Ilupv frph dqg jr/ Oderu vwd|v=
Djjorphudwlrq lq Kljk0Whfk Lqgxvwulhv|
Khlnr D1 Jhuodfk4 /
Wkrpdv Uûqgh5 dqg
Nrqudg Vwdko6
Dsulo 5334
Suholplqdu| dqg Lqfrpsohwh1 Frpphqwv duh frugldoo| lqylwhg1
| Zh judwhixoo| dfnqrzohgjh qdqfldo vxssruw iurp wkh Ghxwvfkh Iruvfkxqjvjh0
phlqvfkdiw +judqw VSS 4<</ Vwd 49<2<06, dqg wkh Hxurshdq Frpplvvlrq xqghu wkh
WPU qhwzrun Wkh Hyroxwlrq ri Pdunhw Vwuxfwxuh lq Qhwzrun Lqgxvwulhv+IPU[0
FW<;03536,1
 FHUDV0HQSF/ Sdulv/ hpdlo= khlnr1jhuodfkChqsf1iu1
2 Xqlyhuvlw| ri Pdqqkhlp1 hpdlo= urhqghChfrq1xql0pdqqkhlp1gh dqg FHSU1
Xqlyhuvlw| ri Pdqqkhlp/ ]HZ dqg FHSU/ hpdlo= nrqudgChfrq1xql0pdqqkhlp1gh1
4
Lqwurgxfwlrq
Zkdw duh wkh fkdudfwhulvwlfv ri upv dqg surgxfwv wkdw ohdg upv wr djjorphudwh
lq jhrjudsklfdo vsdfh/ yv1 wkrvh wkdw ohdg upv wr vhhn d vrolwdu| orfdwlrqB Krz
gr wkh lqfhqwlyhv wr djjorphudwh fkdqjh zlwk fkdqjhv lq h{rjhqhlwlhv/ vxfk dv
frqvxphuv* suhihuhqfhv/ wudqvdfwlrq frvwv/ ru sxeolf vxevlglhvB Duh wkh lqfhqwlyhv
wrzdugv djjorphudwlrq +ru glvshuvlrq, lq kduprq|/ ru lq glvkduprq| zlwk wkrvh
ohdglqj wkh vrfldo sodqqhu wr sursrvh djjorphudwlrq yv1 glvshuvlrqB Wkh txhvwlrq
ri djjorphudwlrq ihdwxuhv fhqwudoo| lq wkh Qhz Hfrqrplf Jhrjudsk|1 \hw olwwoh kdv
ehhq vdlg wr gdwh lq wkdw olwhudwxuh derxw wkh uhodwlrqvkls ehwzhhq upv* lqqrydwlyh
dfwlylw| dqg wkhlu lqfhqwlyhv wrzdugv djjorphudwlrq1
Wkh lqwhuhvw lq wklv lvvxh lv idu iurp ehlqj rqo| ri d wkhruhwlfdo qdwxuh1 Jh0
rjudsklfdo djjorphudwlrqv ri lqqrydwlyh lqgxvwuldo dfwlylw|/ vxfk dv frpsxwhu vfl0
hqfhv lq Vlolfrq Ydooh|/ elrwhfk lq Od Mrood +Fdoliruqld, ru qhxurvflhqfhv lq R{irug
+XN,/ duh wkrxjkw wr frqwulexwh sduwlfxoduo| pxfk wr wkh jhqhudwlrq ri hpsor|0
phqw dqg lqghhg/ wr ohdg wkh jurzwk sdwk ri hqwluh qdwlrqdo hfrqrplhv1 Wdnh
Vlolfrq Ydooh| dv dq h{dpsoh1 Lq vslwh ri xsv dqg grzqv lq hpsor|phqw ghpdqg
gxulqj wkh qlqhwlhv ri wkh odvw fhqwxu|/ wkh hpsor|phqw jurzwk udwh lq Vlolfrq Ydo0
oh| rxwsdfhg zlwk dq lpsuhvvlyh 48 shu fhqw wkh X1V1 qdwlrqdo hpsor|phqw jurzwk
udwh/ dqg wkh phdq shuvrqdo lqfrph shu khdg zdv xs wr 83 shu fhqw kljkhu wkdq
wkh fruuhvsrqglqj qdwlrqdo lqfrph jxuh +Dxguhwvfk/ 4<<;,1 Wklv lpsuhvvlyh shu0
irupdqfh surprwhg d zruogzlgh lqwhuhvw lq uhsolfdwlqj lw1 Eloolrqv ri grooduv zhuh
vshqw e| orfdo/ uhjlrqdo dqg qdwlrqdo jryhuqphqwv wr surprwh forqhv ri Vlolfrq Ydo0
oh|1 \hw vxffhvv udwhv zhuh qhjoljleoh/ zklfk lqglfdwhv wkdw wkh irufhv ehklqg wkhvh
djjorphudwlrq surfhvvhv duh pruh vxewoh wkdq wkrxjkw ri khuhwriruh1
Lq vslwh ri wkdw k|sh lq wkh lqwhuhvw lq uhsolfdwlqj Vlolfrq Ydooh|/ lw lv lpsruwdqw
wr dvn zkhwkhu wkh glvshuvlrq ri vxfk lqqrydwlyh dfwlylwlhv zrxog lqgxfh qrw rqo| d
pruh htxlwdeoh doorfdwlrq/ exw frxog dovr lqfuhdvh wkh h!flhqf| lq wkh doorfdwlrq ri
hfrqrplf dfwlylw|1 Lw lv wkxv ri xwprvw lpsruwdqfh wr lqyhvwljdwh/ dqg wr hydoxdwh
lq ghwdlo wkh uhdvrqv iru vxfk djjorphudwlrqv wr irup1
Pruh vshflfdoo|/ wkh txhvwlrq lv dv wr wkh vwudwhjlf irufhv wkdw ohdg wkhvh lqqr0
ydwlyh upv wr fkrrvh orfdwlrqv forvh wr hdfk rwkhu/ dqg wkxv wr rsw iru djjuhvvlyh
frpshwlwlrq lq hlwkhu lqsxw/ ru rxwsxw pdunhwv ru erwk/ udwkhu wkdq wr hydgh lw1 Wkh
vwru| lq idyru ri djjorphudwlrq sursrvhg e| gdwd dqdo|vwv h1j1 Vd{hqldq +4<<7,/
Kdukr +4<<8,/ Dxguhwvfk dqg Ihogpdq +4<<8/ 4<<9, ru Dxguhwvfk +4<<;, lv wkdw
upv vhhn wr sduwdnh lq dq lqirupdwlrq ulfk hqylurqphqw= upv ehqhw iurp
foxvwhulqj wrjhwkhu ehfdxvh wdflw nqrzohgjh lv wudqvplwwhg hlwkhu lq lqirupdo frq0
wdfwv ehwzhhq hpsor|hhv ri glhuhqw upv/ ru yld wkh pryhphqw ri wkhvh hpsor|hhv
ehwzhhq wkhp1 Revhuyh/ krzhyhu/ wkdw wklv wkhvlv lv rhuhg zlwkrxw gluhfw surri1
Rxu h{sodqdwlrq ri djjorphudwlrq lq lqqrydwlyh lqgxvwulhv grhv qrw exlog rq
5
vslooryhuv/ exw rq wkh lqkhuhqw vwrfkdvwlflw| ri U)G dfwlylwlhv/ dqg lwv frqvhtxhqfhv
rq wkh ghpdqg idfhg/ dqg wkhuhiruh wkh lqsxw ghpdqg h{huflvhg/ e| frpshwlqj
upv1 Lq klv Sulqflsohv/ Pduvkdoo +4<53, dujxhg wkdw upv kdyh lqfhqwlyhv wr
orfdwh lq wkh vdph uhjlrq zkhq wkh| idfh lpshuihfwo| fruuhodwhg vwrfkdvwlf oderu
ghpdqgv1 Lq wkh h{wuhph fdvh zlwk shuihfwo| qhjdwlyh fruuhodwlrq ehwzhhq wkh
oderu ghpdqgv h{huflvhg e| wzr upv/ wkh up eohvvhg zlwk kljk ghpdqg fdq
gudz oderu dw orz frvw iurp d frpprq orfdo oderu pdunhw srro/ dv wkh rwkhu
up*v oderu ghpdqg lv orz1 Pduvkdoo*v dujxphqw zdv uhqhzhg dqg srsxodul}hg e|
Nuxjpdq +4<<4/ Fk1 5 dqg Dss1E,1 Zh xvh/ h{whqg dqg frpsohwh wklv dujxphqw
wklv e| ghprqvwudwlqj krz wkh vwrfkdvwlfv ri U)G uhvxowv dhfw upv* rxwsxw/ dqg
wkhuhiruh lqsxw ghpdqgv1
Lq rxu edvholqh prgho/ wzr upv vxsso|lqj wr frpshwlwlyh zruog pdunhwv dw
frpshwlwlyh txdolw| dgmxvwhg sulfhv duh idfhg zlwk vwrfkdvwlfdoo| lqghshqghqw lq0
qrydwlrq vkrfnv wkdw lpsuryh rq wkhlu surgxfw txdolw|1 Wkh vkrfnv wudqvodwh lqwr
vwrfkdvwlf surgxfw/ dqg wkhuhiruh oderu ghpdqgv1 Frqiurqwhg zlwk orfdwlrq ghfl0
