. KRENGEL
KRENGEL
T. P.
P. HILL
HILL AND
AND U.
344
344
definedby
> kn:
min{min{j>
E defined
byin
PROPOSITION
stop rule
k n:
The stop
rule in
PROPOSITION 2.2.
2.2. The
tnEE ~
4n== min{min{j
R
=
I},
n}
is
optimal,
that
is,
n}
is
that
is,
1},
optimal,
Rij
sup V(tIN
n).
n) == sup
V(inlN
V(tIN== n).
V(tnIN== n)
E En
ttE~
is to
to
In
strategyis
are nn objects,
objects,the
the optimal
optimalstrategy
giventhat
that there
thereare
In other
otherwords,
words,given
observe
n objects
and then
thento
to stop
withthe
the first
first
the first
firstkkn
withoutstopping
stop with
stoppingand
observethe
objectswithout
better than
seen.
object,
ifany,
thatis
is better
than any
seen. It
It is
is well
well known
known
objectpreviously
previously
any object
object,if
any,that
-* e
values
n
records
a
few
that
~
as
n
~
00,
and
the
next
example
records
a
few
typical
values
e
as
and
the
next
that n/k
oo,
typical
example
n/kn
n
of
of n.
n.
-
= O,
= kk4
= 1,
= kk7
= 22 and
1, kk5
EXAMPLE
0, kk3
=
and kg
k8 == kg
k6
EXAMPLE 2.3.
2.3. kk1=
k2
kg=
s == k
i = k
2 =
4 =
6 =
7 =
3 =
= 3.
kk10
lO =
where the
the
be generalized
Next, the
will be
to the
the setting
the above
above notations
notationswill
settingwhere
generalizedto
Next,
are
number of
N is
stop rules
rules are
and randomized
randomizedstop
is aa random
randomvariable
variableand
number
of objects
objects N
clear
allowed.
ofaa fixed
fixedknown
knownnumber
numberof
ofobjects,
it is
is clear
objects,it
allowed.(In
theclassical
classicalsetting
settingof
(In the
that
does
amongthe
the larger
larger
is, in
is also
also optimal
optimalamong
does not
nothelp,
help,that
thatis,
thatrandomization
randomization
fnis
class
class of
ofrandomized
randomizedstop
stoprules.)
rules.)
on
For
positive integer
n denotes
probabilities on
n, TI
denotes the
the set
set of
of probabilities
integer n,
For each
each positive
Hn
=
.,
{1, 2,
...
where
for
all
0
is
of
the
form
2
so P
p
2 ...
..., , n},
{I,
n}, so
p EH
E TIn
is
of
the
form
p
=
(PI'
P2'
...
'
Pn),
where
Pi
~
for
all
P2,
(P1,
pi
Pn)
n
and E'/=IPi
ii and
E=11pi=1.= 1.
are as
as
N is
.J?(N)
R
N
is aa random
randomvariable
variablewith
withdistribution
distribution
R1,...
A(N) EE TIn'
Rn
I-Ing
I , ... , R
n are
randomizedstop
rules
above
of
N and
of N
denotesthe
theset
set of
of randomized
aboveand
and independent
and ~
stoprules
independent
5n denotes
= i}
in the
E En means
{t =
i} is
the IT-algebra
for
R I , ... , R
Rn
means that
that {t
is in
a-algebra genergenerforR1,...,
that is,
is, tt E,~
n , that
.,
variables
...
are
i.i.d.
random
where
ated
by R
,
U
...
,
Ri'~'
where
U
U
...
are
i.i.d.
U[O,
1]
random
variables
U[O,
1]
,
,
,
ated by
...
U1,
U2,
R1,
U1
Ui,
Ri,
I
I
I
2
In other
the
which are
process and
and of
of N.
N. In
otherwords,
words,the
which
are independent
of the
the {R
independentof
{Ri}
i } process
the observed
relative
observer
base his
his selection
selectionrule
rulenot
notonly
on the
observedrelative
observeris
allowedto
to base
onlyon
is allowed
random
ranks, but
but also
event,
aa coin
coinor
or using
usingaa random
say flipping
flipping
on an
an independent
independent
event,say
ranks,
also on
the
number
whichare
are of
of interest
interest(for
(forthe
rules which
Clearlythe
the only
onlystop
stop rules
numbergenerator.
generator.Clearly
goal
best object)
neverstop
withan
an object
of selecting
are those
those which
whichnever
stop with
object
the best
object)are
goal of
selectingthe
which
best seen
be
seen so
so far,
far,so
so every
"reasonable" tt EE ~
whichis
is not
not the
the best
may be
every"reasonable"
En may
= (ql'
.. ., ,qn)
where qi
is the
the probability
that
described
by tt =
probability that
describedby
q2 ...
[0,l]n,
1]n, where
(q1,q2'
qi is
qn) EE [0,
= 11 and
= i,
> i-I.
tt =
Ri =
be assumed
i - 1. Accordingly,
will be
assumed
and tt >
it will
Accordingly,it
that Ri
given that
i, given
The
throughout
that
is essentially
thatonly
suchstop
rulesare
are used,
so ~
essentially[0,
1]n.The
used,so
onlysuch
stoprules
throughout
[0,l]n.
En is
= (ql'
the
. . ., ' qn)
the selection
selectionstrategy
withthe
rule tt =
describesthe
"stop with
strategy"stop
q2 ...
stop
stop rule
(q1, q2'
qn) describes
< ql);
if U
continueobserving
first
probability qi
observing
U1
otherwisecontinue
q1 (i.e.,
(i.e., if
firstobject
objectwith
withprobability
q1); otherwise
I ~
and
better than
probability q2
ifthe
withprobability
and if
the second
is better
than the
the first,
q2 (i.e.,
(i.e.,
first,stop
stopwith
secondobject
objectis
Bather and
and Trustrum
Trustrum
U
... " [see
< q2);
otherwisecontinue,
Abdel-Hamid,Bather
continue,..."
