Expectation Inequalities Associated With Prophet Problems
EXPECTATION INEQUALITIES ASSOCIATED WITH PROPHET PROBLEMS Theodore P. Hill l School of Mathematics
Georgia Institute of Technology
Atlanta, Georgia 30332
ABSTRACT Applications of the original prophet inequalities of Krengel and Sucheston are made to problems of order selection, non-measurable stop rules, look-ahead stop rUles, and iterated maps of random variables. Also, proofs are given of two results of Hill and Hordijk c?ncerning optimal orderings of uniform and exponential d~stributions.
§l.
INTRODUCTION Universal inequalities comparing the two func
tionals M
v
M(X ,X ,···) 2 l and
V(X ,X ,···) l 2
n
sup{EX : t is a stop rule t for
~esearch
E(suP Xn )
xl ,X 2 ,···}
partially supported by NSF Grants DMS-0160 and 01608.
4
het of seque nces of random varia bles are calle d "prop of tion inequ alitie s" becau se of the natur al inter preta ete M as the value to a proph et, or playe r with compl ving fores ight, in an optim al stopp ing proble m invol by random varia bles xl ,X 2 , • • . . First disco vered s have Kreng el and Suche ston [22, 23], these inequ alitie ons tigati been the subje ct of a numbe r of recen t inves [1,2, 4-15 ,17-2 1, 24-27 ]). to In §2, the appli cation s of proph et inequ alitie s or V inequ alitie s involv ing funct ionals other than M l menta are given , with atten tion focuse d on the funda (e.g.,
proph et inequ ality [23J (1)
M
are indep enden t and nonne gative , then 2V, and this bound is sharp .
xl ,X 2 , .•.
If ~
s (Analo gous appli cation s of other proph et inequ alitie to simil ar proble ms are left to the reade r.) Sectio n 3 conta ins proof s of two optim al-ord ering resul ts of Hill and Hordi jk [llJ. APPLICATIONS OF PROPHET INEQU ALITIE S The initia l disco very and appli catio n of proph et inequ alitie s such as (1) were made by Kreng el and semi Suche ston in conju nction with inves tigati ons of In amart s and proce sses with finite value [22, 23J. a this sectio n, other appli catio ns of the basic inequ proble ms lity (1) are given to sever al optim al-sto pping and an iterat ed map proble m. For the main appli cation theore m, wluch follow s §2.
imme diatel y from (1), let
=
,x2 , .•• ) l (More be any (real- value d) funct ional of Xl ' X2'··· . ite forma lly, U is a funct ion from C, the set of infin U
U(X
sequences of probability distributions, to the real numbers. In practice, U is usually Borel measurable, with C endowed with the product topology induced by the total-variation norm topology on the space of probabi lity distributions.) Let X 'x ' ••• be independent nonnegative l 2 random variables. Then (i) V ~ U implies U ::; 2Vi and (E) U ~ M implies M ::; 2lJ.
Theorem 2.1.
~.
Immediate from
(I) •
0
Application to Order Selection Let Us be the value of the sequence xl 'x 2 '··. to a player free to choose the order of observation of the random variables, as well as the time of stopping, that is,
Us
US {X l
,X ,···) ==
2
sup{V{XTI(1)'XTI{2~"'~): TI is a permutation of 1,2, ••• }.
(For a formal definition, including stochastic permuta ions TI, see [9].) Corollary 2.2.
Let X ,X , ••• be independent nonnega l 2 tive random variables. Then (i)
(Hill [9)
Us
~ 2Vi
and
(E) M s 2U ' S
Moreover, the bound in (i) is sharp. E) is a (Whether or not the constant "2" in ( sharp bound is not known to the author.) Inequality b tt than double 1) says that a player may never do e er . of a g1ven d his expected value by rearranging the or er sequence of random variables. Inequality (ii) is imme diate from (I) and the fact that Us ~ Vi only the ques 0
(
tion of its sharpness is of interest.
---- ---- _ _--_. _ . _
....
---~- ----- -~---
Appli cation to Use of Non-M easura ble stop Rules to a Let UN be the value of the seque nce Xl ,X 2 , ... i.e., playe r free to use non-m easura ble stop rules , be any integ er-va lued funct ions s for which {s = j} can ,X . (not neces sarily measu rable) funct ion of xl, ... j ional funct the is UN That is, UN
UN (X l ,X 2 ,···)
sup{EX : s is a "non- measu rable" stop rule} .
s
s (For a forma l defin ition, see [16].)
