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43

FKDSWHU 4 LQJUHGLHQWV RI WKH EDVLF SDUDPHWULF SUREOHP

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44
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45

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Wkhvh wzr hvwlpdwruv duh gl>huhqw1 Rqh lv wkh LY hvwlpdwh iru d xqlyduldwh uhjuhvvlrq/ dqg rqh
lv wkh LY hvwlpdwh iru d pxowlyduldwh uhjuhvvlrq +zkhq wkh lqvwuxphqwv gr qrw vdwlvi| wkh vwdqgdug
ruwkrjrqdolw| uhtxluhphqw,1 Zh frph edfn wr wkhlu uhodwlyh suhflvlrq ehorz1
Hxohu Htxdwlrqv Wkh qh{w wzr sduwv ri H{dpsoh 6 jhqhudwh frqglwlrqdo prphqw uhvwulfwlrqv1 Wkh
edvlf uhihuhqfhv rq Hxohu htxdwlrq hvwlpdwlrq duh Kdoo/4<:;> dqg Kdqvhq dqg Vlqjohwrq/4<;51
Ohw 5 | ' E5• c é é é c 5| ä/ zkhuh 5| duh wkh vwdwh yduldeohv ghwhuplqlqj fxuuhqw sur?wdelolw|1 Sureohp
lv wr fkrrvh ið| E5 | äj zlwk hdfk ð| E5 | ä 5 KE5 | ä/ wr pd{lpl}h
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q ô iZE&ô c %ô ä ý SEðô äj
ô 'f


47

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Jlyhq wklv dvvxpswlrq/ wkh gl>huhqfh lq h{shfwhg glvfrxqwhg ydoxh ri wkh wzr surjudpv/ vd| {E"ä/
pxvw eh ohvv wkdq ru htxdo wr f ru doo " lq d qhljkerxukrrg ri }hur1 Zh frph edfn wr wkh txhvwlrq
ri wkh frqvwudlqwv lpsrvhg e| rqh ehorz/ exw iru qrz dvvxph wklv ihdvlelolw| frqglwlrq zh kdyh
{E"ä ' q ô . EýSEðô ä n qZE&ô E• ý Bä n ðô c %ô n• ä ý q SEðô n• ä
ýdýSEðô n "ä n qZE&ô E• ý Bä n ðô n "c %ô n• ý q SEðô n• ý E• ý Bä"äom5 |

ü

é fé
Vr/ iru h{dpsoh li ô ' f zh kdyh
{E"ä ' SEðf n "ä ý SEðf ä n q.f dZE&• c %• ä ý ZE&• n "c 5• äo ý q.f dSEð• ä ý SEð• n E• ý Bä"äo é f


48
E| dvvxpswlrqv rq wkh sulplwlyhv zh fdq pdnh wklv d gl>huhqwldeoh ixqfwlrq ri "1 Pruhryhu lw htxdov
3 dw " ' f 8 zklfk pxvw eh lwv pd{lpxp1 Khqfh
sdudphwhuv zh zdqw wr hvwlpdwh

Y{
Y" ƒ"'f

' f ru doorzlqj SEüä dqg ZEüä wr ghshqg rq

Sâð Eðf ( wä ý q.f iZ â& E&• c %• ( wä n Sâð Eð• ( wäE• ý Bäj ' f
Dvvxph qrz wkdw E&c %c ðä duh revhuydeoh/ dqg ohw
sEðf c 5• c ð• ( wä ' Sâð Eðf ( wä n qiûE5• ( wä n Sâð Eð• ( wäE• ý Bäjé
Wkhq
.f s Eðf c 5• c ð• ( wä ' f
dw wkh wuxh sdudphwhu ydoxh1 Frqvhtxhqwo|/
.s Eðf c 5• c ð• ( wä}E5f ä ' f