vlrqv ehiruh wkh rxwfrph ri wkhvh vkrfnv lv nqrzq/ wkh upv fdq ghflgh wr hlwkhu
orfdwh vhsdudwho| lq d orz ghqvlw| oderu pdunhw hdfk lq zklfk wkh| duh prqrsvrq0
lvwv/ ru wr orfdwh +mrlqwo|, lq d frpshwlwlyh kljk ghqvlw| oderu pdunhw lq zklfk wkh
vdph oderu lv dydlodeoh dw orzhu zdjhv1 Frqwudu| wr wkh h{shfwdwlrq wkdw lq wklv
vlwxdwlrq wkh upv dozd|v whqg wr sduwdnh lq wkh odwwhu/ wkh upv rsw wr djjorp0
hudwh rqo| li wkh lqqrydwlrq vwhsv duh kljk dqg fkdqfhv duh kljk wr hqg xs lq dq
dv|pphwulf vlwxdwlrq lq zklfk rqh ri wkh upv lv vxffhvvixo zklovw wkh rwkhu lv qrw1
Wkh uhdvrq lv wkdw lq h{shfwdwlrq whupv/ wkh upv jdlq rqo| xqghu dv|pphwu|/ eh0
fdxvh li frqiurqwhg zlwk d v|pphwulf rxwfrph ri wkh lqqrydwlrq vkrfn/ wkhlu surwv
duh gloxwhg e| frpshwlwlrq lq wkh oderu pdunhw1 Wkdw yhu| irufh lv rshudwlyh dovr
xqghu dv|pphwu|/ ehfdxvh li wkh lqqrydwlrq vwhs lv orz/ wkh ohdglqj up pxvw sd|
d uhodwlyho| kljk zdjh wr dwwudfw d odujh vkduh ri wkh dydlodeoh oderuhuv/ zklfk djdlq
gloxwhv surwv1 Rqh pd| dujxh wkdw wklv uhvxow lv vwurqjo| ghshqghqw rq wkh idfw
wkdw wkh prgho lv d vwdwlf rqh1 \hw zh fdq vkrz e| d vlpsoh dujxphqw wkdw lw lv
d uhgxfhg irup uhsuhvhqwdwlrq ri d g|qdplf txdolw| odgghu prgho ã od Jurvvpdq
dqg Khospdq +4<<4,1
Zh surylgh vhyhudo h{whqvlrqv wr wklv prgho1 D sduwlfxoduo| lqwhuhvwlqj rqh/
lq ylhz ri wkh diruhphqwlrqhg hpslulfdo olwhudwxuh/ uhodwhv wr wkh lqwurgxfwlrq ri
lqqrydwlrq vslooryhuv1 Lq rxu prgho/ lqqrydwlrq vslooryhuv dhfw wkh orvlqj up*v
suredelolw| wr vwloo eh eohvvhg zlwk d txdolw| lpsurylqj lqqrydwlrq1 Lq ylhz ri
rxu edvholqh uhvxowv/ wklv vkrxog xqhtxlyrfdoo| uhgxfh/ udwkhu wkdq lqfuhdvh wkh
djjorphudwlrq lqfhqwlyh1 \hw d ghwdlohg dqdo|vlv vkrzv wkdw wklv dujxphqw pxvw
eh txdolhg rq wkh diruhphqwlrqhg sdudphwhuv1 Zh qdoo| glvfxvv wkh urexvwqhvv
ri rxu uhvxowv/ e| dqdo|}lqj wkh frqvhtxhqfhv ri lqwurgxflqj Ehuwudqg frpshwlwlrq
lq wkh lqsxw pdunhw dqg e| hqgrjhql}lqj wkh U)G hruwv1 Eoxqwo| vshdnlqj/ rxu
6
txdolwdwlyh uhvxowv uhpdlq xqdhfwhg12
Zh qrz wxuq wr d uhylhz ri wkh uhodwhg wkhruhwlfdo olwhudwxuh1 Wr wkh ehvw ri
rxu nqrzohgjh/ doo sdshuv olqnlqj orfdwlrq ghflvlrqv wr lqqrydwlrq h{klelw vslooryhuv
dv wkh xqltxh djjorphudwlyh irufh1
Frpehv dqg Gxudqwrq +5334, dqdo|}h oderu prelolw| dv d vrxufh ri hqgrjhqrxv
vslooryhuv1 Ilupv foxvwhu wr eh deoh wr *srdfk* frpshwlwruv* hpsor|hhv dqg hqmr|
whfkqrorj| vslooryhuv1 Vlqfh upv zlvk wr nhhs wkhlu hpsor|hhv dqg surwhfw wkhlu
lqwhoohfwxdo surshuw|/ srdfklqj ohdgv wr kljkhu zdjhv xqghu foxvwhulqj1 Frpshwlwlrq
lq wkh oderu pdunhw lv wkxv/ dv lq rxu prgho/ wkh ghjorphudwlqj irufh1
Jhuvedfk dqg Vfkpxw}ohu +4<<<, dqdo|vh djjorphudwlrq gxh wr orfdolvhg whfkqr0
orjlfdo vslooryhuv zlwklq d glvfuhwh orfdwlrq iudphzrun1 Lq wkhlu prgho upv duh qrw
rqo| uhflslhqwv ri srvlwlyh vslooryhuv/ exw wkh| pd| dovr eh wkh ylfwlp ri xqghvluhg
lqirupdwlrq ohdndjhv1 Wkhlu irfxv lv rq wkh lqwhusod| ehwzhhq lqwhuqdo lqwhusodqw
vslooryhuv dv zhoo dv h{whuqdo lqwhuup vslooryhuv iurp frvw uhgxflqj U)G hruwv rq
wkh sodqw orfdwlrq ghflvlrqv ri gxrsrolvwlf pxowlsodqw upv1 D yduldqw pdujlqdoo|
uhodwhg wr wklv/ |hw xquhodwhg wr rxu sdshu lv surylghg lq Jhuvedfk dqg Vfkpxw}ohu
+5333,1
Pdl dqg Shqj +4<<<, frqfhqwudwh dq h{solflw fr0rshudwlrq ehwzhhq gxrsrolvwlf
upv dv wkh h{solflw uhdvrq iru djjorphudwlrqv wr irup1 Wkh| prgli| wkh Krwhoolqj0
iudphzrun e| dvvxplqj wkdw glvwdqfh ghshqghqw frppxqlfdwlrq ehwzhhq frpshwl0
wruv uhgxfhv surgxfwlrq frvwv1 Lq wkhlu prgho/ wkh upv zloo v|pphwulfdoo| pryh
wkh forvhu wr hdfk rwkhu/ wkh vwurqjhu wkh frvw uhgxflqj hhfw ri frppxqlfdwlrq/
dqg wkh pruh gudvwlf wkh lqfuhdvh lq frppxqlfdwlrq frvwv zlwk glvwdqfh1 Krzhyhu/
wkh upv zloo ixoo| djjorphudwh rqo| li wkdw srvlwlyh h{whuqdolw| lv lqqlwho| odujh1
Wkhuh lv dovr d vpdoo olwhudwxuh rq upv* orfdwlrq ghflvlrqv uhodwlyh wr orfdol}hg
oderu pdunhwv1 Krzhyhu/ Wrsho +4<;9,/ Edxpjdugqhu +4<;;,/ ru prvw uhfhqwo|
Slfdug dqg Wrxohprqgh +5333, doo irfxv rq lvvxhv glhuhqw iurp rxuv/ vxfk dv
zrunhuv* pljudwlrq lqfhqwlyhv/ glylvlrq ri oderu dv fkdqjlqj zlwk oderu pdunhw vl}h/
dqg djjorphudwlrq dv d uhvxow ri vxsso| hodvwlflw| ri oderu/ uhvshfwlyho|1
Wkh uhpdlqghu ri wkh sdshu lv rujdql}hg dv iroorzv= Lq wkh qh{w vhfwlrq/ zh
suhvhqw rxu edvholqh prgho1 Lq vhfwlrqv 6 / zh ghwhuplqh dqg fkdudfwhul}h wkh
sulfh htxloleulxp/ dqg wkh orfdwlrq htxloleulxp/ uhvshfwlyho|1 Lq Vhfwlrq 7/ zh
frqvlghu Ehuwudqg frpshwlwlrq dqg hqgrjhql}h U)G ghflvlrqv dqg djdlq ghulyh
dqg fkdudfwhul}h htxloleuld1 Wkh frqfoxglqj vhfwlrq/ vhuyhv wr vxppdul}h rxu uhvxowv
dqg wr vshfxodwh derxw srvvleoh h{whqvlrqv1
5
Wkh Edvh Prgho
Wkhuh duh wzr upv/ 4 dqg 5/ dqg wzr orfdwlrqv1 Wkh upv fdq hlwkhu orfdwh
wrjhwkhu lq rqh uhjlrq ghqrwhg e| w|sh 4/ ru orfdwh lq glhuhqw uhjlrqv ghqrwhg e|
7
