Should a Team Be Homogeneous? - Columbia University

21 downloads 26285 Views 260KB Size Report
Aug 16, 2000 - ities, the team should be composed of agents of the same type, while, with negative ... The organization (be it a firm, a government agency, a.
5L*_ @ Ai@4 i OL4L}i?iLtq ?_hi@ h@|W wL?_L? 5ULL* Lu ,UL?L4Utc A*Mh} N?iht|)c @?_ ,+_ }t| Sc 2fff Devwudfw Vkrxog dq rujdql}dwlrq kluh shrsoh zlwk vlplodu edfnjurxqgv ru zlwk glhuhqw edfnjurxqgvB Zh irupxodwh wklv txhvwlrq zlwklq wkh iudph0 zrun ri whdp wkhru|1 Wkh whdp lv iruphg e| q djhqwv1 Wkh w|sh ri hdfk djhqw lv hqgrjhqrxv dqg ghwhuplqhv klv lqirupdwlrq vwuxfwxuh dqg klv frvw iru wkh whdp1 Zh vkrz wkdw wkh vljq ri frpsohphqwdulw| ehwzhhq mrev ghwhuplqhv zrunirufh krprjhqhlw|1 Zlwk srvlwlyh frpsohphqwdu0 lwlhv/ wkh whdp vkrxog eh frpsrvhg ri djhqwv ri wkh vdph w|sh/ zkloh/ zlwk qhjdwlyh frpsohphqwdulwlhv/ zrunirufh khwhurjhqhlw| lv rswlpdo1 Wkhvh uhvxowv gr qrw uho| rq uhvwulfwlrqv rq wkh zd| xqfhuwdlqw| lv prghohg ru rq wkh ihdvleoh vhw ri djhqw w|shv= wkh| fdq eh h{sodlqhg lq whupv ri fruuhodwlrq ehwzhhq huuruv frpplwwhg e| glhuhqw djhqwv1 Nh|zrugv= zrunirufh krprjhqhlw|/ whdp wkhru|1 MHO Fodvvlfdwlrq= G54/ O53/ P451



W?|hL_U|L?

Wkh edfnjurxqgv ri wkh shrsoh zkr zrun lq dq rujdql}dwlrq duh qrw h{0 rjhqrxvo| jlyhq1 Wkh rujdql}dwlrq +eh lw d up/ d jryhuqphqw djhqf|/ d At t @ hiti_ ihtL? Lu @ U@T|ih Lu 4) # _ttih|@|L? W @i }hi@|*) Mi?i€|i_ uhL4 _tUttL?t | @t@!L L!c ki??i| hhLc @?_ #4|h V@)@?Lt W |@?! a@U^it h‚i4ihc @|@t #i@|hTL?|c aLt@ B@?tc wt B@hU@?Lc AL4@t Oi**4@??c @hULt wtML@c @* *}hL4c #L*u A@*4@?c 5|iu At @?_ @_i?Uit @| LUUL?c W, E@hUi*L?@c L4Ti 6@Mh@c +LUit|ihc 5|@?uLh_c 5|LU!L*4 5ULL* Lu ,UL?L4Utc Ai @t :c A*Mh}c @?_ AL*Lti E,5, bb. uLh |ih i*Tu* UL44i?|t W @4 @*tL }h@|iu* |L @? i_|Lh @?_ |L @?L?)4Lt hiuihii uLh |ih tiu* t}}it|L?t _ __hittG 5AW,+#c w5,c OL}|L? 5|hii|c wL?_L? `2 2,c Nk( ,4@*G @Th@|9*ti@U! OL4i T@}iG ||TG%%iUL?*ti@U!%t|@g%Th@| W

4

qrqsurw/ hwf1, fkrrvhv zkrp wr kluh1 Wklv sdshu lv frqfhuqhg zlwk rqh glphqvlrq ri wkh klulqj srolf|= wkh ghjuhh ri ydulhw| lq wkh edfnjurxqgv ri wkh shrsoh zkr duh kluhg1 Rq wklv glphqvlrq/ glhuhqw rujdql}dwlrqv fdq dgrsw vwulnlqjo| glhuhqw srolflhv1 Vrph rujdql}dwlrqv sxuvxh d srolf| ri klulqj shrsoh zlwk krpr0 jhqhrxv edfnjurxqgv/ zkloh rwkhu rujdql}dwlrqv dfwlyho| vhhn d ghjuhh ri edfnjurxqg glyhuvlw| lq wkhlu zrunirufh1 Zkdw fkdudfwhulvwlfv ri dq rujdql0 }dwlrq ghwhuplqh lwv rswlpdo ghjuhh ri edfnjurxqg krprjhqhlw|B Fohduo|/ wklv lv d frpsoh{ txhvwlrq dqg rqh frxog dssurdfk lw iurp vhyhudo dqjohv1 Prwlydwlrqdo idfwruv/ lqfhqwlyh lvvxhv/ wkh qhhg iru vhfuhf| doo sod| d uroh lq wklv fkrlfh1 Wklv sdshu/ krzhyhu/ zloo uhvwulfw lwv dwwhqwlrq wr rqh idfwru= lqirupdwlrqdo hflhqf|1 Hyhq zkhq h{dplqhg lq lvrodwlrq/ lqirupdwlrqdo hflhqf| frqvwlwxwhv d kljko| frpsoh{ sureohp1 Fuhphu ^;/ idovh lq vrffhu ehfdxvh ri wkh jrdonhhshu*v vshfldo vwdwxv,1 Dq lqwulqvlfdoo| dv|p0 phwulf sd|r ixqfwlrq zrxog fohduo| eldv uhvxowv lq idyru ri khwhurjhqhrxv djhqw w|shv12 Zlwk wklv dvvxpswlrq/ zh suryh wkh wzr fhqwudo uhvxowv ri wkh sdshu1 Iluvw/ li wkh djhqwv* dfwlrqv duh frpsohphqwv lq wkh sd|r ixqfwlrq/ wkhq wkh vhw ri rswlpdo vroxwlrqv frqwdlqv d vroxwlrq lq zklfk doo djhqwv kdyh wkh vdph w|sh1 Lq wkdw fdvh/ wkh whdp sureohp uhgxfhv wr orrnlqj dw frqjxudwlrqv zlwk rqo| rqh w|sh ri djhqwv1 Vhfrqg/ li wzr dgglwlrqdo dvvxpswlrqv krog +frqfdylw| dqg qrqxqltxhqhvv ri wkh vlqjoh0w|sh rswlpd,/ zh suryh wkdw/  W?

UL?|h@t| |L 4Lt| Lu |i hiUi?| iUL?L4U *|ih@|hi L? Lh}@?3@|L?tc |t T@Tih t ?L| _hiU|*) UL?Uih?i_ | ?Ui?|i ttit `i @tt4i |@| |ihi t ?L 4Lh@* @3@h_ L? |i T@h| Lu @}i?|tG u @? @}i?| t }i? @ _iUtL? u?U|L?c i ** uL**L | 2 ? i@h*ih ihtL? Lu |t T@Tih tLt |@| |i 4@? hit*|t L*_ @*tL ? @ T@h|U*@h @t)44i|hU i?hL?4i?| Eh@| d22o

6

?? ??  Lt|*) W?uLh4@|L? 5|hU|hi uLh }i?|  t}?@* -1 }i?| 

5|@|i Lu |i `Lh*_

   Lt|*) W?uLh4@|L? 5|hU|hi uLh }i?| 2 / / t}?@*

 @U|L?