surylghg 5f lv nqrzq dw | ' f El1h1/ } lv phdvxudeoh zlwk uhvshfw wr wkh j0dojheud jhqhudwhg e|
yduldeohv nqrzq dw gdwh 3,1 Qrz ohw
6E5c wä ' sEðf c %• c ð• ( wä}E5f äé
Wkh fodvvlf Hxohu htxdwlrq hvwlpdwlrq dojrulwkp hpsor|v 6Eüä lq d p1r1p1 hvwlpdwru iru w1
Zh duh xvlqj wkhru| yhu| h{solflwo| khuh/ vr zh gr zhoo wr frqvlghu erwk zkdw wkh wkhru| uhtxluhv/
dqg zkdw lv uhtxluhg iru wkh wkhru|1 Li hlwkhu zh ylrodwh wkh wkhruhwlfdo uhtxluhphqwv/ ru wkh wkhruhwlfdo
uhtxluhphqwv duh qrw olnho| wr eh +dw ohdvw dssur{lpdwho|, vdwlv?hg e| wkh gdwd/ wkhq zh duh qrw olnho|
wr ghulyh ghvludeoh hvwlpdwruv ri wkh sdudphwhuv ri lqwhuhvw1 +Lq wkh sulru h{dpsohv/ zh frxog dozd|v
uhyhuw edfn wr wkh vwdwlvwlfdo lqwhusuhwdwlrq ri wkh ixqfwlrq zh duh hvwlpdwlqj> khuh wkhuh lv qr vxfk
lqwhusuhwdwlrq1 Rq wkh rwkhu kdqg/ lq rughu wr jhw d fdxvdo prgho/ zh duh jrlqj wr qhhg wr dwwdfk d
wkhruhwlfdo lqwhusuhwdwlrq wr wkh vwdwlvwlfdo prghov/ dqg wklv zloo uxq lqwr wkh vdph w|sh ri sureohpv1,
Khuh duh vrph sureohpv wkdw frxog dulvh zlwk Hxohu htxdwlrqv1
Sureohp 4

Wkh dvvxpswlrqv uhtxluhg gr qrw krog iru hdfk revhuydwlrqv1 Frqglwlrqv qhhghg

lqfoxgh= iru doo " vxfk wkdw m"m ÷ V +iru vrph V : f, vxfk wkdw
î Frqglwlrq +d,1 ðWf n " lv ihdvleoh/ dqg


49

FKDSWHU 4 LQJUHGLHQWV RI WKH EDVLF SDUDPHWULF SUREOHP
î Frqglwlrq +e,1 ðW• ý E• ý Bä" lv ihdvleoh z1s141
Ohw wkh fruuhvsrqghqfh gh?qlqj wkh fkrlfh vhw iru ð eh K dqg dvvxph KE5 | ä ' dfc 4ä1 Wkhq li

ðf ' f/ wkh shuwxuedwlrq lv qrw ihdvleoh ehfdxvh frqglwlrq +d, lv qrw vdwlv?hg1 Oxfnlo| zh fdq revhuyh
li ðf ' f vr zh fdq wkurz dzd| wkrvh revhuydwlrqv1 Pruhryhu li zh vhohfw rq ðf : f/ zh vhohfw rq d
yduldeoh nqrzq lq shulrg 3/ dqg khqfh wkh ruwkrjrqdolw| uhvxowv frqwlqxh wr krog1 Frqglwlrq EKä lv
â

â

pruh sureohpdwlf vlqfh zh qhyhu dfwxdoo| vhh ðW• E5 ä iru doo srvvleoh 5 1 Lw vwdwhv wkh glvwulexwlrq ri
ð• ú sxwv qr suredelolw| rq ðú• ' f1 Lw lv wkhuhiruh lqkhuhqwo| xqfkhfndeoh1
Sureohp 5

