W?|h?tU@**) `Lh|*itt MiU|t @t i_@ Lu , U@?}iG
, Tih4i?|@* ,_i?Ui
aL? #g)
#iT@h|4i?| Lu ,UL?L4Ut
N?iht|) Lu ||tMh}
||tMh}c D2Sf N5
Ai*G Ee2 SeH.
6jG Ee2 SeH.b
,4@*G _g)9T||i_
a@U! Ut
#iT@h|4i?| Lu ,UL?L4Ut
N?iht|) Lu ||tMh}
||tMh}c D2Sf N5
Ai*G Ee2 SeH.fS
6jG Ee2 SeH.b
,4@*G LUt9T||i_
At #h@u|G #iUi4Mih 2fff
Devwudfw
Wklv sdshu uhsruwv uhvxowv iurp dq h{shulphqwdo lpsohphqwdwlrq ri Nl|rwdnl dqg Zuljkw*v
+4<;<, vhdufk~wkhruhwlf prgho ri prqh| dv d phglxp ri h{fkdqjh1 Wkh irfxv ri wkh h{shulphqw
lv rq wkh uroh sod|hg e| cwrnhq* remhfwv lq wklv hqylurqphqw1 D wrnhq lv dq remhfw wkdw vhuyhv dv
d vwruh ri ydoxh/ exw zklfk kdv qr lqwulqvlf ydoxh wr dq| lqglylgxdo1 Wkh txhvwlrq zh dgguhvv lv
zkhwkhu vxfk wrnhq remhfwv pd| dovr vhuyh dv phgld ri h{fkdqjh1 Dffruglqj wr wkh wkhru|/ wkhuh
duh sdudphwhul}dwlrqv ri wkh Nl|rwdnl~Zuljkw +4<;<, prgho xqghu zklfk d vlqjoh wrnhq remhfw
vhuyhv dv d phglxp ri h{fkdqjh lq d sxuh vwudwhj| Qdvk htxloleulxp1 Pruhryhu/ dv Dl|djdul
dqg Zdoodfh +4<<5, kdyh vkrzq/ wklv wrnhq remhfw qhhg qrw eh wkh ohdvw frvwo|~wr~vwruh jrrg1
Rxu h{shulphqw lpsohphqwv wkh Nl|rwdnl~Zuljkw +4<;<, hqylurqphqw zlwk d vlqjoh wrnhq remhfw
dqg whvwv zkhwkhu wkhvh htxloleulxp suhglfwlrqv duh urexvw wr wkh g|qdplfv fuhdwhg e| rxw~
ri~htxloleulxp ehkdylru1 Rxu qglqjv vxjjhvw wkdw wkhuh lv vrph vxssruw iru wkh wkhruhwlfdo
suhglfwlrqv> lq sduwlfxodu/ zh qg wkdw vxemhfwv qhduo| dozd|v rhu wr wudgh iru wkh wrnhq remhfw
zkhq vxfk d wudgh orzhuv wkhlu vwrudjh frvwv1 Krzhyhu/ frqwudu| wr wkh wkhru|/ zh dovr qg wkdw
vxemhfwv iuhtxhqwo| uhixvh wr rhu wr wudgh wkh wrnhq remhfw iru pruh frvwo|~wr~vwruh jrrgv zkhq
wkh wkhru| suhglfwv wkh| vkrxog pdnh vxfk wudghv1
Mrxuqdo ri Hfrqrplf Olwhudwxuh Fodvvlfdwlrq Qrv1 F<5/ G;6/ H731
W 6?@?U@* tTTLh| uLh |t ThLiU| @t ThL_i_ M) |i @|L?@* 5Ui?Ui 6L?_@|L? ?_ih }h@?| r 5+bD
f.bSH `i |@?! |i |L @?L?)4Lt hiuihiit @?_ ti4?@h T@h|UT@?|t uLh |ih ?t}|u* UL44i?|t @?_ UL?t|hU|i
t}}it|L?t `i @hi @*tL }h@|iu* |L 5h_@h kL4@?_h @?_ +LMih| L?|Ui**L uLh |ih @ttt|@?Ui ? _ii*LT?} |i
ThL}h@4 ti_ |L U@hh) L| |i i Tih4i?|t Ai h@ _@|@ uhL4 @** i Tih4i?|t hiTLh|i_ ? |t T@Tih @hi @@*@M*i
@|G Pzzf["x[qq&"q>z"
W?|hL_U|L?
E| ghqlwlrq/ prqh| lv dq remhfw wkdw vhuyhv dv d phglxp ri h{fkdqjh1 Dv Rvwur| dqg Vwduu +4<<3,
kdyh hpskdvl}hg/ d phglxp ri h{fkdqjh pxvw dovr eh d vwruh ri ydoxh/ exw qrw doo vwruhv ri ydoxh duh
phgld ri h{fkdqjh1 Wkh ghqlqj fkdudfwhulvwlf ri dq remhfw wkdw lv erwk d vwruh ri ydoxh dqg vhuyhv
dv d phglxp ri h{fkdqjh lv lwv jhqhudo dffhswdelolw| wr lqglylgxdov zkhq rhuhg1 Frqvhtxhqwo|/
d vwrudeoh remhfw wkdw ehfrphv d phglxp ri h{fkdqjh qhhg qrw kdyh ydoxh wr dq|rqh dv hlwkhu d
frqvxpswlrq jrrg ru dv dq lqsxw lqwr d sk|vlfdo surgxfwlrq surfhvv1 Ri frxuvh/ dq lqglylgxdo zkr
dffhswv dq remhfw wkdw kdv qr ydoxh wr wkdw lqglylgxdo dv hlwkhu d frqvxpswlrq jrrg ru dv dq lqsxw
wr d surgxfwlrq surfhvv grhv vr lq wkh eholhi wkdw wkh remhfw zloo eh dffhswdeoh lq h{fkdqjh iru
vrphwklqj ri gluhfw xwlolw| wr wkdw lqglylgxdo1
D vwrudeoh remhfw zlwk qr lqwulqvlf ydoxh lq frqvxpswlrq ru surgxfwlrq lv d _wrnhq%1 Jryhuq0
phqwv pd| pdnh wrnhqv phgld ri h{fkdqjh e| dw/ uhtxlulqj wkdw wkhvh remhfwv eh dffhswhg iru
wkh vhwwohphqw ri doo ghewv sxeolf dqg sulydwh1 Krzhyhu/ zkhq d jryhuqphqw ghvwur|v wkh vwrudjh
ydoxh ri d dw remhfw e| h{fhvvlyh lvvxh/ wkdw remhfw zloo fhdvh wr vhuyh dv d phglxp ri h{fkdqjh/
uhjdugohvv ri wkh jryhuqphqw*v ghfodudwlrq1
Dffruglqj wr wkh wkhru| sursrxqghg e| Nl|rwdnl dqg Zuljkw +4<;<,/ lw lv qrw lpsruwdqw wkdw
dq remhfw kdyh hlwkhu lqwulqvlf ydoxh lq xvh/ ru ydoxh fuhdwhg e| d jryhuqphqw*v dw ghfodudwlrq/ lq
rughu wkdw wkh remhfw vhuyh dv d phglxp ri h{fkdqjh1 Lw lv lpsruwdqw wkdw wkh remhfw eh gxudeoh
dqg qrw wrr frvwo| wr vwruh1 Exw lq wkhlu wkhru|/ dq remhfw zlwk qr rwkhu lqwulqvlfdoo| ydoxdeoh
fkdudfwhulvwlfv fdq dftxluh dqg pdlqwdlq wkh vwdwxv ri d phglxp ri h{fkdqjh vlpso| e| yluwxh ri
wkh idfw wkdw shrsoh frph wr h{shfw lw wr eh jhqhudoo| dffhswhg lq h{fkdqjh iru remhfwv wkdw kdyh
lqwulqvlf ydoxh1 Wkdw lv/ dq remhfw ehfrphv d phglxp ri h{fkdqjh ehfdxvh shrsoh kdyh iruphg d
+udwlrqdo, h{shfwdwlrq wkdw lw zloo frqwlqxh wr vhuyh dv d phglxp ri h{fkdqjh dqg wklv h{shfwdwlrq
qhhg qrw eh vxssruwhg e| dq| surshuw| ri wkh remhfw rwkhu wkdq wkh vrfldo frqyhqwlrq wkdw kdv
hphujhg iurp lwv xvh lq wkh sdvw1
Lq sulru zrun +Gx| dqg Rfkv +4<<<,, zh uhsruwhg wkh uhvxowv ri dq h{shulphqw ghvljqhg wr
whvw wkh lpsolfdwlrqv ri rqh yhuvlrq ri Nl|rwdnl dqg Zuljkw*v +4<;<, prgho/ zkhuh hyhu| remhfw
kdg +lqwulqvlf, frqvxpswlrq ydoxh wr rqh sod|hu w|sh/ exw qr remhfw kdg frqvxpswlrq ydoxh wr doo
sod|hu w|shv1 Lq wkdw yhuvlrq ri wkh Nl|rwdnl~Zuljkw prgho +dv zhoo dv lq wkh yhuvlrq vwxglhg lq wklv
sdshu,/ hdfk sod|hu frxog vwruh rqo| rqh xqlw ri dq| remhfw lq hyhu| shulrg1 Sod|huv zhuh udqgrpo|
4
pdwfkhg sdluzlvh/ dqg frxog hqjdjh lq rqh iru rqh wudghv ri wkh jrrgv wkh| khog lq vwrudjh1 Li
d sod|hu vxffhvvixoo| wudghg iru klv _frqvxpswlrq jrrg/% kh lpphgldwho| surgxfhg d xqlw ri klv
_surgxfwlrq jrrg%/ vr wkdw qr sod|hu zdv hyhu vwrulqj wkh remhfw wkdw |lhoghg klp frqvxpswlrq
ydoxh1 Wkh vwrudjh frvwv ydulhg dfurvv wkh remhfwv1 Lq wkh hqylurqphqw vwxglhg e| Gx| dqg Rfkv
+4<<<,/ hdfk lqglylgxdo zdv iuhtxhqwo| rhuhg d jrrg wkdw kdg qr lqwulqvlf ydoxh dv d frqvxpswlrq
jrrg wr klp2khuvhoi/ wkrxjk lw glg kdyh frqvxpswlrq ydoxh wr d fhuwdlq iudfwlrq ri wkh srsxodwlrq1
Zkhq idfhg zlwk wkh rssruwxqlw| wr wudgh iru vxfk d _qrq0frqvxpswlrq% jrrg/ dq lqglylgxdo frxog
h{klelw hlwkhu ri wzr ehkdylruv= ixqgdphqwdo/ ru vshfxodwlyh1 Ixqgdphqwdo ehkdylru lv h{klelwhg
zkhq dq lqglylgxdo dffhswv wkh rhu li dqg rqo| li wkh rhuhg jrrg kdv d orzhu vwrudjh frvw wkdq wkh
jrrg wkh lqglylgxdo lv fxuuhqwo| vwrulqj1 Vshfxodwlyh ehkdylru lv h{klelwhg li wkh lqglylgxdo dffhswv
vxfk d jrrg hyhq li lw kdv d kljkhu vwrudjh frvw wkdq wkh jrrg wkdw lv fxuuhqwo| khog1 Vshfxodwlrq
lqyroyhv dffhswlqj d jrrg lq dqwlflsdwlrq wkdw wkh kljkhu vwrudjh frvw jrrg zloo doorz wkh kroghu wr
h{hfxwh d vxffhvvixo ixwxuh wudgh iru d jrrg ri lqwulqvlf ydoxh wr wkh kroghu vrrqhu wkdq wkh jrrg
wkdw wkh lqglylgxdo fxuuhqwo| jlyhv lq h{fkdqjh1 Wkh wkhru| suhglfwv wkdw xqghu vrph sdudphwhu
ydoxhv rqo| ixqgdphqwdo ehkdylru zloo eh revhuyhg lq htxloleulxp dqg/ dv d frqvhtxhqfh/ wkh jrrg
zlwk wkh orzhvw vwrudjh frvw zloo hphujh dv d jhqhudoo| dffhswhg phglxp ri h{fkdqjh1 Ri sduwlfxodu
lqwhuhvw/ wkh wkhru| dovr suhglfwv wkdw xqghu vrph sdudphwhu ydoxhv vshfxodwlyh ehkdylru zloo eh
revhuyhg dqg wkdw wklv ehkdylru zloo vxssruw dq htxloleulxp lq zklfk d jrrg wkdw grhv qrw kdyh wkh
plqlpxp vwrudjh frvw zloo hphujh dv d phglxp ri h{fkdqjh1 Rxu sulru h{shulphqw frqfoxghg wkdw
wkh wkhru| zdv sduwldoo| vxffhvvixo lq dffrxqwlqj iru wkh hphujhqfh ri vshflf frpprglwlhv dv phgld
ri h{fkdqjh1 Zkhq wkh hqylurqphqw kdg sdudphwhu ydoxhv iru zklfk wkh wkhru| suhglfwhg wkdw wkh
remhfw zlwk wkh orzhvw vwrudjh frvwv zrxog hphujh dv wkh xqltxh phglxp ri h{fkdqjh/ ehkdylru
whqghg wr frqirup wr wklv suhglfwlrq1 Krzhyhu/ zkhq wkh hqylurqphqw kdg sdudphwhu ydoxhv iru
zklfk wkh wkhru| suhglfwhg wkdw dq remhfw zlwk pruh wkdq wkh plqlpxp vwrudjh frvwv zrxog hphujh
dv d phglxp ri h{fkdqjh/ wkh revhuyhg ehkdylru lqglfdwhg wkdw vshfxodwlyh ehkdylru zdv revhuyhg
pxfk ohvv iuhtxhqwo| wkdq suhglfwhg dqg qr remhfw zklfk kdg ohvv wkdq wkh plqlpxp vwrudjh frvw
hphujhg dv d jhqhudoo| dffhswhg phglxp ri h{fkdqjh1 Wklv qglqj zdv frqvlvwhqw zlwk wkh uhsruw
ri dq h{shulphqw frqgxfwhg e| Eurzq +4<<9,/ zkr xvhg d vrphzkdw glhuhqw h{shulphqwdo ghvljq/
dqg zlwk d vlpxodwlrq vwxg| ri duwlfldoo| lqwhooljhqw djhqwv zkr iroorzhg dq dgdswlyh ohduqlqj
surfhvv wkdw zdv frqgxfwhg e| Pdulprq hw do1 +4<;<,1
Fdq d jrrg wkdw kdv qr ydoxh dv d frqvxpswlrq jrrg wr dq|rqh/ exw zklfk lv ohvv frvwo| wr
5
vwruh wkdq dq| rwkhu gxudeoh jrrg hphujh dv d phglxp ri h{fkdqjh/ dv wkh wkhru| ri Nl|rwdnl
dqg Zuljkw lpsolhvB Wkh idfw wkdw wkh orzhvw frvw jrrg hphujhg dv d phglxp ri h{fkdqjh lq rxu
sulru h{shulphqw vxjjhvwv wkdw d jrrg zlwk qr frqvxpswlrq ydoxh/ exw dovr zlwk wkh orzhvw frvw ri
vwrudjh/ pd|/ lq idfw/ hphujh dv d phglxp ri h{fkdqjh1 Rq wkh rwkhu kdqg/ dffhswlqj d wrnhq remhfw
zlwk qr ydoxh dv d frqvxpswlrq jrrg wr dq| djhqw lv lqkhuhqwo| dq dfw ri vshfxodwlrq1 Wkh revhuyhg
uhoxfwdqfh ri pdq| shrsoh lq wkh sulru h{shulphqwv ri Gx| dqg Rfkv +4<<<, dqg ri Eurzq +4<<9,
wr hqjdjh lq vshfxodwlyh ehkdylru/ vxjjhvwv wkdw d jrrg zlwk qr lqwulqvlf ydoxh pd| qrw hphujh dv
d phglxp ri h{fkdqjh/ hyhq li lw kdv wkh orzhvw vwrudjh frvwv1 Wkh h{shulphqw uhsruwhg ehorz zdv
ghvljqhg wr dqvzhu wklv txhvwlrq dqg/ lq grlqj vr/ wr surylgh d fohdqhu whvw ri wkh Nl|rwdnl0Zuljkw
wkhru|1
Vshflfdoo|/ lq wkh h{shulphqw uhsruwhg ehorz/ d irxuwk remhfw lv dgghg wr wkh hqylurqphqw ri
wkh suhylrxv h{shulphqw1 Wklv remhfw/ jrrg 3/ rffxslhv vwrudjh vsdfh/ lv lq {hg vxsso|/ dqg fdqqrw
eh ghvwur|hg1 Lw kdv qr frqvxpswlrq ydoxh wr dq|rqh1 Vrph lqglylgxdov duh dozd|v kroglqj jrrg
31 Vlqfh jrrg 3 lv qrw d frqvxpswlrq jrrg iru dq|rqh/ li lw lv wr eh dffhswhg lq h{fkdqjh/ wkh
shuvrq zkr dffhswv lw pxvw eh vshfxodwlqj wkdw vrphrqh hovh zloo ydoxh jrrg 3 iru lwv xvh lq ixuwkhu
h{fkdqjhv/ dqg wkdw eholhi lpsolhv wkh ixuwkhu eholhi wkdw |hw rwkhu djhqwv zloo zlvk wr dffhsw wkh
remhfw wr idflolwdwh ixuwkhu h{fkdqjhv/ dg lqqlwxp1 Wklv eholhi lq wkh jhqhudo dffhswdelolw| ri dq
remhfw lv wkh wuxh hvvhqfh ri d phglxp ri h{fkdqjh dqg/ dffruglqj wr wkh Nl|rwdnl0Zuljkw wkhru|/ fdq
ehfrph dwwdfkhg wr dq| vwrudeoh remhfw/ surylghg wkdw lwv frvw ri vwrudjh lv vxflhqwo| orz uhodwlyh
wr rwkhu remhfwv wkdw pljkw vhuyh dv phgld ri h{fkdqjh1 Lqghhg/ dq h{whqvlrq ri wkh Nl|rwdnl0
Zuljkw prgho surylghg e| Dl|djdul dqg Zdoodfh +4<<5,/ d yhuvlrq ri zklfk lv dovr h{dplqhg lq wklv
sdshu/ suhglfwv wkdw xqghu fhuwdlq sdudphwhul}dwlrqv/ dq lqwulqvlfdoo| zruwkohvv remhfw zloo kdyh
vrph pdunhw dffhswdelolw| hyhq zkhq lw lv qrw wkh ohdvw frvwo|0wr0vwruh remhfw1 Wr iruhvkdgrz wkh
uhvxowv ri rxu h{shulphqw/ zh qg wkdw zkhq jrrg 3 lv wkh jrrg zlwk wkh orzhvw vwrudjh frvw/ dqg
wkh sdudphwhuv duh vhw vr dv wr suhglfw wkdw wkh orzhvw vwrudjh frvw jrrg zloo hphujh dv d phglxp
ri h{fkdqjh/ jrrg 3 lv/ lq idfw/ dv zlgho| xvhg dv d phglxp ri h{fkdqjh dv zdv wkh lqwulqvlfdoo|
ydoxhg jrrg zlwk orzhvw vwrudjh lq rxu suhylrxv h{shulphqw1 Lq dgglwlrq/ hyhq zkhq jrrg 3 lv qrw
wkh ohdvw frvwo|~wr~vwruh jrrg/ lw zloo flufxodwh1 Krzhyhu/ lq wklv fdvh/ wkh ghjuhh ri dffhswdelolw|
ri jrrg 3 dqg lwv frqvhtxhqw udwh ri flufxodwlrq lv vljqlfdqwo| ohvv wkdq wkh wkhru| suhglfwv1
Aihi @i Mii? tiih@* L|ih i Tih4i?|@* t|_it ?L*?} |L!i? LMiU|t @t @ t|Lhi Lu @*ic i} ih?@tUL?
@?_ khU!@4T E2fffc w@? @?_ *L|| EbbHc w4c hitUL|| @?_ 5?_ih Ebbec @h4L? @?_ 5?_ih Ebbc bbe
@?_ U@Mi EbHb( #g) EbbH thi)t |t *|ih@|hi N?*!i |iti t|_itc |i k)L|@! @?_ `h}| EbHb
6
2
Ai ,?hL?4i?|
Zh vwxg| yhuvlrqv ri Nl|rwdnl dqg Zuljkw*v +4<;<, vhdufk~wkhruhwlf prgho zlwk d vlqjoh wrnhq
remhfw1 D qlwh srsxodwlrq ri Q djhqwv lv glylghg xs htxdoo| lqwr rqh ri wkuhh w|shv/ uhihuuhg wr dv
w|sh @ 4> 5> 612 Wkhuh duh irxu glhuhqw lqglylvleoh jrrgv/ lqgh{hg e| m @ 3> 4> 5> 61 Jrrg 3 lv wkh
wrnhq remhfw> qr djhqw ghvluhv wr frqvxph wklv jrrg/ dqg iru wklv uhdvrq/ lw lv lqwulqvlfdoo| zruwkohvv1
Wkh vxsso| ri jrrg 3 lv h{rjhqrxv1 W|sh ghvluhv wr frqvxph jrrg m @ exw surgxfhv jrrg . 4
prgxor 61 Wkxv/ w|sh 4 ghvluhv jrrg 4 exw surgxfhv jrrg 5/ w|sh 5 ghvluhv jrrg 5 exw surgxfhv
jrrg 6/ dqg w|sh 6 ghvluhv jrrg 6 exw surgxfhv jrrg 41 Qrwlfh wkdw lq wklv hqylurqphqw wkhuh lv
dq devhqfh ri zkdw Mhyrqv whuphg d _grxeoh~frlqflghqfh~ri~zdqwv% wkdw fdq rqo| eh ryhufrph li
vrph djhqwv duh zloolqj wr wudgh iru jrrgv wkh| gr qrw ydoxh/ exw zklfk wkh| h{shfw wkh| zloo eh
deoh wr odwhu wudgh iru jrrgv wkh| gr ydoxh1 Vxfk jrrgv duh uhjdughg dv phgld ri h{fkdqjh1
Djhqwv kdyh dffhvv wr d vwrudjh whfkqrorj| wkdw hqdeohv wkhp wr vwruh rqh xqlw ri dq| jrrg
lq hyhu| shulrg1 Wkh iudfwlrq ri wkh srsxodwlrq vwrulqj jrrg 3 lv dq h{rjhqrxvo| jlyhq frqvwdqw/
p 5 +3> 4,1 Wkh uhpdlqlqj iudfwlrq ri wkh srsxodwlrq/ 4 p/ kdyh lqwulqvlfdoo| ydoxhg jrrgv lq
vwrudjh1 Vwrudjh/ krzhyhu/ pd| eh frvwo|1 Wkh shu shulrg frvw +lq whupv ri xwlolw|, ri vwrulqj d xqlw
ri jrrg m @ 3> 4> 5> 6 lv ghqrwhg e| f 31 Zh dvvxph wkdw wkhvh irxu vwrudjh frvwv duh frpprq
dfurvv sod|hu w|shv1 Vlploduo|/ iru vlpsolflw|/ zh dvvxph wkdw doo 6 sod|hu w|shv uhfhlyh wkh vdph
xwlolw| iurp frqvxpswlrq x A f dqg kdyh wkh vdph glvfrxqw idfwru 5 ^3> 4`1
Dw wkh ehjlqqlqj ri d wudglqj shulrg doo Q djhqwv duh udqgrpo| sdluhg zlwk doo srvvleoh sdlulqjv
htxdoo| olnho|1 Hdfk djhqw pxvw wkhq ghflgh zkhwkhu wr wudgh wkh jrrg wkdw kh fxuuhqwo| krogv
lq vwrudjh iru wkh jrrg wkdw wkh sod|hu zlwk zkrp kh lv pdwfkhg fxuuhqwo| krogv lq vwrudjh1 D
wudgh rffxuv rqo| li erwk sduwlhv djuhh wr wudgh1 Ehfdxvh jrrgv duh lqglylvleoh dqg djhqwv fdq rqo|
vwruh d vlqjoh xqlw ri dq| jrrg/ h{fkdqjhv dozd|v lqyroyh rqh~iru~rqh vzdsv ri jrrgv lq vwrudjh1e
i?hL?4i?| i i @4?i t L?i ? U 4*|T*ic _h@M*i @?_ ?|h?tU@**) @*i_ }LL_t UL4Ti|i | |i ?^i
|L!i? LMiU| @t t|Lhit Lu @*ic |t ThL_?} @ 4Lhi h}LhLt |it| Lu i|ih @ |L!i? LMiU| 4@) tihi @t @ 4i_4
Lu i U@?}i
2 k)L|@! @?_ `h}| EbHb @tt4i |ihi t @ UL?|?4 Lu @}i?|tc Liih ? Lh i Tih4i?|@* _it}?c i @hi
UL?t|h@?i_ |L UL?t_ih?} @ ?|i TLT*@|L? Lu t3i `i @i 4@_i @? igLh| |L UL?t_ih TLT*@|L? t3it Lu
' H Lh ' 2ec |@| @hi *@h}i M) UL4T@htL? | L|ih i Tih4i?|@* t|_itc ? @? igLh| |L LM|@? @ }LL_
@TThL 4@|L? |L |i UL?|?4 Lu @}i?|t |@| |i |iLh) Thit4it TTi?_ _tUttit |i UL?ti^i?Uit Lu Lh
?|i TLT*@|L? @TThL 4@|L?