[see Abdel-Hamid,
U2
q2); otherwise
2~
if
2.2 says
that if
(1982)].
says that
To relate
relatethis
this to
to the
the classical
classicalproblem,
problem,Proposition
Proposition2.2
(1982)]. To
zeros
N
=
n,
then
an
optimal
stop
rule
is
(0,
...
,0,1,
...
,1),
where
k
zeros
precede
0
where
1,...,
N n, thenan optimalstopruleis (O,... .,
precede
1),
kn
n
... ' qn)
are not
not
rules (ql'
the above
above stop
n
ones. [Formally
n + kkn
stop rules
speaking,the
[Formallyspeaking,
(ql, ...
qn) are
n ones.
1
is
a
relative
rank
less
than
forced
to stop
by
time
n,
but
since
stopping
with
a
relative
rank
less
than
1
is
with
time
but
since
forcedto
n,
stopping
stopby
worth nothing,
nothing, it
aa stop
by time
timenn changes
thatforcing
it is
is easily
seen that
worth
changesnothing.]
nothing.]
forcing
stopby
easilyseen
°
MINIMAXSTRATEGIES
STRATEGIESIN
MINIMAX
INSECRETARY
SECRETARYPROBLEMS
PROBLEMS
345
345
DEFINITION 2.4.
2.4. For
For tt == (ql'
(q 1,....
. .,' qn)
9 and
and pp == (PI'
**,
DEFINITION
' Pn)
fn, the
the
qn) EE ~
(P1, ...
PO) EE TIn'
valueof
ofusing
usingtt given
giventhat
thatthe
thedistribution
value
of
distribution
ofN
N isis p,
p, V(t/p),
V(tlp),is
is given
givenby
by
.
P(t ~< Nand
N and R
V(tlp) == P{t
Rtt == 11 and
V(t/p)
and R
> 11
Rii >
V i E {t
+ 1,t
. . . , N}IJ(N)
ViE
19t+
+292, ...
ft+
p).
N) l(N) == p).
(Recallthe
theassumption
ifthe
assumptionthat
thatif
theobserver
(Recall
j,
observerrejects
the jth
rejectsthe
jth object
and N
N ==jg
objectand
thenhe
he loses.)
loses.)
then
The next
nextlemma
lemmais
is found
foundin
inAbdel-Hamid,
The
Bather
Abdel-Hamid,
Batherand
and Trustrum
Trustrum(1982)
(1982) and
and
is recorded
recordedhere
herefor
forcompleteness.
completeness.
is
(For
the
(For notational
notationalconvenience,
convenience,
the product
product
overan
an empty
setis
is taken
emptyset
takento
to be
be 1.)
over
1.)
LEMMA
2.5. For t = (q19q2
...*
nn
qn) E
V(t/p)
Pjj-l
V(tlp) = Epj-E
E
)=1
j=1
5n and p = (P1,... ,Pn) E Hn
)i
ii-1
-I
i=1
i=
m=
m=1
1qm)·
E qi
n (1(1 -- mqi H
m-1qm).
PROOF. Using
the probability
Using t,t, the
probabilitythat
PROOF.
that all
all of
of the
the first
first m
m objects
objects will
will be
rejected,r(t,
r(t, m),
m), is
is
rejected,
... (1
= (1
r(t, m)
m) =
(1 - ql)(l
ret,
qm/m)
q2/2)) ...
(1 - qm/m)
ql)(1 - q2/2
= j,
and if
if N
N=
the probability
= j)
of winning
and
j, the
probability of
winning with
with this
rule tt is
this rule
is V(tIN
V(tIN =
j)= =
=
i
1).
j-lEj=
Since
V(tlp)
j-1E{=lqir(t,
this yields
yields the
lqir(t, i-I). Since V(tlp) = E
=j), this
the desired
E>1=
desired
lpjV(tlN
j V(tIN =j),
J=IP
equality.
a
equality. D
3.
3. Main
Main theorems
theorems and
and examples.
examples. Recall
Recall that
that S j = E
{= 1i-I and k
k n is
is
Ei=li-l
sj
the
the "cutoff'
"cutoff" for
for the
the optimal
optimal rule
rule in
in the
the classical
classical secretary
secretary
problem
with
n
problemwith n
,objects
"'objects (Definition
(Definition 2.1
2.1 and
and Proposition
Proposition 2.2).
2.2).
DEFINITION
DEFINITION 3.1.
3.1.
Let
Let a,
a 1 == 1,
1, a2
a 2 == 1/2
1/2 and,
and, for
for n
n >> 2,
2,
-
=Sn-
(n
-
Skn-1
kn)/kn + (sn-1
Skn-l)Skn
(See
4, 5,
(See Table
Table 11 for
for an
an', n
n =
= 3,
3,4,
5, 10.)
10.)
Recall
that En
~ is
is the
the set
set of
of randomized
randomized stop
stop rules
rules for
for n
n objects,
objects, Hfn
TI n is
is
Recall also
also that
the
.. ,,n}
n} and
the set
set of
of probabilities
probabilities on
on {1,
{I, ....
and V(tlp)
V(t/p) is
is the
the probability
probability of
of selecting
selecting
the
the best
best object
object using
using t,
t, given
given that
that the
the distribution
distribution of
of the
the number
number of
of objects
objects is
is
p.
p. The
The following
following three
three theorems
theorems are
are the
the main
main results
results of
of this
this paper.
paper.
THEOREM
n infps
THEOREM A.
A. supte
SUPtEY"n
infpElln
V(t/p) =
= aan == infp,=infpElln supret
SUPtEY"n V(tlp).
V(tlp).
T, V(tlp)
REMARK.
REMARK. Although
Although each
each of
of the
th~ terms
terms in
in the
the definition
definition of
of an
a n has
has aa natural
natural
probabilistic
probabilistic interpretation
interpretation (e.g.,
(e.g., si
Si -- s;
Sj is
is the
the expected
expected number
number of
of relative
relative
rank
rank 11 candidates
candidates occurring
occurring between
between the
the ith
ith and
and jth
jth candidates),
candidates), the
the authors
authors
346
346
T. P.
P. HILL
T.