Let Xl ,x 2 , .•. be indep enden t nonne ga Corol lary 2.3. tive random varia bles. Then (i) (Hill and Pestie n [16] ) UN :5 2V; and (ii) H :5 2UW Horeo ver both bound s are sharp . (1); Proof . The inequ alitie s follow imme diatel y from bound the the sharp ness of (i) is in [16]. To see that X be in (ii) is sharp , let Xl be const ant +1, and let 2 (-1) = ( =1-P (X =0). a "long shot" [l2] given by P(X 2 2 0 1. = Us = UN Then M = 2-(, and Appli cation to "Look -Ahea d" Stop Rules to Let UA,k be the value of the seque nce x l ,X 2 , ... g lookin a playe r free to use stop rules s which allow func ahead k steps (i.e., integ er-va lued measu rable tions satisf ying {s = J'}
E
0
0 ) (X l ' •.• , X j +k ' s
UA,k(Xi, X 2 ,···) . SUp{EX s : s is a k-step "look -ahea d" stop rule} Let Xl ,X , ..• be indep enden t nonne ga Corol lary 2.4. 2 integ er. tive random varia bles, and let k be a posit ive Then
(i) UA,k :5 2V; and (ii) M $ 2U ,k. A
1-1ore over, both bound s are sharp . Proof . The inequ alitie s follow immed iately from (1). To see that (i) is sharp , let Xl = const ant +1, X2 = ... = Xk + l = const ant 0, and let Xk + 2 be a "long -1 h s h 0 t w~t P(X k + 2 = E ) = E = 1 P(X k +2 = 0); then UA,k = 2-E and V = 1. To see that (i) is sharp, let Il
•
Xl :: +1, X2 == ' " == Xk + 2 = 0, and let Xk + 3 be the "long and
shot" random varia ble just descri bed; then M = 2-E
o
Thus (i) says that a playe r able to look k steps into the futur e never has optim al expec ted return more and than twice that of a playe r who canno t look ahead, is (ii) says that a proph et's optim al expec ted return a never more than twice that of a playe r who may look 6ixed numbe r of steps into the future . On the other is hand, for a fixed seque nce of random variab les, it clear that lim U k(X l ,x 2 ",,) = M(X l ,x2 ,···)· A' ted Maps Itera ~plication to Let ¢(X,Y ) and w(X,Y) be the random variab les
¢(X,Y) = max{X ,y} and ~(X,Y) = max{X,EY}, and define the random varia bles ¢ n (Xn , ... ,X l ) and ~Il(Xn,···,Xl)
.
~nductively by
k .... oo
92(X ,X ) 2 1
E(max {Y,c}) if X $ E(rnax {Y,c}) and Y > Ci and = 3 otherw ise. [3] Lettin g X and Y be i.i.d . U[O,l ], by Lemma 2.1 of it follow s that (4) is equiv alent to (5)
E[RT( X,aY, c) (X,aY ,c) - ~(aY,X,c) (aY,X ,c)]
~
O.
To see (5), first observ e that (6)
E[RT( X,aY, c) (X,aY ,c) IX
[O,a]]
€
y ) (aX,aY ,c)],
T aX,a ,e m since the distri butio n of X given X € [O,a] is unifor On [O,a], that is, has the same distri butio n as aX. = E (R (
Next calcu late (7)
E(RT( aY,X, c) (aY,X ,c) IX
[O,a]]
) (ay,aX ,c)]
E(R_ = -T(aY ,X,e (aY,aX ,e)], $ E[R-T(aY ,aX,e) €
where the first equal ity follow s as in (6), and the inequ ality since T(aY,a X,C) is the optim al stop rule and (by Lemma 2.1 of (3]) for (aY,aX ,C). Toget her (6) (7)
(8)
imply E[
RT(X, aY,c) (X,aY ,c
)_
~(aY, X,c)
(aY,X ,c)IX
(O,al] € ~
0 a.s.
the ( a, 1] and using , Simil arly, condi tionin g on X € d'tion al distribut~on fact that given X € (a,ll, the con ~
of X is un~604m on (0.,1], one has the following two relations:
(9)
E[RT(x,aY,c) (X,aY,c) IX l
E
E[max{Z,c}] ,
(0.,1]]
and (10)
E[RT(aY,X,c) (aY,X,c)
Ix
E
(0.,1]] E[R
(aY,Z,c)]
T(o.Y,X,c)
~ E[RT(o.Y,Z,c) (aY,Z,c)] E [max{Z ,c}], where Z is uniform (0.,1]
(and independent of Y,X).
From (9) and (10) follows the inequality corres ponding to (8) given that X
E
(0.,1], which together
with (8) yields (5) and completes the proof.
0
Let 0. ,0. "" be a 1 2 sequence of non-increasing positive numbers, and let
Theorem 3.2. (4.6(iii) of [11]).
Xl 'X 2 ' ••. be independent exponentially distributed ran dom variables with means 0. ,0.2"" respectively. Then 1 V(X l
,x 2 ,···)
= sup {V(X IT (1)'X 7T (2)'···: 7T
is a permutation of IN}.
Proof (due to Chris Klaassen).