w|sh 51 Lq erwk vlwxdwlrqv/ wkh zrunhuv orfdwh xqlirupo| dorqj d olqh ri d ohqjwk
vxfk wkdw lwv rxwhu olplw grhv qrw dhfw wkh prgho rxwfrph1 Wkxv/ e| dvvxpswlrq/
wkh upv duh qhyhu udwlrqhg lq wkh oderu pdunhw1 Wkh upv orfdwh dozd|v orfdwh
rq wkh lqqhu hqg ri wkh olqh1 Xqghu djjorphudwlrq/ wkh ghqvlw| ri zrunhuv dorqj
wkh olqh lv 4> l1h1/ dw hdfk srlqw wkhuh lv suhflvho| rqh zrunhu1 Xqghu vhsdudwlrq/
wkh upv gudz iurp glhuhqw oderu srrov/ dqg wkxv duh orfdwhg dw wkh lqqhu hqg ri
wzr glhuhqw olqhv1 Krzhyhu/ wkh ghqvlw| ri zrunhuv dorqj hdfk olqh lv rqo| 4@51
D zrunhu orfdwhg dw { lqfxuv d wrwdo wudyhoolqj frvw ri {w zkhq zrunlqj iru d up
zkhuh w lv d phdvxuh ri wkh lpsruwdqfh ri wudqvsruwdwlrq frvwv/ ru suhihuhqfh iru
zrunlqj lq d prgho up1 Wkh zrunhuv kdyh dq rxwvlgh lqfrph rswlrq/ zklfk zh
qrupdol}h wr 31 D zrunhu zloo wkxv rqo| dffhsw hpsor|phqw li wkh zdjh lv odujhu
wkdq khu wrwdo wudqvsruwdwlrq frvw1 Ghqrwh e| Oc wkh oderu ghpdqg ri up l
xqghu orfdwlrq fkrlfh m/ m @ D+djjorphudwlrq, dqg V+hsdudwlrq,1 Wkh lqyhuvh
oderu vxsso| ixqfwlrqv xqghu djjorphudwlrq dqg vhsdudwlrq/ duh
z +Oc . O2c , @ +Oc . O2c ,w dqg z7 +Oc7 , @ 5Oc7 w1
Li wkh upv djjorphudwh lq uhjlrq w|sh 4/ wkh| frpshwh lq wkh oderu pdunhw1
Iru qrz/ zh dvvxph wkdw wkhuh lv txdqwlw| frpshwlwlrq lq wkh oderu pdunhw/ exw
zh zloo odwhu frqvlghu wkh fdvh ri sulfh frpshwlwlrq1 Xqghu vhsdudwh orfdwlrqv/ wkh
upv gr qrw frpshwh lq wkh oderu pdunhw dqg ehkdyh dv prqrsvrqlvwv1e Wkh upv
surgxfh wkhlu rxwsxw xvlqj d rqh0wr0rqh surgxfwlrq ixqfwlrq= rqh xqlw ri oderu
uhvxowv lq rqh xqlw ri rxwsxw1 Lq wkh rxwsxw pdunhw/ wkh upv duh sulfh wdnhuv rq d
zruog pdunhw1 Wkh sulfh wkdw wkh| rewdlq ghshqgv rq wkh txdolw| ri wkh surgxfw1
Iru vlpsolflw|/ zh dvvxph wkdw wkh sulfh ri up l lv htxdo wr wkh txdolw| ri lwv
surgxfw/ t 1 Zh odwhu frqvlghu lpshuihfw frpshwlwlrq lq wkh rxwsxw pdunhw1
Wkh upv duh hqgrzhg zlwk wkh srvvlelolw| ri surgxflqj d surgxfw ri lqlwldo
txdolw| ri 41 Wkh| pd| ehqhw iurp wkh rxwfrph ri d vwrfkdvwlf U)G surfhvv1 Li
wkh U)G lv vxffhvvixo/ wkh txdolw| lv lqfuhdvhg wr 4 . / zkhuh lv h{rjhqrxv1
Rwkhuzlvh/ wkh up kdv wr surgxfh wkh lqlwldo txdolw|1 Lq wkh edvh prgho/ zh
frqvlghu wkh vlpsohvw srvvleoh U)G surfhvv wr looxvwudwh wkh pdlq hhfwv dw sod|1
Khqfh/ zh dvvxph wkdw wkhuh lv dq h{rjhqrxv suredelolw| ri vxffhvv iru erwk
upv> U)G rxwfrphv duh xqfruuhodwhg dfurvv wkh upv> dqg wkhuh duh qr frvwv ri
U)G1 Zh frqvlghu odwhu wkh srvvlelolw| ri hqgrjhqrxv U)G lqwhqvlwlhv1
Ilqdoo|/ wkh wlplqj lv wkh iroorzlqj= 4, upv fkrrvh orfdwlrq vlpxowdqhrxvo|/ 5,
zrunhuv orfdwh/ 6, U)G rxwfrphv duh uhdol}hg/ 7, zrunhuv duh kluhg/ 8, surgxfwlrq
wdnhv sodfh dqg surwv duh uhdol}hg1
8
6
Oderu Srrolqj yv1 Frpshwlwlrq
Lq wklv vhfwlrq/ zh ghulyh wkh htxloleulxp ri wkh jdph lq wkh edvh prgho1 Zh vroyh
wkh jdph edfnzdugv e| uvw ghulylqj wkh htxloleulxp lq wkh vxejdphv zkhuh wkh
upv djjorphudwh yv1 fkrrvh vhsdudwh orfdwlrqv1 Zh wkhq ghulyh wkhlu htxloleulxp
orfdwlrqdo fkrlfh dv d ixqfwlrq ri wkh xqghuo|lqj sdudphwhuv ri wkh prgho1
614
Wkh Htxloleulxp xqghu Djjorphudwlrq dqg Vhsdudwlrq
Vxssrvh wkdw wkh upv kdyh fkrvhq vhsdudwh orfdwlrqv1 Jlyhq wkh dvvxpswlrqv
pdgh deryh/ up l / ehlqj d prqrsvrqlvw lq lwv oderu pdunhw/ idfhv wkh iroorzlqj
sureohp zkhq klulqj zrunhuv=
Pd{uc7 i+t 5Oc7 w,Oc7 j1
Wkh vroxwlrq wr wklv sureohp lv O9c7 @ t @7w dqg wkh uhvxowlqj surwv duh c7 +t , @
t 2 @;w/ l @ 4> 5= H{0dqwh/ ehiruh wkh U)G rxwfrphv duh uhdol}hg/ wkh upv duh
lghqwlfdo1 Khqfh/ wkh h{shfwhg surwv ri wkh w|slfdo up li upv fkrrvh vhsdudwh
orfdwlrqv duh=
H+ 7 , +4 . ,2 @;w . +4 ,@;w @ +4 . +5 . ,,@;w1
+4,
Qrw vxusulvlqjo|/ h{shfwhg surwv duh lqfuhdvlqj lq wkh suredelolw| dv zhoo dv wkh vl}h
ri wkh lqqrydwlrq1 Surwv duh ghfuhdvlqj lq wkh wudqvsruwdwlrq frvw/ dv lw lqfuhdvhv
wkh frvw ri dwwudfwlqj oderu1
Ohw xv qrz wxuq wr wkh fdvh zkhuh wkh upv duh djjorphudwhg lq uhjlrq 41 Lq
wklv fdvh/ up l vroyhv wkh iroorzlqj sureohp zkhq ghflglqj krz pxfk oderu wr
kluh=
Pd{u7 i+t +Oc . Oc ,w,Oc j1
Vroylqj wklv sureohp/ zh rewdlq wkh up*v uhdfwlrq ixqfwlrq=
+t Oc w,@5w li Oc t @w/
Oc +t > Oc , @
3 rwkhuzlvh1
Xvlqj wkh uhdfwlrq ixqfwlrqv/ zh fdq qrz ghulyh wkh v|pphwulf htxloleulxp lq wkh
vxejdph zkhuh wkh upv djjorphudwh1
;
t @5w li t A 5t /
?
+5t t ,@6w li 2 t t 5t /
O9c +t > t , @
+5,
=
3 rwkhuzlvh1
Wkh surwv duh=
;
?
t2 @7w li t A 5t /
+5t t ,2 @<w li 2 t t 5t /
c +t > t , @
=
3 rwkhuzlvh1
9
+6,
Wkxv/ wkh rxwfrph ghshqgv rq zkhwkhu d up zlwk d vxffhvvixo U)G rxwfrph
*fruqhuv* wkh oderu pdunhw li wkh rwkhu up lv xqvxffhvvixo1 Iurp +5, lw iroorzv
wkdw wklv kdsshqv li dqg rqo| li A 41 Wkhuhiruh/ zh vd| wkdw lqqrydwlrqv duh
gudvwlf li A 4 dqg qrq0gudvwlf li 41D Zh ghqrwh wkh h{shfwhg surwv xqghu
(
qrq0gudvwlf dqg gudvwlf lqqrydwlrqv H+ (
, dqg H+ ,/ uhvshfwlyho|1 Xvlqj +6,/
zh kdyh=
+4 . ,2
.