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gl>huhqwldeoh frvw ri lqyhvwphqw ixqfwlrq ehfrphv sureohpdwlf1 Lq prghov ri fdslwdo dffxpxodwlrq
rqh riwhq zruulhv derxw wkh gl>huhqwldelolw| ri SEðä dw wkh srlqw ð ' f/ dv wklv lv wkh srlqw dw zklfk
vpdoo pryhphqwv fduu| zlwk wkhp wkh gl>huhqfh ehwzhhq vhoolqj dqg sxufkdvlqj dqg lqvwdoolqj/ xqlwv
ri wkh vwrfn1 D frxqw ri wkh qxpehu ri revhuydwlrqv dw zklfk ð lv h{dfwo| }hur lq wkh gdwd rxjkw wr
surylgh vrph lqglfdwlrq ri zkhwkhu wklv lv olnho| wr eh dq lpsruwdqw sureohp lq dq| jlyhq dssolfdwlrq
+dqg lw riwhq lv,1
Qrwh= Li wkhvh dvvxpswlrqv duh qrw vdwlv?hg rqh fdq wr orrn iru d prgl?hg prphqw frqglwlrq1
Wkh sureohpv lq +4, dqg +5, fdq riwhq eh flufxpyhqwhg e| doorzlqj iru wkh srvvlelolw| ri shuwxuedwlrqv
zklfk duh frpshqvdwhg iru dw d udqgrp vwrsslqj wlph zklfk fdq eh pruh wkdq rqh |hdu lq ohqjwk +iru
d glvfxvvlrq vhh Sdnhv/4<<6,1
Sureohp 6

Xs xqwlo qrz zh kdyh dvvxphg wkdw wkhuh lv qr xqrevhuyhg vwdwh yduldeohv lq s Eüä

^ru wkdw zh frxog vxevwlwxwh rxw iru wkhp lq whupv ri revhuydeohv`1 Li wkhuh duh xqrevhuyhg vwdwh
yduldeohv wkhq wkhuh duh d ihz fdvhv wr frqvlghu1
8 Vrphwlphv zh fdq hlwkhu xvh vrphwklqj wkdw pdsv rqh wr rqh zlwk wkh xqrevhuydeoh ru
zh fdq frqvwuxfw d uhodwhg ixqfwlrq zklfk ghshqg rqo| rq revhuydeohv wkdw doorzv xv wr
xvh wkh Hxohu Whfkqltxh1 H1j1/ Free0 Grxjodv Surgxfwlrq ixqfwlrq zkhuh doo lqsxwv exw
fdslwdo duh yduldeoh dqg % lv xqrevhuyhg surgxfwlylw|1 Sur?wv dqg fdslwdo duh revhuyhg1 Lq
wklv fdvh Z â& E&• c %• ( wä ' w& ZE&• c %• ( wä* &• 1

8 Vrphwlphv zh nqrz wkh glvwulexwlrq ri wkh xqrevhuyhg vwdwhv frqglwlrqdo rq wkh revhuyhg
vwdwh yduldeohv1 Wkhq zh duh ohiw rqo| zlwk dq lqwhjudwlrq sureohp1


4:
8 Wkh prvw olnho| fdvh lv wkdw wkh| duh uhodwhg wr wkh revhuyhg vwdwh yduldeohv dqg wkh irup
ri wkh uhodwlrqvkls lv qrw reylrxv +wklv lqfoxghv wkh fdvh ri phdvxuhphqw huuru,1 Wkhq zh
duh lq wurxeoh zlwk Hxohu htxdwlrqv1
Sureohp 7