k)L|@! @?_ `h}| hiuih |L |t ihtL? Lu |ih 4L_i* @t RL_i* W? @?L|ih ihtL?c RL_i* c U t
?L| tL4LhTUc |)Ti ThL_Uit }LL_ 3 4L_*L 5ii @*tL )@}@h @?_ `@**@Ui Ebbc bb2 L }i?ih@*3i
|i 4L_i* |L @**L uLh ? |)Tit @?_ ? }LL_t T*t @ |L!i? LMiU|
e
5ii Liihc 5 EbbD @?_ AhiLt @?_ `h}| EbbD L hi*@ |i ?_tM*|) @tt4T|L? |ihiM) Tih4||?}
ThUit |L Mi i?_L}i?Lt*) _i|ih4?i_ 5U @ _it}? *it Mi)L?_ |i tULTi Lu |t T@Tih
7
Li d sod|hu ri w|sh vxffhvvixoo| wudghv iru jrrg / kh frqvxphv wkdw jrrg/ uhfhlylqj xwlolw| x/
dqg kh lpphgldwho| surgxfhv d qhz xqlw ri klv surgxfwlrq jrrg/ . 4 prgxor 6/ zklfk ehfrphv
wkh jrrg kh krogv lq vwrudjh1 Ixuwkhupruh wkh sod|hu lv lpphgldwho| fkdujhg wkh vwrudjh frvw ri
klv surgxfwlrq jrrg vr wkdw klv qhw sd|r iurp vxffhvvixoo| wudglqj iru klv frqvxpswlrq jrrg lv
x f A 3/ zkhuh m @ . 4 prgxor 61 Li d sod|hu ri w|sh vxffhvvixoo| wudghv iru vrph jrrg m 9@ /
wkh jrrg kh kdv lq vwrudjh lv fkdqjhg wr uhhfw wklv wudglqj ghflvlrq1 Li d wudgh lv qrw pxwxdoo|
djuhhg xsrq/ erwk sod|huv frqwlqxh wr krog wkh vdph jrrgv lq vwrudjh wkdw wkh| khog sulru wr ehlqj
sdluhg1 Lq hlwkhu ri wkhvh odvw wzr fdvhv/ hdfk sod|hu lv fkdujhg wkh vwrudjh frvw ri wkh jrrg kh khog
lq vwrudjh iroorzlqj wkh frpsohwlrq ri wudgh/ vr wkdw klv qhw sd|r iru wkh shulrg lv f / zkhuh m
lv wkh jrrg khog lq vwrudjh dw wkh hqg ri wkh urxqg1 Vlqfh qr w|sh surgxfhv ru frqvxphv jrrg 3/
dqg jrrg 3 fdqqrw eh glvfdughg/ wkh wrwdo vwrfn ri jrrg 3 lq wkh hfrqrp|/ Qp/ grhv qrw fkdqjh
iurp shulrg wr shulrg/ zkloh wkh wrwdo vwrfn ri wkh rwkhu jrrgv pd| ydu| iurp shulrg wr shulrg/
ghshqglqj rq wkh sdlulqj ri djhqwv dqg wkhlu wudglqj ghflvlrqv1
2
L_i* @h@4i|ih3@|L?
Wkh vwrudjh frvwv iru wkh wkuhh lqwulqvlfdoo| ydoxhg jrrgv dozd|v vdwlvi|=
3 f ? f2 ? f ? x=
Dv rxu irfxv lv rq wkh xvh ri wkh lqwulqvlfdoo| zruwkohvv remhfw dv d phglxp ri h{fkdqjh/ zh zloo
frqvlghu wzr glhuhqw fdvhv iru wkh vwrudjh frvw ri wklv remhfw1 Wkh uvw fdvh lv wkh rqh vwxglhg e|
Nl|rwdnl dqg Zuljkw +4<;<,1
Fdvh 4= 3 @ ff ? f
Lq wklv fdvh/ wkh wrnhq remhfw lv erwk frvwohvv wr vwruh dqg wkh ohdvw frvwo|~wr~vwruh remhfw1 Wkh
vhfrqg fdvh sursrvhg e| Dl|djdul dqg Zdoodfh +4<<5,1
Fdvh 5= 3 @ f ? ff ? f2 =
Lq wklv fdvh/ jrrg 3 lv qhlwkhu frvwohvv qru wkh ohdvw frvwo|~wr~vwruh remhfw1 Wkh sduwlfxodu yhuvlrqv
ri wkh Nl|rwdnl~Zuljkw +4<;<, prgho zh vwxg| frpsulvh wzr pdlq sdudphwhul}dwlrqv/ sdudphwhu
vhwv 4 dqg 5/ wkdw duh frqvlvwhqw zlwk fdvhv 4 dqg 5 uhvshfwixoo|1 Wkhvh wzr sdudphwhu vhwv duh
surylghg lq Wdeoh 41 Qrwlfh wkdw wkh ydoxh ri p ydulhv rqo| zlwklq sdudphwhu vhw 41 dqg wkdw
vwrudjh frvwv/ f / wkh xwlolw| ydoxh ri frqvxpswlrq x/ dqg wkh ydoxh ri p doo fkdqjh dv zh pryh
8
Wdeoh 4= Sdudphwhu Vhwv Xvhg lq wkh H{shulphqwv
Sdudphwhu Vhw
4
5
ff
3
4
f
4
3
f2
7
56
f
<
57
x
53
433
31<3
31<3
p
31583/ 31666/ ru 31833
3149:
iurp sdudphwhu vhw 4 wr sdudphwhu vhw 51 Wkhvh sdudphwhu yduldwlrqv zhuh qhfhvvdu| wr hqvxuh wkh
h{lvwhqfh ri wkh wzr glhuhqw vwdwlrqdu| Qdvk htxloleuld zh h{dplqh lq wklv sdshu1
22
5|i@_) 5|@|i i@Lh
Rxu dlp lv wr ghwhuplqh zkhwkhu vxemhfwv dgrsw wudglqj vwudwhjlhv frqvlvwhqw zlwk wkh prgho*v
vwhdg| vwdwh suhglfwlrqv1 Zh qrz glvfxvv wkhvh vwhdg| vwdwh suhglfwlrqv1 D pruh frpsohwh dqdo|vlv
lv surylghg lq Dsshqgl{ 41 Dv rxu irfxv lv rq vwhdg| vwdwh ehkdylru/ zh gurs doo uhihuhqfhv wr wlph1
ghqrwh wkh hqg~ri~shulrg/ h{shfwhg glvfrxqwhg xwlolw| wr sod|hu l ri w|sh iurp vwrulqj
Ohw yc
jrrg m1 Wklv ydoxh lv h{solflwo| ghqhg lq Dsshqgl{ 41 Ohw v&
c 5 ^3> 4` ghqrwh wkh vwudwhj| sod|hg
e| w|sh sod|hu l zkhq vwrulqj jrrg m dqg udqgrpo| sdluhg zlwk d sod|hu vwrulqj jrrg n 9@ m1 Li
&
v&
c @ 4/ wkh w|sh sod|hu rhuv wr wudgh jrrg m iru jrrg n> li vc @ 3 wkh w|sh sod|hu uhixvhv
wr wudgh1D Lq wkh vwhdg| vwdwh/ doo sod|huv ri w|sh / l @ 4> 5> ===> Q@6 sod| dffruglqj wr wkh vdph/
wlph~lqyduldqw vwudwhj|> zh wkhuhiruh gurs wkh l vxevfulsw1 Wklv vwudwhj| lv rswlpdo lq wkh vhqvh
wkdw=
;
A
? i3j
li y A y& /
&
v&
5 A i4j li y ? y /
= ^3> 4` rwkhuzlvh1
Dv lq Nl|rwdnl dqg Zuljkw +4<;<,/ rxu irfxv lv rq vwhdg| vwdwh/ sxuh vwudwhj| htxloleuld/ vr v&
5
i3> 4j/ ; / m/ n1
Dv glvfxvvhg lq Dsshqgl{ 4/ xqghu erwk sdudphwhu vhwv 4 dqg 5/ lw lv dozd|v rswlpdo iru sod|hu
w|sh wr wudgh iru jrrg zkhqhyhu kh phhwv wklv jrrg lq wudgh dv y @ pd{ y iru doo 1 Wkh
udqnlqj ri wkh rwkhu ydoxhv ghshqg rq wkh sduwlfxodu sdudphwhu vhw xvhg1 Vlqfh sod|huv dozd|v wudgh
iru wkhlu frqvxpswlrq jrrg dqg vwudwhjlhv duh dvvxphg wr eh v|pphwulf zh fdq zulwh wkh vwdwlrqdu|
sxuh vwudwhj| yhfwru ri d w|sh sod|hu frpsdfwo| dv
f nc&
>
v @ v ncf > v&c
> v
D
Wu |)Ti T*@)ih | }LL_ 4ii|t @?L|ih T*@)ih t|Lh?} }LL_ c i @tt4i |@| ?L |h@_i LUUht
9
zkhuh . 4 lv w|sh *v surgxfwlrq jrrg dqg n 9@ 3 lv wkh rwkhu lqwulqvlfdoo| ydoxhg jrrg wkdw w|sh
qhlwkhu surgxfhv qru frqvxphv1S
Xqghu sdudphwhu vhw 4/ zkhuh jrrg 3 lv wkh ohdvw frvwo|~wr~vwruh jrrg/ Nl|rwdnl dqg Zuljkw
+4<;<, vkrz wkdw wkhuh h{lvw wzr vwdwlrqdu| sxuh vwudwhj| htxloleuld1 Lq Dsshqgl{ 4 zh yhuli| wkh
h{lvwhqfh ri wkhvh htxloleuld iru rxu oderudwru| lpsohphqwdwlrq ri wkh prgho1 Lq wkh uvw ri wkhvh/
jrrg 3 kdv srvlwlyh ydoxh/ dqg wkh vwudwhj| surohv ri wkh wkuhh sod|hu w|shv duh=
v @ +4> 4> 3,>
v2 @ +4> 4> 4,
v @ +4> 4> 3,
Lq wklv vwhdg| vwdwh/ doo wkuhh w|shv qg lw rswlpdo wr wudgh iru jrrg 3 zkhqhyhu wkh| hqfrxqwhu
lw lq wudgh/ pdnlqj jrrg 3 d jhqhudoo| dffhswhg phglxp ri h{fkdqjh1 Ixuwkhupruh/ lq wklv vwhdg|
vwdwh/ doo sod|huv dgkhuh wr _ixqgdphqwdo/% vwrudjh frvw~uhgxflqj vwudwhjlhv1 Lq sduwlfxodu/ w|sh
4 dqg w|sh 6 sod|huv uhixvh wr wudgh wkhlu surgxfwlrq jrrgv 5 dqg 4 iru wkh pruh frvwo|~wr~vwruh
jrrgv 6 dqg 5/ uhvshfwlyho|/ dqg w|sh 5 sod|huv dozd|v rhu wr wudgh wkhlu surgxfwlrq jrrg 6 iru
wkh ohvv frvwo|~wr~vwruh jrrg 41 Khqfh/ jrrg 4 dovr vhuyhv dv d +olplwhg, phglxp ri h{fkdqjh lq
wklv vwhdg| vwdwh htxloleulxp1
Lq wkh rwkhu vwhdg| vwdwh htxloleulxp/ jrrg 3 idlov wr flufxodwh dv d phglxp ri h{fkdqjh vlpso|
ehfdxvh qr rqh eholhyhv lw kdv ydoxh lq h{fkdqjh/ l1h1 yf @ 3 iru doo 1 Lq wklv htxloleulxp/ doo
sod|huv uhixvh wr dffhsw jrrg 3 lq wudgh exw dozd|v rhu wr dffhsw dq| rwkhu jrrg wkdw kdv d orzhu
vwrudjh frvw wkdq wkh jrrg wkh| duh fxuuhqwo| kroglqj1 Lq sduwlfxodu/ wkh vwudwhj| surohv ri wkh
wkuhh sod|hu w|shv lq wklv vwhdg| vwdwh duh=
v @ +3> 3> 3,>
v2 @ +3> 3> 4,
v @ +3> 3> 3,
Lq wklv vwhdg| vwdwh/ jrrg 4 lv wkh rqo| phglxp ri h{fkdqjh1
Wkh iuhtxhqflhv zlwk zklfk sod|huv rhu wr wudgh iru fhuwdlq jrrgv duh qrw wkh rqo| wkhruhwlfdo
suhglfwlrqv wkdw fdq eh whvwhg1 Lw lv dovr srvvleoh wr whvw wkh prgho*v suhglfwlrqv frqfhuqlqj wkh
vwhdg| vwdwh suredelolw| ri dffhswdqfh ri jrrgv rhuhg lq wudgh dv d ixqfwlrq ri wkh sursruwlrq ri
wkh srsxodwlrq wkdw lv kroglqj wkh wrnhq remhfw/ p1 Dv Nl|rwdnl dqg Zuljkw +4<;<, revhuyh/ wkhvh
dffhswdqfh suredelolwlhv duh d ehwwhu phdvxuh ri _prqh|qhvv% wkdq pruh wudglwlrqdo phdvxuhv vxfk
dv wkh yhorflw| ri flufxodwlrq ri jrrgv vlqfh lq wklv prgho/ jrrgv wkdw duh qrw xvhg dv phgld ri
h{fkdqjh +h1j1 kljk vwrudjh frvw jrrgv, fdq qhyhuwkhohvv kdyh d yhu| kljk yhorflw| ri flufxodwlrq lq
htxloleulxp/ hvshfldoo| dv wkh sursruwlrq ri wkh srsxodwlrq kroglqj wkh wrnhq remhfw/ p/ lqfuhdvhv1
S
&
) Rt)44i|hUc i 4i@? |@| r&
' 3 r :
Wkh dffhswdqfh suredelolw| ri hdfk jrrg lv d ixqfwlrq ri p ehfdxvh wkh fdofxodwlrq ri wklv
suredelolw| xvhv wkh vwhdg| vwdwh sursruwlrqv ri hdfk sod|hu w|sh vwrulqj hdfk w|sh ri jrrg/ dqg
wkhvh sursruwlrqv ghshqg rq wkh ydoxh ri p1 Wkh vwhdg| vwdwh sursruwlrqv iru wkh wkuhh glhuhqw
ydoxhv wkdw zh xvh iru p lq sdudphwhu vhw 4 duh jlyhq lq Dsshqgl{ 41 Xvlqj wkh sursruwlrqv iru
wkh vwhdg| vwdwh lq zklfk jrrg 3 vhuyhv dv d phglxp ri h{fkdqjh/ rqh fdq fdofxodwh wkh qxpehu
ri wlphv hdfk jrrg jhwv rhuhg lq wudgh shu shulrg dqg wkh qxpehu ri wlphv hdfk jrrg lv dfwxdoo|
wudghg shu shulrg1. Wkh vwhdg| vwdwh suredelolw| wkdw d jrrg lv dffhswhg zkhq rhuhg lv wkh udwlr
ri wkh qxpehu ri wlphv lw lv wudghg wr wkh qxpehu ri wlphv lw lv rhuhg lq hdfk shulrg1 Wkhvh vwhdg|
vwdwh dffhswdqfh suredelolwlhv duh jlyhq lq Wdeoh 5 ehorz1
Wdeoh 5= Vwhdg| Vwdwh Suredelolw| wkdw d Jrrg Rhuhg lq Wudgh lv Dffhswhg1
Fdvh 4 +Sdudphwhu Vhw 4, zkhuh Jrrg 3 lv Xvhg dv d Phglxp ri H{fkdqjh
p@
3158
3166
3183
Dffhswdqfh Suredelolw|
Jrrg 3 Jrrg 4 Jrrg 5
4133
318:9
3164:
4133
31854
31646
4133
31783
3163;
ri
Jrrg 6
316<4
316;;
316:;
Vlqfh rxu irfxv khuh lv rq wkh fdvh zkhuh jrrg 3 vhuyhv dv d phglxp ri h{fkdqjh/ zh vhh
wkdw xqghu sdudphwhu vhw 4/ wkh suredelolw| wkdw jrrg 3 lv dffhswhg zkhq rhuhg lv dozd|v 433(
uhjdugohvv ri wkh ydoxh ri p1 Pruh lpsruwdqwo|/ qrwlfh wkdw wkh vwhdg| vwdwh dffhswdqfh suredelolw|
iru jrrg 4/ wkh rqo| rwkhu phglxp ri h{fkdqjh/ lv dovr kljk/ exw wkdw wklv dffhswdqfh suredelolw|
ghfuhdvhv dv p lqfuhdvhv1 Zh qrwh wkdw wkh dffhswdqfh suredelolwlhv iru jrrgv 5 dqg 6 dovr ghfuhdvh
voljkwo| dv p lqfuhdvhv1 Krzhyhu/ qhlwkhu ri wkhvh jrrgv vhuyh dv phgld ri h{fkdqjh lq d vwdwlrqdu|
htxloleulxp1 Zh zloo wkhuhiruh irfxv rxu dwwhqwlrq rq wkh k|srwkhvhv wkdw jrrg 3*v dffhswdelolw|
uhpdlqv xqfkdqjhg dv p lqfuhdvhv/ dqg wkdw jrrg 4*v dffhswdelolw| ghfuhdvhv dv p lqfuhdvhv1
Zh qh{w frqvlghu wkh fdvh zkhuh jrrg 3 lv qrw wkh ohdvw frvwo|~wr~vwruh jrrg/ dv vwxglhg e|
Dl|djdul dqg Zdoodfh +4<<5,1 Xqghu rxu sdudphwhu vhw 5/ zh h{sodlq lq Dsshqgl{ 4 wkdw wkhuh
djdlq h{lvw wzr vwdwlrqdu| sxuh vwudwhj| htxloleuld1 Lq wkh uvw/ jrrg 3 vhuyhv dv d olplwhg phglxp
ri h{fkdqjh/ hyhq wkrxjk lw lv qr orqjhu frvwohvv wr vwruh/ dqg lw lv pruh frvwo| wr vwruh wkdq jrrg
.
At U@? @*tL Mi _L?i uLh |i L|ih i^*Mh4c ihi }LL_ f t ?L| @UUiT|i_ ? |h@_i
;
41 Wkh vwudwhj| surohv ri wkh wkuhh sod|hu w|shv lq wklv vwhdg| vwdwh duh=
v @ +4> 3> 4,>
v2 @ +4> 3> 4,
v @ +3> 4> 3,
Qrwlfh wkdw zkloh w|sh 4 sod|huv dozd|v rhu wkhlu surgxfhg jrrg 5 iru wkh ohvv frvwo|~wr~vwruh
jrrg 3/ lq wklv vwhdg| vwdwh/ wkh| dovr qg lw rswlpdo wr vshfxodwh lq jrrg 6/ uhixvlqj wr wudgh jrrg
6 iru wkh ohvv frvwo| jrrg 31 V|pphwulfdoo|/ wkh| dozd|v rhu wr wudgh jrrg 3 iru jrrg 61 Lq idfw/ d
w|sh 4 sod|hu zloo eh xqdeoh wr wudgh jrrg 3 gluhfwo| iru wkh ghvluhg frqvxpswlrq jrrg 4 zlwkrxw
uvw wudglqj wkdw jrrg 3 iru jrrg 61H W|sh 4 sod|huv dovr sod| wkh vshfxodwlyh vwudwhj| ri wudglqj
wkhlu surgxfhg jrrg 5 iru wkh pruh frvwo|~wr~vwruh jrrg 61b 1 Wkh vwhdg| vwdwh vwudwhj| suroh iru
w|sh 4 sod|huv lv wkh prvw lqwhuhvwlqj suhglfwlrq ri wkh Fdvh 5 hqylurqphqw/ dqg zh irfxv pxfk
dwwhqwlrq rq wkh ehkdylru ri w|sh 4 sod|huv lq wklv hqylurqphqw odwhu lq vhfwlrq 7151
W|sh 5 dqg 6 sod|huv dgkhuh wr ixqgdphqwdo wudglqj vwudwhjlhv dv lq Fdvh 4> Krzhyhu/ wkh
idfw wkdw jrrg 4 lv qrz ohvv frvwo| wr vwruh wkdq jrrg 3/ dowhuv wkhlu wudglqj ehkdylru d olwwoh1 Lq
sduwlfxodu/ w|sh 5 sod|huv zloo frqwlqxh wr rhu wr wudgh wkhlu surgxfhg jrrg 6 iru jrrg 3/ exw
zloo uhixvh wr rhu wr wudgh jrrg 4 iru jrrg 3/ dv jrrg 4 lv qrz frvwohvv wr vwruh1 Iru wkh vdph
uhdvrq/ w|sh 6 sod|huv qrz uhixvh wr rhu wr wudgh wkhlu surgxfhg jrrg 4 iru jrrg 31 W|sh 5 sod|huv
frqwlqxh wr rhu wr wudgh wkhlu surgxfhg jrrg 6 iru jrrg 41
Qrwlfh wkdw lq wklv vwhdg| vwdwh wkhuh duh wkuhh phgld ri h{fkdqjh1 Jrrg 4/ wkh ohdvw frvwo|~wr~
vwruh jrrg/ lv d jhqhudoo| dffhswhg phglxp ri h{fkdqjh1 Ixuwkhupruh/ jrrgv 3 dqg 6 dovr vhuyh dv
olplwhg phgld ri h{fkdqjh> erwk w|shv 4 dqg 5 zloo dffhsw jrrg 3 iurp rqh dqrwkhu> w|sh 4 sod|huv
zloo dffhsw jrrg 3 lq h{fkdqjh iru jrrg 5/ dqg w|sh 5 sod|huv zloo dffhsw jrrg 3 lq h{fkdqjh iru
jrrg 61 Krzhyhu/ w|sh 5 sod|huv zloo uhixvh wr dffhsw jrrg 3 lq h{fkdqjh iru jrrg 41 W|sh 6 sod|huv
zloo qhyhu dffhsw dq|wklqj rwkhu wkdq jrrg 6 lq wudgh/ vr jrrg 3 lv qr orqjhu d jhqhudoo| dffhswhg
phglxp ri h{fkdqjh1 Vlploduo|/ jrrg 6 vhuyhv dv d olplwhg phglxp ri h{fkdqjh lq wklv htxloleulxp/
dv w|sh 4 sod|huv zloo dffhsw jrrg 6 iurp w|sh 5 sod|huv lq h{fkdqjh iru jrrg 3 ru 5/ exw w|sh 5
sod|huv zlwk jrrg 3 ru 4 lq vwrudjh zloo qrw djuhh wr wudgh iru jrrg 61
Wkh rwkhu sxuh vwudwhj| vwhdg| vwdwh lq Fdvh 5 lv rqh zkhuh jrrg 3 kdv qr ydoxh wr dq| w|sh
dqg lv wkhuhiruh uhixvhg lq doo wudghv1 Wkh vwudwhj| surohv ri wkh wkuhh sod|hu w|shv lq wklv vwhdg|
H
Ai) ** Mi ?tUUittu* |h@_?} }LL_ f |L |)Ti 2 Lh |)Ti T*@)iht t|Lh?} }LL_ c @t ML| Lu |iti T*@)ih |)Tit
** @*@)t hiuti tU |h@_itc @t _tUtti_ Mi*L
b
t _tUtti_ ? TTi?_ c ? |t t|i@_) t|@|ic |i @*it |L |)Ti Lu |i uLh }LL_t t@|tu) 2 f
<
vwdwh duh=
v @ +3> 3> 4,>
v2 @ +3> 3> 4,
v @ +3> 3> 3,
Wklv vwudwhj| suroh lv wkh vdph dv wkh _vshfxodwlyh% vwhdg| vwdwh htxloleulxp ghvfulehg lq Nl|rwdnl
dqg Zuljkw +4<;<, iru wkh fdvh ri wkuhh lqwulqvlfdoo| ydoxhg +frpprglw|, remhfwv dqg qr wrnhq remhfw1
Wkh qxphulfdo h{dpsoh suhvhqwhg lq Dl|djdul dqg Zdoodfh +4<<5, lv edvhg rq d sdudphwhul}dwlrq
ri wkh prgho lq zklfk p/ wkh iudfwlrq ri wkh wrwdo srsxodwlrq kroglqj jrrg 3/ lv =358;1 Wklv iudfwlrq
lv wrr vpdoo iru rxu oderudwru| lpsohphqwdwlrq zlwk d qlwh qxpehu ri djhqwv1 Zh kdyh wkhuhiruh
frqvwuxfwhg dqrwkhu qxphulfdo h{dpsoh ri d sxuh vwudwhj| htxloleulxp dorqj wkh olqhv ri Dl|djdul
dqg Zdoodfh*v h{dpsoh/ zkhuh p @ 4@91f Wkh rwkhu sdudphwhu fkrlfhv iru wklv vhfrqg fdvh duh jlyhq
lq sdudphwhu vhw qxpehu 51 Iroorzlqj wkh vdph surfhgxuhv rxwolqhg lq Dl|djdul dqg Zdoodfh +4<<5,
zh yhulhg wkdw rxu sdudphwhu fkrlfhv lq sdudphwhu vhw 5 jlyh ulvh wr wkh wzr sxuh vwudwhj|/ vwhdg|
vwdwh htxloleuld ghvfulehg deryh/ hyhq lq wkh qlwh srsxodwlrq fdvh1 D pruh ghwdlohg ghvfulswlrq ri
wkhvh fdofxodwlrqv lv surylghg lq Dsshqgl{ 41
, Tih4i?|@* #it}?