! +BW~AW " v66

@U|L?

/ ) / }i?| 2

Iljxuh 4= Dq Rujdql}dwlrq zlwk Wzr Djhqwv li wkh djhqwv* dfwlrqv duh vxevwlwxwhv lq wkh sd|r ixqfwlrq/ wkhq wkh vhw ri rswlpdo vroxwlrq frqwdlqv d vroxwlrq zlwk dw ohdvw wzr glvwlqfw w|shv ri djhqwv1 Frpsohphqwdulwlhv duh uhsuhvhqwhg wkurxjk wkh odwwlfh0wkhruhwlf qrwlrqv ri vxshuprgxodu dqg vxeprgxodu ixqfwlrqv +Wrsnlv ^58`/ Ylyhv ^5:`/ Plojurp dqg Urehuwv ^4:`,1 Wkh pdlq dgydqwdjh ri odwwlfh wkhru| lv wkdw lwv ghqlwlrqv duh dssolfdeoh dovr zkhq frqwlqxlw| dqg glhuhqwldelolw| duh qrw dvvxuhg/ zklfk lv wkh fdvh lq wklv sdshu1 Pruhryhu/ surriv duh pdgh vlpsohu dqg vwdwhphqwv pruh lqwxlwlyh1 Lqghhg/ wkh lqwxlwlrq iru wkh pdlq uhvxowv iru wklv sdshu lv fdswxuhg e| wkh iroorzlqj odwwlfh0wkhruhwlf uhvxow/ dovr suryhq lq wkh sdshu1 Frqvlghu d ixqfwlrq wkh dujxphqw ri zklfk lv d yhfwru ri udqgrp yduldeohv1 Zh vkrz wkdw li wkh ixqfwlrq lv vxshuprgxodu/ wkhq wkh h{shfwhg ydoxh ri wkh ixqfwlrq lv kljkhu li wkh udqgrp yduldeohv duh shuihfwo| fruuhodwhg udwkhu wkdq vwrfkdvwlfdoo| lqghshqghqw1 Rq wkh rwkhu kdqg/ li wkh ixqfwlrq lv vxeprgxodu/ wkh h{shfwhg ydoxh lv kljkhu li wkh udqgrp yduldeohv duh vwrfkdvwlfdoo| lqghshqghqw1 Lq wkh sureohp dw kdqg/ djhqwv gr qrw lq jhqhudo kdyh shuihfw lqirupdwlrq1 Wkxv/ wkh| duh erxqg wr ghyldwh iurp wkh ixoo0lqirupdwlrq vroxwlrq1 Li wkhlu dfwlrqv duh frpsohphqwv/ lw lv rswlpdo iru wkhp wr ghyldwh lq d frruglqdwhg pdqqhu/ zklfk rffxuv li wkhlu lqirupdwlrq vwuxfwxuhv duh lghqwlfdo1 Wkxv/ kdylqj djhqwv ri wkh vdph w|sh lv rswlpdo1 Dq dqdorjrxv olqh ri uhdvrqlqj fdq eh iroorzhg zkhq wkh djhqwv* dfwlrqv duh vxevwlwxwhv1 7

Dv wkh lqwxlwlrq lv jhqhudo/ rqh zrxog h{shfw wkh uhvxowv wr krog lq d yhu| jhqhudo vhwwlqj1 Lqghhg wkh| gr1 Lq wklv sdshu/ wkdqnv wr wkh xvh ri odwwlfh wkhru|/ qr sduwlfxodu ixqfwlrqdo irup lv dvvxphg iru wkh sd|r ixqfwlrq1 Pruhryhu/ wkh vhw ri srvvleoh vwdwhv ri wkh zruog dqg wkh suredelolw| glvwulexwlrq rq lw duh ghqhg lq d jhqhudo zd|1 Ilqdoo|/ qr dvvxpswlrq lv pdgh rq wkh vhw ri djhqwv* w|sh/ rq lqirupdwlrq vwuxfwxuhv/ ru rq ghflvlrq ixqfwlrqv1 Wkh uhvxowv duh looxvwudwhg e| wzr h{dpsohv1 Lq wkh uvw/ d surgxfw lv pdgh ri wzr frpsrqhqwv surgxfhg e| wzr glhuhqw glylvlrqv ri wkh vdph up1 Zh vkrz wkdw wkh suhvhqfh ri frpsohphqwdulwlhv lq wkh surgxfwlrq ixqfwlrq sxvkhv wkh whdp wr kdyh glylvlrq pdqdjhuv zlwk wkh vdph w|sh1 Wkh vhfrqg h{dpsoh lv d vhdufk sureohp zlwk wzr uhvhdufkhuv fkrrvlqj lq zklfk gluhfwlrq wr ghyrwh wkhlu vhdufk hruwv1 Zh vkrz wkdw wkh whdp remhfwlyh lv vxeprgxodu dqg khqfh uhvhdufkhuv vkrxog kdyh glhuhqw w|shv1 Wkh sodq ri wkh sdshu lv dv iroorzv1 Vhfwlrq 5 lqwurgxfhv wkh prgho1 Vhfwlrq 6 uhsruwv wkh pdlq uhvxowv1 Vhfwlrq 7 surylghv lqwxlwlrq iru wkh pdlq uhvxowv edvhg rq wkh lghd ri huuru fruuhodwlrq1 Vhfwlrq 8 frqfoxghv1 Uhodwhg Olwhudwxuh Dv vwdwhg deryh/ zh dgrsw wkh whdp0wkhruhwlfdo iudph0 zrun1 Whdp wkhru| zdv ghyhorshg lq wkh Vl{wlhv e| Pduvfkdn dqg Udg0 qhu ^48`1 Lq wkh Vhyhqwlhv lw jdyh ulvh wr d olwhudwxuh rq wkh srvvlelolw| ri ghfhqwudol}lqj ghflvlrq0pdnlqj/ vxfk dv Juryhv dqg Udgqhu ^45` dqg Duurz dqg Udgqhu ^7`1 Diwhu pruh wkdq d ghfdgh ri olplwhg xvh ~ zklfk frlq0 flghg zlwk wkh ghyhorsphqw ri sulqflsdo0djhqw wkhru| ~ whdp wkhru| kdv ehhq h{shulhqflqj d uhqhzhg lqwhuhvw1 Vhyhudo dxwkruv kdyh dssolhg lw wr sureohpv lq rujdql}dwlrq wkhru| wkdw gr qrw vhhp wr qg d vdwlvidfwru| dqvzhu zlwklq wkh sulqflsdo0djhqw iudphzrun1 H{dpsohv duh Drnl ^4/ 5`/ Fuhphu ^:/ ;/ d? > {,