HILL AND
AND U.
U. KRENGEL
KRENGEL
know
of no
no intuitive
know of
be the
minimaxconstant
intuitiveexplanation
explanationwhy
should be
the minimax
why an
constant
an should
appearing
in Theorem
A.
appearingin
TheoremA.
t;
= (qi,
THEOREMB
B (Minimax-optimal
THEOREM
stop
If tn =
... , q;)
is
q *) EE ~
(Minimax-optimal
stop rule).
rule). If
(q1.*,
3n is
defined
definedby
by
q=
q~
J
J
)-1
(ac-1 _s_)(( an - Sj-1
1
1
= 1, k*
,
fiorj
for j = 1, ...
, k n,
< n,
<j ~
fork n <j
forkn
n,
? an
thenV(t;lp)
a,- for
then
for all
all pp EE TIn.
fin
V(t*lp) ~
C (Minimax-optimal
THEOREM C
THEOREM
distributionfor
forN).
(Minimax-optimaldistribution
N).
is defined
by
TI
n is
defined by
fin
pI
Pj=
an(j +
+ 1)-1
(an(i
1)
)
= an(l
- (Sn-l
- Skn_l)-l)
an(1
(Sn-1
skn-1)
= (pi,
If Pn*
Pn* =
If
(pr, ... ,n,P:)
p*) EE
for j <
k n,
< kn,
forj
= k
for j =
forj
kn
n,
for k n <j
< nn
<j <
forkn
o
0
P:
P:
= na
= 1
< 2,
2, pn =
> 2,
1 and
2, p* =
(so for
(so
for nn ~
for nn >
_1)]-1),
and for
then
1)) then
nan[kn(Sn-1
Skn-)I
n[k n(sn-1 - Sk
n
< an
V(tIP
an
V(tIPn*)
n*) ~
all t EE ~.
forallt
for
En.
[Verification
of
p; and
[Verification
ofthe
the above
aboveexpression
expressionfor
forpn*
and of
ofthe
the fact
factthat
that qi*
[0,1]
q?' EE [0,1]
is
is left
leftto
the reader;
to the
the
to the
this requires
reader; this
requiresonly
only elementary
elementaryalgebra
algebra applied
applied to
?
<
1
definitions
of an'
to
the
and
For
show
definitions
of
k
and
Sn.
For
example,
to
show
qi*
~
1,
the
monotonicity
example,
q
monotonicity
an, kn
sn.
n
< (1
and using
+ Sk
of
that it
-1)-1,
and
ofthe
the{Sj}
it is
is enough
to show
showthat
that an
1),
impliesthat
enoughto
(1 +
using
an :::;
Skn{s)} implies
n
a n and
the definition
definitionof
of an
and Sj
this
is
to
the
this
is
equivalent
to
(k
1)(Sn-1
Sk
-1)
~
< nn -equivalent
(kn
1)(snl
n
sj
Skn-1)
n
k n , which
holds.]
which clearly
clearlyholds.]
kn,
stop
Table
the minimax
minimaxvalues
values {an},
and the
the minimax-optimal
rules
Table 11 lists
liststhe
minimax-optimal
stoprules
{fa}, and
and
for
and distributions
forseveral
n.
distributions
severalvalues
values of
of n.
REMARKS.
that
REMARKS. Irle's
Irle's (1980)
ofan
an "unpleasant"
thatis,
(1980) example
exampleof
"unpleasant"distribution,
distribution,
is, aa
distribution
... ,1)
is
distribution for
for which
which no
no stop
rule of
of the
the form
form (0,0, ...
... ,,0,1,1,
0, 1, 1,...,
stop rule
1) is
= (0,0.895,0.001,0.001,
optimal,
p =
is p
he calcu... ,,0.001,0.1)
0.001,0.1) EE TIs,
forwhich
whichhe
calcuoptimal,is
(0, 0.895,0.001,0.001,...
H8, for
lates
be approximately
lates the
ofthe
1,0, 1,1, 1, 1, 1) to
thevalue
value of
the optimal
rule(0,
to be
optimalstop
stoprule
(0, 1,0,1,1,1,1,1)
approximately
TABLE
TABLE 1
n
kn
1 0
22 00
?h
1
!2
ti;
tn*
= (q[,
=
•..
(q *,...qn)
,qi;>
(1)
(1)
(1,
(1,1)1)
P.*
Pl@@P
(1)
(1)
(0,
1)
(0,o1)
(t,o,~)
6
(~,O,O,~)
33 11 ~3
(7(~,1,1)
7,1,1)
4 1 *
(*,1,1,1).
4 2 26
(25
2
29)
60
13
26(26,
1, 1 U .
5,·)t2
(~,~,1,1,1)
(*,-fs,0,0,~)
5 2 ~
, ,
75,49'4
75
(73,75'"75
75)
10
... ,1) (0.139+,0.092+,0.068+,0,0,
... ,0,0.698+)
10 3
3 0.278+
0.278+ (0.278+,0.386+,0.478+,1,1,
(0.278+, 0.386+, 0.478+, 1, 1,...,1)
(0.139+, 0.092+, 0.068+, 0,0,...,0,
0.698+)
-
MINIMAX STRATEGIES
STRATEGIES IN
IN SECRETARY
SECRETARY PROBLEMS
PROBLEMS
MINIMAX
347
347
0.482. Comparison
Comparison of
of this
this value
value with
with those
those in
in Table
Table 1
1 suggests
suggests that
that such
such
0.482.
island distributions
distributions are
are far
far from
from being
being worst-case
worst-case (i.e.,
(Le., minimax-optimal),
minimax-optimal),
island
although aa direct
direct proof
proof of
of this
this is
is not
not known
known to
to the
the authors.
authors.
although
It should
should also
also be
be observed
observed that
that the
the minimax-optimal
minimax-optimal distribution
distribution for
for N
N is
is
It
not one
one of
of the
the other
other "naive-guess"
"naive-guess" distributions
distributions such
such as
as N
N == n
n or
or N
N uniuninot
formly distributed
distributed on
on {1,
2, ...