Zh qrw rqo| qhhg wkdw wkh prphqw frqglwlrq krogv iru hdfk djhqw/ zh uhtxluh wkdw

wkh frqglwlrqdo glvwulexwlrq +frqglwlrqdo rq 5 | ä ri wkdw prphqw frqglwlrq kdv hqrxjk 3lqghshqghqfh4
dfurvv revhuydwlrqv iru dyhudjlqj frqglwlrqv wr zrun1 Wklv lv hqvxuhg li wkh frqglwlrqdo glvwulexwlrq ri
wkh yhfwru ri wkh shulrg | n • vwdwh yduldeohv E5|n• ä ri wkh gl>huhqw djhqwv +frqglwlrqdo rq lqirupdwlrq
nqrzq dw gdwh |ä idfwruv lqwr wkh surgxfw ri wkh glvwulexwlrqv wkdw wkrvh djhqwv dfwxdoo| xvh lq
iruplqj wkhlu rzq h{shfwdwlrqv1 Lq dgglwlrq wr udwlrqdo h{shfwdwlrqv/ wklv uhtxluhv lqghshqghqfh ri
wkh frqglwlrqdo glvwulexwlrqv ri wkh i5|n•cð jù
ð'• 1 Lw lv wklv lqghshqghqfh zklfk lqvxuhv wkdw wkh dyhudjh
ri wkh Hxohu htxdwlrq glvwxuedqfhv/ dyhudjhg ryhu wkh lqglylgxdo*v lq wkh vdpsoh/ frqyhujhv wr }hur
dw wkh wuxh wf / d surshuw| zklfk olhv dw wkh khduw ri wkh frqvlvwhqf| surri zh zloo xvh1
Uhfdoo wkdw 5|n• pxvw lqfoxgh doo yduldeohv zklfk kdyh hlwkhu dq lqghshqghqw h>hfw rq wkh ydoxh ri
wkh Hxohu htxdwlrq/ ru duh ghwhuplqdqwv ri wkh ydoxh ri frqwuro yduldeohv zklfk/ lq wxuq/ duh dujxphqwv
lq wkh Hxohu htxdwlrq1 Iuhtxhqwo| wklv ohdyhv d orw ri urrp iru vwdwh yduldeohv zkrvh uhdol}dwlrqv zloo
hlwkhu eh frpprq/ ru kljko| fruuhodwhg/ dfurvv djhqwv +w|slfdo h{dpsohv lqfoxgh sulfhv/ ohyhov ri
whfkqrorj|/ dqg jryhuqphqwdo srolf| yduldeohv,1
Qrwh wkdw wkh idfw wkdw wkhuh lv ghshqghqfh lq wkh frqglwlrqdo glvwulexwlrq ri wkh 5|n•cð dfurvv
ð grhv qrw qhfhvvdulo| uxoh rxw rxu frqvlvwhqf| frqglwlrq1 Wkdw lv/ wkh vdph udqgrp yduldeoh pd|
d>hfw gl>huhqw djhqwv lq gl>huhqw zd|v/ vr wkdw wkh ghshqghqfh lw lqgxfhv lq wkh uhdol}dwlrq ri wkh
Hxohu htxdwlrq huuruv pd| qrw eh vwurqj hqrxjk wr lqydolgdwh wkh frqyhujhqfh ri wkh vdpsoh dyhudjh

ri wkh wuxh Hxohu htxdwlrq glvwxuedqfhv wr }hur1 Rq wkh rwkhu kdqg/ zkhq wkhuh duh vwdwh yduldeohv
wkdw kdyh lpsruwdqw lpsdfwv rq ehkdylrxu wkdw duh olnho| wr lqgxfh ghshqghqfh dfurvv djhqwv/ wkhq
wkh dujxphqwv zh kdyh ehhq lpsolflwo| uho|lqj rq iru wkh frqvlvwhqf| ri wkh Hxohu htxdwlrq edvhg
hvwlpdwruv duh dw ehvw lqfrpsohwh +dqg pd| eh vhulrxvo| plvohdglqj,> dqg yhu| olwwoh zrun kdv ehhq
grqh rq wkh surshuwlhv ri Hxohu htxdwlrqv xqghu wkhvh flufxpvwdqfhv1 Iru ixuwkhu glvfxvvlrq vhh
Fkdpehuodlq +B ,1 Qrwh khuh wkdw wklv sureohp pd| jr dzd| li wkh wlph glphqvlrq lv orqj hqrxjk1
Sureohp 8

Hxohu htxdwlrqv uhtxluh wkdw zh kdyh wkh wlplqj dvvxpswlrqv mxvw uljkw1 Zh riwhq

?qg wkdw wkh lpsolfdwlrqv ri wkh Hxohu htxdwlrq vhw xs duh qrw vdwlv?hg lq hpslulfdo sureohpv dqg
rqh srvvleoh fxosulw pd| eh vhqvlwlylw| wr wkh wlplqj frqyhqwlrqv +zh duh hvwlpdwlqj r> ri gl>huhqfhv


4;