+4,

zkhuh d 5 D uhsuhvhqwv wkh dfwlrq wdnhq e| djhqw l1 D lv dq rughuhg vhw1 Qr dvvxpswlrq lv pdgh rq wkh irup ri $ h{fhsw wkh iroorzlqj= Dvvxpswlrq 4 +V|pphwu|, Iru doo { 5 [/ $+d > = = = > d? > {, @    @ $+d? > = = = > d > {,

+5,

iru dq| uhduudqjhphqw ri wkh dfwlrq lqglfhv1 Djhqw l lv hqgrzhg zlwk dq lqirupdwlrq vwuxfwxuh  5 / zkhuh  = [ $ \ 1 Zh zloo riwhq uhihu wr dq djhqw*v lqirupdwlrq vwuxfwxuh dv klv _w|sh%1 Wkh dvvxpswlrq wkdw \ grhv qrw ghshqg rq  lv zlwkrxw orvv ri jhqhudolw|1D Wkh ixqfwlrq  lqgxfhv d sduwlwlrq Sw rq wkh vhw [1 Wklv fruuhvsrqgv wr wkh vwdqgdug ghqlwlrq ri lqirupdwlrq vwuxfwxuh +vhh Pduvfkdn dqg Udgqhu ^48/ s1 7;07 = = = > ? +? +{,,> {`j 

cc?

? [ 



fw

Lq wkh uhpdlqghu ri wkh sdshu/ zh dvvxph wkdw wkh whdp sureohp kdv dw ohdvw rqh vroxwlrq1 D

5TTLti |@| tw V _i?L|it |i ti| Lu TLttM*i t}?@*t hiUii_ M) @? @}i?| Lu |)Ti w `i t| ?ii_ |L *i| t ' wMX tw 

:



Bi?ih@* +it*|t #i€??} L4T*i4i?|@h|it

Wr uhsuhvhqw frpsohphqwdulwlhv/ zh dgdsw wkh jhqhudo ghqlwlrq ri vxshu0 prgxodu dqg vxeprgxodu ixqfwlrqv wr wkh sureohp dw kdqg +Vhh iru lqvwdqfh Plojurp dqg Vkdqqrq ^4 = = = > d ? , 5 D? dqg iru doo li/ iru dq| wzr yhfwruv +a d > = = = > d { 5 [/ wkh iroorzlqj krogv a? > {, . $+ d > = = = > d ? > {, $+a d > = = = > d  ,> = = = > plq+a d? > d ? ,> {` . $^pd{+a d > d  ,> = = = > pd{+a d? > d ? ,> {`=  $^plq+a d > d Frqyhuvho|/ $ lv vxeprgxodu lq wkh djhqwv* dfwlrqv li/ jlyhq dq| wzr yhfwruv a? , 5 D? dqg + d > = = = > d ? , 5 D? / iru doo { 5 [/ wkh iroorzlqj krogv +a d > = = = > d a? > {, . $+ d > = = = > d ? > {, $+a d > = = = > d  $^plq+a d > d  ,> = = = > plq+a d? > d ? ,> {` . $^pd{+a d > d  ,> = = = > pd{+a d? > d ? ,> {`= Krz grhv vxshuprgxodulw| uhodwh wr wkh qrwlrq ri frpsohphqwdulw| edvhg rq furvv0ghulydwlyhvB Wkh odwwhu ghqlwlrq lv dssolfdeoh rqo| li wkh ixqf0 wlrq lv wzlfh0glhuhqwldeoh1 Wrsnlv ^58/ Wk1 615` vkrzv wkdw/ li d ixqf0 wlrq lv wzlfh0glhuhqwldeoh/ wkhq wkh ixqfwlrq lv vxshuprgxodu li dqg rqo| li wkh vhfrqg0rughu furvv ghulydwlyhv duh doo qrqqhjdwlyh +lq wkh suhvhqw fdvh= C 2 $@Cd Cd  3 iru l 9@ m,/ zkloh wkh ixqfwlrq lv vxeprgxodu li dqg rqo| li wkh furvv ghulydwlyhv duh doo qrqsrvlwlyh1 Wkxv/ vxshuprgxodulw| lv d jhqhu0 dol}dwlrq ri wkh wudglwlrqdo qrwlrq ri frpsohphqwdulw|1 Lwv xvh ghulyhv iurp wkh idfw wkdw lq pdq| sureohpv wkh vhfrqg0rughu furvv ghulydwlyh lv qrw zhoo0 ghqhg +iru lqvwdqfh/ ehfdxvh wkh djhqw*v dfwlrq lv d glvfuhwh yduldeoh,1 Wkh iroorzlqj uhvxow +suryhq lq wkh dsshqgl{, zloo eh xvhg uhshdwhgo| lq wkh sdshu1 Ohppd 4 Jlyhq dq rughuhg vhw D dqg d yhfwru d @ +d > = = = > d? , 5 D? / ghqh S +d, dv wkh vhw ri doo yhfwruv rewdlqhg e| shupxwlqj hohphqwv ri d1 Frqvlghu i = D? $ ?1 Li i lv vxshuprgxodu +vxeprgxodu,/ wkhq 4 q$ ER

[  ccR? 

M E@

i+s > = = = > s? ,  +,

;

4 [ i+d > = = = > d , q 'cc?

Frqvlghu d ixqfwlrq/ wkh dujxphqwv ri zklfk duh doo ghqhg rq wkh vdph rughuhg vhw1 Wdnh d sduwlfxodu yhfwru ri dujxphqwv1 Li wkh ixqfwlrq lv vxshu0 prgxodu/ wkhq wkh dyhudjh ydoxh ri wkh ixqfwlrq iru doo srvvleoh shupxwdwlrqv ri wkh yhfwru lv vpdoohu ru htxdo wr wkh dyhudjh ydoxh ri wkh ixqfwlrq ri yhf0 wruv lq zklfk doo hohphqwv duh htxdo wr rqh ri wkh dujxphqwv ri wkh lqlwldo yhfwru1 Wkh frqyhuvh krogv li wkh ixqfwlrq lv vxeprgxodu1 Wkh iroorzlqj lv lpphgldwh=S Fruroodu| 4 Vxssrvh i lv vxfk wkdw i+d > = = = > d? , @    @ i+d? > = = = > d , iru dq| uhduudqjhphqw ri wkh yhfwru d1 Li i lv vxshuprgxodu +vxeprgxodu,/ wkhq i+d > = = = > d? ,  +,

2

4 [ i+d > = = = > d , q 'cc?