Wkh vhw ri h{shulphqwv zh uhsruw lq wklv sdshu fdq eh jurxshg lqwr wzr pdlq wuhdwphqwv wkdw pdnh
xvh ri hlwkhu sdudphwhu vhw 4 ru 51 D wrwdo ri 47 h{shulphqwdo vhvvlrqv zhuh frqgxfwhg1 Lq wkh uvw
< vhvvlrqv/ odehohg D4~D6/ E4~E6/ dqg F4~F6/ zh xvhg sdudphwhu vhw 4 dqg ydulhg p/ wkh iudfwlrq
ri wkh srsxodwlrq vwrulqj jrrg 31 Lq vhvvlrqv D4~D6/ zh vhw p @ 4@7/ lq vhvvlrqv E4~E6/ zh vhw
p @ 4@6 dqg lq vhvvlrqv F4~F6 zh vhw p @ 4@51 Lq wkh uhpdlqlqj 8 vhvvlrqv/ odehohg G4~G8/ zh
xvhg sdudphwhu vhw 5 dqg vhw p @ 4@91 Ixuwkhu fkdudfwhulvwlfv ri wkhvh vhvvlrqv duh surylghg lq
Wdeoh 6 dqg duh glvfxvvhg odwhu lq wkh wh{w1
Wkh 47 vhvvlrqv uhsruwhg lq wklv sdshu zhuh frqgxfwhg xvlqj wkh Xqlyhuvlw| ri Slwwvexujk*v
h{shulphqwdo hfrqrplfv frpsxwhu oderudwru|1 Iru hdfk vhvvlrq/ d jurxs ri 4; ru 57 vxemhfwv +dv
lqglfdwhg lq Wdeoh 6, zdv uhfuxlwhg1 Lq fkrrvlqj srsxodwlrq vl}hv zh zhuh frqvwudlqhg e| wkh qxpehu
ri frpsxwhu zrunvwdwlrqv dydlodeoh lq wkh oderudwru| dqg e| wkh uhtxluhphqw wkdw srsxodwlrq vl}hv
eh pxowlsohv ri 9 dv zh zdqwhg wr kdyh htxdo qxpehuv ri hdfk ri wkh wkuhh w|shv dqg zh zdqwhg doo
sod|huv wr eh sdluhg zlwk rqh dqrwkhu lq hdfk urxqg ri wkh jdph1 Vxemhfwv zhuh sulpdulo| Xqlyhuvlw|
f
|ih ULUit uLh 6 @hi TLttM*i Ai ULUi Lu 6 4t| Mi h@|ih t4@** uLh |ihi |L Mi @ Thi t|h@|i}) @t
i^*Mh4 ihi |i |L!i? LMiU| t ti_ @t @ 4i_4 Lu i U@?}i @?_ t |i tiUL?_ *i@t| ULt|*)|Lt|Lhi LMiU|
Ai hit|hU|L?t L? |i @*i uLh 6 _iTi?_ L? |i L|ih T@h@4i|iht |@| @hi ULti? uLh |i 4L_i*
43
Wdeoh 6= Fkdudfwhulvwlfv ri H{shulphqwdo Vhvvlrqv
5ittL?
4Mih
2
2
2
#
#2
#
#e
#DS
N|*|) V@*i
Lu L?t4T|L?
2f
2f
2f
2f
2f
2f
2f
2f
2f
ff
ff
ff
ff
ff
,^*Mh4
A)Ti@
6
6
6
6
6
6
6
6
6
5
5
5
5
5
4Mih Lu
5MiU|t
2e
2e
2e
H
H
H
2e
2e
H
H
H
H
H
H
ihUi?|@}i |
AL!i? MiU| E6
2Df
2Df
2Df
Dff
Dff
Dff
S.
S.
S.
S.
S.
4Mih Lu
B@4it
S
S
f
2
.
.
H
H
.
4Mih Lu
+L?_tK
f2
fH
2f
f
f2
fe
f.
b2
bH
S.
f.
HS
bb
bb
@
6'6?_@4i?|@*c 5'5TiU*@|i
AL|@* ?4Mih Lu |h@_?} hL?_t uhL4 @** }@4it T*@)i_
S
5MiU|t ? |t tittL? ihi ThL_i_ | @__|L?@* ?uLh4@|L? UL?Uih??} |i TLT*@|L?_i _t
|hM|L? Lu t|h@|i}it T*@)i_
K
ri Slwwvexujk xqghujudgxdwhv1 Hdfk vxemhfw zdv uhfuxlwhg iru d vlqjoh <3 plqxwh vhvvlrq1 Qr vxemhfw
sduwlflsdwhg lq pruh wkdq rqh vhvvlrq dqg qr vxemhfw kdg dq| sulru h{shulhqfh zlwk wkh jdph1
Dw wkh ehjlqqlqj ri hdfk vhvvlrq/ vxemhfwv zhuh udqgrpo| dvvljqhg wr sod| wkh uroh ri rqh ri wkh
wkuhh sod|hu w|shv1 Wkh| zhuh lqvwuxfwhg wkdw wkhlu w|sh zrxog qrw fkdqjh iru wkh gxudwlrq ri wkh
vhvvlrq1 Vxemhfwv uhfhlyhg zulwwhq lqvwuxfwlrqv frqfhuqlqj wkh jdph wkh| zhuh derxw wr sod|1 D
frs| ri wkh lqvwuxfwlrqv xvhg lq wkh _G% vhvvlrqv lv surylghg lq Dsshqgl{ 5 +rwkhu lqvwuxfwlrqv duh
vlplodu,1 Wkh lqvwuxfwlrqv zhuh wkh vdph iru doo sod|huv dqg wklv idfw zdv uhlqirufhg e| uhdglqj wkh
lqvwuxfwlrqv dorxg1 Wkxv/ sod|huv zhuh lqiruphg ri wkh remhfwlyhv ri hdfk ri wkh wkuhh sod|hu w|shv1
Lq sduwlfxodu/ wkh| ohduqhg wkdw hdfk w|sh ghvluhg wkh jrrg fruuhvsrqglqj wr wkhlu rzq w|sh dqg
wkdw qr w|sh ghvluhg wr frqvxph jrrg 31
Diwhu wkh zulwwhq lqvwuxfwlrqv zhuh uhdg dorxg dqg txhvwlrqv zhuh dqvzhuhg d vlqjoh sudfwlfh
jdph zdv sod|hg wr foduli| wkh lqvwuxfwlrqv dqg wr doorz vxemhfwv wr ehfrph idploldu zlwk wkh
frpsxwhu lqwhuidfh1 Iroorzlqj wkh sudfwlfh urxqg/ vxemhfwv ehjdq sod|lqj iru prqh|/ surylglqj
uhvsrqvhv rq wkhlu frpsxwhu whuplqdov zkhq surpswhg1
Wkh frpsxwhu surjudp wkdw zh ghyhorshg iru wklv h{shulphqw shuirupv wkh udqgrp pdwfklqj
44
ri vxemhfwv/ uhsruwv edfn wr vxemhfwv rq lqirupdwlrq uhohydqw wr wkhlu wudglqj ghflvlrqv dqg nhhsv
wudfn ri lqirupdwlrq vxfk dv wudglqj ghflvlrqv/ jrrgv lq vwrudjh/ dqg srlqwv hduqhg e| hdfk vxemhfw1
Lw dovr froohfwv/ xsgdwhv dqg uhyhdov wr vxemhfwv lqirupdwlrq rq wkh klvwrulfdo dyhudjh sursruwlrqv ri
hdfk w|sh vwrulqj hdfk ri wkh irxu glhuhqw jrrgv1 Wklv lqirupdwlrq vhuyhv dv d vxflhqw vwdwlvwlf
iru wkh klvwrulfdo glvwulexwlrq ri vwudwhjlhv wkdw kdyh ehhq sod|hg1
Wkh 47 h{shulphqwdo vhvvlrqv lqyroyhg dq dyhudjh ri 43 jdphv hdfk/ dqg hdfk jdph frqvlvwhg ri
dq dyhudjh ri 43 wudglqj urxqgv1 Wkxv/ hdfk vhvvlrq lqyroyhg dq dyhudjh ri 433 wudglqj urxqgv1 Wkh
wrwdo qxpehu ri jdphv dqg wudglqj urxqgv iru hdfk vhvvlrq lv uhsruwhg lq Wdeoh 61 Hdfk wudglqj
urxqg fdq eh ghvfulehg dv iroorzv1
Dw wkh vwduw ri wkh wudglqj urxqg wkh frpsxwhu surjudp udqgrpo| sdluv doo vxemhfwv lq wkh
vhvvlrq1 Rqfh sdluhg/ wkh uvw frpsxwhu vfuhhq rq hdfk sod|hu*v prqlwru uhyhdov lqirupdwlrq derxw
wkh sod|hu zlwk zkrp wkh| duh pdwfkhg ~ l1h1 wkh w|sh ri wklv rwkhu sod|hu dqg wkh jrrg wklv
rwkhu sod|hu kdv lq vwrudjh ~ vhh Iljxuh 41 Sod|huv duh dovr uhplqghg ri wkhlu rzq w|sh/ wkh jrrg
wkh| fxuuhqwo| kdyh lq vwrudjh/ wkh vwrudjh frvwv lq srlqwv ri wkh irxu jrrgv/ 3/ 4/ 5 dqg 6/ dqg
wkh qxpehu ri srlqwv wkh| hduq zkhq wkh| rewdlq wkh jrrg fruuhvsrqglqj wr wkhlu w|sh1 Ryhu wkh
frxuvh ri d jdph sod|huv dffxpxodwh srlqwv udwkhu wkdq fdvk edodqfhv1 Hdfk sod|hu*v srlqw wrwdo iru
wkh fxuuhqw jdph/ dv ri wkh odvw wudglqj urxqg lv lqglfdwhg rq wklv uvw wudglqj vfuhhq1 Wkh pdqqhu
lq zklfk srlqwv duh frqyhuwhg lqwr fdvk sd|phqwv lv glvfxvvhg ehorz1
Sod|huv duh dvnhg zkhwkhu wkh| zdqw wr wudgh wkh jrrg wkh| kdyh lq vwrudjh iru wkh jrrg ri wkh
sod|hu zlwk zkrp wkh| duh pdwfkhg1 Wkh| uhvsrqg wr wklv txhvwlrq e| w|slqj hlwkhu \ iru |hv ru
Q iru qr dqg wkh| kdyh dq rssruwxqlw| wr fkdqjh wkhlu ghflvlrq ehiruh lw lv qdol}hg1 Wudgh rffxuv
li dqg rqo| li erwk sduwlhv djuhh wr d wudgh> doo wudghv frqvlvw ri rqh~iru~rqh h{fkdqjhv ri wkh wzr
jrrgv wkh sdluhg sod|huv kdyh lq vwrudjh1
^Iljxuh 4 khuh1`
Ehiruh pdnlqj wkhlu ghflvlrqv/ sod|huv frxog frqvlghu vrph dgglwlrqdo lqirupdwlrq wkdw zdv
surylghg rq wkh uvw wudglqj urxqg vfuhhq ~ uhihu djdlq wr Iljxuh 41 Lq wkh plggoh ri wklv vfuhhq lv d
edu fkduw lqglfdwlqj wkh fxpxodwlyh suredelolw| wkdw wkh fxuuhqw jdph zloo hqg 4 wr 43 urxqgv iurp
wkh fxuuhqw urxqg1 Wkh fxpxodwlyh suredelolwlhv lq wklv edu fkduw uhhfw wkh frqvwdqw/ rqh~lq~whq
fkdqfh +143 suredelolw|, wkdw wkh jdph zloo hqg iurp dq| rqh urxqg wr wkh qh{w1 Zh fkrvh wr kdyh
d udqgrp vwrsslqj uxoh iru hdfk jdph iru wzr uhdvrqv1 Iluvw/ wkh Nl|rwdnl~Zuljkw prgho hqylvlrqv
45
wkdw djhqwv duh lqqlwho|~olyhg1 Vhfrqg/ wkh prgho dovr vshflhv wkdw djhqwv glvfrxqw wkh ixwxuh
xvlqj d frqvwdqw glvfrxqw idfwru 5 +3> 4, wkdw lv frpprq dfurvv w|shv1 Wkh udqgrp vwrsslqj
uxoh surylghv xv zlwk d phfkdqlvp iru lpsohphqwlqj erwk ri wkhvh ihdwxuhv ri wkh Nl|rwdnl~Zuljkw
hqylurqphqw1 Dv lqglfdwhg lq Wdeoh 4/ lv dozd|v vhw wr =<3 lq erwk sdudphwhu vhwv 4 dqg 51 Wr
lpsohphqw glvfrxqwlqj ri ixwxuh sd|rv e| <3(/ zh fkrvh wr hqg hdfk jdph zlwk suredelolw| 4 /
l1h1 zlwk suredelolw| 43( dw wkh hqg ri dq| urxqg wkdw kdv ehhq uhdfkhg1 Iroorzlqj wkh frpsohwlrq
ri d wudglqj urxqg/ wkh frpsxwhu surjudp gudzv d udqgrp qxpehu iurp d xqlirup glvwulexwlrq
ryhu ^3> 4`1 Li wklv udqgrp gudz lv ohvv wkdq ru htxdo wr =<3/ wkh jdph frqwlqxhv rq zlwk dqrwkhu
urxqg1 Rwkhuzlvh/ wkh mxvw frpsohwhg urxqg lv ghfoduhg wr eh wkh odvw wudglqj urxqg ri wkh jdph1
Vxemhfwv zhuh wrog wkdw wkhuh zdv d frqvwdqw/ rqh~lq~whq fkdqfh wkdw hdfk jdph zrxog hqg iurp
rqh urxqg wr wkh qh{w1 Wkxv/ wkh h{shfwhg ohqjwk ri hdfk jdph zdv
3q
@ 43 urxqgv1 Wr dyrlg
ryhuo| orqj jdphv/ zh fkrvh wr hqg dq| jdph wkdw h{fhhghg 73 wudglqj urxqgv1 Vxemhfwv zhuh qrw
lqiruphg wkdw wklv xsshu erxqg ri 73 urxqgv zdv lq hhfw> wkh| zhuh rqo| wrog ri wkh 143 suredelolw|
wkdw wkh jdph zrxog hqg iurp rqh urxqg wr wkh qh{w1 Zh qrwh wkdw wkh xsshu erxqg ri 73 urxqgv
zdv uhdfkhg lq mxvw 5 ri wkh 46; jdphv sod|hg lq doo 47 h{shulphqwdo vhvvlrqv1
Dw wkh erwwrp ri wkh uvw wudglqj urxqg vfuhhq vxemhfwv zhuh suhvhqwhg zlwk d wdeoh lqglfdwlqj
wkh shufhqwdjh ri hdfk w|sh ri djhqw vwrulqj hdfk w|sh ri jrrg dv ri wkh hqg ri wkh odvw wudglqj
urxqg1 Wkhvh shufhqwdjhv duh dyhudjh shufhqwdjhv edvhg rq wkh hqwluh klvwru| ri wkh fxuuhqw jdph
xs wkurxjk wkh odvw wudglqj urxqg dqg duh xsgdwhg iroorzlqj wkh frpsohwlrq ri hyhu| wudglqj urxqg1
Vxemhfwv zhuh lqiruphg ri wklv lqirupdwlrq dqg krz lw lv fdofxodwhg lq wkh lqvwuxfwlrqv1 Wkh| zhuh
ixuwkhu lqvwuxfwhg wkdw wkh| pljkw zdqw wr xvh wklv lqirupdwlrq lq iruplqj dq hvwlpdwh ri krz orqj
lw pljkw wdnh wkhp wr phhw d sod|hu zkr erwk kdg wkh jrrg fruuhvsrqglqj wr wkhlu w|sh dqg zkr
zrxog eh zloolqj wr rhu wklv jrrg lq wudgh +vhh wkh lqvwuxfwlrqv lq Dsshqgl{ 5,1 Wkh lqirupdwlrq
rq wkh sursruwlrq ri hdfk w|sh vwrulqj hdfk jrrg lv dvvxphg wr eh frpprq nqrzohgjh lq wkh
wkhruhwlfdo Nl|rwdnl~Zuljkw hqylurqphqw1 Uhyhdolqj wklv lqirupdwlrq wr vxemhfwv zdv rxu zd| ri
lpsohphqwlqj wklv frpprq nqrzohgjh dvvxpswlrq1
Rxu lpsohphqwdwlrq ri glvfrxqwlqj dqg wkh lqirupdwlrq zh surylgh sod|huv derxw wkh lqyhqwru|
glvwulexwlrqv ri jrrgv e| w|sh duh wzr ihdwxuhv zklfk glhuhqwldwh wklv vwxg| dqg rxu hduolhu vwxg|/
Gx| dqg Rfkv +4<<<,/ iurp suhylrxv vwxglhv ri dgdswlyh ehkdylru lq wkh Nl|rwdnl~Zuljkw +4<;<,
prgho1 +Vhh/ h1j1 Pdulprq hw do1 +4<;<, dqg Eurzq +4<<9,,1 Zh eholhyh wkdw wkhvh wzr ihdwxuhv ri
rxu ghvljq vkrxog idflolwdwh wkh sod| ri vshfxodwlyh vwudwhjlhv +lq hqylurqphqwv zkhuh vshfxodwlrq
46
pdnhv vhqvh, lq wkdw wkh| hqfrxudjh vxemhfwv wr frqvlghu qrw rqo| wkh vwrudjh frvw ri wkh irxu jrrgv/
exw dovr wkh uhodwlyh olnholkrrg wkdw wkh| zloo eh deoh wr phhw d sod|hu zkr erwk kdv wkh jrrg wkh|
ghvluh dqg zkr lv zloolqj wr hqjdjh lq wudgh1 Ixuwkhupruh/ lqirupdwlrq frqfhuqlqj wkh sursruwlrq ri
lqglylgxdov vwrulqj jrrg 3 pd| idflolwdwh wkh dgrswlrq ri wklv wrnhq remhfw dv d phglxp ri h{fkdqjh1
Diwhu doo sod|huv* wudglqj ghflvlrqv kdyh ehhq pdgh/ dq| pxwxdoo| djuhhg xsrq wudghv duh lp0
sohphqwhg dqg wkh uhvxowv ri hdfk sdlu*v wudglqj ghflvlrqv duh uhyhdohg wr vxemhfwv rq d vhfrqg
wudglqj urxqg vfuhhq zklfk lv dovr looxvwudwhg lq Iljxuh 41 Dw wkh erwwrp ri wklv vhfrqg vfuhhq/ wkh
lqirupdwlrq rq wkh hfrqrp|~zlgh klvwrulfdo dyhudjh shufhqwdjh ri hdfk w|sh ri djhqw vwrulqj hdfk
w|sh ri jrrg kdv ehhq xsgdwhg wr wdnh dffrxqw ri zkdw kdv rffxuuhg lq wkh mxvw frpsohwhg urxqg
ri sod|1
Rqfh wkh uhvxowv ri d wudglqj urxqg kdyh ehhq pdgh nqrzq wr doo sod|huv/ wkh frpsxwhu gudzv d
udqgrp qxpehu wr ghwhuplqh zkhwkhu wkh jdph zloo frqwlqxh rq zlwk dqrwkhu urxqg1 Li wkh udqgrp
gudz lv vxfk wkdw wkh jdph frqwlqxhv +zklfk rffxuv zlwk suredelolw| 1<3,/ sod|huv vhh d phvvdjh rq
wkhlu vhfrqg wudglqj urxqg vfuhhq lqglfdwlqj wkdw wkh jdph zloo frqwlqxh1 Lq wklv fdvh/ wkh| fduu|
wkh jrrgv wkh| kdyh lq vwrudjh ryhu wr wkh qhz wudglqj urxqg1 Rwkhuzlvh/ +zlwk suredelolw| 143,
wkh mxvw frpsohwhg urxqg lv ghfoduhg wr eh wkh odvw urxqg ri wkh jdph/ dqg sod|huv vhh d phvvdjh
dw wkh erwwrp ri wkh vhfrqg wudglqj urxqg vfuhhq lqglfdwlqj wkdw wkh jdph kdv mxvw hqghg1
Li wkh jdph frqwlqxhv/ sod|huv duh udqgrpo| sdluhg rqfh djdlq dqg duh dvnhg wr pdnh dqrwkhu
wudglqj ghflvlrq1 Li wkh jdph kdv hqghg/ wkh sod|huv* hqg~ri~jdph srlqw wrwdo lv vwruhg e| wkh
frpsxwhu dqg/ ghshqglqj rq wkh wlph dydlodeoh/ d qhz jdph pd| ehjlq1
Sod|huv ehjlq hyhu| qhz jdph +lqfoxglqj wkh uvw, zlwk d wrwdo ri 433 srlqwv1 Srlqwv duh wkhq
dgghg ru vxewudfwhg iurp hdfk sod|hu*v srlqw wrwdov lq dffrugdqfh zlwk wkhlu wudglqj ghflvlrqv1 Wkh
jrrg wkdw hdfk sod|hu kdv lq vwrudjh dw wkh vwduw ri hyhu| qhz jdph ghshqgv rq wkhlu w|sh1 D fhuwdlq
iudfwlrq/ 4 p/ ri hdfk w|sh ehjlqv d qhz jdph zlwk wkh jrrg wkhlu w|sh surgxfhv lq vwrudjh1 Wkh
uhpdlqlqj iudfwlrq/ p/ ri hdfk w|sh ehjlqv d qhz jdph zlwk jrrg 3 lq vwrudjh1 Wkh ghflvlrq dv wr
zkhwkhu d sod|hu ehjlqv d qhz jdph zlwk wkhlu surgxfwlrq jrrg ru jrrg }hur lq vwrudjh lv udqgrp1
W? #g) @?_ Ut Ebbb i UL?t_ihi_ |L _gihi?| tUi4it uLh |i ?|@* _t|hM|L? Lu }LL_t Lih |)Tit
Ai ht| @t t4*@h |L |i L?i ti_ ihic ? |@| i@U |)Ti Mi}@? i@U }@4i | |ih ThL_U|L? }LL_ ? t|Lh@}i
E|ihi ihi ?L ?|h?tU@**) Lh|*itt |L!i? LMiU|t W? |i tiUL?_ tUi4ic |i ?|@* _t|hM|L? Lu }LL_t Lih
|)Tit @t 4@_i @t U*Lti @t TLttM*i |L |i t|i@_) t|@|i @t i^*Mh4 _t|hM|L? `i uL?_ |@| |ihi @t
?L t}?U@?| _gihi?Ui ? tMiU| Mi@Lh uhL4 t?} i|ih L?i Lu |iti |L ?|@*3@|L? tUi4it AihiuLhic i
ULti |L UL?t_ih L?*) L?i ?|@*3@|L? tUi4i ? |i i Tih4i?| hiTLh|i_ ihi|i L?i ? U i@U |)Ti Mi}?t
i@U }@4i | i|ih |ih ThL_U|L? }LL_ Lh }LL_ f ? t|Lh@}i
47
Vlqfh wkhuh duh htxdo qxpehuv ri hdfk sod|hu w|sh/ dqg vlqfh jrrg 3 lv qrw surgxfhg/ frqvxphg ru
glvfdughg/ wkh iudfwlrq p dovr uhsuhvhqwv wkh iudfwlrq ri wkh hqwluh vxemhfw srro wkdw lv vwrulqj jrrg
3 dw dq| prphqw lq wlph1 Rxu sdudphwhu fkrlfhv iru p duh jlyhq lq Wdeoh 41
Qrwh wkdw zlwk wklv lqlwldol}dwlrq vfkhph/ sod|huv pd| vwduw hdfk qhz jdph zlwk d jrrg lq vwrudjh
wkdw lv glhuhqw iurp wkh jrrg wkh| khog lq vwrudjh dw wkh hqg ri wkh suhylrxv jdph1 Pruhryhu/ dw
wkh ehjlqqlqj ri hyhu| qhz jdph/ wkh wdeoh lqglfdwlqj wkh shufhqwdjh ri hdfk w|sh ri sod|hu vwrulqj
hdfk w|sh ri jrrgv lv uhlqlwldol}hg wr uhhfw wkh lqlwldo glvwulexwlrq ri jrrgv e| w|sh lq wkh qhz
jdph1 Wkxv/ zlwk hdfk qhz jdph/ vrph ri wkh sod|huv* dydlodeoh lqirupdwlrq fkdqjhv1 Krzhyhu/
vlqfh d sod|hu*v w|sh uhpdlqv xqfkdqjhg ryhu dq hqwluh vhvvlrq dv grhv wkh sdudphwhul}dwlrq ri wkh
jdph lwvhoi dqg wkh srro ri vxemhfw sduwlflsdqwv/ hdfk sod|hu fdq gudz xsrq klv ru khu rzq sdvw
h{shulhqfh zkhq pdnlqj wudglqj ghflvlrqv lq wkh qhz jdph1
Wkh wrwdo qxpehu ri jdphv wkdw zhuh sod|hg zdv ghwhuplqhg e| xv dffruglqj wr wkh wlph
dydlodeoh1 Vxemhfwv zhuh uhfuxlwhg iru <3 plqxwh vhvvlrqv/ exw zhuh qrw wrog lq dgydqfh krz pdq|
jdphv zrxog eh sod|hg1 Ehfdxvh rxu jdphv hqghg udqgrpo|/ wkh wrwdo qxpehu ri jdphv dqg wudglqj
urxqgv sod|hg ydulhg vrphzkdw dfurvv vhvvlrqv dv fdq eh vhhq lq Wdeoh 61 Rxu dlp zdv wr rewdlq
durxqg 433 wudglqj urxqgv shu vhvvlrq/ dqg dv qrwhg deryh/ wklv zdv/ lq idfw/ wkh dyhudjh qxpehu
ri wudglqj urxqgv sod|hg lq doo vhvvlrqv1
Zkhq zh ghflghg wkdw wkh odvw jdph kdg ehhq sod|hg/ zh dvnhg rxu surjudp wr fkrrvh rqh
jdph dw udqgrp iurp doo ri wkh jdphv sod|hg lq wkdw vhvvlrq1 Wkh vxemhfwv* qdo srlqw wrwdov iurp
wkh fkrvhq jdph zhuh frqyhuwhg lqwr d suredelolw| ri zlqqlqj d '43 sul}h wkdw zdv lq dgglwlrq wr
wkh '43 sd|phqw wkdw hdfk vxemhfw zdv jxdudqwhhg wr uhfhlyh iru sduwlflsdwlqj lq wkh <3~plqxwh
h{shulphqwdo vhvvlrq1 Hdfk dgglwlrqdo srlqw d vxemhfw hduqhg deryh wkh lqlwldo 433 srlqwv wkh|
zhuh jlyhq lq hdfk jdph lqfuhdvhg wkhlu suredelolw| ri zlqqlqj wkh dgglwlrqdo '43 sul}h e| wkh
vdph dprxqw li wkdw jdph zdv wkh rqh fkrvhq dw udqgrp1 Wkh '43 sul}hv zhuh dzdughg edvhg rq
d udqgrp fkrlfh iru wkh fxw~r suredelolw|1 Wklv phfkdqlvp iru wkh dzduglqj ri dgglwlrqdo '43
sul}hv zdv fduhixoo| h{sodlqhg wr vxemhfwv lq wkh lqvwuxfwlrqv dqg zdv lqwhqghg wr pd{lpl}h wkhlu
shuirupdqfh lq hyhu| urxqg ri sod|1 Vxemhfwv zhuh ixuwkhu lqvwuxfwhg wkdw wkh| zhuh qrw frpshwlqj
zlwk rqh dqrwkhu iru wkh '43 sul}hv1 Wkh| zhuh wrog wkdw hdfk sod|hu lq wkh vhvvlrq frxog zlq d '43
sul}h li kh ru vkh hduqhg hqrxjk dgglwlrqdo srlqwv lq wkh jdph fkrvhq/ uhodwlyh wr wkh pd{lpxp
qxpehu ri srlqwv d sod|hu ri wkhlu w|sh frxog kdyh ehhq h{shfwhg wr hduq lq wkh jdph fkrvhq1 Wkh
pd{lpxp h{shfwhg qxpehu ri srlqwv wdnhv lqwr dffrxqw d sod|hu*v w|sh/ dqg wkh qxpehu ri wudglqj
48
urxqgv sod|hg1 Rxu xvh ri d Urwk~Pdorxi +4<:<, elqdu| orwwhu| wr ghwhuplqh dfwxdo fdvk sd|phqwv
zdv lqwhqghg wr frqwuro iru vxemhfwv* glhulqj dwwlwxghv wrzdugv ulvn1
e
, Tih4i?|@* +it*|t
Zh qrz wxuq wr wkh uhvxowv ri rxu h{shulphqwdo vhvvlrqv/ irfxvlqj rq wkh djjuhjdwh ehkdylru e| doo
sod|huv ri d jlyhq w|sh1 Wdeoh 7 uhsruwv wkh djjuhjdwh iuhtxhqf| zlwk zklfk hdfk w|sh rhuv wr
Wdeoh 7= Iuhtxhqflhv Zlwk Zklfk Hdfk W|sh Rhuv wr Wudgh iru wkh Jrrg Fruuhvsrqglqj wr Wkhlu
W|sh Ryhu Hdfk Kdoi ri d Vhvvlrq
5ittL?