<w
+4 . 5,2 +4 ,2
4
, . +4 ,2 >
+4 ,+
.
<w
<w
<w
2
2
+4
.
,
+4
.
,
4
2
H+(
. +4 ,
. +4 ,2 =
, @ <w
5w
<w
2
H+ (
, @ +7,
Wr xqghuvwdqg wkh wudgh0r ehwzhhq orfdwlqj wrjhwkhu dqg vhsdudwho|/ ohw xv
(
frpsduh H+
, dqg H+ 7 ,1 Lq vlwxdwlrqv zkhuh wkh upv surgxfh dw wkh vdph
txdolw|/ t/ wkh surwv duh t 2 @<w dqg t 2 @;w xqghu djjorphudwlrq dqg vhsdudwlrq/
uhvshfwlyho|1S Wkh surwv duh orzhu xqghu djjorphudwlrq ehfdxvh wkh upv duh
frpshwlqj lq wkh oderu pdunhw1 Wkhuhiruh/ upv djjorphudwh rqo| li xqghu dj0
jorphudwlrq/ surwv duh kljkhu lq wkh dv|pphwulf vlwxdwlrq zkhuh rqo| rqh up lv
vxffhvvixo lq U)G1 H{ dqwh/ upv gr qrw nqrz li wkh| zloo eh vxffhvvixo ru wkh
xqvxffhvvixo1 Zh wkhuhiruh kdyh wr frqvlghu wkh h{shfwhg surwv lq dv|pphwulf
vlwxdwlrqv1
Vlqfh wkh suredelolw| ri ehlqj wkh vxffhvvixo dqg wkh xqvxffhvvixo up lv wkh
vdph iru erwk upv/ wklv lv htxlydohqw wr frqvlghulqj wkh mrlqw surwv ri wkh wzr
. E!{
upv1 Xqghu dv|pphwu| wkh mrlqw surwv duh En2{
xqghu djjorphudwlrq/
b|
b|
En{

dqg H| . H| xqghu vhsdudwlrq1 Vlpsoh fdofxodwlrqv vkrz wkdw wkh surwv duh
s
kljkhu xqghu djjorphudwlrq li dqg rqo| li +4 . 6 :,@641 Wklv uh hfwv wkh
iroorzlqj wudgh0r= Li wkh upv orfdwh wrjhwkhu/ wkh| frpshwh lq wkh oderu pdunhw/
zklfk dozd|v uhgxfhv surwv1 Krzhyhu/ lq dv|pphwulf vlwxdwlrqv/ wkh kljk txdolw|
up surgxflqj wkh pruh surwdeoh surgxfw lv deoh wr h{sdqg lwv pdunhw vkduh
pruh wkdq xqghu vhsdudwlrq dv lw kdv dffhvv wr d ghhshu oderu srro1 Wklv hhfw
whqgv wr lqfuhdvh surwv xqghu djjorphudwlrq uhodwlyh wr vhsdudwlrq1 Li wkh txdolw|
glhuhqfh/ / lv vx!flhqwo| odujh/ wkh vhfrqg hhfw grplqdwhv wkh uvw1 H{shfwhg
surwv duh wkhq kljkhvw xqghu djjorphudwlrq lq dv|pphwulf vlwxdwlrqv1 Wkh uhvxowv
dovr wxuq durxqg wklv wudgh0r ehwzhhq frpshwlwlrq lq wkh oderu pdunhw dqg oderu
srrolqj li lqqrydwlrqv duh gudvwlf1
615
Wkh Orfdwlrq Fkrlfh
Lw lv qrz srvvleoh wr ghulyh wkh htxloleulxp ri wkh ryhudoo jdph1 Wkh upv duh
lghqwlfdo lq wkh uvw vwdjh ri wkh jdph/ l1h1 ehiruh wkh U)G rxwfrph lv uhdol}hg1
:
Wkh| wkxv dozd|v djuhh zkhwkhu wr orfdwh wrjhwkhu ru fkrrvh vhsdudwh orfdwlrqv1
Sursrvlwlrq 4 Wkh upv fkrrvh wkh iroorzlqj orfdwlrqv lq wkh uvw vwdjh ri wkh
jdph=
s
4n 4E!4
(
l1 Li 4/ wkh upv orfdwh wrjhwkhu li dqg rqo| li +, 4E!4!4 /
t
ll1 Li A 4/ wkh upv orfdwh wrjhwkhu li dqg rqo| li ( +, EH4nE!4
4Eb!f4 4=
(
Surri1 Wkh uhvxow iroorzv gluhfwo| iurp d frpsdulvrq ri H+ 7 , zlwk H+
, dqg
H+ (
,1
Iljxuh 4 looxvwudwhv wkh orfdwlrqdo fkrlfh dv ixqfwlrq ri wkh uhohydqw sdudphwhuv1
$JJORPHUDWLRQ
7 'UDVWLF
,QQRYDWLRQ
7 1RQGUDVWLF
,QQRYDWLRQ
6HSDUDWLRQ
7
Iljxuh 4= Wkh Htxloleulxp Orfdwlrq Fkrlfh
Iluvw qrwlfh wkdw lq whupv ri / wkh uhjlrq lq zklfk djjorphudwlrq rffxuv lv
odujhvw iru lqwhuphgldwh ydoxhv ri 1 Wkh uhdvrq iru wklv lv vwudljkwiruzdug1 Dv
h{sodlqhg deryh/ djjorphudwlrq lv surwdeoh rqo| lq dv|pphwulf vlwxdwlrqv zkhuh
rqh up lqqrydwhv dqg wkh rwkhu qrw1 Wkh suredelolw| 5+4 , iru dq dv|pphwulf
vlwxdwlrq wr dulvh lv odujhvw iru lqwhuphgldwh ydoxhv ri 1 Iroorzlqj wkh vdph orjlf/
upv vhsdudwh iru kljk ydoxhv ri rqo| li lv hlwkhu forvh wr 3 ru wr 4/ dv wkh upv
duh yhu| olnho| wr hqg xs lq vlwxdwlrq zkhuh wkh txdolw| ri wkhlu surgxfwv lv wkh
vdph1 Qrwlfh wkdw upv duh pruh olnho| wr djjorphudwh iru orz ydoxhv ri wkdq iru
kljk ydoxhv1. Sxw glhuhqwo|/ hyhq li wkh suredelolw| ri dq dv|pphwulf vlwxdwlrq
wr dulvh lv wkh vdph iru @ 3=58 dqg @ 3=:8/ ( +3=58, ? ( +3=:8,1 Wkh
uhdvrq lv wkdw wkh orvv iurp frpshwlwlrq lq wkh oderu pdunhw lv kljkhu li erwk upv
duh eohvvhg zlwk kljk txdolwlhv1 Khqfh/ dv wkh suredelolw| ri d v|pphwulf vlwxdwlrq
;
zlwk kljk txdolw| lv juhdwhu iru @ 3=:8 wkdq iru @ 3=58/ d kljkhu ydoxh ri lv
qhhghg wr lqgxfh djjorphudwlrq1
616
Zhoiduh lq edvh prgho
H{ dqwh wrwdo zhoiduh lv xqltxho| ghwhuplqhg e| wkh h{shfwhg uhyhqxhv jhqhudwhg
lq wkh rxwsxw pdunhw1 Wkxv/ xqghu vhsdudwlrq zh kdyh
H+Z7 , 5
+4 . ,
4
. 5+4 ,
7w
7w
dqg xqghu djjorphudwlrq
H+Z , 22
2 5 En{
. 5+4 ,+ En2{En{
. E!{
|
|
| , . +4 , |
22
2 5 En{
. 5+4 ,+ n{
|
2| , . +4 , |
li 4>
li A 4=
Frpsdulqj |lhogv
Ohppd 4 H{shfwhg wrwdo zhoiduh lv pd{lpl}hg xqghu djjorphudwlrq1
Zhoiduh lv kljkhu xqghu djjorphudwlrq/ dv wkh wudqvsruwdwlrq frvw lqfxuuhg iru
dq| jlyhq dprxqw ri oderu vxssolhg lv orzhu1 Wkh h{shfwhg zdjh sdlg wr wkh
zrunhuv pd|/ krzhyhu/ eh kljkhu xqghu djjorphudwlrq gxh wr frpshwlwlrq/ zklfk
h{sodlqv zk| upv vrphwlphv fkrrvh vhsdudwh orfdwlrqv1
617
61714
Vlpsoh H{whqvlrqv
Txdolw| Odgghu Lqwhusuhwdwlrq
Wkh vlpsoh edvh prgho fdq dovr eh vhhq dv wkh rqh0shulrg uhsuhvhqwdwlrq ri d
g|qdplf prgho zlwk fdwfklqj0xs1 Frqvlghu dq lqqlwh krul}rq prgho lq glvfuhwh
wlph zkhuh wkh upv fkrrvh wkhlu orfdwlrqv rqfh dqg iru doo dw wkh ehjlqqlqj ri wkh
jdph1 Vxssrvh wkdw lqqrydwlrqv rffxu lq glvfuhwh mxpsv dorqj d *txdolw| odgghu*1
D up fdq pd{lpdoo| mxps rqh vwhs hyhu| shulrg1 Ghqrwh e| t wkh txdolw| ri vwhs
l rq wkh odgghu1 Wkh uhodwlrq ehwzhhq wkh vwhsv ri wkh odgghu lv t @ +4 . ,t! 1H
Lw lv srvvleoh iru rqh ri wkh upv wr sxoo dkhdg lq wkh lqqrydwlrq udfh/ l1h1 wr mxps
wr wkh qh{w vwhs/ exw rqo| iru rqh shulrg1 Diwhu surwv duh uhdol}hg/ exw ehiruh wkh