5Ui?| L?_|L? uLh |i T|4@*|) Lu `Lh!uLhUi OL 4L}i?i|)

Lw lv qrz srvvleoh wr vwdwh wkh pdlq uhvxow ri wklv sdshu1 Surylghg wkdw wkh whdp*v sureohp kdv dw ohdvw rqh rswlpdo vroxwlrq dqg surylghg wkdw Dvvxpswlrq 4 krogv/ zh kdyh wkh iroorzlqj= Sursrvlwlrq 4 Li $ lv vxshuprgxodu lq wkh djhqwv* dfwlrqv/ wkhq wkh vhw ri rswlpdo vroxwlrqv wr wkh whdp sureohp frqwdlqv dw ohdvw rqh frqjxudwlrq lq zklfk  @ = = = @ ? 1 Surri Vxssrvh $ lv vxshuprgxodu lq wkh djhqwv* dfwlrqv1 Zh zloo suryh wkdw/ iru dq| frqjxudwlrq lq zklfk qrw doo djhqwv kdyh wkh vdph w|sh/ zh fdq qg d frqjxudwlrq zklfk jlyhv d juhdwhu ru htxdo h{shfwhg sd|r dqg lq zklfk doo djhqwv kdyh wkh vdph w|sh1 Frqvlghu d ihdvleoh fkrlfh ri w|shv/ ghqrwhg e|  > = = = > ? / dqg d ihdvleoh fkrlfh ri ghflvlrq ixqfwlrqv/ ghqrwhg e|  +,> = = = > ? +,1 Qrwlfh wkdw $^ + +{,,> = = = > ? +? +{,,> {` S

LhL**@h)  @t Mii? ThLi? ? @ _hiU| @) M) i)ih @?_ LL!ihii dSc hLTLt|L? o AL |i Mit| Lu 4) !?L*i_}ic wi44@  U t Lu ?_iTi?_i?| ?|ihit| t ?i Ai LhL**@h) t tUi?| |L ThLi |i hit*|t ? 5iU|L?t 2 @?_   Ai wi44@ t ?ii_i_ uLh 5iU|L? e


= = = > ? +? +{,, iru doo { 5 [1 E| Fruroodu| 4 dqg Dvvxpswlrq 4/ zh kdyh wkdw/ iru doo { 5 [/ 4 i$^ + +{,,> = = = >  + +{,,> {` q . = = = . $^? +? +{,,> = = = > ? +? +{,,> {`j  $^ + +{,,> = = = > ? +? +{,,> {` Wdnh h{shfwdwlrqv ryhu wkh vhw ri srvvleoh vwdwhv dqg dgg wkh wrwdo frvw ri zdjhv rq erwk vlghv 4 +Hi$^ + +{,,> = = = >  + +{,,> {`j . qfw q . = = = . Hi$^? +? +{,,> = = = > ? +? +{,,> {`j . qfw? ,  Hi$^ + +{,,> = = = > ? +? +{,,> {`j .

? [ 



fw

Wkhq iru dw ohdvw rqh ri wkh w|shv/ vd| & / Hi$^& +& +{,,> = = = > & +& +{,,> {`j . qfw&  Hi$^ + +{,,> = = = > ? +? +{,,> {`j .

? [ 



fw

Wkxv/ iru dq| ihdvleoh fkrlfh ri w|shv dqg ri ghflvlrq ixqfwlrqv/ wkhuh h{0 lvwv d ihdvleoh fkrlfh ri w|shv dqg ghflvlrq ixqfwlrqv lq zklfk  @ = = = @ ? zkr grhv dw ohdvw dv zhoo1 Wkhq/ li wkh vhw ri vroxwlrqv wr wkh whdp sure0 ohp lv qrw hpsw|/ dv zh kdyh dvvxphg/ lw pxvw frqwdlq d vroxwlrq lq zklfk  @ = = = @ ? 1 Wkh lqwxlwlrq ehklqg Sursrvlwlrq 4 lv wkdw li wkh sd|r ixqfwlrq lv vx0 shuprgxodu wkhq wkh whdp lv ehwwhu r li djhqwv frpplw fruuhodwhg huuruv udwkhu wkdq xqfruuhodwhg huuruv1 Zh zloo h{soruh wklv wkhph lq Vhfwlrq 71. Sursrvlwlrq 4 vd|v wkdw/ dprqj doo wkh srvvleoh vroxwlrqv wr wkh whdp*v sureohp/ wkhuh lv dw ohdvw rqh lq zklfk doo djhqwv kdyh wkh vdph w|sh1 Lw . 5TTLti ?t|i@_ |@| |i T@)Lg u?U|L? t t|hU|*) tTih4L_*@hc |@| tc |@|  < hiT*@Uit $< ? |i _i€?|L? Lu tTih4L_*@h u?U|L? }i? ? 5iU|L? e Ai?c | t i@t) |L tii |@|c uLh @?) tL*|L? ? U w '    ' w? _Lit ?L| L*_c | t @*4Lt| @*@)t TLttM*i |L €?_ @| *i@t| L?i tL*|L? ? U w '    ' w? @?_ U )i*_t @ t|hU|*) }ih i TiU|i_ T@)Lg Ai?c |i ti| Lu LT|4@* tL*|L?t ** }i?ihU@**) UL?|@? L?*) UL?€}h@|L?t ? U w '    ' w? 

43

grhv qrw lghqwli| zklfk/ dprqj doo wkh vroxwlrqv ri wkdw nlqg/ lv wkh rswlpdo rqh1 Krzhyhu/ d sursrvlwlrq olnh 4 juhdwo| vlpsolhv wkh wdvn ri ghvljqlqj dq rswlpdo rujdql}dwlrqdo vwuxfwxuh1 Wkh rujdql}dwlrq ghvljqhu fdq/ zlwkrxw orvv ri jhqhudolw|/ irfxv rq vroxwlrqv lq zklfk doo djhqwv kdyh wkh vdph w|sh1 Iru h{dpsoh/ li wkhuh duh 53 djhqwv dqg 7 lqirupdwlrq vwuxfwxuhv/ wkh qxpehu ri srvvleoh frqjxudwlrqv ~ frxqwlqj v|pphwulf vwuxfwxuhv rqo| rqfh ~ lv ;;881 Krzhyhu/ e| dsso|lqj Sursrvlwlrq 4/ wkh rujdql}dwlrq ghvljqhu nqrzv wkdw wkh qxpehu ri frqjxudwlrqv vkh qhhgv wr fkhfn lv mxvw 71 Wkh iroorzlqj h{dpsoh looxvwudwhv wkh xvh ri Sursrvlwlrq 4= H{dpsoh 4 +D Surgxfw Pdgh ri Wzr Frpsrqhqwv,= Frqvlghu d up pdgh ri wzr glylvlrqv1 Djhqw 4 lv wkh pdqdjhu ri Glylvlrq 4 dqg Djhqw 5 lv wkh pdqdjhu ri Glylvlrq 51 Wkh qdo surgxfw ri wkh up lv rewdlqhg e| dvvhpeolqj d frpsrqhqw surgxfhg e| Glylvlrq 4 dqg d frpsrqhqw surgxfhg e| Glylvlrq 51 Djhqw 4 ghflghv d / wkh txdqwlw| surgxfhg e| Glylvlrq 4/ dqg Djhqw 5 ghflghv d2 / wkh txdqwlw| surgxfhg e| Glylvlrq 51 Ehfdxvh hdfk surg0 xfw qhhgv erwk frpsrqhqwv/ wkh qxpehu ri lwhpv surgxfhg lv plq+d > d2 ,1 Wkh up idfhv dq lqhodvwlf ghpdqg fxuyh1 Lw fdq vhoo xs wr { surgxfwv dw d xqlw sulfh s1 Li lw surgxfhv pruh wkdq { surgxfwv/ wkh h{fhvv zloo eh xqvrog1 Wkhuhiruh/ wkh qxpehu ri surgxfwv vrog lv wkh plqlpxp ehwzhhq wkh qxpehu ri surgxfwv surgxfhg dqg wkh qxpehu ri surgxfwv ghpdqghg= plq+d > d2 > {,1 Ghpdqg ghshqgv rq wkh vwdwh ri wkh zruog uhsuhvhqwhg e| wkh uhdo udqgrp yduldeoh { zlwk d jlyhq suredelolw| glvwulexwlrq s+{,1 Wkh xqlw frvw lv wkh vdph iru erwk frpsrqhqwv= n +ohw xv dvvxph wkdw n ? 3=8s,1 Wkh sd|r ixqfwlrq ri wkh up lv $+d > d2 > {, @ s plq+d > d2 > {,  n+d . d2 ,