FKDSWHU 4 LQJUHGLHQWV RI WKH EDVLF SDUDPHWULF SUREOHP

lq srolflhv ryhu wlph/ zklfk duh xvxdoo| vpdoo dqg pd| eh vzdpshg e| rwkhu vrxufhv ri huuru,1 Grlqj
shuwxuedwlrqv ryhu d orqjhu krul}rq wkhq rqh |hdu pd| lq idfw whqg wr zrun ehwwhu hpslulfdoo|/ exw L
kdyh qrw vhhq pxfk zrun wkdw h{dplqhv wklv1
Qdvk Htxloleuld Lq wkhvh h{dpsohv zkdw wkh prgho vshfl?hv lv d vhw ri ixqfwlrqv ûð G U_? $ Uû/
vxfk wkdw
.iûð E5• c é é é c 5? ( wäm5•• c é é é c 5•? j ' fé
Zh fdq dozd|v frqvwuxfw }ð dv d ixqfwlrq ri E5•• c é é é c 5•? ä dqg wkhq i6ð Ewä ' ûð Ewä û }ð j?ð'• / vxfk wkdw
•[
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CEwä '
? ð'•
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Krzhyhu/ wkh uhdol}dwlrqv ri wkh 6ð Eüc wä ri wkh gl>huhqw djhqwv duh qrz uhodwhg wr rqh dqrwkhu/ vr
wkhuh lv d txhvwlrq ri frqglwlrqv wkdw zloo ohdg wr wkh frqvlvwhqf| dqg dv|pswrwlf qrupdolw| ri dq
hvwlpdwru wkdw plqlpl}hv
•[
nC? Ewän ' n
6ð Eüc wäné
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Qdvk Htxloleulxp lv d vshfldo fdvh ri wklv1 Wdnh iru h{dpsoh d vwdqgdug sulflqj htxloleulxp lq d
pdunhw zlwk ? surgxfwv hdfk surgxfhg e| gl>huhqw ?upv zkrvh vwdwh yduldeohv duh ghqrwhg e| 51
Wkhq wkh htxloleulxp dvvxpswlrq lv Rð vdwlv?hv
4@ .dZ ð ER• c é é é c R? c 5• c é é é c 5? ( wämR• c é é é c R? c 5•• c é é é c 5•? oé
Rð

Jlyhq dssursuldwh uhjxodulw| wklv lpsolhv
.d

YZ ð Eü( wä
mR• c é é é c R? c 5•• c é é é c 5•?j ' fc
YRð

zkhuh wkh 5•æ duh wkh yduldeohv nqrzq zkhq wkh sulfhv kdyh wr eh vhw1 Li zh revhuyh YZ ð Eü( wä*YRð
+zklfk zrxog eh wkh fdvh li/ iru h1j1/ zh revhuyhg sur?wv/ dqg wkh ghulydwlyh ri sur?wv fdq eh h{suhvvhg
dv d vlpsoh ixqfwlrq ri sur?wv,/ dqg fdq frpsxwh wkh dssursuldwh h{shfwdwlrq/ wkhq
YZ ð Eü( wä
YZ ð Eü( wä
' .d
mR• c é é é c R? c 5•• c é é é c 5•? o n "ð

YRð
YRð
zkhuh .d"ð mR• c é é é c R? c 5• c é é é c 5? o ' f/ iru ð ' •c é é é c ?/ vr rxu prphqw frqglwlrqv duh


4<
é

è
YZ ð Eü( wä
YZ ð Eü( wä
6ð Eüc wä '
ý .d
mR• c é é é c R? c 5•• c é é é c 5•? o}ð Eüä é
YRð
YRð
Dowhuqdwlyho| zh pljkw kdyh d prgho iru
.d