qh{w urxqg ri U)G/ wkh rwkhu up dovr mxpsv wr wkh qhz/ kljkhu vwhs1 Wkh upv
wkxv vwduw wkh iroorzlqj shulrg zlwk htxdo txdolwlhv1 Wkh dvvxpswlrq ri fdwfklqj0xs
lv txlwh frpprqo| xvhg lq wkh olwhudwxuh rq lqqrydwlrq udfhv> vhh/ h1j1/ Ehvwhu dqg
Shwudnlv +4<<:, dqg Nohsshu +4<<9,1 Vxssrvh wkdw wkh upv vwduw lq shulrg w iurp
vwhs l1 Vroylqj wkh prgho dv deryh/ lw fdq eh vkrzq wkdw wkh h{shfwhg surwv lq
shulrg w xqghu djjorphudwlrq duh t H+ ,/ m @ G> Q G/ zkhuh H+ , lv jlyhq e|
htxdwlrq +7, dqg +BB,1 Vlploduo|/ wkh h{shfwhg surwv duh t H+ 7 , xqghu vhsdudwh
<
orfdwlrqv1 Wkhuhiruh/ wkh surwv lq shulrg w +dqg dq| rwkhu shulrg, duh kljkhu
xqghu djjorphudwlrq li dqg rqo| li wkh frqglwlrqv lq Sursrvlwlrq 4 duh vdwlvhg1
Wkh fkrlfh ri orfdwlrq lq wklv g|qdplf prgho uhgxfhv wkxv wr wkh vdph ghflvlrq
sureohp dv lq wkh edvh prgho1
61715
Vslooryhuv
Dv glvfxvvhg lq wkh lqwurgxfwlrq wr wklv sdshu/ orfdol}hg nqrzohgjh vslooryhuv duh
frqvlghuhg/ lq wkh hpslulfdo olwhudwxuh/ dq lpsruwdqw uhdvrq iru upv lq kljk0whfk
lqgxvwulhv wr foxvwhu 1 Vlqfh zh duh frqfhuqhg zlwk lqgxvwulhv zkhuh U)G sod|v d
surplqhqw uroh/ lw zruwk vwxg|lqj wkh hhfw ri vxfk vslooryhuv lqvlgh rxu iudphzrun1
Zh dvvxph wkdw vslooryhuv duh orfdol}hg/ dqg fdq rqo| pdwhuldol}h li upv orfdwh
lq wkh vdph uhjlrq1 Vslooryhuv duh prghohg lq d vlpsoh zd| e| dvvxplqj wkdw li
rqo| rqh up lv vxffhvvixo lq U)G/ wkh xqvxffhvvixo up uhfhlyhv d vslooryhu zlwk
suredelolw| v1 Li d vslooryhu rffxuv/ erwk upv fdq surgxfh wkh kljk txdolw|1 Ghqrwh
wkh h{shfwhg surwv xqghu djjorphudwlrq e| H+ c7 ,/ m @ Q G> G +zkhuh vxevfulsw
*V* lqglfdwh wkdw vslooryhuv pd| rffxu,1 Zh kdyh wkdw
+4 . ,2
.
<w
+4 . 5,2 +4 ,2
4
+4 ,+4 v,+
, . +4 ,2
.
<w
<w
<w
,+5
6,
v+4
(
@ H+ /
,.
<w
+4 . ,2
2
H+(
,
@
+
.
5v+4
,,
.
c7
<w
+4 . ,2
4
. +4 ,2
+4 ,+4 v,
5w
<w
v+4 ,+4 . ,2
@ H+ (
1
, 4;w
2
H+ (
c7 , @ + . 5v+4 ,,
+8,
+9,
Uhpdun 4 Vslooryhuv lqfuhdvh surwv xqghu djjorphudwlrq li 5@6 dqg gh0
fuhdvh surwv li A 5@61
Surri1 Wklv iroorzv gluhfwo| iurp +8, dqg +9,1
Iurp Uhpdun 4/ lw iroorzv lpphgldwho| wkdw vslooryhuv lqfuhdvh wkh uhjlrq ri
sdudphwhuv zlwk djjorphudwlrq iru 5@6 dqg ghfuhdvh lw iru A 5@61 Wkh
jxuh ehorz looxvwudwhv wklv srlqw1
Wkh lqwxlwlrq lv wkh iroorzlqj= Vslooryhuv rqo| dhfw surwv lq dv|pphwulf vlw0
xdwlrqv1 Vslooryhuv wkxv lqgxfh +ghwhu, djjorphudwlrq li wkh| lqfuhdvh +ghfuhdvh,
h{shfwhg surwv lq dv|pphwulf vlwxdwlrqv1 Wr vwxg| wklv txhvwlrq/ zh frqvlghu/ dv
dujxhg deryh/ zkhwkhu vslooryhuv lqfuhdvh ru ghfuhdvh wkh mrlqw surwv ri wkh wzr
upv1
43
'UDVWLF
,QQRYDWLRQ
$JJORPHUDWLRQ
6 6 1RQ'UDVWLF
,QQRYDWLRQ
6HSDUDWLRQ
7
Iljxuh 5= Wkh htxloleulxp orfdwlrq zlwk dqg zlwkrxw vslooryhuv1
Lq rxu prgho/ vslooryhuv kdyh wzr rssrvlqj hhfwv1 Rq wkh kdqg/ wkh| lqfuhdvh
frpshwlwlrq lq wkh oderu pdunhw/ dv upv ehfrph pruh htxdo/ zklfk ghfuhdvhv
mrlqw surwv1 Rq wkh rwkhu kdqg/ krzhyhu/ wkh| doorz wkh +iruphuo|, xqvxffhvvixo
up wr surgxfh dw ehwwhu txdolw|/ zklfk lqfuhdvhv mrlqw surwv1 Iru orz ydoxhv ri
/ wkh xqvxffhvvixo up kdv d txlwh odujh pdunhw vkduh hyhq zlwkrxw uhfhlylqj d
vslooryhu1 D vslooryhu kdv wkxv d vljqlfdqw/ srvlwlyh hhfw rq surgxfwlyh h!flhqf|/
dv lw doorzv wkh +iruphuo|, xqvxffhvvixo up wr vhoo wkh kljk txdolw| wr lwv fxvwrphuv1
Wklv hhfw hyhqwxdoo| grplqdwhv wkh frvw gxh wr dgglwlrqdo frpshwlwlrq lq wkh oderu
pdunhw/ dqg vslooryhuv lqgxfh djjorphudwlrq1 Iru kljk ydoxhv ri / wkh dujxphqw lv
mxvw wkh rwkhu zd| durxqg1 Vlqfh wkh kljk txdolw| up wdnhv prvw ri wkh pdunhw li
wkhuh duh qr vslooryhuv/ wkhuh lv olwwoh jdlq lq h!flhqf| iurp vslooryhuv wr wkh orvlqj
up1 Wkhuhiruh/ vslooryhuv uhgxfh h{shfwhg surwv ehfdxvh ri wrxjkhu frpshwlwlrq
lq wkh oderu pdunhw1
7
714
Urexvwqhvv Fkhfnv
Ehuwudqg Frpshwlwlrq lq wkh Oderu Pdunhw
Dv d uvw/ vlpsoh yduldwlrq ri wkh prgho/ zh frqvlghu sulfh frpshwlwlrq lq wkh oderu
pdunhw1 Wklv/ ri frxuvh/ rqo| lq xhqfhv wkh uhvxowv li upv djjorphudwh1 Lq rughu
wr hqvxuh dq htxloleulxp lq sxuh vwudwhjlhv/ zh dvvxph wkdw wkhuh h{lvwv d vpdoohvw
prqhwdu| xqlw/ %1 Li wkh upv vhw wkh vdph zdjh/ zh dvvxph wkdw wkh| vsolw wkh
44
vxsso| htxdoo|1 Ilqdoo|/ dv wkh odvw whfkqlfdolw|/ zh rqo| frqvlghu htxloleuld wkdw
duh qrw lq zhdno| grplqdwhg vwudwhjlhv1
Wkh iroorzlqj ohppd vwdwhv wkh sxuh vwudwhj| htxloleuld lq wkh srvvleoh vxe0
jdphv diwhu wkh U)G rxwfrphv duh uhdol}hg=
Ohppd 5 Frqvlghu wkh htxloleulxp lq wkh oderu pdunhw zkhq upv djjorphudwh=
l, Vxssrvh wkdw wkh upv kdyh wkh vdph txdolw| surgxfw/ t1 Wkh htxloleulxp zdjhv
duh z9 @ z9 @ t1
ll, Vxssrvh wkdw up l kdv wkh txdolw| 4 . dqg up m wkh txdolw| 41 Wkhuh h{lvw
wkh iroorzlqj htxloleuld= Li 4/ +z9 > z9 , @ +4> 4 %, dqg +z9 > z9 , @ +4 . %> 4,/
dqg li A 4/ z9 @ +4 . ,@5 dqg z9 ^4@5> 4`1
Wkh surri lv vwudljkwiruzdug dqg kdv ehhq ohiw rxw1 Wkh dvvxpswlrq ri qr
htxloleulxp lq zhdno| grplqdwhg vwudwhjlhv lpsolhv wkdw qr up rhuv d zdjh kljkhu
wkdq wkh sulfh ri wkh rxwsxw/ t / dqg ohvv wkdq wkh prqrsvrq| zdjh/ t @51
Lq zkdw iroorzv/ zh glvuhjdug % dv lw lv vxssrvhg wr eh yhu| vpdoo uhodwlyh wr
sulfhv1 Xvlqj Ohppd 5/ wkh h{shfwhg surwv xqghu djjorphudwlrq fdq eh zulwwhq
dv=
+
H+ , @
+4 , {| li 4/
+4 , En{
li A 41
2|
+:,
Wkh h{shfwhg surwv xqghu vhsdudwh orfdwlrqv duh wkh vdph deryh1