+6,

Wkh vwuxfwxuh ri wkh sureohp vxjjhvwv wkdw wkh djhqwv* dfwlrqv duh frp0 sohphqwv1 Lqghhg/ lw fdq eh yhulhg wkdw wkh sd|r ixqfwlrq +6, lv vxshu0 prgxodu lq d dqg d2 iru doo { +Vhh wkh Dsshqgl{ iru d irupdo yhulfdwlrq,1 Wkhuhiruh/ zh fdq dsso| Sursrvlwlrq 4= wkh vhw ri rswlpdo vroxwlrqv frqwdlqv dw ohdvw rqh vroxwlrq lq zklfk Djhqw 4 dqg Djhqw 5 kdyh wkh vdph lqirupd0 wlrq vwuxfwxuh1 Wklv uhvxow lv lqghshqghqw ri wkh glvwulexwlrq ri { dqg ri wkh ihdvleoh vhw ri w|shv1

44



5Ui?| L?_|L? uLh |i T|4@*|) Lu `Lh!uLhUi Oi| ihL}i?i|)

Wklv vxevhfwlrq suhvhqwv d sduwldo sdudooho ri Sursrvlwlrq 4 iru zrunirufh khwhurjhqhlw|1 Wzr dgglwlrqdo dvvxpswlrqv duh qhfhvvdu|1 Iluvw/ Dvvxpswlrq 5 +Frqfdylw| ri wkh Sd|r Ixqfwlrq, Wkh vhw D lv frq0 yh{ dqg wkh sd|r ixqfwlrq $ lv frqfdyh lq d > d2 > = = = > d? 1 Dvvxpswlrq 5 jxdudqwhhv wkdw wkh whdp lv ulvn0dyhuvh1 D ulvn0orylqj pdqdjhu pd| zdqw wr kluh krprjhqhrxv djhqwv lq rughu wr frruglqdwh rq ulvnlhu dfwlrqv1 Wkh vhfrqg dgglwlrqdo dvvxpswlrq h{foxghv vlwxdwlrqv lq zklfk rqh w|sh ri zrunhu lv vxshulru wr doo rwkhu w|shv= Ghqlwlrq 5 D rqh0w|sh rswlpxp lv d vroxwlrq wr pd{

t 

w W c kW  'cc?

Hi$^W +W +{,,> = = = > W? +W +{,,> {`j  qW

Dvvxpswlrq 6 +Qrqxqltxhqhvv ri Rqh0W|sh Rswlpd, Wkhuh h{lvw wzr glvwlqfw ydoxhv  W dqg WW dqg wzr vhwv ri ghflvlrq ixqfwlrqv iW j'cc? dqg j'cc? vxfk wkdw +W > iW j'cc? , dqg WW > iWW j'cc? , duh erwk rqh0 iWW   w|sh rswlpd1 Dvvxpswlrq 6 frqvlghuv d uhvwulfwhg sureohp1 Vxssrvh wkh whdp fdq rqo| kluh djhqwv ri rqh w|sh= zklfk w|sh ri djhqwv zrxog lw kluhB Wkh dvvxpswlrq uhtxluhv wkdw wkhuh duh dw ohdvw wzr rswlpdo w|shv1 Zlwkrxw wklv dvvxpswlrq/ lw frxog eh wkh fdvh wkdw d w|sh ri djhqw lv vwulfwo| cehwwhu* wkdq wkh rwkhuv1 Wklv zrxog lpso| wkdw zrunirufh krprjhqhlw| lv rswlpdo lq d wulyldo zd|1 Rq wkh frqwudu|/ lq uhdolw|/ wkh oderu vxsso| lv khwhurjhqhrxv dqg/ iru dq| jlyhq suroh/ lw frpsulvhv vhyhudo w|shv ri zrunhuv/ qrqh ri zklfk fohduo| grplqdwhv wkh rwkhu1H Ri frxuvh/ Dvvxpswlrqv 5 dqg 6 gr qrw lpso| wkdw khwhurjhqhlw| lv wkh rs0 wlpdo vroxwlrq1 Lq sduwlfxodu/ li wkh sd|r ixqfwlrq lv vwulfwo| vxshuprgxodu/ wkhq doo rswlpdo frqjxudwlrqv vwloo uhtxluh ixoo krprjhqhlw|/ dv suhglfwhg e| Sursrvlwlrq 41 Zlwk Dvvxpswlrqv 4 wkurxjk 6/ wkh iroorzlqj krogv= H 6Lh ?t|@?Uic u @ _iT@h|4i?| @?|t |L hi @ u@U*|)c | U@? ULLti uhL4 }h@_@|it Lu @hLt }h@_@|i tULL*t Aihi ** Mi tiih@* tULL*t | t4*@h h@?!?}t OLiihc |? |i ti| Lu tULL*t ? |i t@4i h@?!?}c |ihi 4@) Mi *@h}i _gihi?Uit ? |ih4t Lu uLUt Lh t|)*i