?4Mih
2
2
2
#
#2
#
#e
#D
A)Ti giht |L
Ah@_i uLh BLL_
6ht| O@*u 5iUL?_ O@*u
bb
bb
bb
bS
bD
b.
bD
H.
b
b.
bb
ff
b.
b2
bD
bf
ff
bb
bD
bS
b.
ff
ff
ff
ff
ff
be
be
** 5ittL?t@
@
b.
bS
A)Ti 2 giht |L
Ah@_i uLh BLL_ 2
6ht| O@*u 5iUL?_ O@*u
b.
bH
bH
b2
b.
Hb
ff
be
ff
be
b
bS
b2
H2
ff
bb
ff
b.
b
ff
bb
be
ff
ff
bH
ff
.e
HD
bS
be
A)Ti giht |L
Ah@_i uLh BLL_ 6ht| O@*u 5iUL?_ O@*u
be
bb
b.
bb
ff
ff
b.
b
ff
ff
bf
ff
ff
be
ff
ff
bH
ff
bH
ff
bb
ff
bH
ff
bS
b
.e
Hb
bS
b.
`i}|i_ ih@}i
wudgh iru wkh jrrg fruuhvsrqglqj wr wkhlu w|sh ryhu wkh uvw dqg vhfrqg kdoi ri doo 47 h{shulphqwdo
vhvvlrqv12 Vlqfh vxemhfwv zhuh lqiruphg wkdw wkh| hduqhg d srvlwlyh qhw sd|r lq srlqwv rqo| iurp
dftxlulqj wkh jrrg fruuhvsrqglqj wr wkhlu w|sh/ wkh rhu iuhtxhqflhv uhsruwhg lq Wdeoh 7 vkrxog doo
eh 433(1 Lqghhg/ zh vhh wkdw lq prvw fdvhv/ wkhvh iuhtxhqflhv duh hlwkhu dw 433(/ ru yhu| forvh wr
433(1 Wklv qglqj vhuyhv wr frqup wkdw prvw vxemhfwv xqghuvwrrg wkh uroh wkh| zhuh dvvljqhg
wr sod| dv rqh ri wkh wkuhh glhuhqw w|shv dqg zhuh vxflhqwo| zhoo prwlydwhg e| wkh lqfhqwlyh
vwuxfwxuh ri rxu h{shulphqwdo ghvljq1
2
Wu @ tittL? UL?tt|i_ Lu ff |h@_?} hL?_tc |i ht| @*u Lu |i tittL? UL?tt|t Lu |i ht| Df |h@_?} hL?_t @?_
|i tiUL?_ @*u Lu |i tittL? UL?tt|t Lu |i hi4@??} Df |h@_?} hL?_t
49
e
Mtihi_ Mi@Lh i? }LL_ f @t 3ihL t|Lh@}i ULt|
Uhfdoo wkdw zkhq jrrg 3 lv ohdvw frvwo|~wr~vwruh/ wkhuh lv rqh vwhdg| vwdwh lq zklfk jrrg 3 lv
xqlyhuvdoo| dffhswhg lq wudgh +e| doo wkuhh w|shv, dqg dqrwkhu vwhdg| vwdwh/ lq zklfk qr rqh eholhyhv
lw kdv ydoxh/ dqg wkhuhiruh qr w|sh rhuv wr wudgh iru lw1 Lq hlwkhu vwhdg| vwdwh/ doo wkuhh w|shv duh
suhglfwhg wr sod| _ixqgdphqwdo% vwudwhjlhv> vshfxodwlyh vwudwhjlhv vkrxog qrw eh revhuyhg1
Wdeohv 8def uhsruw wkh djjuhjdwh iuhtxhqflhv zlwk zklfk hdfk w|sh rhuv wr wudgh iru jrrg 3 ryhu
wkh uvw dqg vhfrqg kdoi ri wkh < vhvvlrqv lq zklfk jrrg }hur kdg qr vwrudjh frvw/ dqg zdv wkhuhiruh
wkh ohdvw frvwo|~wr~vwruh jrrg +Vhvvlrqv D4~D6/ E4~E6/ dqg F4~F6,1 Wkh rhu iuhtxhqflhv lq wkhvh
wdeohv duh ixuwkhu glvdjjuhjdwhg dffruglqj wr wkh jrrg wkdw hdfk w|sh zdv rhulqj lq h{fkdqjh iru
jrrg 31 Dv wkh wdeohv uhyhdo/ zkhq jrrg 3 lv ohdvw frvwo| wr vwruh +kdv }hur vwrudjh frvw,/ lw lv
zlgho|/ wkrxjk qrw txlwh xqlyhuvdoo|/ dffhswhg lq h{fkdqjh1
Wkh rhu iuhtxhqflhv lq Wdeohv 8def/ duh lq prvw fdvhv idu iurp }hur/ dqg e| wkh vhfrqg kdoi
ri wkh vhvvlrqv/ duh lq pdq| fdvhv dssurdfklqj 4133/ wkh suhglfwlrq iru wkh vwhdg| vwdwh lq zklfk
jrrg 3 vhuyhv dv d phglxp ri h{fkdqjh1 Wkhuh duh/ krzhyhu/ d ihz h{fhswlrqv1 Lq sduwlfxodu/ lq
vhvvlrqv D6 dqg E5/ w|sh 4 sod|huv/ ehfrph pruh uhoxfwdqw wr wudgh jrrg 6 iru jrrg 3 iurp wkh
uvw wr wkh vhfrqg kdoi ri wkhvh vhvvlrqv1 Wklv ehkdylru vxjjhvwv wkdw dw ohdvw vrph w|sh 4 sod|huv
duh _vshfxodwlqj% lq jrrg 6 +e| uhixvlqj wr wudgh jrrg 6 iru jrrg 3,/ hyhq wkrxjk vxfk vshfxodwlyh
ehkdylru qhyhu frpsulvhv d vwhdg| vwdwh ehvw uhvsrqvh lq wklv sdudphwhul}dwlrq ri wkh prgho1e
Qrwlfh dovr wkdw e| wkh vhfrqg kdoi ri doo < vhvvlrqv/ w|sh 5 sod|huv duh/ lq wkh djjuhjdwh/ pruh
zloolqj wr wudgh jrrg 6 iru jrrg 3 wkdq wkh| duh zloolqj wr wudgh jrrg 4 iru jrrg 3> wklv glhuhqfh
lq ehkdylru lv frqvlvwhqw zlwk ixqgdphqwdo vwrudjh frvw frqvlghudwlrqv/ exw lq htxloleulxp/ wkhuh
zrxog eh qr vxfk glhuhqfh1
Wkh srvlwlyh iuhtxhqflhv zlwk zklfk doo w|shv rhu wr wudgh iru jrrg 3/ dv uhsruwhg lq Wdeohv
8def/ lqglfdwh wkdw sod|huv eholhyh jrrg 3 kdv ydoxh1 Li jrrg 3 lv d phglxp ri h{fkdqjh lq wklv
hqylurqphqw/ wkhq sod|huv zkr krog lw vkrxog rqo| rhu wr wudgh lw iru wkhlu frqvxpswlrq jrrg/
dqg qrw iru dq| rwkhu jrrg1 Lqghhg/ dv wkh ohiw froxpqv ri Wdeohv 9def uhyhdo/ e| wkh vhfrqg kdoi
ri hdfk vhvvlrq/ zlwk ihz h{fhswlrqv/ sod|huv duh uduho| rhulqj wr wudgh jrrg 3 iru wkh jrrg wkh|
`i __ ?L| UL?t_ih U@tit ? U ML| T*@)iht @_ }LL_ f ? t|Lh@}i @t |ihi t ?L |iLhi|U@* Thi_U|L? ?
|t U@ti
e
#t@}}hi}@|L? Lu |i _@|@ EuhL4 @** tittL?t hii@*t |@| 4Lt| tMiU|t @hi T*@)?} tL4i|?} U*Lti |L @ Thic
h@|ih |@? @ 4 i_ t|h@|i}) i? 4ii|?} @ }LL_ |@| _giht uhL4 |i }LL_ |i) Uhhi?|*) L*_ ? t|Lh@}i W| t
|ihiuLhi @TThLTh@|i |L hiuih |L |i tTiU*@|i Mi@Lh i LMtihi @t @ht?} uhL4 |i Mi@Lh Lu RtL4i |)Ti
T*@)iht
4:
A@M*i D@G 6hi^i?Uit `| `U A)Ti giht |L Ah@_i 6Lh BLL_ fc 5ittL?t
giht |L Ah@_i uLh f
giht |L Ah@_i 2 uLh f giht |L Ah@_i uLh f
5ittL? ' 6ht| O@*u 5iUL?_ O@*u 6ht| O@*u 5iUL?_ O@*u 6ht| O@*u 5iUL?_ O@*u
2f
Hb
bH
bf
bb
Hf
.D8
2
2f
S
H
S.
HS
e
ef
2f
..
H
.S
HS
H
DS
2f
bb
ff
bH
ff
ff
ff
2
2f
HD
Hb
He
b.
ff
De
2f
bf
ff
b
ff
.
ff8
2f
He
HH
HD
HH
Sf
_
2
2f
H
b
HD
be
Sb
H
2f
.H
b2
Hb
b2
H
ff8
A@M*i DMG 6hi^i?Uit `| `U A)Ti 2 giht |L Ah@_i 6Lh BLL_ fc 5ittL?t
2 giht |L Ah@_i uLh f
2 giht |L Ah@_i uLh f 2 giht |L Ah@_i uLh f
5ittL? ' 6ht| O@*u 5iUL?_ O@*u 6ht| O@*u 5iUL?_ O@*u 6ht| O@*u 5iUL?_ O@*u
2f
b2
b.
HH
bf
be
ff
2
2f
H2
Hf
.
.
Hb
He
2f
H.
HH
SH
.S
ff
bS
2f
bD
bS
b
b
b.
ff
2
2f
b
HH
H.
.
b.
b.
2f
Hb
b
bD
bf
HS
b
2f
.S
S.
Se
DD
.b
.2
2
2f
b
bS
.b
b
bS
bH
2f
HH
ff
.e
ff
bS
ff
A@M*i DUG 6hi^i?Uit `| `U A)Ti giht |L Ah@_i 6Lh BLL_ fc 5ittL?t
giht |L Ah@_i uLh f
giht |L Ah@_i uLh f giht |L Ah@_i 2 uLh f
5ittL? ' 6ht| O@*u 5iUL?_ O@*u 6ht| O@*u 5iUL?_ O@*u 6ht| O@*u 5iUL?_ O@*u
2f
H
bD
H
bD
Hf
_
2
2f
..
H2
b
HH
eH
Sf
2f
D
SH
D.
.2
.
H
2f
b2
HD
be
H.
Hf
Df8
2
2f
ff
ff
ff
ff
ff8
_
2f
S
.f
Se
.
Df
S.
2f
b2
be
b
b.
bf
D.
2
2f
.D
Db
Hf
SH
e.
fS
2f
SD
.b
S.
.b
ef
.D8
8 Aihi ihi *itt |@? D LMtih@|L?t uLh |t |h@_?} t|@|L?
_ Aihi ihi ?L LMtih@|L?t uLh |t |h@_?} t|@|L?
4;
qhlwkhu frqvxph qru surgxfh1
A@M*i S@G 6hi^i?Uit `| `U A)Ti giht |L Ah@_i 6Lh BLL_ c 5ittL?t
giht |L Ah@_i f uLh giht |L Ah@_i 2 uLh 5ittL? ' 6ht| O@*u 5iUL?_ O@*u 6ht| O@*u 5iUL?_ O@*u
2f
ff
fD
.
2
2f
fS
ff
f
D
2f
S
f
2D
2f
ff
fS
2e
2.
2
2f
S2
H
e2
eH
2f
fS
ff
2
e
2f
e
ff
b
ff
2
2f
f
ef
2
2f
ff
22
2
A@M*i SMG 6hi^i?Uit `| `U A)Ti 2 giht |L Ah@_i 6Lh BLL_ c 5ittL?t
2 giht |L Ah@_i f uLh 2 giht |L Ah@_i uLh
5ittL? ' 6ht| O@*u 5iUL?_ O@*u 6ht| O@*u 5iUL?_ O@*u
2f
S
f
bS
bD
2
2f
fe
H2
.
2f
D
f.
bH
be
2f
f
ff
ff
2
2f
fb
fH
ff
ff
2f
f
fe
bf
be
2f
De
b
H
HD
2
2f
2S
fe
He
ff
2f
ff
bS
ff
A@M*i SUG 6hi^i?Uit `| `U A)Ti giht |L Ah@_i 6Lh BLL_ 2c 5ittL?t
giht |L Ah@_i f uLh 2 giht |L Ah@_i uLh 2
5ittL? ' 6ht| O@*u 5iUL?_ O@*u 6ht| O@*u 5iUL?_ O@*u
2f
f
ff
e
fS
2
2f
2
D
2.
2f
S
fb
2
S
2f
b
fe
b
f
2
2f
ff
ff
fD
ff
2f
H
2H
ef
2f
e
fS
e
f
2
2f
2
fS
e
D
2f
fD
fH
fb
fD
Wkh uljkw~prvw froxpqv ri Wdeohv 9def uhyhdo wkh iuhtxhqflhv zlwk zklfk hdfk w|sh lv rhulqj
wr wudgh wkhlu surgxfwlrq jrrg iru wkh rwkhu/ qrq~wrnhq/ qrq~frqvxpswlrq jrrg1 Uhfdoo wkdw/
uhjdugohvv ri zkhwkhu jrrg 3 kdv ydoxh/ wkh vwhdg| vwdwh suhglfwlrq lv wkdw doo wkuhh w|shv dgkhuh
wr _ixqgdphqwdo/% vwrudjh~frvw~uhgxflqj vwudwhjlhv lq vxfk wudglqj hqfrxqwhuv1 Iru wkh prvw sduw/
vxemhfwv dsshdu wr eh dgrswlqj vxfk vwudwhjlhv/ hvshfldoo| e| wkh vhfrqg kdoi ri hdfk vhvvlrq1 Lq
sduwlfxodu/ w|sh 4 dqg 6 sod|huv duh uhixvlqj wr wudgh wkhlu surgxfwlrq jrrgv 5 dqg 4 iru wkh
pruh frvwo|~wr~vwruh jrrgv 6 dqg 5/ uhvshfwlyho|/ dqg w|sh 5 sod|huv duh rhulqj wr wudgh wkhlu
4<
surgxfwlrq jrrg 6 iru wkh ohvv frvwo|~wr~vwruh jrrg 4/ zklfk dovr vhuyhv dv d phglxp ri h{fkdqjh lq
wklv hqylurqphqw1 Krzhyhu/ wkhuh duh vrph h{fhswlrqv wr wklv qglqj> wkhuh lv shuvlvwhqw vshfxodwlrq
e| vrph w|sh 4 sod|huv lq jrrg 6 lq vhvvlrqv E4/ E5/ dqg F6/ dqg shuvlvwhqw vshfxodwlrq e| vrph
w|sh 6 sod|huv lq jrrg 5 lq vhvvlrqv D5 dqg E61 Vlploduo|/ lq vhvvlrq D5/ wkhuh lv vrph ghwhulrudwlrq
lq wkh iuhtxhqf| zlwk zklfk w|sh 5 sod|huv wudgh jrrg 6 iru jrrg 41D
Wkh rqo| wuhdwphqw yduldeoh wkdw ydulhg dfurvv vhvvlrqv D4~D6/ E4~E6 dqg F4~F6 zdv wkh
ydoxh ri p/ wkh iudfwlrq ri wkh srsxodwlrq vwrulqj jrrg 31 Uhfdoo wkdw wkh suredelolw| wkdw jrrg
3 lv dffhswhg zkhq rhuhg lv suhglfwhg wr eh lqyduldqw wr fkdqjhv lq p/ exw wkdw wkh dqdorjrxv
dffhswdqfh suredelolw| iru jrrg 4 lv suhglfwhg wr ghfuhdvh dv p lqfuhdvhv1 Wr dvvhvv zkhwkhu wkhvh
suhglfwlrqv krog/ zh fdofxodwhg wkh qxpehu ri wlphv jrrg 3 zdv rhuhg lq wudgh dqg wkh qxpehu ri
wlphv wkdw jrrg 3 zdv erwk rhuhg dqg dffhswhg/ l1h1 wkh qxpehu ri wlphv jrrg 3 zdv h{fkdqjhg1S
Wkh udwlr ri wkh odwwhu qxpehu wr wkh iruphu qxpehu lv wkh iuhtxhqf| zlwk zklfk jrrg 3 lv dffhswhg
lq wudgh zkhq rhuhg1 Zh fdofxodwhg wklv vdph dffhswdqfh iuhtxhqf| iru jrrg 41 Wkhvh dffhswdqfh
iuhtxhqflhv iru wkh uvw dqg vhfrqg kdoyhv ri vhvvlrqv D4~D6/ E4~E6 dqg F4~F6 duh suhvhqwhg lq
Wdeoh :1 Zh kdyh dovr lqfoxghg lq Wdeoh : wkh wkhruhwlfdo/ htxloleulxp dffhswdqfh iuhtxhqflhv iru
jrrgv 3 dqg 4 zklfk duh uhsurgxfhg iurp Wdeoh 5 dqg duh irxqg xqghu wkh khdglqj Htxlo1 Iuht1
Lq Wdeoh : zh revhuyh wkdw/ e| wkh vhfrqg kdoi ri hdfk vhvvlrq/ wkh iuhtxhqf| zlwk zklfk jrrg 3 lv
Wdeoh := Iuhtxhqflhv Zlwk Zklfk Jrrgv 3 dqg 4 duh Dffhswhg Zkhq Rhuhg/
Fdvh 4 +Sdudphwhu Vhw 4,/ Vhvvlrqv D4~F61
5ittL?
2
2
2
6'
2D
2D
2D
Df
Df
Df
BLL_ f UUiT|i_ `i? gihi_
6ht| O@*u 5iUL?_ O@*u ,^* 6hi^
HD
bD
ff
.e
HD
ff
.f
H2
ff
b.
b2
ff
b.
bf
ff
.S
b
ff
Hf
H
ff
H
H2
ff
.b
HS
ff
BLL_ UUiT|i_ `i? gihi_
6ht| O@*u 5iUL?_ O@*u ,^* 6hi^
.f
S
DH
.