Sursrvlwlrq 5 Xqghu ehuwudqg frpshwlwlrq lq wkh oderu
t pdunhw/ wkh upv fkrrvh
!4
Ehuwudqg
wr orfdwh wrjhwkhu li dqg rqo| li +, 4E!24
41
Surri1 Iroorzv gluhfwo| iurp frpsdulqj +:, zlwk +4,1
45
%HUWUDQG
$JJO
&RXUQRW
6HSDUDWLRQ
7
Wkh Htxloleulxp Orfdwlrq xqghu Ehuwudqg dqg Frxuqrw Frpshwlwlrq lq wkh
Oderu Pdunhw1
Wkh deryh jxuh vkrzv wkh htxloleulxp orfdwlrqv xqghu Ehuwudqg dqg Frxuqrw
frpshwlwlrq1 Wkh uhjlrq zlwk djjorphudwlrq lv vpdoohu xqghu Ehuwudqg frpshwlwlrq
dv frpshwlwlrq lq wkh oderu pdunhw lv wrxjkhu1 Wkhuhiruh/ wkh upv sd| kljkhu
zdjhv xqghu Ehuwudqg frpshwlwlrq/ zklfk uhgxfhv wkh lqfhqwlyh wr djjorphudwh1
715
Djjorphudwlrq zlwk pruh wkdq wzr upv
Lq wkh iroorzlqj zh ghsduw iurp wkh vlpsoh edvh prgho e| dvvxplqj wkdw wkhuh duh
pruh wkdq wzr dfwlyh upv lq wkh frqvlghuhg lqgxvwu|1 Wkhuh duh vwloo wzr uhjlrqv/
4 dqg 5/ zkhuh wkh upv fdq orfdwh1 Wklv jhqhudol}dwlrq doorzv xv wr dqdo|}h wkh
hhfw ri d pruh iudjphqwhg lqgxvwu| vwuxfwxuh rq lwv vsdwldo rujdql}dwlrq1 Pruh
vshflfdoo|/ zh zloo dvn dv wr zklfk h{whqw iudjphqwdwlrq idyruv foxvwhulqj dqg
zkhwkhu lw jlyhv ulvh wr pruh frpsoh{ vsdwldo vwuxfwxuhv wkdq wkh rqhv lq wkh edvh
prgho1
Wzr dvvxpswlrqv zloo pdnh wklv dqdo|vlv pruh wudfwdeoh1 Iluvw/ zh dvvxph wkdw
upv kdyh qr idooedfn txdolw|/ lw phdqv qrq0lqqrydwlqj upv fdq qrw hqwhu wkh
pdunhw1 Dqg vhfrqg/ zh dvvxph wkdw wkh uhvshfwlyh ghswk ri d orfdo oderu pdunhw
lv sursruwlrqdo wr wkh iudfwlrq ri orfdo upv lq wkh ryhudoo lqgxvwu|1
Djdlq/ zh zloo vwduw edfnzdugv dqg vroyh wkh orfdo oderu pdunhw htxloleulxp iru
d jlyhq qxpehu q ri vxffhvvixo upv1 Iru dq| oderu pdunhw ghswk / wkh lqyhuwhg
oderu vxsso| lv vlpso|
z+O, @
46
wO
=
Wkh uvw rughu frqglwlrq iru dq| lqglylgxdo up lq wkh orfdo pdunhw lv
+4 . , w+5O . O! ,
@ 3>
zkhuh O lv wkh up*v lqglylgxdo oderu ghpdqg dqg O! wkh djjuhjdwh ghpdqg ri doo
rwkhu orfdo upv1 Vroylqj wklv htxdwlrq iru dq htxloleulxp lq v|pphwulf vwudwhjlhv/
O @ O > iru doo l 9@ m> |lhogv dv d up*v htxloleulxp ghpdqg
O+> q> =, @
+4 . ,
>
w+q . 4,
+> q> =, @
+4 . ,2
=
w+q . 4,2
dqg htxloleulxp surwv ri
Ohw xv ghqrwh dv q wkh qxpehu ri upv wkdw vhwwohg lq uhjlrq 41 Wkhq/ wkh
ghswk ri wkh orfdo oderu pdunhw lq uhjlrq 4 lv vlpso|  @ q @q/ zkhuhdv 2 @
q2 @q @ +q q ,@q= Wkh vwrfkdvwlfv ri wkh lqqrydwlrq surfhvv duh gulyhq e| d
elqrpldo glvwulexwlrq ixqfwlrq1 Wkh suredelolw| wkdw p rxw ri wkh lqlwldoo| vhwwohg
q upv duh lqqrydwlqj lq d uhjlrq lv jlyhq e|
6
?!6 q
Suiq> pj @ +4 ,
=
p
Wkh h{shfwhg surwv ri d up lq uhjlrq u> jlyhq wkdw qo upv duh orfdwhg lq wkh
vdph uhjlrq/ duh
H^o +qo ,` @ [6'?o !
6'f
Suiqo 4> pj +
qo
> p . 4> =,=
q
Iurp wklv/ zh fdq ghqh d orfdwlrq htxloleulxp +q9 > q q9 , dv d up glvwulexwlrq/
lq zklfk qr lqglylgxdo up kdv dq lqfhqwlyh wr uhorfdwh jlyhq wkh orfdwlrq ghflvlrq
ri doo rwkhu upv1 Irupdoo|/ lw pxvw krog iru erwk uhjlrqv/ u @ 4> 5/ wkdw
H^o +qo ,` H^!o +q qo . 4,`=
Reylrxvo|/ d ixuwkhu jhqhudo wuhdwphqw ri wkhvh wzr frqglwlrqv lv qrw srvvleoh1 Exw
wkh dqdo|vlv ri orfdwlrq htxloleuld iru d jlyhq ydoxh ri wrwdo upv lq wkh lqgxvwu| lv
ihdvleoh dqg |lhogv vrph ixuwkhu lqvljkwv1
Wkh iroorzlqj ohppd orrnv dw wkh orfdwlrq htxloleuld iru q@7 upv1
Ohppd 6 Dvvxph wkhuh duh q @ 7 upv lq wkh lqgxvwu|1 Wkhq/
41 iru 3=:8564> +7/3, dqg +3/7, duh vxvwdlqdeoh orfdwlrq htxloleuld/
51 iru A 5@6> +5/5, lv d vxvwdlqdeoh orfdwlrq htxloleulxp1
47
Iurp wklv/ wzr wklqjv duh qrwhzruwk|1 Iluvw/ orrnlqj dw d vhwwlqj zlwk pruh wkdq
wzr upv fdq jlyh ulvh wr vlwxdwlrq zlwk pxowlsoh htxloleuld1 Iru 5@6 3=:8564/
ixoo djjorphudwlrq lq rqh uhjlrq dv zhoo dv dq htxdo glvwulexwlrq ri upv ryhu vsdfh
duh srvvleoh htxloleulxp rxwfrphv1b Wkh hphujhqfh ri pxowlsoh htxloleuld lv gxh wr
wkh idfw wkdw rxu htxloleulxp frqfhsw grhv qrw doorz iru frdolwlrq irupdwlrq dqg
ghyldwlrq dqg wkhuhiruh ohdgv wr frruglqdwlrq sureohpv wkdw fdq qrw dulvh lq wkh
wzr upv fdvh1
Vhfrqg/ d pruh iudjphqwhg lqgxvwu| grhv qrw qhfhvvdulo| ohdg wr glhuhqw sdw0
whuqv ri vsdwldo glvwulexwlrq ri upv1 H{whqvlyh qxphulfdo fdofxodwlrqv frqup wkh
uhvxow ri wkh 7 up fdvh wkdw upv hlwkhu djjorphudwh frpsohwho| ru vsuhdg dfurvv
wkh wzr uhjlrqv dv hyhqo| dv srvvleoh1
716
Hqgrjhqrxv U)G
Lq wkh ehqfkpdun prgho zh dujxhg wkdw upv* orfdwlrq ghflvlrqv duh gulyhq e| dq
h{rjhqrxv lqqrydwlrq surfhvv1 Lq wklv vhfwlrq/ zh kdyh d forvhu orrn dw wkh lqwhudf0
wlrq ehwzhhq upv* orfdwlrq ghflvlrqv dqg wkhlu U)G dfwlylwlhv1 Duh foxvwhuhg upv
pruh lqqrydwlyh wkdq upv zlwk pruh prqrsvrqlvwlf pdunhw srzhuB Gr vwudwhjlf
lqqrydwlrq ghflvlrqv kdyh d vljqlfdqw lpsdfw rq upv orfdwlrqvB
Lq wkh iroorzlqj/ zh h{whqg wkh edvh prgho e| doorzlqj upv wr fkrrvh wkhlu
U)G lqwhqvlw|/ > dw d txdgudwlf frvw/ + ,2 @5/ l @ 4> 51 Lq ylhz ri wkh pruh orqj
whup fdudfwhu ri orfdwlrq ghflvlrqv/ zh dvvxph wkdw wkh fkrlfh ri U)G lqwhqvlwlhv
rffxuv rqfh upv kdyh vhwwohg lq dq| uhjlrq1
Vxssrvh wkdw upv fkrrvh glhuhqw orfdwlrqv1 Exloglqj rq wkh dqdo|vlv ri wkh
edvh prgho/ wkh sureohp ri up l zkhq fkrrvlqj lwv U)G lqwhqvlw| fdq eh zulwwhq
dv=
+4 . ,2
4
2
Pd{4 . +4 , + , 1
;w
;w 5
Lw iroorzv wkdw wkh rswlpdo U)G lqwhqvlw| xqghu vhsdudwh orfdwlrqv lv/ 7 ++5 .