45

Sursrvlwlrq 5 Li $ lv vxeprgxodu/ wkhq wkh vhw ri rswlpdo vroxwlrqv frq0 wdlqv dw ohdvw rqh vroxwlrq lq zklfk lw lv qrw wuxh wkdw  @ = = = @ 2 1 Wr looxvwudwh wkh vfrsh ri Sursrvlwlrq 5/ frqvlghu wkh iroorzlqj= H{dpsoh 5 +D Vhdufk Sureohp,= Frqvlghu d whdp ri wzr uhvhdufkhuv= 4 dqg 51 Wkhuh duh wzr srvvleoh hogv ri uhvhdufk= Ohiw dqg Uljkw1 Rqo| rqh ri wkh hogv lv surplvlqj/ exw wkh uhvhdufkhuv gr qrw nqrz zklfk1 Ohw } eh d udqgrp yduldeoh zklfk fdq dvvxph wkh ydoxhv 3 dqg 4 zlwk htxdo suredelolw|1 Li } @ 3/ Ohiw lv surplvlqj1 Li } @ 4/ Uljkw lv surplvlqj1 Hdfk djhqw zrunv iru d xqlwdu| dprxqw ri wlph1 Djhqw l fkrrvhv d 5 ^3> 4`/ zklfk uhsuhvhqwv wkh shufhqwdjh ri klv zrun wlph wkdw kh ghyrwhv wr vhdufklqj Ohiw1 Wkh shufhqwdjh ri wlph wkdw kh ghyrwhv wr vhdufklqj Uljkw lv jlyhq e| 4  d 1 Zlwklq d uhvhdufk hog/ wkhuh duh ghfuhdvlqj uhwxuqv wr vfdoh1 Iru lq0 vwdqfh/ dvvxph wkdw wkh suredelolw| ri vxffhvv lv sursruwlrqdo wr wkh vtxduh urrw ri wkh wrwdo uhvhdufk wlph vshqw lq wkdw hog1 Li wkh uhvhdufk lv vxf0 fhvvixo/ wkh whdp uhfhlyhv d sd|r ri T1 Wkh whdp*v sd|r fdq eh zulwwhq dv= s s 5  d  d2 d . d2 s . T+4  }, s +7, $+d > d2 > }, @ T} 5 5 s zkhuh wkh ghqrplqdwru 5 lv xvhg wr qrupdol}h wkh suredelolwlhv1 Vxssrvh wkdw wkhuh duh wzr w|shv ri uhvhdufkhuv= wkrvh hgxfdwhg lq xql0 yhuvlw| K +zlwk lqirupdwlrq vwuxfwxuh  @ k, dqg wkrvh hgxfdwhg lq xqlyhu0 vlw| N +zlwk  @ n,1 Erwk w|shv revhuyh | zlwk vrph huuru/ exw wkh huuruv duh xqfruuhodwhg dfurvv w|shv1 Ohw wkh vwdwh ri wkh zruog eh { @ +|> M > g ,/ zkhuh M dqg g duh lqghshqghqwo| xqlirupo| glvwulexwhg rq ^3> 4`1 Wkh wzr srvvleoh lqirupdwlrq vwuxfwxuhv duh ghqhg dv iroorzv= +

k+}, @ dqg

+

n+}, @

} 4}

li M  s li M  s

} 4}

li g  s li g  s

zkhuh s 5 +3=8> 4` ghqrwhv wkh suhflvlrq ri wkh vljqdo1 Lw lv hdv| wr fkhfn wkdw wkh ixqfwlrq lq +7, lv vxeprgxodu lq d dqg d2 1 Pruhryhu/ Dvvxpswlrqv 5 dqg 6 krog dv zhoo1 Wkhuhiruh/ e| Sursrvlwlrq 5/ wkh vhdufk sureohp ghvfulehg khuh dozd|v kdv dq rswlpdo vroxwlrq lq zklfk rqh djhqw lv ri w|sh k dqg wkh rwkhu lv ri w|sh n1 46

Vhdufk sureohpv w|slfdoo| hqwdlo vxeprgxodu sd|r ixqfwlrqv1 Wkh pruh rqh djhqw vhdufkhv lq d gluhfwlrq/ wkh pruh wkh rwkhu djhqwv vkrxog vhdufk lq rwkhu gluhfwlrqv1 Ri frxuvh/ wkh whdp fdq dozd|v rughu wkh djhqwv wr vsuhdg htxdoo| rq doo srvvleoh gluhfwlrqv1 Krzhyhu/ wklv ghflvlrq ixqfwlrq lv fohduo| qrw rswlpdo ehfdxvh lw grhv qrw wdnh lqwr dffrxqw wkh djhqwv* vljqdov1 Wkh ehvw wklqj wkh whdp ghvljqhu fdq gr lv wr kluh djhqwv zkr uhfhlyh xqfruuhodwhg vljqdov/ vr wkdw djhqwv fdq vsuhdg rq glhuhqw gluhfwlrqv zlwkrxw uhqrxqflqj wkhlu vljqdov1b

e

,hhLh LLh_?@|L?

Wklv vhfwlrq grhv qrw gluhfwo| uhihu wr wkh fhqwudo wkhph ri wkh sdshu1 Krz0 hyhu/ lw looxvwudwhv d surshuw| ri vxshuprgxodu dqg vxeprgxodu ixqfwlrqv wkdw lv xvhixo lq lqwhusuhwlqj wkh uhvxowv ri wklv sdshu1 Frqvlghu d ixqfwlrq ri udqgrp yduldeohv1 Vxssrvh wkh udqgrp yduldeohv fdq eh hlwkhu shuihfwo| frghshqghqw ru pxwxdoo| lqghshqghqw1 Wklv vhfwlrq suryhv wkdw/ li wkh ixqf0 wlrq lv vxshuprgxodu/ wkh h{shfwhg ydoxh ri wkh ixqfwlrq lv kljkhu zkhq wkh yduldeohv duh shuihfwo| frghshqghqw/ zkloh/ li wkh ixqfwlrq lv vxeprgxodu/ lw lv kljkhu zkhq wkh yduldeohv duh pxwxdoo| lqghshqghqw1 Lq wkh oljkw ri wklv uhvxow/ zh fdq surylgh vrph lqwxlwlrq rq wkh uhvxowv suhvhqwhg lq Vhfwlrq 61 Frqvlghu wzr udqgrp yhfwruv= | U @ +| > |2 >    > |? , |( @ +|f > |f >    > |f , zkhuh |f > | > |2 >    > |? duh lghqwlfdoo| glvwulexwhg/ pxwxdoo| lqghshqghqw udqgrp yduldeohv1 Frqvlghu d ixqfwlrq i = ?? $ ?1 Wkh iroorzlqj fdq eh suryhq1f Sursrvlwlrq 6 Li i lv vxshuprgxodu/ wkhq H^i +| ( ,`  H^i+| U ,`/ zkloh li i lv vxeprgxodu/ wkhq H^i +| ( ,`  H^i+| U ,`1 Li d ixqfwlrq lv vxshuprgxodu/ wkhq wkh h{shfwhg ydoxh lv kljkhu lq wkh fdvh ri fruuhodwhg huuruv wkdq lq wkh fdvh ri xqfruuhodwhg huuruv1 Wkh rssrvlwh b 6Lh @ _tUttL? Lu |i hL*i Lu ?ULhhi*@|i_ ?uLh4@|L? ? ti@hU ThLM*i4tc tii @*tL @tt@? @?_ 5U@ht? dDo Ai) UL?t_ih @ U*@tt Lu 4*|@}i?| ti@hU ThLM*i4t @?_ _i4L? t|h@|i |i @*i Lu i|ihL}i?i|) M@ti_ L? |i _i@ Lu i Tih4i?|@|L? i |ih?@*|it f hLTLt|L? @t UL?iU|hi_ M) *}hL4 @?_ +LMih|t dHo Ai ThLLu t ? |i TTi?_ 

47

krogv li wkh ixqfwlrq lv vxeprgxodu1 Dowkrxjk Sursrvlwlrq 6 lv qrw xvhg wr suryh Sursrvlwlrqv 4 dqg 5/ lw surylghv lqwxlwlrq iru wkrvh uhvxowv1 Vxssrvh wkh djhqwv ri dq rujdql}dwlrq duh erxqg wr frpplw huuruv1 Wkh |*v fdq eh lqwhusuhwhg dv wkh dfwlrqv ri wkh djhqwv1 Wkh dfwlrqv duh udqgrp ehfdxvh wkh djhqw*v vljqdo kdv d udqgrp glvwxuedqfh1 Dvvxph wkdw wkh rujdql}dwlrq ghvljqhu fdqqrw uhgxfh wkh hqwlw| ri huuruv/ exw fdq fkrrvh zkhwkhu wkh huuruv duh shuihfwo| frghshqghqw ru pxwxdoo| lqghshqghqw1 Li wkh whdp sd|r ixqfwlrq lv vxshuprgxodu/ wkh rujdql}dwlrq ghvljqhu zloo zdqw wkh huuruv wr eh shuihfwo| frghshqghqw1 Zrunirufh krprjhqhlw| lv d ghylfh wr pdnh wkh huuruv shuihfwo| frghshqghqw1 Rq wkh rwkhu kdqg/ li wkh whdp sd|r ixqfwlrq lv vxeprgxodu/ wkh rujdql}dwlrq ghvljqhu zloo zdqw wkh huuruv wr eh pxwxdoo| lqghshqghqw dqg khwhurjhqhlw| lv d ghylfh wr pdnh huuruv pxwxdoo| lqghshqghqw1

D

L?U*tL?