....,,n}
or N
N == 1
1 with
with probability
probability p
p and
and =
=n
n with
with
n} or
{1, 2,
formly
probability 1
1 -- p.
p. As
As far
far as
as the
the authors
authors know,
know, this
this Pn*
Pn* is
is aa new
new distribution
distribution on
on
probability
n points.
points.
n
Theorem A
A follows
follows from
from Theorems
Theorems B
Band
C. No
No direct
direct proof
proof that
that
Clearly Theorem
and C.
Clearly
supinf
= inf
infsup
is known
known to
to the
the authors;
authors; although
although V(tlp)
is linear
linear in
in p
p and
and
V(tlp) is
sup is
inf=
sup
TIn
is convex
convex and
and compact,
compact, V(tlp)
is neither
neither convex
convex nor
nor concave
concave in
in t,
t, and
and
V(tlp) is
fln is
known generalizations
generalizations of
of the
the classical
classical minimax
minimax theorem
theorem of
of game
game theory
theory do
do not
not
known
seem to
to apply.
apply. (The
(The results
results in
in this
this paper
paper may
may also
also be
be interpreted
interpreted as
as aa zero-sum
zero-sum
seem
two-person game
game as
as follows.
follows. Player
Player II picks
picks the
the distribution
distribution of
of N,
N, and
and player
player II
II
two-person
N objects,
picks the
the stop-rule
stop-rule or
or selection-strategy
selection-strategy t;
t; if
if tt selects
selects the
the best
best of
of the
the N
objects,
picks
then player
player II pays
pays player
player II
II one
one dollar;
dollar; and
and otherwise
otherwise no
no money
money changes
changes
then
of
hands. The
The constant
constant an
then represents
represents the
the value
value of this
this game.)
game.)
hands.
an then
are
C are
4. Proof
Proof of
Theorem B.
B. The
The conclusions
conclusions of
B and C
TheoremsBand
of Theorems
4.
of Theorem
=
will
n
trivial
for
n
="
1
and
easy
for
n
=
2,
so
for
the
remainder
of
this
paper,
n
will
of
this
paper,
n
=
for
the
remainder
2, so
trivialfor n 1 and easy for
be aa fixed
fixed integer
integer strictly
bigger than
than 2,
to simplify
notation, k
k =
= kn,
kn,
and to
simplifynotation,
2, and
strictlybigger
be
cases
`= En
the
>
7=
~ and
TI
=
TIn.
(Observe
that
n
>
2
precludes
the
degenerate
cases
n
2
=
degenerate
that
precludes
II
(Observe
and
Hn.
where
= 0;
2.3.)
see Example
Example 2.3.)
where kk =
0; see
are positive
and k are
and jj and
numbers, and
Suppose
positive
{a iY11 are
are real
real numbers,
Suppose {aJi=l
? jj ~2 k.
integers
satisfying nn ~
If both
both
k. If
integerssatisfying
LEMMA
4.1.
LEMMA 4.1.
(1)
(1)
(a1 +
kak)/, 2 (a1 +
.a
.+aj)/
and
and
(2)
(2)
aam2am+l
m ~ am + 1
then (a1 +
...
for all m E {k
forallmE{+k
? (a1 +
+aj)/j
...
+ 2, ...
... ,,n),
+ 1,
1, k +
n },
+an)/n.
= n,
< n.
Then condin. Then
condiPROOF.
so assume
assume jj <
is trivial,
trivial,so
conclusionis
If jj =
the conclusion
n, the
PROOF. If
tions
imply
imply
and (2),
tions(1)
respectively,
(2), respectively,
(1) and
(a
+
(a1l +
...
+aj)/j
~
+
2 (ak+l
*.
+aj)/j
(ak+1 +
*.
+aj)/(j
--k k)
+aj)/(j
? (a
~
+
j +l +
(aj+l
*
+an)/(n
?+an)/(n --j),j),
E
so (a
+ ... +an)/n.
+a
+ ... +aj)/j
+ aj)/j ~
+an)/n. D
2 (a
(a, l +
(a, l +
fROPOSITION
4.2.
PROPOSITION 4.2.
sup
inf V(tlp)
V(tlp) =
sup inf
ttE
E Y
pEn
5 PeHII
max
max
min
min
tt=(q1,...,qn)E
= (q 1, ... , q n) E Y:
£ = 1 't/ i > k
jjE{1,2,...,k,n)
E {I, 2, ... , k, n}
[7: qq=1=Vi>k
V(tIN
=j).
V(tIN =j).
T.
U. KRENGEL
KRENGEL
T. P.
P. HILL
HILL AND
AND U.
348
348
7 and
PROOF. Since
Since V(tlp)
V(t/p) is
is continuous
continuous in
in both
both tt and
and p,
p, and
and since
since !T
and H
n are
are
PROOF.
compact, the
the sup
sup and
and inf
inf are
are attained.
attained. Moreover
Moreover
compact,
inf V(tlp)
V(t/p) == minV(tIN=j),
minV(t/N =j),
inf
(3)
(3)
pEn
pElI
H
j5:n
<n
. .,, n}.
since V(tlp)
V(t/p) is
is linear
linear in
in p,
p, and
and H
n isis the
the set
set of
of all
all probabilities
probabilities on
on {1,
{I, 2,
2, ....
n}.
since
The proof
proof of
of the
the optimality
optimality of
of the
the backward
backward induction
induction procedure
procedur~ implies
implies
The
that if
if tt is
is any
any stopping
stopping time
time for
for an
an adapted
adapted sequence
sequence of
of a-algebras
IT-algebras ~
-1 cCc ~ Cc
that
...