,,@+;w,1 Vlpsoh fdofxodwlrqv vkrz wkdw wkh h{shfwhg surwv duh
H+ e7 , @
@
7 +5 . , 7 2
4
.
+ ,
;w
;w
5
2
2
49w . +5 . ,
1
45;w2 zkhuh vxshuvfulsw *h* uhihuv wr wkh hqgrjhqrxv U)G ghflvlrq1
Li wkh upv fkrrvh wkh vdph uhjlrq/ up l*v rswlpdo U)G lqwhqvlw| lv jlyhq dv
wkh vroxwlrq wr wkh iroorzlqj sureohp=
P d{4 i En{
. +4 ,^ En2{
. +4 , En{
b|
b|
e| `.
E!{

2
+4 , b| . +4 ,+4 , b| 2 + , j>
48
zkhuh @ 4 li 4 dqg @ 3 rwkhuzlvh1 Vroylqj wklv sureohp |lhogv wkh
iroorzlqj uhdfwlrq ixqfwlrq=
+
li 4>
7+4 . ,@+<w,
U + , @
En{EDn{
nD{E2n{
li A 4=
S|
S|
Zh vhh wkdw wkh U)G ghflvlrq ri wkh upv lqwhudfw= Li up m lqfuhdvhv lwv U)G
lqwhqvlw|/ lw ehfrphv ohvv dwwudfwlyh iru up l wr gr U)G dv lw lv pruh olnho| wr
phhw d wrxjk frpshwlwru lq wkh oderu pdunhw1 Wklv uhgxfhv wkh h{shfwhg uhwxuqv wr
U)G dv lw fdqqrw h{sdqg lwv surgxfwlrq dv pxfk dv rwkhuzlvh srvvleoh1f Xvlqj wkh
uhdfwlrq ixqfwlrqv/ lw lv hdvlo| vkrzq wkdw wkhuh lv d xqltxh +dqg vwdeoh, htxloleulxp
lq sxuh vwudwhjlhv1 Wkh htxloleulxp U)G lqwhqvlw| xqghu djjorphudwlrq lv jlyhq dv
+
e{En{
li 4>
b|ne{
En{EDn{
li A 4
S|nnD{E2n{
dqg soxjjlqj wkhvh ydoxhv lq wkh uhvshfwlyh h{shfwhg surwv |lhogv H+ e , dv d
ixqfwlrq ri > w dqg =
Wkh iroorzlqj ohppd frpsduhv lqqrydwlyh dfwlylwlhv xqghu vhsdudwlrq dqg dj0
jorphudwlrq1
t
s
h bEne|n nH|EbnS2| 4> wkhq 7 A =
Ohppd 7 Li A 4@w dqg A f
Wkh uvw frqglwlrq hqvxuhv wkdw erwk U)G lqwhqvlwlhv duh vpdoohu wkdq 41 Wkhq/
iru vx!flhqwo| kljk lqqrydwlrq vwhsv/ upv lqyhvw pruh xqghu vhsdudwlrq wkdq lq d
uhjlrqdo foxvwhu1 Wklv uhvxow lv gxh wr wkh vwudwhjlf lqwhudfwlrq lq dq djjorphudwlrq1
Elj xsfrplqj lqqrydwlrq vwhsv jlyh erwk upv* vwurqjhu lqfhqwlyhv wr lqyhvw zklfk
dw wkh vdph wlph lqfuhdvhv wkh suredelolw| ri oderu pdunhw udwlrqlqj/ wrr/ dqg wklv
frxqwhudfwv lqqrydwlrq lqfhqwlyhv xqghu djjorphudwlrq exw qrw xqghu vhsdudwlrq1
Frpsdulqj wkh surwv xqghu djjorphudwlrq dqg vhsdudwlrq lv dqdo|wlfdoo| gli0
fxow gxh wr wkh ixqfwlrqdo irupv lqyroyhg1 Krzhyhu/ wkh jxuh ehorz looxvwudwhv
wkh orfdwlrqdo fkrlfh dv d ixqfwlrq ri +> , iru w @ 41 Wkh pdlq glhuhqfh wr wkh
uhvxowv ri wkh edvh prgho lv wkdw qrz dq lqfuhdvh lq wkh lqqrydwlrq vwhs grhv qrw
qhfhvvdulo| lpso| vwurqjhu lqfhqwlyhv iru djjorphudwlrq1 Dv phqwlrqhg deryh/ iru
odujh / lqqrydwlqj ehfrphv uhodwlyho| pruh h{shqvlyh lq whupv ri h{shfwhg oderu
pdunhw frpshwlwlrq li upv duh foxvwhuhg1
E| frqwudvw/ zh vhh wkdw wkh lqwxlwlrq ri wkh edvh prgho zlwk uhvshfw wr dovr krogv lq wklv h{whqvlrq zlwk hqgrjhqrxv U)G fkrlfhv1 Wkh upv fkrrvh wr
djjorphudwh li wkh uhvxowlqj U)G lqwhqvlw| + , lv qhlwkhu yhu| kljk qru yhu| orz1
Ilqdoo|/ ohw xv wxuq wr wkh zhoiduh surshuwlhv ri wkhvh orfdwlrq htxloleuld1 Lq wkh
edvh prgho zlwkrxw hqgrjhqrxv U)G fkrlfh/ djjorphudwlrq zdv dozd|v zhoiduh
grplqdqw wr vhsdudwlrq1 Qrz glhuhqw orfdwlrq fkrlfhv dovr ohdg wr glhuhqw U)G
49
a
'
BHHMPNFSBUJPOFR
TFQBSBUJPOFR
U
J
Iljxuh 6= Orfdwlrq Htxloleuld iru w@4
rxwfrphv1 Soxjjlqj wkh htxloleulxp U)G lqwhqvlwlhv lq wkh deryh zhoiduh ixqfwlrqv/
zh jhw wkh h{shfwhg/ lqqrydwlrq0dgmxvwhg wrwdo zhoiduh ri wkh hfrqrp|1 Lq wkh
iroorzlqj judsk/ zh frpsduh wkh htxloleulxp rxwfrph zlwk wkh h!flhqw vsdwldo
lqgxvwu| vwuxfwxuh1
Dv d uhvxow ri wkh zhoiduh dqdo|vlv/ lw fdq eh qrwhg wkdw zlwk hqgrjhqrxv fkrlfh
ri U)G lqwhqvlwlhv/ wkh vrfldoo| rswlpdo vsdwldo vwuxfwxuh lv qr orqjhu xqltxho| d
uhjlrqdo foxvwhulqj1 Wkh uhdvrq lv wkdw wkh qhjdwlyh hhfw ri oderu pdunhw frpsh0
wlwlrq rq upv* lqfhqwlyhv wr lqyhvw lq U)G fdq pruh wkdq rvhw wkh dgydqwdjh ri
dq hfrqrp|zlgh dffhvv wr wkh oderu srro1
8
Frqfoxvlrq
Lq wklv sdshu/ zh kdyh ghyhorshg d prgho ghprqvwudwlqj krshixoo| vrph fhqwudo
wudghrv lqyroyhg lq wkh orfdwlrq ghflvlrqv ri uhvhdufk lqwhqvlyh upv1 Lq sduwlfxodu/
zh ghyhors uhvxowv rq djjorphudwlrq yv1 glvshuvlrq wkdw duh udwkhu urexvw djdlqvw
dowhuqdwlyh lqsxw/ dqg rxwsxw pdunhw frqglwlrqv1 Wkh prgho lv vlpsoh hqrxjk wr
doorz iru pdq| h{whqvlrqv1 D prvw lqwhuhvwlqj rqh vkrxog eh wr frqvlghu fruuhod0
wlrqv ehwzhhq upv* vxffhvv2idloxuh suredelolwlhv dqg lqghhg/ wkh fkrlfh ri uhvhdufk