Zh kdyh frqvlghuhg wkh sureohp ri d whdp wkdw pxvw ghflgh wkh lqirupdwlrq vwuxfwxuh ri lwv djhqwv1 Wklv sdshu kdv hvwdeolvkhg d jhqhudo frqqhfwlrq eh0 wzhhq frpsohphqwdulwlhv dfurvv djhqwv dqg wkh rssruwxqlw| ri klulqj djhqwv zlwk vlplodu fkdudfwhulvwlfv1 Li wkh sd|r ixqfwlrq lv vxshuprgxodu/ djhqwv vkrxog eh krprjhqhrxv1 Li wkh sd|r ixqfwlrq lv vxeprgxodu/ djhqwv vkrxog eh khwhurjhqhrxv1 Zkloh wkh dqdo|vlv suhvhqwhg khuh kdv ehhq sxuho| wkhruhwlfdo/ lwv pdlq lghdv fdq eh dssolhg wr lpsruwdqw rujdql}dwlrqdo lvvxhv1 Wklv sdshu suhglfwv wkdw wkh zrunirufh krprjhqhlw| ri d frpsdq| lv ghwhuplqhg e| wkh w|sh ri lqwhudfwlrq ehwzhhq lwv djhqwv1 Wkhuhiruh/ dfwlylwlhv iru zklfk jrrg w ehwzhhq ydulrxv xqlwv lv wkh uvw frqfhuq zloo kdyh d krprjhqhrxv zrun0 irufh lq rughu wr pd{lpl}h frruglqdwlrq1 Rq wkh rwkhu kdqg/ dfwlylwlhv wkdw uhyroyh durxqg h{sorlwdwlrq ri qhz rssruwxqlwlhv zloo kdyh d pruh khwhur0 jhqhrxv zrunirufh lq rughu wr pd{lpl}h wkh fkdqfh ri ghyhorslqj vxffhvvixo lqqrydwlrqv1 Shukdsv wkh prvw olplwlqj dvvxpswlrq ri wklv sdshu lv wkdw djhqwv fdqqrw frppxqlfdwh zlwk hdfk rwkhu ehwzhhq wkh wlph wkh| revhuyh wkhlu vljqdov dqg wkh wlph wkh| wdnh wkhlu dfwlrqv1 Ri frxuvh/ li wkhuh lv dq h{rjhqrxv ohyho ri frppxqlfdwlrq/ wkh suhvhqw prgho fdq hdvlo| eh h{whqghg wr dsso| wr wkh wkdw sduw ri lqirupdwlrq zklfk kdv qrw ehhq frppxqlfdwhg1 Krzhyhu/ wkh uhdo fkdoohqjh lv wr ohw frppxqlfdwlrq eh hqgrjhqrxv1 Duurz ^6/ s1 8908 z> = = = > z , . t+z> w> = = = > w ,  t+z> = = = > z, . t+ w> = = = > w, Frqvlghu qrz t = V ,n6 $ ? dqg dvvxph wkdw t lv vxshuprgxodu lq doo lwv dujxphqwv1 Wkhq/ iru dq| yhfwru +} > = = = > }, , 5 V , / dq| w 5 V/ dqg dq| z 5 V/

3  |4it } € ~

3€  |4it ~

6

}

6

t+} > = = = > }, > w> z> = = = > z , . t+} > = = = > }, > z> w> = = = > w , 6 |4it } € ~

}

6

|4it € ~

 t+} > = = = > }, > w> = = = > w, . t+} > = = = > }, > z> = = = > z,

+8,

Li zh dsso| +8, wr wkh sureohp dw kdqg/ zh kdyh wkdw iru dq| n @ 5> 6> = = = > q dqg iru dq| +s > = = = > s& , 5 D& / ?

3 & n€ |4it ~

}

}

?

3 & n€ |4it~

i+s > = = = > s&32 > s&3 > s& > = = = > s& , . i+s > = = = > s&32 > s& > s&3 > = = = > s&3 ,

3 & n€2 |4it~  i +s > = = = > s&32 > s& > = = = > s& , . i+s > = = = > s&32 > s&3 > = = = > s&3 , ?

3 & n€2 |4it ~

}

}

?

+9,

Ohw S& +d, @ i+s > = = = > s& ,m+s > = = = > s? , 5 S +d,j1 Ohw s& @ +s > = = = > s& ,1 E| +9,/ ?

[ 7

R&

M



3 & n€ |4it ~

}

}

?

3 & n€ |4it~

^i +s > = = = > s&32 > s&3 > s& > = = = > s& ,.i +s > = = = > s&32 > s& > s&3 > = = = > s&3 ,`

E 

& @

[ 7

R&

M

?

3 & n€2 |4it ~

}

}

?

3 & n€2 |4it~

^i +s > = = = > s&32 > s& > = = = > s& , . i+s > = = = > s&32 > s&3 > = = = > s&3 ,`

E 

& @

+:, 49

Krzhyhu/ lw lv hdv| wr vhh wkdw ?

[ 7

R&

M

E 

& @

[

@

7

R&

M

3 & n€ |4it ~

}

i+s > = = = > s&32 > s&3 > s& > = = = > s& ,

3 & n€ |4it~ i+s > = = = > s&32 > s& > s&3 > = = = > s&3 , ?

}

E 

& @

dqg [ 7

R&

@

M

E 

& @

[ 7

R&

M

E 

?

3 & n€2 |4it ~

}

i +s > = = = > s&32 > s& > = = = > s& ,

3 & n€2 |4it~ i +s > = = = > s&32 > s&3 > = = = > s&3 , }

?

& @

vr wkdw +:, ehfrphv= [ 7

R&



M

E 

& @

[ 7

R&

M

E 

?

3 & n€ |4it ~

}

i +s > = = = > s&3 > s& > = = = > s& ,

3 & n€2 |4it~ i +s > = = = > s&32 > s&3 > = = = > s&3 , }

?

+;,

& @

Qrwlfh/ krzhyhu/ wkdw qrz s& grhv qrw dsshdu lq wkh uljkw0kdqg vlgh ri +;,1 Wkxv/ iru dq| +s > = = = > s&3 , 5 S&3 +d,/ wkh vxppdwlrq frqwdlqv q  n . 4 lghqwlfdo hohphqwv fruuhvsrqglqj wr wkh srvvleoh ydoxhv ri s& 1 Khqfh/ +;, ehfrphv [ 7

R&

M

E 

?