By Proposition 4.5 of
[11] and renormalizing, it suffices to show
-c c
-~-~ x c (11) xe x + e -c
Georgia Institute of Technology
Atlanta, Georgia 30332
ABSTRACT Applications of the original prophet inequalities of Krengel and Sucheston are made to problems of order selection, non-measurable stop rules, look-ahead stop rUles, and iterated maps of random variables. Also, proofs are given of two results of Hill and Hordijk c?ncerning optimal orderings of uniform and exponential d~stributions.
§l.
INTRODUCTION Universal inequalities comparing the two func
tionals M
v
M(X ,X ,···) 2 l and
V(X ,X ,···) l 2
n
sup{EX : t is a stop rule t for
~esearch
E(suP Xn )
xl ,X 2 ,···}
partially supported by NSF Grants DMS-0160 and 01608.
4
het of seque nces of random varia bles are calle d "prop of tion inequ alitie s" becau se of the natur al inter preta ete M as the value to a proph et, or playe r with compl ving fores ight, in an optim al stopp ing proble m invol by random varia bles xl ,X 2 , • • . . First disco vered s have Kreng el and Suche ston [22, 23], these inequ alitie ons tigati been the subje ct of a numbe r of recen t inves [1,2, 4-15 ,17-2 1, 24-27 ]). to In §2, the appli cation s of proph et inequ alitie s or V inequ alitie s involv ing funct ionals other than M l menta are given , with atten tion focuse d on the funda (e.g.,
proph et inequ ality [23J (1)
M
are indep enden t and nonne gative , then 2V, and this bound is sharp .
xl ,X 2 , .•.
If ~
s (Analo gous appli cation s of other proph et inequ alitie to simil ar proble ms are left to the reade r.) Sectio n 3 conta ins proof s of two optim al-ord ering resul ts of Hill and Hordi jk [llJ. APPLICATIONS OF PROPHET INEQU ALITIE S The initia l disco very and appli catio n of proph et inequ alitie s such as (1) were made by Kreng el and semi Suche ston in conju nction with inves tigati ons of In amart s and proce sses with finite value [22, 23J. a this sectio n, other appli catio ns of the basic inequ proble ms lity (1) are given to sever al optim al-sto pping and an iterat ed map proble m. For the main appli cation theore m, wluch follow s §2.
imme diatel y from (1), let
=
,x2 , .•• ) l (More be any (real- value d) funct ional of Xl ' X2'··· . ite forma lly, U is a funct ion from C, the set of infin U
U(X
sequences of probability distributions, to the real numbers. In practice, U is usually Borel measurable, with C endowed with the product topology induced by the total-variation norm topology on the space of probabi lity distributions.) Let X 'x ' ••• be independent nonnegative l 2 random variables. Then (i) V ~ U implies U ::; 2Vi and (E) U ~ M implies M ::; 2lJ.
Theorem 2.1.
~.
Immediate from
(I) •
0
Application to Order Selection Let Us be the value of the sequence xl 'x 2 '··. to a player free to choose the order of observation of the random variables, as well as the time of stopping, that is,
Us
US {X l
,X ,···) ==
2
sup{V{XTI(1)'XTI{2~"'~): TI is a permutation of 1,2, ••• }.
(For a formal definition, including stochastic permuta ions TI, see [9].) Corollary 2.2.
Let X ,X , ••• be independent nonnega l 2 tive random variables. Then (i)
(Hill [9)
Us
~ 2Vi
and
(E) M s 2U ' S
Moreover, the bound in (i) is sharp. E) is a (Whether or not the constant "2" in ( sharp bound is not known to the author.) Inequality b tt than double 1) says that a player may never do e er . of a g1ven d his expected value by rearranging the or er sequence of random variables. Inequality (ii) is imme diate from (I) and the fact that Us ~ Vi only the ques 0
(
tion of its sharpness is of interest.
---- ---- _ _--_. _ . _
....
---~- ----- -~---
Appli cation to Use of Non-M easura ble stop Rules to a Let UN be the value of the seque nce Xl ,X 2 , ... i.e., playe r free to use non-m easura ble stop rules , be any integ er-va lued funct ions s for which {s = j} can ,X . (not neces sarily measu rable) funct ion of xl, ... j ional funct the is UN That is, UN
UN (X l ,X 2 ,···)
sup{EX : s is a "non- measu rable" stop rule} .
s
s (For a forma l defin ition, see [16].)