YZ ð Eü( wä
mR• c é é é c R? c 5•• c é é é c 5•? o
YRð

lq whupv ri revhuyhg dqg xqrevhuyhg yduldeohv dqg hvwlpdwh gluhfwo| r> ri wkdw +wklv lv krz prvw ri
wkh L1R1 olwhudwxuh zrunv,1
Qrwh wkdw doo ri wkh sureohpv lq wkh ?uvw fdvh uhdsshdu khuh/ exw wkhuh uhodwlyh lpsruwdqfh gl>huv1
Lq sduwlfxodu wkh ghshqghqfh sureohp uhdsshduv lq vsdghv1 Iluvw/ li wkh i"ð j uhsuhvhqw dq|wklqj
exw phdvxuhphqw huuru wkh| duh=
8 olnho| wr kdyh frpprq ghwhuplqdqwv dfurvv ?upv +ghpdqg ru vxsso| vkrfnv,1
8 hyhq li wkhuh zdv qr ghshqghqfh lq wkh i"ð j wkh dgmxvwphqw surfhvv wkdw uhvxowv iurp wkh

" uhdol}dwlrqv lv olnho| wr fdxvh wkh prphqw frqglwlrq iru wkh ð|û ?up wr ghshqg rq wkh
" ri wkh æ |û ?up1 Qrwh wkdw li wkhuh duh uhdo 3vwuxfwxudo glvwxuedqfhv4 +lq frqwudvw wr
phdvxuhphqw huuru, wkh h{shfwdwlrq/ .dYZ ð Eü( wä*YRð mR• c é é é c R? c 5•• c é é é c 5•? äo}ð Eüä/ lv ryhu
wkh i"æ j1 Ri frxuvh zh frxog dozd|v lqwhjudwh rxw ryhu wklv glvwulexwlrq/ exw wklv uhtxluhv
wkh zkroh glvwulexwlrq ri wkh i"æ j dqg qrw mxvw wkh frqglwlrqdo phdq dvvxpswlrq/ wklv
glvwulexwlrq pljkw uhtxluh dq dgglwlrqdo vhw ri dgmxvwphqw sureohpv/ dqg lw pljkw dovr
uhtxluh fdofxodwlqj d frpsolfdwhg lqwhjudo1
Wkh vhfrqg sureohp wkdw rffxuv khuh lv wkh sureohp ri xqrevhuyhg vwdwhg yduldeohv wkdw duh
nqrzq wr wkh djhqw exw qrw wkh hfrqrphwulfldq1 Ohw 5•ð ' E%ð c # ð ä/ zlwk # ð uhsuhvhqwlqj surgxfwlylw|
ru surgxfw txdolw| gl>huhqfhv wkdw duh nqrzq wr wkh djhqw exw qrw wkh hfrqrphwulfldq1 Wkhq rxu
prphqw frqglwlrq lv
.d6ð E5• c 52 c é é é c 5? äm%• c # • c %2 c #2 c é é é c %? c # ? o ' fc
vr qrz lq rughu wr irup wkh prphqw frqglwlrq dv d ixqfwlrq ri revhuydeohv zh kdyh wr pdnh dq
dvvxpswlrq rq wkh glvwulexwlrq ri # frqglwlrqdo rq %/ dqg lqwhjudwh rxw zlwk uhvshfw wr wklv glvwule0
xwlrq1 Wklv lv sduwlfxoduo| sureohpdwlf li/ dv lv olnho|/ % lv d ixqfwlrq ri #1 Hyhq zkhq lw lv qrw/ li wkh
# gr qrw hqwhu wkh ghulydwlyh olqhduo| +dv wkh| gr lq EOS,/ zh duh olnho| wr hqfrxqwhu d qrqdqdo|wlf
lqwhjudo wkdw zh zloo kdyh wr wdnh dffrxqw ri1


53

FKDSWHU 4 LQJUHGLHQWV RI WKH EDVLF SDUDPHWULF SUREOHP

H{dpsoh 71 Vlpxodwlrq
Wdnh dq| ri wkh h{dpsohv deryh dqg dvvxph wkdw wkhuh lv dq dgglwlrqdo vrxufh +ru vrxufhv, ri xqre0
vhuyhg udqgrpqhvv/ vr wkdw wkh revhuyhg ixqfwlrq/ 6E5c wä/ lv fdofxodwhg dv
]
6E5c wä ì • E#c 5c wäå E_#m5äc
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