SS
DH
.2
SS
DH
SH
D
D2
D
Se
D2
Se
.H
D2
S
D2
eD
S2
DD
eD
S
D.
eD
dffhswhg zkhq rhuhg lv dozd|v juhdwhu wkdq ;3(/ exw qhyhu h{fhhgv <8(1 Wkhvh kljk/ exw ohvv wkdq
D
54*@h @?L4@*Lt Mi@Lh @t LMtihi_ M) #g) @?_ Ut Ebbb L i @4?i_ |t t@4i i Tih4i?|@*
i?hL?4i?| M| |L| |i |L!i? }LL_ f }@?c _t@}}hi}@|L? hii@*t |@| | t @ Uih|@? uh@U|L? Lu T*@)iht L
@hi Tihtt|i?|*) T*@)?} Thic tTiU*@|i t|h@|i}itc @?_ ?L| @** T*@)iht @_LT|?} tL4i !?_ Lu 4 i_ t|h@|i})
S
`i i U*_i_ uhL4 |iti U@*U*@|L?t U@tit ihi ML| T@h|it |L @? i U@?}i @_ }LL_ f ? t|Lh@}i
53
xqlyhuvdo dffhswdqfh iuhtxhqflhv duh d uhhfwlrq ri wkh kljk rhu iuhtxhqflhv uhsruwhg lq Wdeohv 8def1
Zkdw lv lqwhuhvwlqj wr qrwh lv wkdw/ frqwudu| wr wkh wkhru|/ wkhuh dsshduv wr eh vrph hylghqfh wkdw
wkh dffhswdelolw| ri jrrg 3 glplqlvkhv dv p lv lqfuhdvhg1 Xvlqj d vlpsoh qrqsdudphwulf udqn rughu
whvw/ zh qg qr vljqlfdqw glhuhqfh lq dffhswdqfh iuhtxhqflhv iru jrrg 3 ehwzhhq wkh vhfrqg kdoi ri
vhvvlrqv D4~D6 dqg E4~E6/ dqg ehwzhhq wkh vhfrqg kdoi ri vhvvlrqv D4~D6 dqg F4~F61 Krzhyhu/
zh qg wkdw wkh dffhswdqfh iuhtxhqflhv lq wkh vhfrqg kdoi ri vhvvlrqv F4~F6/ zkhuh p @ =83 duh
vljqlfdqwo| orzhu +s @ =38, wkdq wkh fruuhvsrqglqj iuhtxhqflhv iru vhvvlrqv E4~E6 zkhuh p @ =661
Rxu gdwd wkxv surylgh d pl{hg uhvxow zlwk uhvshfw wr wkh lqyduldqfh ri wkh dffhswdelolw| ri jrrg 3
wr fkdqjhv lq wkh ydoxh ri p1
Dffruglqj wr wkh wkhru|/ wkh dffhswdqfh ri jrrg 4 dv d phglxp ri h{fkdqjh lv vwulfwo| ohvv wkdq
433(/ dqg wkh qhhg iru jrrg 4 dv d orz frvw phglxp ri h{fkdqjh ghfuhdvhv dv p lqfuhdvhv1 Wkh gdwd
lq Wdeoh : lqglfdwh wkdw e| wkh vhfrqg kdoi ri hdfk vhvvlrq/ wkh dffhswdelolw| ri jrrg 4 lv lqghhg ohvv
wkdq wkdw ri jrrg 31 Krzhyhu/ lq wzr rxw ri wkuhh sdlu~zlvh frpsdulvrqv ryhu wuhdwphqwv/ vhvvlrqv
D40D6 yhuvxv E4~E6 dqg vhvvlrqv E4~E6 yhuvxv F4~F6/ urexvw udqn rughu whvwv uhyhdo wkdw zh
fdqqrw uhmhfw wkh qxoo k|srwkhvlv wkdw lqfuhdvhv lq p kdyh qr hhfw rq wkh vhfrqg kdoi dffhswdqfh
iuhtxhqflhv iru jrrg 41 Ixuwkhupruh/ lq doo exw rqh vhvvlrq/ +vhvvlrq E4, wkh vhfrqg kdoi dffhswdqfh
iuhtxhqflhv iru jrrg 4 h{fhhg wkh suhglfwhg htxloleulxp dffhswdqfh iuhtxhqflhv1 Wkhvh glhuhqfhv/
krzhyhu/ pljkw vlpso| eh dwwulexwhg wr wkh idfw wkdw wkh vwhdg| vwdwh lv qhyhu shuihfwo| dfklhyhg/
dv uhyhdohg lq Wdeohv 8 dqg 91
Lq vxppdu|/ zkhq wkh wrnhq remhfw kdv }hur vwrudjh frvw dqg lv wkh ohdvw frvwo|~wr~vwruh jrrg/
zh revhuyh wkdw lw lv zlgho| xvhg dv d phglxp ri h{fkdqjh/ dv suhglfwhg e| wkh wkhru|1 Krzhyhu/
wkhuh lv d vxflhqw yroxph ri xqsuhglfwhg vshfxodwlyh ehkdylru wr fdoo lqwr txhvwlrq wkh vkdushu
suhglfwlrqv ri wkh wkhru|1 Wkh wrnhq remhfw lv qrw xqlyhuvdoo| dffhswhg dqg lw lv qrw fohdu wkdw
lqfuhdvhv lq wkh vxsso| ri wklv remhfw glplqlvk djhqwv* uholdqfh rq rwkhu jrrgv dv phgld ri h{fkdqjh1
e2
Mtihi_ Mi@Lh i? }LL_ f t ULt|*)|Lt|Lhi
Lq h{shulphqwdo vhvvlrqv G4~G8/ zh xvhg sdudphwhu vhw 5 lq zklfk jrrg 3 lv pruh frvwo|~wr~vwruh
wkdq jrrg 4 exw lv pxfk ohvv frvwo|~wr~vwruh wkdq hlwkhu jrrg 5 ru jrrg 61. Wkh sdudphwhu ydoxhv
.
W? tittL?t ##ec tMiU|t ihi }i? |i i @U| t@4i ?uLh4@|L? @t ? tittL?t W? tittL? #Dc i
ThL_i_ tMiU|t | @__|L?@* ?uLh4@|L? L? |i t|LhU@* uhi^i?Uit | U i@U |)Ti Lgihi_ |L |h@_i
}LL_ uLh }LL_ & ' ? |i Uhhi?| }@4i Ai ThLtL? Lu |t @__|L?@* t|h@|i}U ?uLh4@|L? @TTi@ht |L @i
@_ *||*i igiU| L? |i @}}hi}@|i L|UL4it @t ** MiUL4i i_i?| ? A@M*it .@MU Mi*L
54
fkrvhq lpso| wkdw lq wkh vwhdg| vwdwh zkhuh jrrg 3 lv ydoxhg +zklfk zh irfxv rq khuh, sod|hu w|shv
4 dqg 5 zloo dozd|v wudgh wkhlu surgxfwlrq jrrgv 5 dqg 6/ uhvshfwlyho|/ iru hlwkhu jrrg 3 ru jrrg 4
vr dv wr uhgxfh wkhlu vwrudjh frvwv1 W|sh 5 sod|huv zkr kdyh wudghg iru jrrg 4 zloo eh xqzloolqj wr
wudgh lw iru jrrg 3 dqg w|sh 6 sod|huv/ zkr surgxfh jrrg 4/ zloo uhixvh wr wudgh iru dq| jrrg rwkhu
wkdq jrrg 61 Khqfh d w|sh 4 sod|hu zkr krogv jrrg 3/ fdq rqo| rewdlq jrrg 4 e| uvw wudglqj jrrg
3 iru jrrg 6 +d vshfxodwlyh wudgh, dqg wkhq wudglqj jrrg 6 wr d w|sh 6 sod|hu lq h{fkdqjh iru jrrg
41 Lq wklv yhuvlrq ri wkh prgho/ lw lv dovr d vwhdg| vwdwh ehvw uhvsrqvh iru w|sh 4 sod|huv wr wudgh
wkhlu surgxfwlrq jrrg 5 gluhfwo| iru jrrg 6 +dqrwkhu vshfxodwlyh wudgh,/ wkxv e|sdvvlqj wkh xvh ri
jrrg 3 dowrjhwkhu1 Lq hlwkhu fdvh/ rqfh d w|sh 4 kdv wudghg iru jrrg 6/ kh vkrxog qrw rhu wr wudgh
jrrg 6 iru jrrg 31
Wdeohv ;def glvsod| wkh uhodwlyh iuhtxhqflhv zlwk zklfk hdfk sod|hu w|sh zkr idfhg wkh rssru0
wxqlw| wr wudgh iru jrrg 3 dfwxdoo| rhuhg wr pdnh vxfk d wudgh ryhu wkh uvw dqg vhfrqg kdoi ri
vhvvlrqv G4~G81 Frqvlvwhqw zlwk wkh vwhdg| vwdwh suhglfwlrq/ erwk w|sh 5 dqg w|sh 6 sod|huv qhduo|
dozd|v uhixvh wr rhu wr wudgh jrrg 4 iru jrrg 3/ hvshfldoo| e| wkh vhfrqg kdoi ri d vhvvlrq1 Krzhyhu/
frqwudu| wr wkh vwhdg| vwdwh suhglfwlrq/ lq wkh vhfrqg kdoi ri irxu ri wkh yh vhvvlrqv/ 5:~68( ri
doo w|sh 5 sod|huv zkr khog jrrg 6 dqg phw jrrg 3 lq wudgh uhixvhg wr rhu wr wudgh1 Dv iru wkh
ehkdylru ri w|sh 4 sod|huv/ wkh gdwd lq Wdeoh ;d vkrz wkdw w|sh 4 rhuv wr wudgh jrrg 5 iru jrrg 3
ghfuhdvh vrphzkdw iurp wkh uvw wr wkh vhfrqg kdoi ri irxu ri wkh yh vhvvlrqv/ dqg duh qhyhu htxdo
wr wkh suhglfwhg iuhtxhqf| ri 4131 Dovr frqwudu| wr wkh vwhdg| vwdwh suhglfwlrq/ lq wkh vhfrqg kdoi
ri wkuhh ri wkh yh vhvvlrqv/ rqh~wklug ru pruh ri w|sh 4 sod|huv zkr kdg jrrg 6 lq vwrudjh dqg
phw jrrg 3 lq wudgh zhuh zloolqj wr rhu wr wudgh wkhlu jrrg 6 iru jrrg 31
Ilqdoo|/ Wdeohv <def uhsruw wkh iuhtxhqflhv zlwk zklfk hdfk w|sh rhuhg wr wudgh iru wkh rwkhu/
qrq~wrnhq/ qrq~frqvxpswlrq jrrg lq vhvvlrqv G4~G81 Uhfdoo wkdw wkh vwhdg| vwdwh suhglfwlrq lv
wkdw w|sh 4 zloo sod|huv zloo sod| wkh vshfxodwlyh vwudwhj| ri rhulqj wr wudgh jrrg 3 ru jrrg 5 iru
wkh pruh frvwo|~wr~vwruh jrrg 6/ zkloh w|shv 5 dqg 6 frqwlqxh wr dgkhuh wr ixqgdphqwdo/ vwrudjh
frvw uhgxflqj vwudwhjlhv1 Dv vkrzq lq Wdeoh <d/ e| wkh vhfrqg kdoi ri hdfk vhvvlrq/ rqo| 4; wr 73(
ri w|sh 4 sod|huv zkr khog jrrg 3 lq vwrudjh dqg idfhg dq rssruwxqlw| wr wudgh iru jrrg 6 rhuhg
wr pdnh wkh wudgh1 Wkxv wkh pdmrulw| ri w|sh 4 sod|huv uhixvhg wr sod| wkh vshfxodwlyh vwudwhj| lq
wklv wudglqj vlwxdwlrq/ ghvslwh wkh idfw wkdw e| wkh vhfrqg kdoi ri hdfk vhvvlrq +dv uhyhdohg hduolhu lq
Wdeohv ;e dqg ;f,/ qhduo| doo w|sh 5 dqg 6 sod|huv uhixvhg wr rhu wr wudgh jrrg 4/ wkh jrrg ghvluhg
e| w|sh 4 sod|huv/ iru jrrg 31 E| frqwudvw/ w|sh 4 sod|huv glg vkrz d surqrxqfhg zloolqjqhvv
55
A@M*i H@G 6hi^i?Uit `| `U A)Ti giht |L Ah@_i 6Lh BLL_ fc 5ittL?t ##D
5ittL?
#
#2
#
#e
#D
'
ff
ff
ff
ff
ff
giht |L Ah@_i 2 uLh f
6ht| O@*u 5iUL?_ O@*u
b2
HD
b2
.b
b2
b2
.2
De
He
H
giht |L Ah@_i uLh f
6ht| O@*u 5iUL?_ O@*u
ff8
2b
2b
S.8
Hf
S.8
ef
S
2
A@M*i HMG 6hi^i?Uit `| `U A)Ti 2 giht |L Ah@_i 6Lh BLL_ fc 5ittL?t ##D
5ittL?
#
#2
#
#e
#D
'
ff
ff
ff
ff
ff
2 giht |L Ah@_i uLh f
6ht| O@*u 5iUL?_ O@*u
fb
2D
ff
2e
ff
2
ff
f.
fS
2 giht |L Ah@_i uLh f
6ht| O@*u 5iUL?_ O@*u
bf
.
ff
ff
.H
.
bf
.f
.
SD
A@M*i HUG 6hi^i?Uit `| `U A)Ti giht |L Ah@_i 6Lh BLL_ fc 5ittL?t ##D
5ittL?
#
#2
#
#e
#D
'
ff
ff
ff
ff
ff
giht |L Ah@_i uLh f
6ht| O@*u 5iUL?_ O@*u
e
ff
f2
ff
f2
ff
f2
ff
2
giht |L Ah@_i 2 uLh f
6ht| O@*u 5iUL?_ O@*u
ff8
_
ff8
_
_
_
.
ff8
ef
_
8 Aihi ihi *itt |@? D LMtih@|L?t uLh |t |h@_?} t|@|L?
_ Aihi ihi ?L LMtih@|L?t uLh |t |h@_?} t|@|L?
56
wr vshfxodwh e| wudglqj jrrg 5 iru jrrg 6/ d pryh wkdw zrxog lqfuhdvh wkhlu vwrudjh frvwv rqo|
voljkwo|1H Wkxv/ wkh djjuhjdwh sdwwhuq ri ehkdylru h{klelwhg e| wkh pdmrulw| ri w|sh 4 sod|huv zkr
idfhg dq rssruwxqlw| wr wudgh iru jrrg 6 zdv wr sod| ixqgdphqwdo h{fhsw zkhq wkh| frxog jhw jrrg
6 dw olwwoh lqfuhphqwdo frvw1
A@M*i b@G 6hi^i?Uit `| `U A)Ti giht |L Ah@_i 6Lh BLL_ c 5ittL?t ##D
5ittL?
#
#2
#
#e
#D
'
ff
ff
ff
ff
ff
giht |L Ah@_i f uLh 6ht| O@*u 5iUL?_ O@*u
ef
D
.
ff
H
f
2
fD
giht |L Ah@_i 2 uLh 6ht| O@*u 5iUL?_ O@*u
Df
.f
D.
bf
f
DD
.
ee
DD
A@M*i bMG 6hi^i?Uit `| `U A)Ti 2 giht |L Ah@_i 6Lh BLL_ c 5ittL?t ##D
5ittL?
#
#2
#
#e
#D
'
ff
ff
ff
ff
ff
2 giht |L Ah@_i f uLh
6ht| O@*u 5iUL?_ O@*u
Hb
ff
ff
ff
bD
ff
ff
ff
b2
ff
2 giht |L Ah@_i uLh
6ht| O@*u 5iUL?_ O@*u
HH
be
ff
ff
bD
b
HH
ff
H
He
A@M*i b@G 6hi^i?Uit `| `U A)Ti giht |L Ah@_i 6Lh BLL_ 2c 5ittL?t ##D
5ittL?
#
#2
#
#e
#D
'
ff
ff
ff
ff
ff
giht |L Ah@_i f uLh 2
6ht| O@*u 5iUL?_ O@*u
ff
ff
fH
ff
ff
ff
ff
22
e
giht |L Ah@_i uLh 2
6ht| O@*u 5iUL?_ O@*u
f
ff
fD
f2
fS
ff
f
f2
fb
Wdeohv <e dqg <f uhyhdo wkdw w|shv 5 dqg 6 zhuh erwk dgkhulqj wr wkh suhglfwhg ixqgdphqwdo
vwudwhjlhv lq wkh yh G vhvvlrqv1 Lq sduwlfxodu/ w|sh 5 sod|huv zhuh qhduo| dozd|v rhulqj wr wudgh
jrrgv 6 dqg 3 iru wkh ohdvw frvwo|~wr~vwruh jrrg 4/ dqg w|sh 6 sod|huv zhuh xqzloolqj wr wudgh jrrgv
4 dqg 3 iru wkh pruh frvwo|~wr~vwruh jrrg 51
e
`ihi @)Lg W?Ui?|it 5Ui?|*) 5@*i?|q
Rqh pljkw txhvwlrq zkhwkhu wkh lqfhqwlyhv iru w|sh 4 sod|huv wr vshfxodwh lq jrrg 6 xqghu sdudphwhu
vhw 5 zhuh vxflhqwo| odujh1 Rqh zd| wr dqvzhu wklv txhvwlrq lv wr vxssrvh wkdw doo wkuhh w|shv zhuh
H
+iU@** |@| ? T@h@4i|ih ti| 2c |i _gihi?Ui ? t|Lh@}i ULt| Mi|ii? }LL_t 2 @?_ t t| L?i TL?| ES2 ' 2
TL?|tc S ' 2e TL?|t Ai .fI @?_ }ih h@|it Lu tTiU*@|L? |@| i LMtihi ? |hii Lu |i i # tittL?t @hi
|i }it| h@|it Lu tTiU*@|L? |@| @i Mii? hiTLh|i_ ? @** Lu |i i Tih4i?|@* t|_it Lu |i k)L|@!`h}|
EbHb 4L_i* At ?_?} t}}it|t |@| | t TLttM*i |L LM|@? tTiU*@|i Mi@Lh ? |i *@MLh@|Lh) ThL_i_ |@|
t|Lh@}i ULt| _gihi?Uit @hi tUi?|*) t4@**
57
sod|lqj ixqgdphqwdo vwudwhjlhv lq wklv hqylurqphqw1 Lq wkdw hyhqw/ wkh vwhdg| vwdwh suredelolw| wkdw
hdfk w|sh vxffhvvixoo| wudghv iru klv frqvxpswlrq jrrg lq dq| urxqg lv 42< +1444,1 Krzhyhu/ li w|sh
4 sod|huv sod| dffruglqj wr wkh _vshfxodwlyh% vwhdg| vwdwh htxloleulxp suhglfwlrq/ dqg vshfxodwh lq
jrrg 6/ zkloh w|shv 5 dqg 6 dgkhuh wr ixqgdphqwdo wudglqj vwudwhjlhv/ wkhq wkh suredelolw| wkdw
hdfk w|sh vxffhvvixoo| wudghv iru klv frqvxpswlrq jrrg lq dq| urxqg lv 148:/ dq lqfuhdvh ri 74(1b
D vhfrqg zd| lq zklfk wr dgguhvv wklv txhvwlrq lv wr ghwhuplqh krz pdq| srlqwv vxemhfwv
zrxog kdyh hduqhg li wkh| kdg sod|hg dffruglqj wr wkh suhglfwhg sxuh vwudwhjlhv dqg idfhg wkh vdph
vhtxhqfh ri udqgrp pdwfkhv wkh| h{shulhqfhg lq wkh oderudwru| vhvvlrqv1 Wklv dssurdfk lqyroyhv d
vlpsoh vlpxodwlrq h{huflvh/ zklfk zh qrz ghvfuleh1 Iru hdfk vhvvlrq D4~G8/ zh vlpxodwhg wkh srlqw
wrwdov wkdw hdfk vxemhfw zrxog kdyh hduqhg dw wkh hqg ri hdfk jdph kdg wkh| sod|hg dffruglqj
wr wkh vwhdg| vwdwh vwudwhj| surohv iru wkh htxloleulxp lq zklfk jrrg 3 vhuyhv dv d phglxp ri
h{fkdqjh +dv glvfxvvhg lq vhfwlrq 515,1 Lq hdfk vlpxodwlrq zh xvhg wkh vdph qxpehu ri vxemhfwv/
4; ru 57/ wkdw zhuh suhvhqw lq d vlqjoh h{shulphqwdo vhvvlrq1 Wkhvh vlpxodwhg vxemhfwv zhuh hdfk
hqgrzhg zlwk wkh vdph lqlwldo jrrg wkdw wkh uhdo vxemhfwv kdg lq vwrudjh lq wkh uvw urxqg ri hyhu|
jdph1 Wkh vlpxodwhg vxemhfwv dovr idfhg wkh vdph vhtxhqfh ri udqgrp pdwfklqjv wkdw zhuh idfhg
e| wkhlu uhdo frxqwhusduwv1 Wkh ohqjwk ri hdfk jdph zdv wkhuhiruh wkh vdph dv lq wkh h{shulphqwdo
vhvvlrq1 Zh fdofxodwhg wkh vlpxodwhg srlqw wrwdov dw wkh hqg ri hdfk jdph iru hdfk sod|hu/ dv lw zdv
hqg ri jdph wrwdov wkdw ghwhuplqhg sod|huv* hduqlqjv1 Zh wkhq fdofxodwhg wkh udwlr ri wkh dfwxdo
hqg0ri0jdph srlqw wrwdo hduqhg e| hdfk uhdo vxemhfw/ wr wkh vlpxodwhg hqg0ri0jdph srlqw wrwdo iru
wkdw sod|hu1 Ilqdoo|/ zh fdofxodwhg wkh dyhudjh udwlr iru doo sod|huv ri d jlyhq w|sh ryhu doo jdphv
sod|hg lq d vhvvlrq1 Wkhvh dyhudjh udwlrv duh uhsruwhg lq Wdeoh 431 Qrwlfh wkdw lq vhvvlrqv D40D6/
E40E6 dqg F40F6/ wkhvh dyhudjh udwlrv duh yhu| forvh wr 413/ lqglfdwlqj wkdw wkh uhdo vxemhfwv* srlqw
wrwdov zhuh/ rq dyhudjh/ yhu| forvh wr zkdw zrxog kdyh ehhq suhglfwhg li wkh| kdg dozd|v dgkhuhg
wr wkh ixqgdphqwdo wudglqj vwudwhjlhv/ dqg dozd|v rhuhg wr wudgh iru jrrg 31
Qrwh dovr wkdw wkhuh lv d vxevwdqwldo ghwhulrudwlrq lq wkhvh udwlrv iru vhvvlrqv G40G71 Wkh orzhu
udwlrv lq wkh G vhvvlrqv duh sulpdulo| d frqvhtxhqfh ri wkh odfn ri vshfxodwlyh sod| e| W|sh 4 sod|huv
b
L|i uh|ih |@| u @** |hii |)Tit T*@) u?_@4i?|@* t|h@|i}it ? |t U@ti ihi }LL_ t *itt ULt|*) |L t|Lhi
|@? }LL_ fc |i? L?i U@? tL |@| ? @ t|i@_) t|@|ic |)Ti T*@)iht i?_ T L*_?} |i i?|hi t|LU! Lu }LL_ fW? |t U@tic |)Ti T*@)iht @hi ?iih @M*i |L |h@_i }LL_ f uLh }LL_ ?Lh @hi |i) iih **?} |L |h@_i }LL_ f uLh
@?) L|ih }LL_ 5?Ui |)Ti tMiU|t i@h?i_ @ TLt|i ?i| T@)Lg L?*) i? |i) tUUittu**) |h@_i_ uLh }LL_ c |
tii4t ?*!i*) |@| |i) L*_ @i Mii? UL?|i?| |L L*_ L?|L }LL_ f uLh ih) *L?} u |i) uL?_ |i) ihi ?@M*i
|L _hiU|*) |h@_i }LL_ f uLh }LL_ 6Lh |iti hi@tL?tc i Mi*ii |@| |ihi ihi t|hL?} ?Ui?|it uLh |)Ti T*@)iht
|L @_LT| tTiU*@|i t|h@|i}it ? @** R# tittL?t
58
A@M*i fG +@|L Lu 5MiU| L?| AL|@*t |L O)TL|i|U@* L?| AL|@*t uhL4 *@)?}
UULh_?} |L 5|i@_) 5|@|i hi 5|h@|i}itG ih@}it uLh ,@U *@)ih A)Ti
@*U*@|i_ Nt?} L?| AL|@*t @| |i ,?_ Lu ** B@4it *@)i_ ? @ 5ittL?