Let Xl ,x 2 , .•. be indep enden t nonne ga Corol lary 2.3. tive random varia bles. Then (i) (Hill and Pestie n [16] ) UN :5 2V; and (ii) H :5 2UW Horeo ver both bound s are sharp . (1); Proof . The inequ alitie s follow imme diatel y from bound the the sharp ness of (i) is in [16]. To see that X be in (ii) is sharp , let Xl be const ant +1, and let 2 (-1) = ( =1-P (X =0). a "long shot" [l2] given by P(X 2 2 0 1. = Us = UN Then M = 2-(, and Appli cation to "Look -Ahea d" Stop Rules to Let UA,k be the value of the seque nce x l ,X 2 , ... g lookin a playe r free to use stop rules s which allow func ahead k steps (i.e., integ er-va lued measu rable tions satisf ying {s = J'}
E
0
0 ) (X l ' •.• , X j +k ' s
UA,k(Xi, X 2 ,···) . SUp{EX s : s is a k-step "look -ahea d" stop rule} Let Xl ,X , ..• be indep enden t nonne ga Corol lary 2.4. 2 integ er. tive random varia bles, and let k be a posit ive Then
(i) UA,k :5 2V; and (ii) M $ 2U ,k. A
1-1ore over, both bound s are sharp . Proof . The inequ alitie s follow immed iately from (1). To see that (i) is sharp , let Xl = const ant +1, X2 = ... = Xk + l = const ant 0, and let Xk + 2 be a "long -1 h s h 0 t w~t P(X k + 2 = E ) = E = 1 P(X k +2 = 0); then UA,k = 2-E and V = 1. To see that (i) is sharp, let Il
•
Xl :: +1, X2 == ' " == Xk + 2 = 0, and let Xk + 3 be the "long and
shot" random varia ble just descri bed; then M = 2-E
o
Thus (i) says that a playe r able to look k steps into the futur e never has optim al expec ted return more and than twice that of a playe r who canno t look ahead, is (ii) says that a proph et's optim al expec ted return a never more than twice that of a playe r who may look 6ixed numbe r of steps into the future . On the other is hand, for a fixed seque nce of random variab les, it clear that lim U k(X l ,x 2 ",,) = M(X l ,x2 ,···)· A' ted Maps Itera ~plication to Let ¢(X,Y ) and w(X,Y) be the random variab les
¢(X,Y) = max{X ,y} and ~(X,Y) = max{X,EY}, and define the random varia bles ¢ n (Xn , ... ,X l ) and ~Il(Xn,···,Xl)
.
~nductively by
k .... oo
92(X ,X ) 2 1
E(max {Y,c}) if X $ E(rnax {Y,c}) and Y > Ci and = 3 otherw ise. [3] Lettin g X and Y be i.i.d . U[O,l ], by Lemma 2.1 of it follow s that (4) is equiv alent to (5)
E[RT( X,aY, c) (X,aY ,c) - ~(aY,X,c) (aY,X ,c)]
~
O.
To see (5), first observ e that (6)
E[RT( X,aY, c) (X,aY ,c) IX
[O,a]]
€
y ) (aX,aY ,c)],
T aX,a ,e m since the distri butio n of X given X € [O,a] is unifor On [O,a], that is, has the same distri butio n as aX. = E (R (
Next calcu late (7)
E(RT( aY,X, c) (aY,X ,c) IX
[O,a]]
) (ay,aX ,c)]
E(R_ = -T(aY ,X,e (aY,aX ,e)], $ E[R-T(aY ,aX,e) €
where the first equal ity follow s as in (6), and the inequ ality since T(aY,a X,C) is the optim al stop rule and (by Lemma 2.1 of (3]) for (aY,aX ,C). Toget her (6) (7)
(8)
imply E[
RT(X, aY,c) (X,aY ,c
)_
~(aY, X,c)
(aY,X ,c)IX
(O,al] € ~
0 a.s.
the ( a, 1] and using , Simil arly, condi tionin g on X € d'tion al distribut~on fact that given X € (a,ll, the con ~
of X is un~604m on (0.,1], one has the following two relations:
(9)
E[RT(x,aY,c) (X,aY,c) IX l
E
E[max{Z,c}] ,
(0.,1]]
and (10)
E[RT(aY,X,c) (aY,X,c)
Ix
E
(0.,1]] E[R
(aY,Z,c)]
T(o.Y,X,c)
~ E[RT(o.Y,Z,c) (aY,Z,c)] E [max{Z ,c}], where Z is uniform (0.,1]
(and independent of Y,X).
From (9) and (10) follows the inequality corres ponding to (8) given that X
E
(0.,1], which together
with (8) yields (5) and completes the proof.
0
Let 0. ,0. "" be a 1 2 sequence of non-increasing positive numbers, and let
Theorem 3.2. (4.6(iii) of [11]).
Xl 'X 2 ' ••. be independent exponentially distributed ran dom variables with means 0. ,0.2"" respectively. Then 1 V(X l
,x 2 ,···)
= sup {V(X IT (1)'X 7T (2)'···: 7T
is a permutation of IN}.
Proof (due to Chris Klaassen).
By Proposition 4.5 of
[11] and renormalizing, it suffices to show
-c c
-~-~ x c (11) xe x + e -c