... cc 9j
and t*
t* the
the optimal
optimal stopping
stopping time,
time, then
then V(t')
V(t') 2~ V(t)
V(t) if
if t'
t' is
is obtainobtain9 and
ed from
from tt by
by stopping
stopping at
at time
time ii on
on an
an arbitrary
arbitrary Si-measurable
~-measurable subset
subset
ed
{t >> i, t*
t* == i}. Hence,
Hence, by
by Proposition
Proposition 2.2
2.2 replacing
replacing an
an arbitrary
arbitrary tt ==
of {t
57results
(ql' ... ' qn) E !T
by f=(q1,...,qk,1,1,...,l)E
f = (ql' ... ' qk' 1, 1, ... ,1) E !T
results in
in at
at least
least as
as
7 by
(ql,...,qn)ES
(deterministic)
high aa probability
probability of
of selecting
selecting the
the best
best object
object for
for any
any given
given (deterministic)
high
number of
of objects
objects (<
(~ n),
n), that
that is,
is,
number
V(fIN =j) 2
~ V(tIN=j)
V(tIN =j) forall
for all jj <~ n.
n.
V(tIN=j)
(4)
(4)
Together, (3),
(3), (4)
(4) and
and the
the compactness
compactness of
of !T
imply
57 imply
Together,
(5)
(5)
sup inf
infV(tlp)
=
V(tlp) =
sup
pEn
GE
GEpH
,- p
ttE.r
max
max
t=(ql' ...
. . ' qn)E.r: qi= 1 't/
t=(q1,.
qn).E7:qi=1Vi>k
=j).
minV(tIN=j).
minV(tIN
i>k j<n
j 5:n
To complete
complete the
the proof
proof of
of the
the proposition,
proposition, it
it is
is enough
enough to
to show
show that
that for
for all
all
To
1},
n
...
+
1,
Ek
+
{k
all
E_
for
and
= (ql'
...
,qk'
1,
...
,1)
E
!T,
and
for
all
j
E
{k
+
1,
k
+
2,
...
,
n
I},
,
1,...
j
..
1)
I7,
29
tt =
,qkg
(q19.
V(tIN
=j) ~
= k),
V(tIN = n)}.
n)}.
k),V(tIN=
min{V(tIN=
2 min{V(tIN
V(tIN=j)
(6)
(6)
folas folnumbers{ai}i=l
real numbers
definereal
and define
[0,l]n and
,1) EE1]=[0,
{ai}in> as
Fix
= (ql'
... ,1)
,' qk'
1,...
...
(q1,...
Fix tt =
qk, 1,
1.
>
i
for
m-lqm)
lows:
a
=
ql
and
ai
=
qin~2l(1
m-lqm)
for
i
>
1.
=
=
lows:a,l q1 and ai qi~lH-11(1
all j,
forall
[0, 1] for
Since
j,
> kk and
= 1
and qj
1 for
all ii >
forall
Since qi
qj EE [0,1]
qi =
(7)
(7)
am
> a m+ l
am>am+l
forallm>k.
for
all m > k.
To
n}. To
{1,2,...
all jj EE {I,
for all
By
2, ... , n}.
** +aj)/j
2.5, V(tIN
Lemma 2.5,
By Lemma
V(tIN==j)j) == (a
+aj)/j for
(a,l + ...
is,
that
=
establish
(6),
suppose
V(tIN
=
j)
~
V(tIN
=
k),
that
is,
<
k),
=
V(tIN
establish(6), supposeV(tIN j)
(a
(a,l +
(8)
(8)
...
? (a
*. +ak)/k
+ak)/k ~
+aj)lj.
(a,l + ... +aj)/j.
By
(withk == k),
4.1 (with
Lemma4.1
and Lemma
(8) and
and (8)
(7) and
By(7)
k)g
V(tIN
V(tIN== n),
n) ,
* +an)/n
?+an)/n == V(tIN
*D +aj)/j
=j) == (a
(a,l + ...
V(tIN =j)
+aj)/j ~? (a
(a,l + ...
which
(6). Da
establishes(6).
whichestablishes
LEMMA
4.3.
4.3.
LEMMA
A,
1)E=E !T,
. .. ,,1)
....,,qk'
For
all tt== (ql'
Forall
19 ...
19 1,
qk9 1,
(q19...
kn-{(
V(tIN
sn-l -- Sk-)(1
Sk-l)
n) == kn
V(tIN== n)
[(Sn-1
(J + 1)
V(tlN=j))
1)-lV(tIN
(1 - :~>j'+
j) )
-
E
=
j.
k).
Sk-1 --1)V(tlN
-(Sn-l
- Sk-l
1)V(tIN == k)
-(Sn--
MINIMAXSTRATEGIES
STRATEGIESIN
INSECRETARY
MINIMAX
SECRETARYPROBLEMS
PROBLEMS
349
349
PROOF. First
Firstititwill
willbe
be shown
shownthat
that
PROOF.
(9)
(9)
(9)
)i
n (1-m-lqm)
(1 - m- qm) = -1 1
m=1
m=l
)-1
j-1
LE
m=1
m=1
(m
V (t/N = m)
+ 1)-lV(tlN=
(m? +
m)
--V(tIN=j)
V(tIN =j) for
forall
all j~k.
j k.
The proof
proofof
of(9)
(9) is
is by
by induction
1,(1
inductionon
on j.
j. For
For jj == 1,
The
(1 -- q1)
V(tIN == 1)
1) by
by
q1) == 11 -- V(tIN
Lemma2.5.
2.5. Assume
Assumethat
thatthe
theequality
equalityin
in (9)
(9) holds
Lemma
holdsfor
forall
all jj ~< k
k and
and calculate
calculate
k+1
k+l
kk
k
(1
- 1q m) ==
- 1q m) -- (k
+1
(1
- 1q m)
(1 -- m
1 (1(1 -- mmlqm)
mlqm)
(k +
+ 1)
1)- 1qk
(1 -- m
qk+l
mlqm)
nH
m=1
(10)
(10)
nH
n
m=1
m=1
k-1
k-l
= 1= -1- L
(t/N =
= k)
= m)
,(m(m + 1)-l
1) - VV(tlN
m) -- V(t/N
V(tJN=
k)
m=1
m=1
- V(t/N= k + 1) + k(k + 1)
A
""
-1
"
V(t/N= k)
k
= 1-
L
(m + 1)-lV (tIN = m) - V(tIN =
k + 1),
m=1
wherethe
the second
secondequality
equalityin
in (10)
(10) follows
followsby
bythe
theinduction
inductionhypothesis
where
and
hypothesis
and the
the
fact(from
(fromLemma
Lemma2.5)
2.5) that
that m=1
fact
(m= 1
V(tIN= k +
+ 1)
1) =
= (k
+ 1)
V(t/N
-l qk
(ik+
1)1q~
k
H (1
nIi
1) -lV(tIN
= k),
(11- m- 1qm) + k(k ++1)t
V (t/N =I),
-m
m=1
q m) +1(s
which
which establishes
establishes (9).