5ittL?
2
**
2
**
2
** #
#2
#
#e
#D
** #
A)Ti
fbD
fe
fH
fb
fbD
fb
f2
fbS
fb
fb.
fb.
fbS
fHb
fHb
fDb
f.b
fSe
f.f
A)Ti 2
ff
fbe
fb.
fb.
fb.
f
fb.
fbb
fbD
fb.
fb.
fb.
fH
fHb
fS
f.D
fS
fH
A)Ti fbb
fb.
fb
fb.
fb.
ff
fbb
fbb
fbD
fb.
fbH
fb.
fb
fb.
fHb
fHb
f..
fHb
lq wkhvh vhvvlrqv1 Zkhq W|sh 4 sod|huv sod|hg wkh vshfxodwlyh vwudwhjlhv ri wudglqj jrrgv 3 dqg 5
iru jrrg 6 prvw iuhtxhqwo|/ dv lq vhvvlrq G5/ wkh udwlrv lq Wdeoh 43 duh dw wkhlu kljkhvw ohyhov/ dqg
zkhq W|sh 4 sod|huv sod|hg wkhvh vshfxodwlyh vwudwhjlhv ohdvw riwhq/ dv lq vhvvlrq G6/ wkhvh udwlrv
duh dw wkhlu orzhvw ohyhov1 Wkh lqhflhqw wudglqj ehkdylru ri W|sh 4 sod|huv kdv vslooryhu hhfwv iru
W|sh 5 dqg 6 sod|huv dv zhoo/ uhvxowlqj lq orzhu udwlrv iru wkhvh sod|hu w|shv1
Vlqfh wkh suredelolw| ri zlqqlqj wkh dgglwlrqdo '43 sul}h zdv d olqhduo| lqfuhdvlqj ixqfwlrq ri
wkh qxpehu ri srlqwv hduqhg lq rqh udqgrpo| fkrvhq urxqg/ wkh udwlrv lq Wdeoh 43 dovr surylgh
xv zlwk hvwlpdwhv ri wkh dyhudjh h{shfwhg orvv iurp qrw sod|lqj dffruglqj wr wkh suhglfwhg sxuh
vwudwhjlhv lq hdfk vhvvlrq/ +ru ryhu doo vhvvlrqv ri d wuhdwphqw,1 Iru lqvwdqfh/ lq doo G vhvvlrqv/ W|sh
4 sod|huv hduqhg rq dyhudjh/ :3( ri wkh srlqwv wkh| frxog kdyh hduqhg kdg wkh| sod|hg wkh vwhdg|
vwdwh sxuh vwudwhj| suroh1 Wkh dyhudjh h{shfwhg orvv zdv wkhuhiruh 63( ri '43133 ru '61331
D
L?U*tL?
Zh kdyh uhsruwhg wkh uhvxowv ri dq h{shulphqw wkdw zdv ghvljqhg wr lpsohphqw Nl|rwdnl dqg
Zuljkw*v +4<;<, prgho ri prqh| dv d phglxp ri h{fkdqjh zlwk wkuhh jrrgv dqg d vlqjoh wrnhq
59
remhfw1 Wkh pdlq txhvwlrq zh vrxjkw wr dqvzhu zdv zkhwkhu dq lqwulqvlfdoo| zruwkohvv wrnhq remhfw
wkdw zdv qrw lqyhvwhg zlwk ydoxh e| jryhuqphqw ghfuhh/ frxog qhyhuwkhohvv vhuyh dv d phglxp ri
h{fkdqjh lq dffrugdqfh zlwk wkh wkhru|1 Wkh h{shulphqwdo hylghqfh vxjjhvwv wkdw vxemhfwv zloo
lqghhg rhu wr wudgh iru d wrnhq remhfw hyhq wkrxjk wklv remhfw kdv qr lqwulqvlf ydoxh/ lv qrw ohjdoo|
uhtxluhg/ dqg pd| qrw hyhq eh wkh ohdvw frvwo|~wr~vwruh jrrg1 Wkh htxloleulxp suhglfwlrqv ri wkh
Nl|rwdnl~Zuljkw prgho frph forvhvw wr ehlqj ixooohg zkhq doo w|shv* ehvw uhvsrqvh lv wr sod|
ixqgdphqwdo vwudwhjlhv/ dv lq vhvvlrqv D4~F61
Qhyhuwkhohvv/ hyhq lq wkhvh vhvvlrqv/ wkh iuhtxhqflhv zlwk zklfk jrrgv 3 dqg 4 vhuyh dv phgld ri
h{fkdqjh glhu wr vrph h{whqw iurp wkh vwhdg| vwdwh htxloleulxp iuhtxhqflhv1 Lq sduwlfxodu/ Jrrg
3 lv dffhswhg zkhq rhuhg ohvv iuhtxhqwo| dqg Jrrg 4 lv dffhswhg zkhq rhuhg pruh iuhtxhqwo|
wkdq suhglfwhg lq wkh vwhdg| vwdwh1 Ixuwkhupruh/ lq frqwudvw wr wkh wkhru|/ wkh dffhswdelolw| ri
wkh wrnhq remhfw/ jrrg 3/ dsshduv wr eh vhqvlwlyh wr wkh vxsso| ri jrrg 3/ dqg wkh dffhswdelolw| ri
wkh lqwulqvlfdoo| ydoxhg jrrg 4 pd| qrw ghfuhdvh zlwk lqfuhdvhv lq wkh vxsso| ri jrrg 31 Ilqdoo|/
vxemhfwv kdyh glfxow| uhfrjql}lqj zkhq vshfxodwlyh vwudwhjlhv duh wkhlu ehvw uhvsrqvh lq vhvvlrqv
G4~G8/ ghvslwh dghtxdwh prqhwdu| lqfhqwlyhv1 Lq sduwlfxodu/ pdq| w|sh 4 sod|huv zkr wudgh iru
jrrg 3 idlo wr uhfrjql}h wkdw wkh| zloo eh xqdeoh wr wudgh jrrg 3 gluhfwo| iru wkh jrrg wkh| ghvluh/
l1h1 jrrg 4/ wkh ohdvw frvwo|~wr~vwruh jrrg1 Wkh pdmrulw| ri w|sh 4 vxemhfwv uhixvh wr wudgh jrrg
3 iru wkh pruh frvwo|~wr~vwruh jrrg 6 dqg wr jhw wkhlu frqvxpswlrq jrrg 4/ wkh| pxvw djuhh wr
vxfk wudghv1 Zh frqfoxgh wkdw wkh wrnhq remhfw ~jrrg 3~ grhv vhuyh dv d phglxp ri h{fkdqjh lq
rxu lpsohphqwdwlrq ri wkh wkhruhwlfdo hqylurqphqw/ exw lw dsshduv wr kdyh d pruh olplwhg uroh dv d
phglxp ri h{fkdqjh wkdq wkhru| suhglfwv1
5:
TTi?_
Lq wklv dsshqgl{ zh ghqh dqg ghprqvwudwh wkh h{lvwhqfh ri wkh vwdwlrqdu| sxuh vwudwhj| htxloleuld
glvfxvvhg lq wkh wh{w xvlqj wkh sdudphwhu vhwv zh frqvlghuhg lq wkh h{shulphqw dv uhsruwhg lq Wdeoh
41 Wkh oderudwru| hqylurqphqw glhuv iurp Nl|rwdnl dqg Zuljkw*v +4<;<, wkhruhwlfdo hqylurqphqw
lq rqh lpsruwdqw uhvshfw= lqvwhdg ri kdylqj d frqwlqxxp ri djhqwv ri hdfk w|sh zh kdyh d qlwh
srsxodwlrq ri Q djhqwv zlwk Q@6 ri hdfk w|sh/ zkhuh Q @ 57 ru 4;1 Zh pxvw wkhuhiruh ghprqvwudwh
wkdw wkh Qdvk htxloleuld lq wkh wkhruhwlfdo hqylurqphqw zlwk d frqwlqxxp ri djhqwv duh dovr Qdvk
htxloleuld lq wkh hqylurqphqw zh vwxg| zlwk d qlwh srsxodwlrq ri djhqwv1
Zh ehjlq zlwk wkh ghqlwlrq ri dq htxloleulxp/ zklfk lv vlplodu wr wkdw irxqg lq Nl|rwdnl dqg
Zuljkw +4<;<, dqg Dl|djdul dqg Zdoodfh +4<<5,1 Dv ghvfulehg lq wkh wh{w/ qr jrrg pd| eh iuhho|
glvsrvhg ri dqg doo h{fkdqjhv frqvlvw ri rqh~iru~rqh vzdsv ri jrrgv lq lqyhqwru|1 Zkhq dq djhqw
vxffhvvixoo| wudghv iru klv frqvxpswlrq jrrg/ kh lpphgldwho| frqvxphv wkdw jrrg dqg surgxfhv d
xqlw ri klv surgxfwlrq jrrg1 Ohw lqgh{ d sduwlfxodu sod|hu w|sh/ 4/ 5 ru 6/ dqg ohw uhsuhvhqw wkh
vhw ri sod|hu w|shv1 Ohw v&
c +w, 5 ^3> 4` ghqrwh wkh vwudwhj| ri d w|sh sod|hu zlwk lqgh{ qxpehu
l @ 4> 5> ===> Q@6/ zkr lv vwrulqj jrrg m dqg lv udqgrpo| sdluhg zlwk d sod|hu vwrulqj jrrg n 9@ m
lq urxqg w1 Ohw vc +w, ghqrwh wkh yhfwru ri sod|hu l*v vwudwhjlhv dw w1 Lq htxloleulxp/ doo l sod|huv
ri w|sh zloo kdyh wkh vdph vwudwhj| yhfwru/ ghqrwhg v +w,1 Ohw wkh vhw ri vxfk vwudwhj| yhfwruv
eh ghqrwhg e| v+w,1 Iroorzlqj Nl|rwdnl dqg Zuljkw/ zh irfxv rq sxuh vwudwhj| htxloleuld/ zkhuh
&
&
v&
+w, 5 i3> 4j1 Li v @ 4/ wkh w|sh sod|hu rhuv wr wudgh jrrg m iru jrrg n> li v @ 3/ wkh w|sh sod|hu uhixvhv wr wudgh jrrg m iru jrrg n1 Ohw s +w, ghqrwh wkh sursruwlrq ri sod|huv ri w|sh zkr
duh vwrulqj jrrg m dw wkh vwduw ri urxqg w/ dqg ohw s+w, ghqrwh wkh yhfwru ri doo vxfk sursruwlrqv1
Ghqh wkh whup
d +w, @ s +w, s +w,
[[
# '&
&
s&# +w,v&
+w,v# +w, .
[[
# '&
&
s& +w,s# +w,v&
+w,v# +w,>
+D=4,
zklfk uhsuhvhqwv wkh iudfwlrq ri w|sh djhqwv zlwk jrrg m lpphgldwho| iroorzlqj wudglqj dw wlph
w1 Wkh uvw whup rq wkh uljkw kdqg vlgh ri wklv h{suhvvlrq lv wkh iudfwlrq ri w|sh djhqwv zlwk
jrrg m sulru wr wudglqj/ zkloh wkh qh{w wzr whupv uhsuhvhqw wkh rxwrz dqg lqrz/ uhvshfwlyho|/
ri vxfk djhqwv lpphgldwho| iroorzlqj wudglqj1 Wkh vhtxhqfh is+w,j hyroyhv dffruglqj wr s+w . 4, @
I ^s+w,> v+w,`/ zkhuh hohphqwv ri wkh pds I duh ghqhg dv iroorzv=
s +w . 4, @
;
A
?
3 li m @ d +w, . d +w, li m @ . 4 prgxor 6>
A
=
9 / . 4 prgxor 6=
d +w, li m @
+D=5,
zkhuh d +w, uhsuhvhqwv qhz surgxfwlrq ri jrrg .4 prgxor 61 W|sh djhqwv zkr vxffhvvixoo| wudgh
iru jrrg lpphgldwho| surgxfh dS
xqlw ri wkhlu surgxfwlrq
jrrg . 4 prgxor 6/ khqfh s +w, @ 3
S n
f
;w1 Wkh lqlwldo frqglwlrqv duh wkdw s +4, @ p/ dqg s +4, @ 4 p/ zkhuh p uhsuhvhqwv wkh
h{rjhqrxvo| jlyhq iudfwlrq ri wkh hqwluh srsxodwlrq wkdw lv hqgrzhg zlwk jrrg 31
S Vlqfh jrrg 3 lv
qhlwkhu surgxfhg qru iuhho| glvsrvdeoh/ lq dgglwlrq wr +D140D15, zh dovr uhtxluh sf +w, @ p ;w1
Dq htxloleulxp vwduwlqj iurp s+4, lv d vhtxhqfh is+w . 4,> v+w,j vxfk wkdw
41 s+w . 4, @ I ^s+w,> v+w,`/
51 vc +w, lv lqglylgxdoo| rswlpdo iru dq| sod|hu l jlyhq ivc3 +w,/ v# +w,/ s+w . 4,j1
zkhuh vc3 +w, lv xqghuvwrrg wr eh wkh wlph w vwudwhj| sxuvxhg e| doo rwkhu w|sh sod|huv h{foxglqj
sod|hu l1 Sduw 5 ri wklv ghqlwlrq glhuv iurp wkdw ri Nl|rwdnl dqg Zuljkw zkr rqo| uhtxluhg wkdw
v +w, eh rswlpdo iru dq| djhqw ri w|sh 1 Wkh glhuhqfh lv wkdw/ lq rxu iudphzrun/ d vlqjoh sod|hu l
5;
lv qrq~dwrplvwlf/ dqg khqfh pd| dhfw wkh sursruwlrqv s+w, e| klv fkrlfh ri vwudwhj|1 Zh zloo irfxv
rq vwdwlrqdu| sxuh vwudwhj| htxloleuld dv lq Nl|rwdnl dqg Zuljkw1
Wr dvvhvv zkhwkhu vwudwhjlhv duh lqglylgxdoo| rswlpdo/ lw zloo eh xvhixo wr lqwurgxfh vrph dg0
glwlrqdo qrwdwlrq1 Ohw yc
ghqrwh wkh h{shfwhg xwlolw| wr lqglylgxdo l ri w|sh vwrulqj jrrg m
lpphgldwho| iroorzlqj wudglqj dw wlph w1 Lq sduwlfxodu/ ghqh
+w, @
yc
;
A
A
A
A
A
A
?
A
A
A
A
A
A
=
k
S S
&c
&
s
+w
.
4,
vc&
# & #c& c +w . 4,v# +w . 4,yc +w . 4,l
&c
9 >
. 4 vc&
c +w . 4,v# +w . 4, yc +w . 4, li m @
k
S S
nc&
&c
n
&
+w . 4,yc +w . 4,
x . # & s#c& +w . 4, vc +w . 4,v#
l
nc&
n +w . 4, y n +w . 4,
li m @ =
. 4 vc +w . 4,v&c
#
c
f . +D=6,
Uhfdoo wkdw lq rxu hqylurqphqw/ wkh xwlolw| iurp frqvxpswlrq x/ lv frpprq dfurvv jrrgv dqg w|shv
zkloh vwrudjh frvwv glhu dfurvv jrrgv m @ 3> 4> 5> 6/ exw qrw dfurvv w|shv1 Ixuwkhupruh/ wkhuh duh qr
frqvxpswlrq2surgxfwlrq ghflvlrqv iru djhqwv wr pdnh/ qru lv wkhuh dq| iuhh glvsrvdo1 Wkh lqglylgxdo
rswlpl}lqj frqglwlrqv fdq eh ghqhg dv=
v&
c +w, 5
;
A
? i3j
A
=
i4j
^3> 4`
li
li
& +w,/
yc
+w, A yc
& +w,/
+w, ? yc
yc
rwkhuzlvh1
Iroorzlqj Nl|rwdnl dqg Zuljkw/ zh zloo qrw frqvlghu wkh odvw fdvh/ zkhuh lqglylgxdov sod| pl{hg
vwudwhjlhv/ dqg zloo lqvwhdg irfxv rq vwhdg| vwdwh +wlph0lqyduldqw,/ sxuh vwudwhj| htxloleuld1 Dffrug0
lqjo|/ zh zloo gurs doo ixuwkhu uhihuhqfhv wr wlph1 Vlqfh sod|huv dozd|v wudgh iru wkhlu frqvxpswlrq
jrrg v
c +w, @ 4 iru doo w/ dqg vwudwhjlhv duh dvvxphg wr eh v|pphwulf zh fdq zulwh wkh vwdwlrqdu|
sxuh vwudwhj| yhfwru ri d w|sh sod|hu frpsdfwo| dv
f nc&
>
v @ v ncf > v&c
> v
zkhuh . 4 lv w|sh *v surgxfwlrq jrrg dqg n 9@ 3 lv wkh rwkhu lqwulqvlfdoo| ydoxhg jrrg wkdw w|sh
qhlwkhu surgxfhv qru frqvxphv1
Iru rxu sdudphwhul}dwlrq ri fdvh 4/ zkhuh 3 @ ff ? f ? f2 ? f / zh fodlp wkdw/ dv lq Nl|rwdnl
dqg Zuljkw*v +4<;<, prgho zlwk d frqwlqxxp ri hdfk djhqw w|sh/ wkhuh h{lvw rqo| wzr vwdwlrqdu|
sxuh vwudwhj| htxloleuld lq rxu prgho zlwk d qlwh srsxodwlrq ri hdfk djhqw w|sh1 Lq rqh ri wkhvh
vwhdg| vwdwhv/ jrrg 3 lv xvhg dv d jhqhudoo| dffhswhg phglxp ri h{fkdqjh/ jrrg 4 dovr vhuyhv dv d
phglxp ri h{fkdqjh dqg wkh vwudwhj| yhfwruv duh v @ +4> 4> 3,/ v2 @ +4> 4> 4,/ v @ +4> 4> 3,1 Lq wkh
rwkhu vwdwlrqdu| htxloleulxp/ jrrg 3 lv qhyhu dffhswhg lq wudgh/ vr wkdw v @ +3> 3> 3,/ v2 @ +3> 3> 4,/
v @ +3> 3> 3,1 Lq wklv vwhdg| vwdwh/ jrrg 4 frqwlqxhv wr vhuyh dv d phglxp ri h{fkdqjh1 Iru rxu
sdudphwhul}dwlrq ri fdvh 5/ zkhuh 3 @ f ? ff ? f2 ? f / zh fodlp wkhuh h{lvwv d vwdwlrqdu| sxuh
vwudwhj| htxloleulxp zkhuh v @ +4> 3> 4,/ v2 @ +4> 3> 4,/ v @ +3> 4> 3,1 Lq wklv htxloleulxp/ w|sh 4
sod|huv vshfxodwh lq wkh prvw frvwo|0wr0vwruh jrrg 6/ zklfk vhuyhv dv d olplwhg phglxp ri h{fkdqjh
dorqj zlwk jrrgv 4 dqg 31
Rxu vwudwhj| iru ghprqvwudwlqj wkh h{lvwhqfh ri vxfk htxloleuld lv dv iroorzv1 Zh uvw yhuli|
wkdw rxu sdudphwhu vhwv 4 dqg 5 lpso| wkh h{lvwhqfh ri wkh vwdwlrqdu| sxuh vwudwhj| htxloleuld
uhihuuhg wr deryh/ xqghu wkh dvvxpswlrq ri d frqwlqxxp ri djhqwv ri hdfk w|sh dv lq Nl|rwdnl dqg
Zuljkw +4<;<, dqg Dl|djdul dqg Zdoodfh +4<<5,1 Zh wkhq wxuq rxu dwwhqwlrq wr wkh fdvh ri d
qlwh srsxodwlrq ri Q djhqwv1 Zh dvvxph wkdw Q 4 ri wkhvh djhqwv sod| dffruglqj wr wkh sxuh
vwudwhjlhv wkdw fkdudfwhul}h rqh ri wkh vwdwlrqdu| htxloleuld ri wkh prgho zlwk frqwlqxxp ri djhqwv1
Wkh orqh uhpdlqlqj +Q | , djhqw sod|v vrph rwkhu vwudwhj|1 Zh wkhq qxphulfdoo| ghwhuplqh wkh
5<
h{shfwhg srsxodwlrq sursruwlrq ri djhqwv zkr duh w|sh @ 4> 5> 6 dqg vwrulqj jrrg m @ 3> 4> 5> 6/
s +d sur{| iru wkh vwhdg| vwdwh sursruwlrqv,1 Ghqrwh wkhvh h{shfwhg ydoxhv e| sa 1 Xvlqj wkhvh
h{shfwhg ydoxhv/ zh wkhq fdofxodwh wkh h{shfwhg glvfrxqwhg xwlolw| wr wkh ghyldqw djhqw l ri w|sh iurp kroglqj jrrg m/ yc
1 Zh wkhq udqn wkh ydoxhv yc
dqg yhuli| wkdw wklv udqnhg olvw zrxog lpso|
wkdw wkh ghyldqw djhqw zrxog gr ehwwhu e| sod|lqj dffruglqj wr wkh sxuh vwudwhj| wkdw lv ehlqj
sod|hg e| wkh rwkhu w|sh djhqwv/ zklfk lv wkh fdqglgdwh vwdwlrqdu| htxloleulxp vwudwhj| yhfwru1
Zh frqgxfw wklv h{huflvh iru d vlqjoh ghyldqw djhqw ri hdfk ri wkh wkuhh sod|hu w|shv/ iru doo ydoxhv
ri Q dqg p wkdw zh xvhg lq wkh h{shulphqw1 Wklv surfhgxuh doorzv xv wr frqup wkh h{lvwhqfh ri
d vwdwlrqdu| sxuh vwudwhj| Qdvk htxloleulxp lq wkh qlwh srsxodwlrq fdvh1
@h@4i|ih 5i| r
Zh uvw yhulhg wkdw rxu sdudphwhu vhw 4/ zkhuh 3 @ ff ? f ? f2 ? f / zdv frqvlvwhqw zlwk
wkh h{lvwhqfh ri d sxuh vwudwhj| htxloleulxp zlwk d frqwlqxxp ri djhqwv ri hdfk w|sh/ zkhuh doo
wkuhh w|shv sod| ixqgdphqwdo vwudwhjlhv/ dqg zkhuh jrrg 3/ wkh ohdvw frvwo|~wr~vwruh jrrg/ vhuyhv
dv d jhqhudoo| dffhswhg phglxp ri h{fkdqjh1 Xvlqj wkh vwudwhjlhv v @ +4> 4> 3,/ v2 @ +4> 4> 4,/ dqg
v @ +4> 4> 3, lq +D14, zh vroyhg iru wkh vwhdg| vwdwh sursruwlrqv s +w, @ s ;w xvlqj +D15, iru hdfk
ri wkh wkuhh glhuhqw ydoxhv ri p lq sdudphwhu vhw 41 Wkhvh vwhdg| vwdwh sursruwlrqv duh jlyhq lq
Wdeoh D4 ehorz1 Uhfdoo wkdw s dqg p uhsuhvhqw sursruwlrqv ri wkh hqwluh srsxodwlrq ri sod|huv/
udwkhu wkdq sursruwlrqv zlwklq d fhuwdlq w|sh +dv lq wkh ruljlqdo Nl|rwdnl dqg Zuljkw +4<;<, sdshu,1
Xvlqj wkhvh vwhdg| vwdwh sursruwlrqv wrjhwkhu zlwk wkh wkuhh sxuh vwudwhj| yhfwruv dqg wkh prgho
A@M*i G 5|i@_) 5|@|i W?i?|Lh) hLTLh|L?tG
Ai @ti `ihi BLL_ f t wi@t| Lt|*) |L 5|Lhic t Nti_ @t L?i) @?_
Aihi t @ L?|?4 Lu ,@U *@)ih A)Ti
@ti E@h@4i|ih 5i|
6'
2D
Df
R7f
fSS
fb2
Df
R72
2S.
2e
H
R7
fff
fff
fff
R7f2
fH2
fb
S
R72
f
fHe
fD2
R72
eH
ef
H
R7f
f2
2
HS
R7
2
2f
e.