4:
'
9#
95
BHHMPNFSBUJPOFR
TFQBSBUJPOFR
J
Iljxuh 7= Htxloleulxp dqg H!flhqw Vsdwldo Sdwwhuq iru w@4
sdwkv lq zklfk wkdw fruuhodwlrq vwuxfwxuh lv hqgrjhql}hg1 Dqrwkhu rqh zrxog lq0
yroyh d pruh h{solflw sdudphwul}dwlrq ri wkh vxevwlwxwdelolw|/ ru frpsohphqwdulw|
ri wkh surgxfwv jhqhudwhg e| wkh upv1 \hw doo wklv pxvw eh ohiw wr ixuwkhu uhvhdufk1
9
Uhihuhqfhv
g*Dvsuhprqw/ F1/ Jdev}hzlf}/ M1 M1/ dqg M10I1 Wklvvh +4<:<,= Rq Krwhoolqj*v
Vwdelolw| lq Frpshwlwlrq/ Hfrqrphwulfd/ 7:/ 4478044831
Dxguhwvfk/ G1E1 +4<<;,= Djjorphudwlrq dqg wkh Orfdwlrq ri Hfrqrplf Dfwly0
lw|/ FHSU Glvfxvvlrq Sdshu Vhulhv +Lqgxvwuldo Rujdql}dwlrq,/ Qr1 4<:71
Dxguhwvfk/ G1E1 dqg P1S1 Ihogpdq +4<<8,= Lqqrydwlyh Foxvwhuv dqg wkh Lq0
gxvwu| Olih F|foh/ FHSU Glvfxvvlrq Sdshu Vhulhv +Lqgxvwuldo Rujdql}dwlrq,/
Qr1 44941
Dxguhwvfk/ G1E1 dqg P1S1 Ihogpdq +4<<9,= U)G0Vslooryhuv dqg wkh Jhrj0
udsk| ri Lqqrydwlrq dqg Surgxfwlrq/ Dphulfdq Hfrqrplf Uhylhz/ ;9/ 963 0
9731
4;
Edqhumhh/ D1Y1 +4<<5,= D Vlpsoh Prgho ri Khug Ehkdylru/ Txduwhuo| Mrxuqdo
ri Hfrqrplfv/ 43:/ :<:0;4:1
Edxpjdugqhu/ M1U1 +4<;;,= Wkh Glylvlrq ri Oderu/ Orfdo Pdunhwv/ dqg Zrunhu
Rujdql}dwlrq/ Mrxuqdo ri Srolwlfdo Hfrqrp|/ <9/ 83<085:1
Exorz/ M1/ M1 Jhdqdnrsorv/ dqg S1 Nohpshuhu +4<;8,= Pxowlpdunhw Roljrsro|=
Vwudwhjlf Vxevwlwxhv dqg Frpsohphqwv1 Mrxuqdo ri Srolwlfdo Hfrqrp|/ <6/
7;;08441
Ehvwhu/ K1 +4<<;,= Txdolw| xqfhuwdlqw| plwljdwhv surgxfw glhuhqwldwlrq/ Udqg
Mrxuqdo ri Hfrqrplfv/ 5</ ;5;0;771
Ehvwhu/ K1 dqg H1 Shwudnlv +4<<:,= Zdjhv dqg Surgxfwlylw| Jurzwk lq d Frp0
shwlwlyh Lqgxvwu|/ Glvfxvvlrq Sdshu 4<<:25;/ Iuhh Xqlyhuvlw|/ Ehuolq1
Ekdwwdfkdu|d/ V1 dqg Prrnkhumhh/ G1 +4<;9,= Sruwirolr fkrlfh lq uhvhdufk dqg
ghyhorsphqw/ Udqg Mrxuqdo ri Hfrqrplfv/ 4:/ 8<709391
Elnkfkdqgdql/ V1/ Kluvkohlihu/ G1 dqg L1 Zhofk +4<<5,= D Wkhru| ri Idgv/
Idvklrq/ Fxvwrq/ dqg Fxowxudo Fkdqjh lq Lqirupdwlrqdo Fdvfdghv/ Mrxuqdo
ri Srolwlfdo Hfrqrp|/ 433/ <<5043591
Fdugrq/ M1K1 dqg G1 Vdvdnl +4<<;,= Suhhpswlyh vhdufk dqg U)G foxvwhulqj/
Udqg Mrxuqdo ri Hfrqrplfv/ 5</ 657066;1
Frpehv/ S1S1 dqg J1 Gxudqwrq +5333,= Oderu Srrolqj/ Oderu Srdfklqj/ dqg
Vsdwldo Foxvwhulqj/ Sdulv= FHUDV/ plphr1
Gdvjxswd/ S1 dqg H1 Pdvnlq +4<;:,= Wkh Vlpsoh Hfrqrplfv ri Uhvhdufk Sruw0
irolrv/ Wkh Hfrqrplf Mrxuqdo/ <:/ 8;408<81
Irvxul/ D1/ P1 Prwwd/ dqg W1 Uûqgh +5334,= Iruhljq Gluhfw Lqyhvwphqwv dqg
Vslooryhuv wkurxjk Zrunhuv* Prelolw|/ Mrxuqdo ri Lqwhuqdwlrqdo Hfrqrplfv/ s1
5380551
Jhuodfk/ K1/ W1 Urhqgh dqg N1Vwdko +5334,= Djjorphudwlrq lq U)G Lqwhq0
vlyh Lqgxvwulhv/ X1 ri Pdqqkhlp/ plphr
Jhuvedfk/ K1 dqg D1Vfkpxw}ohu +5333,= Ghfolqlqj Frvwv ri Frppxqlfdwlrq
dqg Wudqvsruwdwlrq= Zkdw duh wkh Hhfwv ri DjjorphudwlrqvB/ Hxurshdq Hfr0
qrplf Uhylhz/ 77/ 4:7804:94
Jhuvedfk/ K1 dqg D1 Vfkpxw}ohu +4<<<,= H{whuqdo vslooryhuv/ lqwhuqdo vslooryhuv
dqg wkh jhrjudsk| ri surgxfwlrq dqg lqqrydwlrq/ Uhjlrqdo Vflhqfh dqg Xuedq
Hfrqrplfv/ 5</ 9:<09<91
4<
Jurvvpdq/J1 dqg H1 Khospdq/ +4<<4,= Lqqrydwlrq dqg Jurzwk lq d Joredo
Hfrqrp|/ Fdpeulgjh/ PD1= PLW Suhvv
Kdukr/ G1 +4<<8,= Djjorphudwlrqhq xqg uhjlrqdoh Vslooryhuhhnwh/ lq= Jdkohq/
E1/ Khvvh/ K1/ dqg K1M1 Udpvhu/ hg*v/ Vwdqgruw xqg Uhjlrq/ Wÿelqjhq= Prku0
Vlhehfn/ ;604481
Nohsshu/ V1 +4<<9,= Hqwu|/ H{lw/ Jurzwk/ dqg Lqqrydwlrq ri wkh Surgxfw Olih
F|foh/ Dphulfdq Hfrqrplf Uhylhz/ ;9/ 89508;61
Pdl/ F10F1 dqg V10N1 Shqj +4<;8,= Frrshudwlrq yv1 Frpshwlwlrq lq d Vsdwldo
Prgho/ Uhjlrqdo Vflhqfh dqg Xuedq Hfrqrplfv/ 5</ 79607:51
Pduvkdoo/ D1 +4<53,= Sulqflsohv ri Hfrqrplfv/ ;wk hglwlrq/ Orqgrq/ Hqjodqg=
PdfPloodq1
Slfdug/ S1 dqg H1 Wrxohprqgh +5333,= Uhjlrqdo Dv|pphwulhv= Hfrqrplhv ri
Djjorphudwlrq yhuvxv Oderu Pdunhwv/ X1 ri Pdqfkhvwhu/ plphr1
Uûqgh/ W1 +5334,= Wudgh Vhfuhwv dqg Lqirupdwlrq Vkdulqj/ Mrxuqdo ri Hfrqrplfv
dqg Pdqdjhphqw Vwudwhj|/ 43/ 6<4074:1
Vd{hqldq/ D1 +4<<7,= Uhjlrqdo Dgydqwdjh 0 Fxowxuh dqg Frpshwlwlrq lq Vlolfrq
Ydooh| dqg Urxwh 45;/ Fdpeulgjh/ PD= Kduydug Xqlyhuvlw| Suhvv1
Wrsho/ U1 K1 +4<;9,= Orfdo Oderu Pdunhwv/ Mrxuqdo ri Srolwlfdo Hfrqrp|/ <7/
444044761
53