(9).
Since qj
> k,
qj =
= 11 for
for all
all jj >
k, Lemma
Lemma 2.5
2.5 and
and the
the definition
definition of
of {Sj}
imply that
that
Since
{sj} imply
- k
k
-I
ii-1
1
= n-1 iJ;:lqifll(1-m-lqm)
h
V(tIN=n)
V(tlN- n) =nE
qi
(1 - m-lqm)
[
+k(sn-1
+k(Sn-l -- Sk-1)
Sk-l)
k (1 mQl
m-1qm)
mlqm)J.
l
But
But E
L7=lqin~21(1
m- 1qm) == kV(tlN
kV(tiN == k)
k) (Lemma
(Lemma 2.5
2.5 again),
again), so
so (9)
(9) (with
(with
=lqitlim 1(1 -- m-lqm)
k) and
and (11)
(11) yield
yield the
the desired
desired equality.
equality. m
D
jj == k)
Heuristics.
Heuristics. Although
Although aa direct
direct calculus-based
calculus-based proof
proof of
of Theorem
Theorem B
B should
should be
be
possible,
the proof
proof given
given below
below is
is greatly
greatly facilitated
facilitated by
by Proposition
Proposition 4.2
4.2 and
and
possible, the
Lemma
Lemma 4.3,
4.3, which
which both
both also
also serve
serve as
as heuristics
heuristics for
for the
the structure
structure of
of the
the
minimax-optimal
minimax-optimal stop
stop rule.
rule. For
For example,
example, Proposition
Proposition 4.2
4.2 says
says that
that any
any general
general
stop
rule can
can be
be replaced
replaced by
by aa stop
stop rule
rule with
with qi
q i == 11 for
for all
all ii >> k,
k, and
and that
that with
with
stop rule
such
stop
rules,
the
critical
such stop rules, the critical values
values occur
occur when
when N
N == jj for
for some
some jj in
in
{1,
{1, 2,...,
2, ... , k,
k, n};
n}; that
that is,
is, if
if N
N == jj eE {k
{k +
+ 1, ... , nn -- 1},
1}, the
the observer's
observer's probabilprobabil1....
ity
ity of
of selecting
selecting the
the best
best object
object is
is at
at least
least as
as high
high as
as the
the minimum
minimum of
of the
the other
other
possible
possible values
values for
for j.j. (Incidentally
(Incidentally this
this also
also suggests
suggests why
why the
the minimax-optimal
minimax-optimal
distribution
in Theorem
Theorem C
C places
places no
no mass
mass on
on {k
{k ++ 1,
1, ... , n
n -- 1}.
1}. For
For fixed
fixed tt of
of
distribution in
the
the form
form known
known to
to be
be optimal
optimal (i.e.,
(i.e., qi
qi == 11 for
for all
all ii >> k),
k), Lemma
Lemma 4.3
4.3 implies
implies
,
...
,
T.
U. KRENGEL
KRENGEL
T. P.
P. HILL
HILL AND
AND U.
350
350
that V(tlN
n) is
is aa decreasing
decreasing function
function of
of V(tIN
V(tlN == j)
j) for
for jj <~ k.
k. Together
Together
that
V(tIN== n)
with Proposition
Proposition 4.2,
4.2, this
this suggests
suggests via
via aa "Robin
"Robin Hood
Hood principle"
principle" (shifting
(shifting
with
mass to
to decrease
decrease the
the maximum
maximum and
and increase
increase the
the minimum)
minimum) that
that the
the extremal
extremal
mass
case occurs
occurs when
when V(tIN
V(tlN == 1)
1) == V(tIN
V(tlN == 2)
2) == *
... == V(tIN
V(tlN == k)
k) =
= V(tIN
V(tlN == n).
n).
case
.. ,, qk
Solving this
this set
set of
of k
k equations
equations for
for the
the k
k unknowns
unknowns q1,
q l' ....
q k leads
leads to
to the
the
Solving
minimax-optimal stop
stop rule
rule in
in Theorem
Theorem B.
B. Once
Once the
the correct
correct extremal
extremal stop
stop rule
rule
minimax-optimal
is guessed,
guessed, of
of course
course it
it is
is then
then much
much easier
easier to
to prove
prove directly
directly that
that it
it is
is in
il) fact
fact
is
without justifying
justifying the
the derivation
derivation of
of the
the guess.
guess.
optimal, without
optimal,
PROOF OF
OF THEOREM
THEOREM B.
B.
PROOF
By (6)
(6) it
it suffices
suffices to
to show
show that
that
By
V(t,iIN=j)=a
forj={1,2,
... ,k,n}.
n}.
j ={1 ,2 ,...,
V(t*lN =j) = a n for
(12)
(12)
To establish
establish (12),
(12), first
first check
check by
by induction
induction that
that qqj*n~-=ll(1
m-lq~) =
= a?
an for
for
To
lHi-11(1 - m-lq,*)
< k.
all jj <~ k,
k, so
so Lemma
Lemma 2.5
2.5 implies
implies that
that V(t,iIN
= j)
j) =
= an
an for
for all
all jj ~
k. To
To check
check
V(t*lN =
all
=
(sn1 - skl)
V(t*IN == n)
that V(t,iIN
n) =
= as,
an' use
use Lemma
Lemma 4.3
4.3 and
and the
the fact
fact that
that (Sn-l
Sk-l) =
that
k-1an(n - k)(1
k)(1 -- ansk)-l
to calculate
calculate
k-lan(n
anSk)-1 to
V(t;:'IN
V(tn*N=
n -lk[(
Sn-l - Skl)(11
Sk-l)
1k[(Sn-1
= n) =
= n
n)
(1 - a<~>j
1)i)-1)
A (j + 1)
+
-an
-(Sn-l
- Sk-l
-- 1)a
sk-1
1)an] n ]
-(Sn-1
(1 - an j~/-l) + an ]
=
-k [(sn-1
=
n -lk[(
Sn-l - Sk-l)
Sk-1) (1
-
-an
Ji)
+ an]
= an.