R72
fff
fff
fff
sdudphwhuv jlyhq lq sdudphwhu vhw 4/ zh xvhg +D16, wr fdofxodwh wkh vwdwlrqdu| +h{shfwhg, ydoxh
wr hdfk w|sh ri vwrulqj hdfk jrrg iru hdfk ydoxh ri p1 Lw lv uhdglo| yhulhg wkdw lq doo fdvhv/ wkh
udqnlqjv ri wkhvh ydoxhv vdwlvi|=
y ? y2 ? yf ? y >
y2 ? y2 ? y2f ? y22 >
y2 ? y ? yf ? y =
Wkhvh udqnlqjv lpso| wkdw wkh vwudwhj| yhfwruv v @ +4> 4> 3,/ v2 @ +4> 4> 4, dqg v @ +4> 4> 3, duh
rswlpdo iru hdfk w|sh/ jlyhq s dqg p1
Iroorzlqj wkh vdph surfhgxuh/ zh yhulhg wkdw wkh fdvh zkhuh jrrg 3 lv qhyhu dffhswhg lq
h{fkdqjh dqg sod|huv dgkhuh wr wkh vwudwhjlhv v @ +3> 3> 3,/ v2 @ +3> 3> 4, dqg v @ +3> 3> 3, lv
dovr d vwdwlrqdu| sxuh vwudwhj| htxloleulxp xqghu sdudphwhu vhw 41 Wdeoh D5 jlyhv wkh vwhdg|
vwdwh sursruwlrqv lq wklv fdvh iru wkh wkuhh glhuhqw ydoxhv ri p irxqg lq sdudphwhu vhw 41 Lq rxu
h{shulphqwdo hqylurqphqw/ p zdv lqlwldoo| glvwulexwhg hyhqo| ryhu wkh wkuhh w|shv/ khqfh sf @ p@6
; 1 Djdlq/ rqh fdq xvh wkhvh vwhdg| vwdwh sursruwlrqv/ wkh wkuhh vwudwhj| yhfwruv/ dqg wkh sdudphwhu
63
A@M*i 2G 5|i@_) 5|@|i W?i?|Lh) hLTLh|L?tG
@ti `ihi BLL_ f t L| Nti_ @t L?i) @?_ Aihi t @ L?|?4 Lu ,@U *@)ih A)Ti
@ti E@h@4i|ih 5i|
6'
2D
Df
R7f
fH
S.
R72
2Df
222
S.
R7
fff
fff
fff
R7f2
fH
S.
R72
2D
fH
R72
2D
fH
R7f
fH
S.
R7
2Df
222
S.
R72
fff
fff
fff
fkrlfhv lq sdudphwhu vhw 4 lq +D16, wr yhuli| wkdw iru hdfk ydoxh ri p/ yf @ 3 ; / dqg=
y ? y2 ? y >
y2 ? y2 ? y22 >
y2 ? y ? y =
Zh qh{w qhhg wr yhuli| wkdw lq qlwh srsxodwlrqv zlwk ; ru 9 sod|huv ri hdfk w|sh/ zkhuh doo
sod|huv exw rqh sod| dffruglqj wr wkhvh ixqgdphqwdo vwudwhjlhv/ wkh h{shfwhg glvwulexwlrq ri jrrgv
ryhu sod|hu w|shv frqwlqxhv wr lpso| wkh vdph udqnlqj ri wkh ydoxhv y 1 Wklv odvw vwhs fdqqrw
eh grqh dqdo|wlfdoo| +dv idu dv zh nqrz,1 Frqvhtxhqwo| zh uhvruwhg wr qxphulfdo vlpxodwlrqv1 Lq
sduwlfxodu/ zh frqvlghuhg srsxodwlrqv ri vl}h Q @ 57 ru 4; sod|huv +; ru 9 sod|huv ri hdfk w|sh,1
Zh uhtxluhg Q 4 sod|huv wr pdgh wudglqj ghflvlrqv dffruglqj wr wkh fdqglgdwh ixqgdphqwdo
htxloleulxp wudglqj vwudwhjlhv1 Wkh uhpdlqlqj sod|hu sod|hg dffruglqj wr rqh ri wkh hljkw srvvleoh
sxuh vwudwhj| yhfwruv iru hdfk sod|hu w|sh1 Wklv vhw ri hljkw vwudwhj| yhfwruv lqfoxghv wkh fdqglgdwh
htxloleulxp vwudwhj| dv zhoo dv : rwkhu srvvleoh _ghyldqw% vwudwhjlhv1
Lq wkh vlpxodwlrqv/ diwhu wkh Q sod|huv vxeplwwhg rhuv/ dq| pxwxdoo| djuhhdeoh wudghv zhuh lp0
sohphqwhg dqg wkh uhvxowlqj fkdqjhv lq lqyhqwru| glvwulexwlrqv zhuh uhfrughg1 Lq wkhvh vlpxodwlrqv/
dv lq wkh h{shulphqw/ d iudfwlrq p ri hdfk sod|hu w|sh lqlwldoo| kdv jrrg 3 lq vwrudjh> wkh uhpdlqlqj
iudfwlrq ri hdfk sod|hu w|sh lqlwldoo| kdv wkhlu surgxfwlrq jrrg lq vwrudjh1 Hdfk vlpxodwlrq uxq
frqvlvwv ri 43 urxqgv/ wkh h{shfwhg ohqjwk ri d jdph lq rxu h{shulphqw1 Dw wkh hqg ri hdfk uxq
zh fdofxodwhg wkh dyhudjh vlpxodwhg sursruwlrqv sa ryhu doo 43 urxqgv sod|hg +vr wkhvh ydoxhv wdnh
lqwr dffrxqw rxu lqlwldol}dwlrq surfhgxuh,1 Zh frqgxfwhg d wrwdo ri 43 vlpxodwlrq uxqv +dv hdfk
h{shulphqwdo vhvvlrq frqvlvwhg ri dssur{lpdwho| 43 jdphv, iru hdfk ydoxh ri p/ srsxodwlrq vl}h/
Q @ 57 ru 4;/ dqg iru wkh fdvh zkhuh rqh sod|hu sod|hg dffruglqj wr rqh ri wkh hljkw srvvleoh sxuh
vwudwhjlhv iru klv w|sh zkloh wkh rwkhu Q 4 sod|huv sod|hg dffruglqj wr wkh fdqglgdwh htxloleulxp
vwudwhjlhv1 Dv dq looxvwudwlrq/ zh uhsruw lq Wdeoh D5/ wkh phdq dqg +vwdqgdug ghyldwlrq, ri 43
sa ydoxhv iru hdfk ydoxh ri p dqg Q / iru wkh fdvh zkhuh d w|sh 4 ghyldqw ~ sod|hu qxpehu 4 ~
sod|v rqh ri wkh ; srvvleoh sxuh vwudwhjlhv v / dqg wkh rwkhu Q 4 sod|huv dgkhuh wr wkh fdqglgdwh
htxloleulxp +ixqgdphqwdo, vwudwhjlhv1
Qrwh wkdw wkh uvw _ghyldqw% vwudwhj| olvwhg iru hdfk sdudphwhul}dwlrq/ v @ +4> 4> 3, lv lq idfw/
wkh fdqglgdwh ixqgdphqwdo sxuh vwudwhj|> wkh sursruwlrqv lq wklv fdvh fdq eh frpsduhg zlwk wkh
wkhruhwlfdo vwhdg| vwdwh sursruwlrqv jlyhq lq Wdeoh D41 Wkh glhuhqfh ehwzhhq wkh sursruwlrqv
lq Wdeohv D6 dqg D4 duh txlwh vpdoo dqg duh gxh wr wkh qlwhqhvv ri wkh srsxodwlrq1 Wkh rwkhu
: vwudwhjlhv wuxo| uhsuhvhqw ghyldqw vwudwhjlhv sod|hg e| rqh w|sh 4 ghyldqw sod|hu1 Zkhq wkhvh
vwudwhjlhv duh sod|hg/ zh dovr revhuyh wkdw wkhuh lv qrw pxfk fkdqjh lq wkh phdq sursruwlrqv uhodwlyh
wr wkh wkhruhwlfdo vwhdg| vwdwh sursruwlrqv1
64
6
'
2D
Df
A@M*i G i@? R @?_ E5|@?_@h_ #i@|L? 6hL4 f 54*@|L? +?t
@ti E@h@4i|ih 5i| 6?|i LT*@|L? Lu 2e Lh H *@)ihtc 6 ' 2Dc Lh Df
#i@?|
'
r
Rf
R2
R
Rf2
R2
R2
Rf
R
2e
Eccf
f.f
2S
fff
fHD
fb.
D
fbD
2H
Ef2 Ef2 Efff EfS EfD EfS Ef2 Ef2
Efcfcf
fSD
2SH
fff
fb
ff
e2
fbe
2b
EfD EfD Efff Ef2b Ef2D Ef. Ef22 Ef22
Efcfc
fDD
2SH
ff
fHb
ff
ee
fS
22.
Eff Ef Eff Efb Ef22 Ef2 Ef2 Ef2
Efccf
fDH
2.D
fff
fH2
fH
e
f
22
Ef2 Ef2 Efff Ef2H Ef2b Ef. Ef2e Ef2e
Efcc
f.
2Df
f2
fH
f
eb
fb.
2S
Ef EfS Effb Efb Efe Ef2f Ef22 Ef22
Ecfcf
fDH
2.D
fff
fbD
fHH
Df
fbH
2D
Ef Ef Efff EfS EfH Ef Ef22 Ef22
Ecfc
fDD
2S
f.
fHH
D
f
f.
22S
Ef Ef. Ef2 EfS Ef2f Efe Ef2 Ef2
Ecc
fS
2S2
ff
fH.
fS
ef
f2
2
Ef. Efb Effb EfH Ef2b Ef Ef2S Ef2S
H
Eccf
fb2
2e
fff
2
f.
eH
2b
2fe
Efb Efb Efff Efe Ef2 Ef2S Ef2H Ef2H
Efcfcf
fHD
2eH
fff
S
f.2
eD
2
2f
Efb Efb Efff Efe Ef2 Ef2D Efe Efe
Efcfc
fH2
2
f2f
2D
fSb
b
2S
2f.
Ef22 Ef2. Ef. Ef2 EfS Ef2S Ef2 Ef2
Efccf
f.D
2DH
fff
2.
f..
2b
2f2
Ef22 Ef22 Efff Ef22 Ef2D Ef Ef2H Ef2H
Efcc
f.2
2D2
ffb
2e
fHH
2
.
bS
Ef2 Ef2e EffS Ef2e Ef2 Ef22 Eff Eff
Ecfcf
fbe
2b
fff
f2
fHH
e
.
bS
Efb Efb Efff Efe Ef2b Ef2D Ef2e Ef2e
Ecfc
fH
2f
f2
f.
fHf
eS
H
2D
Ef Ef2 Ef2 Ef2 Efb Ef2H Ef2. Ef2.
Ecc
fbH
2H
f.
fS
fH
eS
2b
2fe
EfD EfH Ef Ef2D Ef2H Ef2D Ef2e Ef2e
H
Eccf
e.
HS
fff
Sb
fe
2
He
eb
Ef2e Ef2e Efff Ef22 Ef22 Ef2f Ef2b Ef2b
Efcfcf
b
be
fff
.D
fe
D
HS
e.
Ef2D Ef2D Efff Ef2. Ef2b Ef2b Ef2. Ef2.
Efcfc
f
.H
f2D
Hf
fef
bf
e
Efb Ef2b EfH EfH Ef2. Ef Ef2 Ef2
Efccf
S
b.
fff
b
feH
fb2
.
S2
Ef2 Ef2 Efff Efb Ef Ef22 Ef2D Ef2D
Efcc
eS
.D
f2
Sb
fD
HD
eH
Ef22 Ef2b Ef Ef2 Efe Ef2f Eff Eff
Ecfcf
eS
HH
fff
S
fef
b
e2
Ef. Ef. Efff Ef2 Efb Efb Ef2D Ef2D
Ecfc
D
SS
fe
SH
fD
e
.b
De
Eff EfH Ef2 Ef2b Ef2 Ef Ef2b Ef2b
Ecc
De
S
fS
.
fee
S
.
Sf
Ef Efb Efe Ef2b Ef2 Ef2f Eff Eff
R2
fff
Efff
fff
Efff
fff
Efff
fff
Efff
fff
Efff
fff
Efff
fff
Efff
fff
Efff
fff
Efff
fff
Efff
fff
Efff
fff
Efff
fff
Efff
fff
Efff
fff
Efff
fff
Efff
fff
Efff
fff
Efff
fff
Efff
fff
Efff
fff
Efff
fff
Efff
fff
Efff
fff
Efff
Iru hdfk ghyldqw vwudwhj| dqg vhw ri phdq sursruwlrqv uhsruwhg lq Wdeoh D5 zh xvhg h{suhvvlrqv
D40D5 wr fdofxodwh wkh ydoxh wr wkh w|sh 4 ghyldqw ri wkh irxu jrrgv1 Lq fdofxodwlqj wkhvh ydoxhv/
65
zh dvvxphg wkdw wkh rwkhu Q 4 sod|huv dgkhuhg wr wkh fdqglgdwh htxloleulxp vwudwhjlhv1 Lw lv
vlpsoh exw whglrxv wr yhuli| wkdw lq doo fdvhv wkh ydoxhv wr wkh w|sh 4 ghyldqw ri wkh irxu jrrgv
dozd|v vdwlvi|=
2
f
y
? y
? y
? y
lqglfdwlqj wkdw wkh ghyldqw*v rswlpdo vwudwhj| lv wr sod| wkh htxloleulxp vwudwhj| iru w|sh 4 sod|huv/
v @ +4> 4> 3,1
Ri frxuvh/ zh kdyh dovr uhshdwhg wklv vdph h{huflvh iru wkh fdvh ri d vlqjoh w|sh 5 ru w|sh 6
ghyldqw/ wkrxjk zh gr qrw uhsruw wkh uhvxowv khuh1 Zkloh wkhvh ghyldqw sod|huv dovr dhfw wkh
sursruwlrqv ri sod|huv vwrulqj wkh ydulrxv jrrgv/ wkh rswlpdo vwudwhj| iru wkh w|sh 5 dqg 6 sod|huv
qhyhuwkhohvv uhpdlq wkh ixqgdphqwdo htxloleulxp vwudwhjlhv v2 @ +4> 4> 4, dqg v @ +4> 4> 3,/ dv lv
uhyhdohg lq dq h{huflvh vlplodu wr wkh rqh frqgxfwhg deryh iru wkh w|sh 4 ghyldqw sod|hu1
Ilqdoo| zh kdyh uhshdwhg wkh vdph surfhgxuhv ghvfulehg deryh iru wkh rwkhu sxuh vwudwhj|
htxloleulxp/ zkhuh jrrg 3 lv qhyhu dffhswhg lq h{fkdqjh/ vr wkdw wkh vwudwhj| yhfwruv duh v @
+3> 3> 3,/ v2 @ +3> 3> 4, dqg v @ +3> 3> 3,1 Zh irxqg wkdw lq rxu srsxodwlrqv ri Q @ 57ru4;djhqwv/
d vlqjoh ghyldqw sod|hu ri hdfk w|sh/ sod|lqj doo srvvleoh vwudwhjlhv glg qrw dowhu wkh h{shfwhg
lqyhqwru| sursruwlrqv lq vxfk d zd| dv wr uhyhuvh wkh udqnlqjv lq wkh ydoxhv wr vwrulqj hdfk jrrg
wkdw zrxog suhydlo lq wkh vwdwlrqdu| htxloleulxp zkhuh doo sod|huv zhuh sod|lqj dffruglqj wr wkh
fdqglgdwh htxloleulxp vwudwhjlhv1 Khqfh/ v @ +3> 3> 3,/ v2 @ +3> 3> 4, dqg v @ +3> 3> 3, dovr frpsulvh
d vwdwlrqdu| Qdvk htxloleulxp lq rxu hqylurqphqw zlwk d qlwh srsxodwlrq ri sod|huv1
@h@4i|ih 5i| r2
Zh qh{w yhuli| wkdw rxu sdudphwhu vhw 5/ zkhuh 3 @ f ? ff ? f2 ? f / lv frqvlvwhqw zlwk wkh
h{lvwhqfh ri d vwdwlrqdu| sxuh vwudwhj| htxloleulxp zkhuh d frqwlqxxp ri djhqwv ri hdfk w|sh sod|
dffruglqj wr wkh vwudwhjlhv v @ +4> 3> 4,/ v2 @ +4> 3> 4, dqg v @ +3> 4> 3,/ dqg zkhuh jrrg 3 vhuyhv
d olplwhg uroh dv d phglxp ri h{fkdqjh hyhq wkrxjk lw kdv kljkhu vwrudjh frvw wkdq jrrg 41 Lq
wklv htxloleulxp/ jrrg 6 dovr kdv d olplwhg uroh dv d phglxp ri h{fkdqjh dqg jrrg 4 vhuyhv dv d
jhqhudoo| dffhswhg phglxp ri h{fkdqjh1
Iroorzlqj wkh vdph surfhgxuhv dv ghvfulehg deryh zh xvhg wkh wkuhh fdqglgdwh vwudwhj| yhfwruv
dqg rxu fkrlfh ri p @ 4@9 lq sdudphwhu vhw 5 wr fdofxodwh wkh vwhdg| vwdwh lqyhqwru| sursruwlrqv
zklfk duh jlyhq lq Wdeoh D71 Xvlqj wkhvh vwhdg| vwdwh sursruwlrqv/ wkh wkuhh vwudwhj| yhfwruv dqg
A@M*i eG 5|i@_) 5|@|i W?i?|Lh) hLTLh|L?tG
Ai @ti `ihi BLL_ f t L| wi@t| Lt|*) |L 5|Lhi M| t Nti_ @t L?i)
@?_ Aihi t @ L?|?4 Lu ,@U *@)ih A)Ti
@ti 2 E@h@4i|ih 5i| 2
6'
S.
R7f
f.
R72
D.
R7
fSb
wkh sdudphwhul}dwlrq ri wkh prgho jlyhq lq
yhuli| wkdw=
y2 ?
y2 ?
y2 ?
R7f2
fSf
R72
HS
R72
fH.
R7f
fff
R7
R72
fff
sdudphwhu vhw 5/ rqh fdq xvh wkh ghqlwlrqv lq +D16, wr
yf ? y ? y >
y2f ? y2 ? y22 >
yf ? y ? y =
Wkhvh udqnlqjv lpso| wkdw wkh vwudwhj| yhfwruv v @ +4> 3> 4,/ v2 @ +4> 3> 4, dqg v @ +3> 4> 3, zrxog
eh rswlpdo iru hdfk w|sh/ jlyhq s dqg p1
Zh wkhq uhshdwhg wkh h{huflvh zh frqgxfwhg iru fdvh 4/ wr yhuli| wkdw wklv vwdwlrqdu| htxloleulxp
uhpdlqv dq htxloleulxp zlwk d qlwh srsxodwlrq ri Q @ 4; djhqwv +9 ri hdfk w|sh,1 Lq sduwlfxodu/
66
zh frqvlghuhg wkh srvvlelolw| wkdw rqh djhqw ri hdfk w|sh sod|hg hdfk ri wkh hljkw srvvleoh vwudwhjlhv
iru klv w|sh/ zkloh wkh rwkhu Q 4 djhqwv dgkhuhg wr wkh fdqglgdwh htxloleulxp vwudwhjlhv1 Zh
fdofxodwhg wkh h{shfwhg lqyhqwru| sursruwlrqv dqg xvhg wkhvh wr ghwhuplqh wkh udqnlqjv ri wkh
ydoxh ixqfwlrqv iru wkh ghyldqw sod|hu1 Lq doo fdvhv zh irxqg/ rqfh djdlq/ wkdw wkh udqnlqjv ri wkhvh
ydoxh ixqfwlrqv lpsolhg wkdw wkh ghyldqw sod|hu zrxog qg lw rswlpdo wr dgkhuh wr wkh fdqglgdwh
htxloleulxp vwudwhj| iru klv w|sh1 Zh wkhuhiruh frqfoxgh wkdw wkh vwudwhj| yhfwru v @ +4> 3> 4,/
v2 @ +4> 3> 4, dqg v @ +3> 4> 3, frqvwlwxwhv d vwdwlrqdu| Qdvk htxloleulxp lq rxu hqylurqphqw zlwk
d srsxodwlrq ri mxvw Q @ 4; sod|huv1 Zh qrwh wkdw/ dqdorjrxv wr fdvh 4/ wkhuh lv dqrwkhu vwdwlrqdu|
Qdvk htxloleulxp xqghu sdudphwhu vhw 5 zkhuh wkh wrnhq remhfw lv qrw dffhswhg lq h{fkdqjh/ dqg
wkh vwudwhj| yhfwruv duh v @ +3> 3> 4,/ v2 @ +3> 3> 4, dqg v @ +3> 3> 3,1 Zh kdyh dovr yhulhg wkh
h{lvwhqfh ri wklv htxloleulxp zlwk d qlwh srsxodwlrq ri djhqwv iroorzlqj wkh vdph surfhgxuhv dv
rxwolqhg deryh iru wkh rwkhu fdvhv1
67
TTi?_ 2
Wklv dsshqgl{ surylghv wkh zulwwhq lqvwuxfwlrqv wkdw zhuh jlyhq wr vxemhfwv lq h{shulphqwdo vhvvlrqv
G4~G71 Wkh lqvwuxfwlrqv iru rwkhu vhvvlrqv duh vlplodu1 Wkhvh zulwwhq lqvwuxfwlrqv zhuh uhdg dorxg
sulru wr wkh ehjlqqlqj ri sod|1 Lq dgglwlrq/ d vlqjoh sudfwlfh wudglqj urxqg zdv sod|hg lq rughu wr
idploldul}h vxemhfwv zlwk wkh frpsxwhu lqwhuidfh1
LQVWUXFWLRQV
Jhqhudo
\rx duh derxw wr sduwlflsdwh lq dq h{shulphqw lq hfrqrplf ghflvlrq pdnlqj1 Ixqglqj iru wklv h{0
shulphqw kdv ehhq surylghg e| wkh Qdwlrqdo Vflhqfh Irxqgdwlrq1 Sohdvh uhdg wkhvh lqvwuxfwlrqv
fduhixoo|1 Li |rx kdyh dq| txhvwlrqv sohdvh ihho iuhh wr dvn1 Zh dvn wkdw |rx qrw wdon zlwk rqh
dqrwkhu gxulqj wkh h{shulphqw1
Wklv h{shulphqwdo vhvvlrq lv glylghg xs lqwr d qxpehu ri jdphv1 Hdfk jdph frqvlvwv ri d qxpehu
ri urxqgv1
Sduwlflsdqwv lq wklv vhvvlrq kdyh ehhq glylghg xs htxdoo| lqwr rqh ri wkuhh glhuhqw w|shv= w|sh
4/ w|sh 5 ru w|sh 61 \rxu w|sh zdv fkrvhq udqgrpo| dw wkh ehjlqqlqj ri wkh vhvvlrq dqg zloo eh
lqglfdwhg lq wkh wrs ohiw fruqhu ri |rxu frpsxwhu vfuhhq1 \rxu w|sh zloo qrw fkdqjh iru wkh gxudwlrq
ri wkh vhvvlrq1
Lq dgglwlrq wr wkh wkuhh glhuhqw w|shv ri sod|huv/ wkhuh duh irxu glhuhqw w|shv ri jrrgv= jrrg 3/
jrrg 4/ jrrg 5/ dqg jrrg 61
Hdfk sod|hu ehjlqv hdfk jdph zlwk 433 srlqwv1 Lq dgglwlrq/ dw wkh vwduw ri d jdph hdfk sod|hu kdv
hlwkhu rqh xqlw ri jrrg 3 ru rqh xqlw ri wkh jrrg wkdw kh ru vkh _surgxfhv% lq vwrudjh1 Hdfk sod|hu
w|sh surgxfhv d jrrg wkdw lv glhuhqw iurp klv ru khu w|sh=
Sod|hu W|sh
4
5
6
Surgxfhv
Jrrg 5
Jrrg 6
Jrrg 4
Lq wklv vhvvlrq/ 4 sod|hu ri hdfk w|sh vwduwv d jdph zlwk jrrg 3 lq vwrudjh1 Wkh uhpdlqlqj sod|huv
vwduw d jdph zlwk wkh jrrg wkh| _surgxfh% lq vwrudjh1
Sod|huv pd| frqwlqxh wr krog jrrg 3 ru wkh jrrg wkh| surgxfh lq vwrudjh/ ru wkh| pd| h{fkdqjh
wkhvh jrrgv iru rqh xqlw ri wkh jrrg khog lq vwrudjh e| dqrwkhu sod|hu1 H{fkdqjhv ri jrrgv duh
dozd|v rqh iru rqh1
\rxu Remhfwlyh
\rxu remhfwlyh lq hyhu| urxqg ri hyhu| jdph lv wr jhw dv pdq| srlqwv dv srvvleoh1 \rx hduq d
srvlwlyh qxpehu ri srlqwv rqo| zkhq |rx rewdlq wkh jrrg fruuhvsrqglqj wr |rxu w|sh1 Wkxv/ w|sh 4
sod|huv zloo zdqw wr rewdlq dv pdq| xqlwv dv srvvleoh ri jrrg 4/ w|sh 5 sod|huv zloo zdqw wr rewdlq d
pdq| xqlwv dv srvvleoh ri jrrg 5/ dqg w|sh 6 sod|huv zloo zdqw wr rewdlq dv pdq| xqlwv dv srvvleoh
ri jrrg 61 Vwrulqj fhuwdlq jrrgv zloo frvw |rx srlqwv1 Vwrulqj jrrg 3 frvwv |rx 4 srlqw shu urxqg/
vwrulqj jrrg 5 frvwv |rx 56 srlqwv shu urxqg dqg vwrulqj jrrg 6 frvwv |rx 57 srlqwv shu urxqg1
68
Wkhuh lv qr frvw iru vwrulqj jrrg 41 Wkh pruh srlqwv |rx hduq wkh juhdwhu lv |rxu suredelolw| ri
zlqqlqj dq dgglwlrqdo sul}h ri '43 +iru d wrwdo ri '53,1 Vlqfh hdfk sod|hu surgxfhv d jrrg wkdw lv
glhuhqw iurp klv ru khu w|sh/ sod|huv pxvw wudgh wr jhw wkh jrrg wkdw fruuhvsrqgv wr wkhlu w|sh1
Qrwlfh wkdw qr sod|hu ri w|sh m surgxfhv wkh jrrg ghvluhg e| wkh sod|hu w|sh zkr surgxfhv jrrg m1
Wkhuhiruh/ wr jhw wkh jrrg fruuhvsrqglqj wr |rxu w|sh/ |rx pd| kdyh wr hqjdjh lq pruh wkdq rqh
wudgh1
Hdfk jdph frqvlvwv ri d vhtxhqfh ri urxqgv1 Zh zloo qrz ghvfuleh krz d urxqg lv sod|hg1
Wkh Sod| ri D Urxqg
Lq hdfk urxqg/ sod|huv ri doo w|shv duh udqgrpo| sdluhg zlwk rqh dqrwkhu1 Rq wkh uvw vfuhhq wkdw
|rx vhh/ |rx duh wrog wkh w|sh ri wkh sod|hu zlwk zkrp |rx duh pdwfkhg lq wkh fxuuhqw wudglqj
urxqg dqg wkh w|sh ri jrrg wkdw wklv rwkhu sod|hu kdv lq vwrudjh1 \rx duh dovr uhplqghg ri wkh
jrrg |rx fxuuhqwo| krog lq vwrudjh1 Li wkh rwkhu sod|hu djuhhv/ |rx pd| wudgh wkh jrrgv wkdw |rx
erwk kdyh lq vwrudjh1 \rx duh dvnhg=
Gr |rx zdqw wr wudgh |rxu jrrg iru wkh rwkhu sod|hu*v jrrgB +\2Q,=
W|sh \ iru \hv/ wr lqglfdwh wkdw |rx zdqw wr wudgh wkh jrrg |rx fxuuhqwo| kdyh lq vwrudjh iru wkh
jrrg wkh rwkhu sod|hu kdv lq vwrudjh ru w|sh Q iru Qr/ wr lqglfdwh wkdw |rx gr qrw zdqw wr pdnh
wkh wudgh1 \rx zloo wkhq eh dvnhg zkhwkhu |rx zdqw wr fkdqjh |rxu ghflvlrq1 W|sh \ rqo| li |rx
zdqw wr fkdqjh |rxu ghflvlrq1 W|sh Q rwkhuzlvh1
Wkh Rxwfrph ri d Urxqg
Rqfh doo sod|huv kdyh pdgh wkhlu wudglqj ghflvlrqv/ d vhfrqg vfuhhq dsshduv wkdw whoov |rx wkh
rxwfrph ri wkh odvw urxqg1 Wkhuh duh wkuhh srvvleoh rxwfrphv iru hdfk wudglqj urxqg=
41 \rx sursrvhg wr wudgh/ |rxu sursrvdo zdv dffhswhg/ dqg |rx uhfhlyhg wkh jrrg fruuhvsrqglqj
wr |rxu w|sh1 Lq wklv fdvh/ |rx hduq d srvlwlyh qhw sd|r lq srlqwv dv ghwhuplqhg lq Wdeoh 4
ehorz1
W|sh
4
5
6
Wdeoh 4
Srlqwv iru Rewdlqlqj Jrrg
Vwrudjh Frvw ri
Fruuhvsrqglqj wr W|sh
Jrrg Surgxfhg
433
433
433
0
0
0
56
57
3
Qhw Sd|r
@
@
@
:: srlqwv
:9 srlqwv
433 srlqwv
\rx uhfhlyh 433 srlqwv iru rewdlqlqj wkh jrrg wkdw fruuhvsrqgv wr |rxu w|sh plqxv wkh vwrudjh
frvw iru vwrulqj d xqlw ri wkh jrrg |rx surgxfh1 Zkhqhyhu |rx uhfhlyh wkh jrrg fruuhvsrqglqj
wr |rxu w|sh/ |rx lpphgldwho| surgxfh d qhz xqlw ri |rxu surgxfwlrq jrrg vr wkdw dw wkh
hqg ri wkh urxqg/ wkh jrrg |rx kdyh lq vwrudjh lv |rxu surgxfwlrq jrrg1
51 \rx sursrvhg wr wudgh/ |rxu sursrvdo zdv dffhswhg dqg |rx uhfhlyhg lq wudgh d jrrg wkdw grhv
qrw fruuhvsrqg wr |rxu w|sh1 Lq wklv fdvh/ |rx orvh d fhuwdlq qxpehu ri srlqwv fruuhvsrqglqj
wr wkh vwrudjh frvw ri wkh jrrg |rx uhfhlyhg iurp wkh rwkhu sod|hu dv ghwhuplqhg lq Wdeoh 5
dw wkh wrs ri wkh qh{w sdjh1 Wkh jrrg |rx qrz kdyh lq vwrudjh lv wkh jrrg |rx uhfhlyhg iurp
wkh rwkhu sod|hu1
69
Jrrg
3
4
5
6
Wdeoh 5
Vwrudjh Frvw Shu Urxqg
4 srlqw
3 srlqwv
56 srlqwv
57 srlqwv
61 \rx ru wkh sod|hu zlwk zkrp |rx zhuh pdwfkhg fkrvh qrw wr wudgh1 Lq wklv fdvh qr wudgh
rffxuv1 Wkh jrrg |rx qrz kdyh lq vwrudjh grhv qrw fkdqjh iurp wkh suhylrxv urxqg1 \rx orvh
d fhuwdlq qxpehu ri srlqwv fruuhvsrqglqj wr wkh vwrudjh frvw ri wkh jrrg |rx krog lq vwrudjh/
dv ghwhuplqhg e| Wdeoh 51
Qrwh wkdw dw wkh hqg ri hyhu| urxqg ri hyhu| jdph |rx dozd|v kdyh lq vwrudjh rqh xqlw ri d jrrg
wkdw grhv qrw fruuhvsrqg wr |rxu w|sh1 Wkhuhiruh/ lq hyhu| urxqg/ |rx dozd|v orvh d fhuwdlq qxpehu
ri srlqwv fruuhvsrqglqj wr wkh vwrudjh frvw ri wkh jrrg |rx kdyh lq vwrudjh dv ghwhuplqhg e| Wdeoh
51 \rx hduq d srvlwlyh qhw sd|r lq srlqwv/ dv ghwhuplqhg lq Wdeoh 4/ rqo| zkhq |rx rewdlq wkh
jrrg wkdw fruuhvsrqgv wr |rxu w|sh1
Zkhq grhv d jdph frqwlqxh dqg zkhq grhv lw hqgB
Diwhu wkh rxwfrph ri wkh suhylrxv urxqg kdv ehhq uhyhdohg wr doo sod|huv/ d udqgrp qxpehu lv gudzq
iurp wkh lqwhuydo 4 wr 4331 Li wklv udqgrp qxpehu lv ohvv wkdq ru htxdo wr <3/ wkh jdph frqwlqxhv
rq zlwk dqrwkhu urxqg1 Li wkh udqgrp qxpehu lv juhdwhu wkdq <3/ wkh mxvw frpsohwhg urxqg zloo eh
wkh odvw urxqg ri wkh jdph1 Wkxv/ diwhu hyhu| urxqg wkhuh lv d rqh~lq~whq fkdqfh wkdw wkh jdph
zloo qrw frqwlqxh lqwr wkh qh{w urxqg1 Wkh edu fkduw lq wkh plggoh ri wkh wudglqj vfuhhq uhhfwv
wkh suredelolw| wkdw wkh jdph zloo hqg 4~43 urxqgv iurp wkh fxuuhqw urxqg1
Zkhq d jdph hqgv/ |rx zloo vhh d phvvdjh rq |rxu vfuhhq1 \rx zloo eh wrog |rxu srlqw wrwdo iru
wkdw jdph1 Wkhq/ ghshqglqj rq wkh wlph dydlodeoh/ d qhz jdph zloo ehjlq1 \rx zloo vwduw wkh qhz
jdph zlwk 433 srlqwv dqg hlwkhu jrrg }hur ru wkh jrrg |rxu w|sh surgxfhv lq vwrudjh1
Vwudwhjlf Frqvlghudwlrqv
Ehiruh pdnlqj |rxu wudglqj ghflvlrqv/ |rx pd| zdqw wr wdnh dffrxqw ri vrph ri wkh lqirupdwlrq
wkdw lv dydlodeoh wr |rx rq wkh uvw vfuhhq wkdw |rx vhh1
Dw wkh wrs ri wklv vfuhhq/ |rx duh uhplqghg ri |rxu w|sh/ wkh urxqg qxpehu/ dqg |rxu qhz srlqw
wrwdo iru wkh fxuuhqw jdph/ dv ri wkh odvw wudglqj urxqg1 \rx duh dovr uhplqghg wkdw |rx uhfhlyh
433 srlqwv iru wkh jrrg fruuhvsrqglqj wr |rxu w|sh/ dqg wkh frvwv lq srlqwv iru vwrulqj d xqlw ri
hdfk ri wkh irxu jrrgv lq hyhu| urxqg1 \rx pd| zdqw wr wdnh wkhvh vwrudjh frvwv lqwr dffrxqw zkhq
ghflglqj zkhwkhu ru qrw wr wudgh wkh jrrg |rx fxuuhqwo| kdyh lq vwrudjh1
Lq wkh plggoh ri |rxu vfuhhq lv d edu fkduw lqglfdwlqj wkh fxpxodwlyh suredelolw| wkdw wkh jdph zloo
hqg 4 wr 43 urxqgv iurp wkh fxuuhqw urxqg1 Wklv fkduw uhhfwv wkh 4 lq 43 fkdqfh wkdw wkh jdph
zloo hqg iurp rqh urxqg wr wkh qh{w1 Revhuyh wkdw wklv suredelolw| lv lqfuhdvlqj/ lqglfdwlqj wkdw
lw lv lqfuhdvlqjo| olnho| wkh jdph zloo hqg 4 wr 43 urxqgv iurp wkh fxuuhqw urxqg1 Vlqfh wkh jdph
pd| hqg vrrq/ |rx pd| qrw eh deoh wr phhw zlwk d sod|hu zkr lv zloolqj wr wudgh iru wkh jrrg |rx
fxuuhqwo| kdyh lq vwrudjh1 Wkhuhiruh/ lq dgglwlrq wr frqvlghulqj vwrudjh frvwv/ |rx pd| dovr zdqw
wr frqvlghu wkh wlph lw zloo wdnh |rx wr wudgh iru wkh jrrg wkdw fruuhvsrqgv wr |rxu w|sh/ zkhq
fkrrvlqj zkhwkhu ru qrw wr wudgh wkh jrrg |rx fxuuhqwo| kdyh lq vwrudjh1
6:
Dw wkh erwwrp ri |rxu vfuhhq |rx zloo vhh d wdeoh olvwlqj wkh shufhqw ri hdfk w|sh ri sod|hu wkdw
lv vwrulqj hdfk w|sh ri jrrg1 Wkhvh shufhqwdjhv duh dyhudjh shufhqwdjhv ryhu doo urxqgv wkdw kdyh
ehhq sod|hg lq wkh fxuuhqw jdph/ dqg duh xsgdwhg dw wkh hqg ri hyhu| wudglqj urxqg1 \rx pd|
zdqw wr xvh wklv lqirupdwlrq lq pdnlqj dq hvwlpdwh ri krz orqj lw zloo wdnh |rx wr phhw d sod|hu
zkr erwk kdv wkh jrrg fruuhvsrqglqj wr |rxu w|sh/ dqg zkr zloo zdqw wr wudgh lw iru wkh jrrg |rx
fxuuhqwo| kdyh lq vwrudjh/ ru iru vrph rwkhu jrrg |rx frxog jhw lq wudgh1
Hduqlqjv
Doo vxemhfwv zkr frpsohwh wklv <3 plqxwh h{shulphqwdo vhvvlrq duh jxdudqwhhg wr uhfhlyh d '43
sd|phqw1 Ghshqglqj rq krz pdq| srlqwv |rx hduq/ lw lv srvvleoh iru |rx wr hduq dq dgglwlrqdo '43
sul}h iru d wrwdo ri '531
Iroorzlqj wkh odvw jdph ri wkh vhvvlrq/ |rxu srlqw wrwdo iurp rqh jdph/ fkrvhq dw udqgrp iurp doo
ri wkh jdphv wkdw |rx sod|hg/ zloo eh frqyhuwhg lqwr d suredelolw| ri zlqqlqj wkh dgglwlrqdo '43
sul}h1 \rx zloo kdyh d srvlwlyh suredelolw| ri zlqqlqj wkh '43 sul}h li |rxu srlqw wrwdo iurp wkh jdph
fkrvhq h{fhhgv wkh lqlwldo wrwdo ri 433 srlqwv wkdw |rx duh jlyhq dw wkh ehjlqqlqj ri hyhu| jdph1 Li
|rxu srlqw wrwdo lq d jdph idoov ehorz wkh lqlwldo 433 srlqw wrwdo/ |rxu suredelolw| ri zlqqlqj wkh
'43 sul}h lv 3 li wkdw jdph lv wkh rqh fkrvhq dw udqgrp1
\rxu suredelolw| ri zlqqlqj wkh '43 sul}h ghshqgv rqo| rq krz pdq| dgglwlrqdo srlqwv |rx zhuh
deoh wr rewdlq lq wkh jdph fkrvhq uhodwlyh wr wkh pd{lpxp qxpehu ri srlqwv wkdw zhuh rewdlqdeoh
lq wklv vdph jdph iru d sod|hu ri |rxu w|sh1 \rx duh qrw frpshwlqj zlwk rwkhu sod|huv iru wkh '43
sul}h1 Hdfk ri |rx frxog zlq wkh '43 sul}h li |rx hduqhg hqrxjk dgglwlrqdo srlqwv lq wkh jdph
fkrvhq1 Qrwh wkdw hdfk dgglwlrqdo srlqw |rx hduq ryhu wkh 433 lqlwldo srlqwv |rx duh jlyhq lqfuhdvhv
|rxu suredelolw| ri zlqqlqj wkh '43 sul}h e| vdph dprxqw1 Wkxv/ wkh pruh srlqwv |rx hduq lq d
jdph ryhu wkh lqlwldo srlqw wrwdo ri 433/ wkh juhdwhu lv |rxu suredelolw| ri zlqqlqj wkh '43 sul}h/
li wkdw jdph lv fkrvhq dw udqgrp1 Vlqfh |rx gr qrw nqrz dw wkh rxwvhw ri sod| zkdw jdph zloo
eh fkrvhq dw udqgrp/ |rxu remhfwlyh lq hyhu| jdph vkrxog eh wr rewdlq dv pdq| srlqwv dv lv srvvleoh1
DUH WKHUH DQ\ TXHVWLRQV EHIRUH ZH EHJLQB
6;
+iuihi?Uit
Dl|djdul/ V1U1 dqg Q1 Zdoodfh +4<<4,/ _H{lvwhqfh ri vwhdg| vwdwhv zlwk srvlwlyh frqvxpswlrq lq
wkh Nl|rwdnl~Zuljkw prgho/% Uhylhz ri Hfrqrplf Vwxglhv 8;/ <34~<491
Dl|djdul/ V1U1 dqg Q1 Zdoodfh +4<<5,/ _Ildw prqh| lq wkh Nl|rwdnl~Zuljkw prgho/% Hfrqrplf
Wkhru| 5/ 77:~7971
Ehuqdvfrql/ P1 dqg R1 Nlufkndps +5333,/ _Zk| gr prqhwdu| srolflhv pdwwhuB Dq h{shulphqwdo
vwxg| ri vdylqj dqg lqdwlrq lq dq ryhuodsslqj jhqhudwlrqv prgho/% Mrxuqdo ri Prqhwdu|
Hfrqrplfv 79/ 648~6761
Eurzq/ S1P1 +4<<9,/ _H{shulphqwdo hylghqfh rq prqh| dv d phglxp ri h{fkdqjh/% Mrxuqdo ri
Hfrqrplf G|qdplfv dqg Frqwuro 53/ 8;6~9331
Gx|/ M1 +4<<;,/ _Prqhwdu| wkhru| lq wkh oderudwru|/% Ihghudo Uhvhuyh Edqn ri Vw1 Orxlv Uhylhz
;3 +Vhswhpehu2Rfwrehu,/ <~591
Gx|/ M1 dqg M1 Rfkv +4<<<,/ _Hphujhqfh ri prqh| dv d phglxp ri h{fkdqjh= dq h{shulphqwdo
vwxg|/% Dphulfdq Hfrqrplf Uhylhz ;</ ;7:~;::1
Kroodqg/ M1K1 +4<;9,/ _Hvfdslqj eulwwohqhvv= Wkh srvvlelolwlhv ri jhqhudo~sxusrvh ohduqlqj dojr0
ulwkpv dssolhg wr sdudooho uxoh~edvhg v|vwhpv/% lq= U1V1 Plfkdovnl/ M1J1 Fduerqhoo dqg W1P1
Plwfkhoo/ +hgv1,/ Pdfklqh Ohduqlqj= Dq Duwlfldo Lqwhooljhqfh Dssurdfk/ yro1 5/ Orv Dowrv/
FD= Prujdq Ndxipdqq/ 8<6~9561
Nl|rwdnl/ Q1 dqg U1 Zuljkw +4<;<,/ _Rq prqh| dv d phglxp ri h{fkdqjh/% Mrxuqdo ri Srolwlfdo
Hfrqrp| <:/ <5:~871
Oldq/ S1 dqg F1U1 Sorww +4<<;,/ _Jhqhudo htxloleulxp/ pdunhwv/ pdfurhfrqrplfv dqg prqh| lq d
oderudwru| h{shulphqwdo hqylurqphqw/% Hfrqrplf Wkhru| 45/ 54~:81
Olp/ V1V1/ H1F1 Suhvfrww/ dqg V1 Vxqghu +4<<7,/ _Vwdwlrqdu| vroxwlrq wr wkh ryhuodsslqj jhqhudwlrqv
prgho ri dw prqh|= h{shulphqwdo hylghqfh/% Hpslulfdo Hfrqrplfv 4</ 588~5::1
Pdulprq/ U1/ H1 PfJudwwdq dqg W1M1 Vdujhqw +4<<3,/ _Prqh| dv d phglxp ri h{fkdqjh lq dq
hfrqrp| zlwk duwlfldoo| lqwhooljhqw djhqwv/% Mrxuqdo ri Hfrqrplf G|qdplfv dqg Frqwuro 47/
65<~:61
Pdulprq/ U1 dqg V1 Vxqghu +4<<6,/ _Lqghwhuplqdf| ri htxloleuld lq d k|shulqdwlrqdu| zruog=
h{shulphqwdo hylghqfh/% Hfrqrphwulfd 94/ 43:6~443:1
Pdulprq/ U1 dqg V1 Vxqghu +4<<7,/ _H{shfwdwlrqv dqg ohduqlqj xqghu dowhuqdwlyh prqhwdu|
uhjlphv= dq h{shulphqwdo vwxg|/% Hfrqrplf Wkhru| 7/ 464~4951
PfFdeh/ N1D1 +4<;<,/ _Ildw prqh| dv d vwruh ri ydoxh lq dq h{shulphqwdo pdunhw/% Mrxuqdo ri
Hfrqrplf Ehkdylru dqg Rujdql}dwlrq 45/ 548~5641
Rvwur|/ M1P1 dqg U1P1 Vwduu +4<<3,/ _Wkh wudqvdfwlrqv uroh ri prqh|/% Fkdswhu 4 ri E1P1 Iulhgpdq
dqg I1K1 Kdkq +hgv1,/ Kdqgerrn ri Prqhwdu| Hfrqrplfv yro1 4/ Qhz \run= Qruwk~Kroodqg/
6~951
Urwk/ D1H1 dqg P1Z1N1 Pdorxi +4<:<,/ _Jdph wkhruhwlf prghov dqg wkh uroh ri lqirupdwlrq lq
edujdlqlqj= Dq h{shulphqwdo vwxg|/% Sv|fkrorjlfdo Uhylhz ;9/ 8:7~<71
Vkl/ V1 +4<<8,/ _Prqh| dqg sulfhv= d prgho ri vhdufk dqg edujdlqlqj/% Mrxuqdo ri Hfrqrplf Wkhru|
9:/ 79:07<91
Wuhmrv/ D1 dqg U1 Zuljkw +4<<8,/ _Vhdufk/ edujdlqlqj/ prqh| dqg sulfhv/% Mrxuqdo ri Srolwlfdo
Hfrqrp| 436/ 44;~4741
6<
(KIWTG+NNWUVTCVKQPQHVJG6TCFKPI5ETGGPU7UGFKPVJG'ZRGTKOGPV
#6TCFKPI4QWPF5ETGGP
You are a type 2 player. This is Round
Your Point Total:
5.
151.
<RXHDUQSRLQWVSHUXQLWRIJRRG\RXREWDLQ
,WFRVWVSRLQWSHUURXQGIRUVWRULQJJRRG
,WFRVWV0 points per round for storing good 1.
,WFRVWVSRLQWVSHUURXQGIRUVWRULQJJRRG
,WFRVWVSRLQWVSHUURXQGIRUVWRULQJJRRG
You currently have good 0 in storage.
<RXDUHPDWFKHGZLWKDSOD\HURIW\SH
7KLVSOD\HUKDVJRRGLQVWRUDJH
'R\RXZDQWWRWUDGH\RXUJRRGIRUWKHRWKHUSOD\HU
VJRRG"<1 Probability the game will end:
1
0.75
0.5
0.25
0
1
2
3
4
5
6
7
8
9
10
Rounds from now.
3HUFHQWRI(DFK7\SHRI3OD\HU6WRULQJ(DFK7\SHRI*RRG
+LVWRULFDO$YHUDJHDVRI5RXQG
7\SH*RRG*RRG*RRG*RRG
0.97
0.00
0.00
$6TCFKPI4QWPF5ETGGP
You are a type 2 player. This is Round
Your Point Total:
5.
227.
<RXHDUQSRLQWVSHUXQLWRIJRRG\RXREWDLQ
,WFRVWVSRLQWSHUURXQGIRUVWRULQJJRRG
,WFRVWVSRLQWVSHUURXQGIRUVWRULQJJRRG
,WFRVWV 23 points per round for storing good 2.
,WFRVWVSRLQWVSHUURXQGIRUVWRULQJJRRG
,QWKHODVWURXQG\RXKDGJRRGin storage.
<RXZHUHPDWFKHGZLWKDSOD\HURIW\SH
7KLVSOD\HUKDGJRRGLQVWRUDJH
<RXSURSRVHGWRWUDGH\RXUJRRGIRUWKHRWKHUSOD\HU
VJood 2.
7KHRWKHUSOD\HUDJUHHGWRWUDGH
<RXUHFHLYHGDQHWSD\RIIRISRLQWVIRUWKHURXQG
<RXUQew point total is 227.
You now have good 3 in storage.
Percent of Each Type of Player Storing Each Type of Good
+LVWRULFDO$YHUDJHDVRI5RXQG
7\SH*RRG*Rod 1
Good 2
Good 3
0
7KHJDPHZLOOFRQWLQXH3OHDVHZDLWIRUWKHQH[WURXQG
© Copyright 2026 Paperzz