= n-1k[ank-1(n
=
+
+ an]
(1 - anSk)
k)(1 - anSk)-l(la?n
anSk)
n1k[sank 1(n -- k)(Ian] =
anSkl)
D
?
4.2 suggests
5.
mentionedabove,
As mentioned
suggests
Proposition4.2
Theorem C.
C. As
above,Proposition
5. Proof
Proof of
of Theorem
no
that
(worst-case
places
distribution
the observer)
forthe
places no
observer)distribution
thatany
minimax-optimal
(worst-casefor
any minimax-optimal
mass
to aa guess
Hood principle
leads to
+ 1,
1,...
and again
guess
... ,, nn -- I},
Robin Hood
principleleads
on {k
again aa Robin
1), and
mass on
{k +
which
values
... , k,
n}. For
For example,
foreach
clearly
example,clearly
each jj in
k,n}.
{1,2,
2,...,
whichhas
break-even
valuesfor
in {I,
has break-even
= (1,1,
pi
since otherwise
otherwisetaking
(1 1 ...
1) yields
was
> an.
As was
V(tlPn*)
yieldsV(tIP
takingtt =
an As
...,,1)
Pr ~< an'
a?, since
n*) >
been
has been
the
Gistribution
P
distribution
onceaa worst-case
worst-case
forthe
theoptimal
thecase
case for
rule,once
stoprule,
optimalstop
Pn*
n* has
of
Thus moat
guessed,
in fact
mucheasier.
easier.Thus
mostof
factminimax
minimaxis
is then
thecheck
checkthat
thatit
it is
is in
thenmuch
guessed,the
forP
the
theguess
in the
the heuristics
heuristicswhich
whichgenerated
guessfor
generatedthe
thework
workwas
was hidden
hiddenin
Pn*.
n*.
FORMAL
ARGUMENT.
to show
ARGUMENT. It
It is
show
is enough
FORMAL
enoughto
(13)
(13)
< an
V(tIP
V(tlPn*)
an
n*) ~
= (ql'
for all
all tt =
1,1,1, ...
* * ,1)
1)
qk 1,
for
...
(q9 *...,' qk'
E :T,
H7
E
for all
< V«q1'
1,......,,1)lp)
all {qi}
[0, 1]
l)Ip) for
by (4), V«ql'
V((q1,... ... ', qn)lp)
since by
~
V((q, ... , qk'
qk, 1,
qn)Ip)
{qJ} EE [0,1]
and
and all
H.
all pp EEiII.
[In
which
whichsays
withequality
says
that(13).
holdswith
throughout,
equalitythroughout,
itwill
willbe
be seen
seenthat
(13).holds
fact,it
[In fact,
with
that
*,
all"
reasonable"
stop
rules,
Le.,
all
stop
rules
with
rules
intuftively
all
all
"reasonable"
stop
rules,
i.e.,
thatagainst
stop
intuitively
againstP
Pn*,
n
= 1
qi
withthe
thesame
sameprobability.]
1 for
thebest
forall
all ii > k,
selectthe
bestobject
probability.]
objectwith
k, select
qi =
351
351
SECRETARY PROBLEMS
PROBLEMS
IN SECRETARY
STRATEGIES IN
MINIMAX STRATEGIES
MINIMAX
= (ql'
Fix tt =
e [0, l]n
calculate
and calculate
. .. ,1)
. . . ' qk'
1]n and
12...
X1) E
qk, 1, 1,
(ql ...
=
V(tIP
n*) =
V(tlPn*)
1 i
(k-1
I:L
i-1
+ 1»
(1 -anC~:
(j(j
+
i~l
qi
1))
(j(j
qj H (1
an
i=i
m=l
j=l
-1 E
E
mlqm)
m
-lqm)
k
i-1
k
i=1
m=i
m=i
i-I
k
)
l k
+ m~l
+k1 i~l
E (1 --m-lqm)
(- - m-lqm)
+km-lqm) +
m-lqm)
H (1
E qi
qj l!l
i-i
i-I
k-1 k-1
k-Ik-I
JI (1(1 - m -lqm)
i1 -- (j
+ 1)
E (i-I
= an
-l)qi H
(j +
1)1)qi
anE i~l Ei
(
=
+k 1l
+k-
kk
i-1
i-I
kk
i=i
m=i
m=i
)
H (1
+ l~[l
(1 -m-lqm)
H (1 --mqm)
i~l qi
- m -lqm) +
m -lqm)
qj l~[l
i-1
k
=
=
m-1qm)
-
H (1 (i- -k-1)qi
anL~l (i-I
- k-l)qi l:L
-
~m=l
z=1
m-lqm)
m-lqm)
kk
i-1
i-I
kk
)
l i~l qi l!l
H (1 - m-lqm)
+ J}l
+k
m-lqm)
+k- m-lqm) +
E qj H (1 -m-lqm)
i=i
m=i
m=i
IiI
k
i-1
k
an[i~l
= n Ei1qjj(1
+ mQl (1
mlqm)j
(1 -- m
m1qm)
=
i-lqi
(1 -- m
-lqm) +
-lqm)]
m=l
m=l
i=1
[k
an[i~l
{(i-lqi
(ijlqi
i=1
= a[E
=
i-1
-- 1)
1) lil
m=l
(1
fII(1
i-1
--
H (1
m -lqm)
+ lil
(1
qm) +
m
m=l
+
+
k
=
=
i -1
H (1- m-lqm)
m q ) {H(i
n
an[i~l
{~~(
i=1
m=l
-
-
iJlH
m=l
mlqm)}
-- m
-lqm)}
k
(1 -H (1
mQl
m=i
lq m )]
m-lqm)
m-
m lq
(1 - mqm)}
(1
m)}
k
lq m )]
+ mQl
+
m-lqm)
H1(1 -- mm=i
= aEng
t
=
sinceL
E j:I L i = l1 =
secondsince
the second
2.5, the
Lemma2.5,
by Lemma
followsby
equalityfollows
where
firstequality
wherethe
the first
This
k. This
forii == k.
disappearsfor
and the
the third
thirdsince
sincethe
the first
firstsummand
summanddisappears
L
7:l L j: l and
E*-1lE*i
O
theorem.D
the theorem.
completes
and the
(13) and
of(13)
proofof
theproof
completesthe
A
TheoremA
of TI
[nIn
subsetof
as aa subset
viewedas
be viewed
can be
6.
Since TIn
6. Asymptotics.
Asymptotics. Since
[In can
n++ 11,, Theorem
using
A
direct
check
nonincreasing.
is nonincreasing.
n} is
A direct check using
shows,
that
sequence{a
thesequence
thatthe
indirectly,
shows,indirectly,
{an}
knn
knn (e.g.,
(e.g.,k
knn++ 11 is
eitherk
about k
a n and
observations
is either
generalobservations
the
of
about
and general
the definition
definition
of an
n.
in
decreasing
strictly
are
the {an}
in fact
factthe
or
+ 1)
thatin
or k
showsthat
1) shows
{an} are strictly decreasing in n.
kit
n+
352
352
T. P.
P. HILL
HILL AND
T.
ANDU.
U. KRENGEL
KRENGEL
Sincesn
n and kkn
Since
e-11 (where
meanslimn
(wherean
lognand
1),
a
limn-+00 an/b
sn ~ log
n ~ nne
n means
an ~ bbn
n === 1),
itfollows
followseasily
easilythat
that
it
an
(log n)
atn~ (logn)-l,
q..7*
J
and
and
~
j :::;;
e-1n,
(f(logn-logj)1
(log n - log j) - 1 for
forj
<e-1n,
11for
1\
for jj > e-1n
e-1n
log n ) - 1
f(((j(j + 1)1)logn)1
for
forjj :::;;< e-1n,
e-1n
Pj*
0
pJ*~ NO
{
2(logn)-1
t2(log
n)
for
n,
fore-1n
<j << n,
e-1n <j
for
j == n.
forj=n.
= 0,
In particular,
limn
particular,lim
in contrast
In
classical
contrastto
to the
0, in
the well-known
well-known
classicalresult
result
n -+ oo an ==
that for
forthe
the deterministic
N = n,
deterministic
case N
that
case
of
n, the
the probability
probability
of selecting
selectingthe
the best
best
an optimal
-* 00.
decreasesmonotonically
object
to
object(using
to ee-11 as
(usingan
as nn ~
optimalstrategy)
strategy)decreases
oc.
monotonically
The optimal
optimal"stopping-probabilities"
"stopping-probabilities"
The
{qj*}
which
{qji} are
are nondecreasing,
nondecreasing,
whichis
is also
also
intuitively
plausible,since
sinceif
ifit
it is
is optimal
at
optimalto
intuitively
plausible,
to stop
stopwith
withaa certain
certainprobability
probability
at
= 1),
timeii (given
(givenR'i
thenat
1), then
at later
latertimes
time
accrued
timeswith
witheven
evenmore
moreinformation
information
R-i=
accrued
it should
shouldbe
be optimal
optimalto
to stop
stop with
withat
at least
it
if
least as
as high
highas
as probability
probability
if aa rank
rank 11
objectis
is observed.
observed.
object
The following
ofthe
following
the asymptotic
alternativepossible
resultaan
possiblederivation
The
alternative
result
n ~
derivationof
asymptotic
1
(log
has
been
n)
given by
(log n)-l
has been given
by Samuels
Samuels (1989).
(1989). Since
Since the
the expected
expectednumber
number of
of
relatively
ones ("
relatively best
best ones
records") will
will be
be about
N, this
this suggests
that the
the rule
rule
("records")
aboutlog
log N,
suggeststhat
stop
at each
each of
stop with
with probability
probability l/log
first log
will succeed
with
1/lognn at
thefirst
ofthe
log nn records
recordswill
succeedwith
probability
probability about
about 1/log
l/log n
n no
no matter
matter what
what the
the distribution
distribution of
of N
is.
(A
formal
N is. (A formal
derivation
derivation using
using this
this approach
approach seems
seems to
to require
require more
more information
information about
the
about the
actual distribution
",actual
distribution of
of the
the number
number of
of records
records than
than just
just its
its expectation.)
expectation.)
Acknowledgments.
Acknowledgments. The
The questions
questions answered
answered in
in this
this paper
paper arose
arose in
in aa
discussion
discussion following
following aa lecture
lecture by
by Professor
Professor S.
S. Samuels
Samuels at
at the
the American
American MatheMathematical
matical Society
Society meeting
meeting in
in Atlanta
Atlanta in
in January
January 1988.
The authors
authors are
are also
also
1988. The
grateful
to H.
H. J.
J. D6ring
Doring for
for aa number
number of
of stimulating
stimulating conversations
conversations and
and
grateful to
suggestions,
suggestions, to
to the
the Mathematics
Mathematics Research
Research Institute
Institute at
at Oberwolfach
Oberwolfach for
for hospihospitality
tality during
during aa visit
visit in
in which
which this
this research
research was
was begun
begun and
and to
to the
the Associate
Associate
Editor
Editor and
and referee
referee for
for several
several suggestions.
suggestions.
REFERENCES
REFERENCES
ABDEL-HAMID, A.,
A., BATHER,
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