3 & n€ |4it ~

}

i+s > = = = > s&3 > s& > = = = > s& ,

& @

 +q  n . 4,

[ 7

R&3

M

3 & n€2 |4it~ i+s > = = = > s&32 > s&3 > = = = > s&3 , }

E 

&3 @

E| dsso|lqj + = = = > s?32 > s?3 > s? ,

? @

4:

?

+ = = = > s?32 > s?3 > s?3 ,

11 1

 4  5      +q  n,

[ 7

R&

11 1

11 1

 +q  4,$

[ 7



M E@

M

?

3 & n€ |4it ~

}

i+s > = = = > s&3 > s& > = = = > s& ,

E 

& @

|4it € ~ i +s > = = = > s ,

}

?

+43,

Qrwlfh wkdw wkh ohiw0kdqg vlgh ri +43, lv htxdo wr [ ER ccR? M E@

dqg wkdw

[ 

'cc?

i +s > = = = > s? ,

i+d > = = = > d , @

[ 7



M E@

}

?

|4it € ~

i +s > = = = > s ,

Khqfh/ +43, ehfrphv [ ER ccR? M E@

[

i+s > = = = > s? ,  +q  4,$ 

'cc?

i +d > = = = > d ,

E| glylglqj erwk vlghv ri +9, e| q$/ wkh sursrvlwlrq lv suryhq iru i vxshu0 prgxodu1 Li i lv vxeprgxodu/ wkh surri jrhv wkurxjk lq wkh vdph idvklrq zlwk vzlwfkhg lqhtxdolw| vljqv1 Yhulfdwlrq Wkdw wkh Sd|r Ixqfwlrq lq Ht1 +6, Lv Vxshuprgxodu Frqvlghu dq| wzr yhfwruv ri vwudwhjlhv +d > d2 , dqg +d > d2 ,1 E| Ghqlwlrq 4/ zh kdyh wr suryh wkdw/ iru doo {/ $^plq+d > d ,> plq+d2 > d2 ,> {` . $^pd{+d > d ,> pd{+d2 > d2 ,> {` +44,  $+d > d2 > {, . $+d > d2 > {, Dvvxph zlwkrxw orvv ri jhqhudolw| wkdw d  d 1 Li dovr d2 A d2 / wkhq +44, krogv dv dq htxdolw|1 Khqfh/ vxssrvh wkdw d2  d2 1 Wkhq/ +44, ehfrphv plq+d > d2 > {, . plq+d > d2 > {,  plq+d > d2 > {, . plq+d > d2 > {, 4;

+45,

Jlyhq wkdw d  d dqg d2  d2 / zlwkrxw orvv ri jhqhudolw|/ zh fdq dvvxph wkdw d  d2 +li lw kdsshqv wkdw d ? d2 / zh fdq vzlwfk wkh vx{hv  dqg  dqg dovr vzlwfk wkh lqgh{hv 4 dqg 5,1 Wkhuh duh wkuhh srvvleoh fdvhv= d  d2  d2  d d  d2  d  d2 d  d  d2  d2 Iru doo wkuhh fdvhv lw lv hdv| wr yhuli| wkdw +45, krogv1 Wkhuhiruh/ wkh sd|r ixqfwlrq lq Ht1 +6, lv vxshuprgxodu1 Wr suryh Sursrvlwlrq 5 wkh iroorzlqj lv xvhixo Ohppd 5 Xqghu Dvvxpswlrq 5/ wkh vhw ri rqh0w|sh rswlpd lqfoxghv d vr0 oxwlrq lq zklfk  +|, @ = = = @  +|, Ohppd 5 uhihuv wr wkh sureohp lq zklfk djhqwv duh duwlfldoo| uhvwulfwhg wr eh ri wkh vdph w|sh1 Lq wkdw fdvh/ frqyh{lw| ri wkh dfwlrq vsdfh dqg frqfdylw| ri wkh sd|r ixqfwlrq duh wrjhwkhu d vxflhqw frqglwlrq wr kdyh dq rswlpdo vroxwlrq lq zklfk djhqwv doo djhqwv kdyh wkh vdph ghflvlrq ixqfwlrq1 Surri ri Ohppd 5 Vxssrvh doo djhqwv kdyh wkh vdph w|sh 1 Iru doo | 5 \ / wkh whdp fkrrvhv d uxoh ri dfwlrq zklfk pd{lpl}hv j^ +|,> = = = > ? +|,` @ Hi$^d > = = = > d? > {`m+{, @ |j

+46,

Vxssrvh $ lv v|pphwulf dqg frqfdyh lq d1 Ehfdxvh wkh h{shfwdwlrq lv d olqhdu rshudwru/ dovr j lv v|pphwulf dqg frqfdyh lq d1 Dvvxph wkh uxoh ri dfwlrq dW @ +dW > dW2 >    > dW? , lv d pd{lpxp ri j+d> |,1 E| v|pphwu|/ doo uhrughulqjv ri dW duh pd{lpd/ wrr1 Li zh ghqh d W @

S?  S?  W S?' dW dW ' d  > > = = = > ' 

q

q

q

wkhq/ e| frqfdylw| ri j/ zh jhw j+ dW > |,  j+dW > |,1 Wklv krogv iru doo srvvleoh W vljqdov |1 Pruhryhu/ d  lv d v|pphwulf uxoh ri dfwlrq1 Wkhq/ li wkhuh h{lvwv dq rswlpdo uxoh ri dfwlrq ~ dv kdv ehhq dvvxphg wkurxjkrxw wklv sdshu ~ wkhq wkhuh dovr h{lvwv d v|pphwulf rswlpdo uxoh ri dfwlrq1 4
 , 5 dujpd{k ck2 cw cw2 $^ + +{,,> = = = > ? +? +{,,> {`  fw  fw2 Vxssrvh wkdw  lv wkh rswlpdo uxoh ri dfwlrq dvvrfldwhg wr  dqg  lv wkh rswlpdo uxoh ri dfwlrq dvvrfldwhg wr  1 E| Ohppd 5/ iru  wkhuh h{lvwv dq rswlpdo vhw ri ghflvlrq ixqfwlrqv  +, @    @ ? +, @  +, dqg iru  wkhuh h{lvwv dq rswlpdo vhw ri ghflvlrq ixqfwlrqv  +, @    @ ? +, @  +,1 Vxssrvh $ lv vxeprgxodu1 E| Fruroodu| 4/ & djhqwv ?3& djhqwv € ~ } € ~       $i ^ +{,`> = = = >  ^ +{,`>  ^ +{,`> = = = >  ^ +{,`j  ^nfw . +q  n,fw `

}



? djhqwv € ~ n }   $i ^ +{,`> = = = >  ^ +{,`j  nfw q ? djhqwv € ~ q  n }   . $i ^ +{,`> = = = >  ^ +{,`j  +q  n,fw q







iru doo { 5 [ dqg iru dq| n @ 4> 5> = = = > q1 Wdnh h{shfwdwlrqv dqg uhfdoo wkdw   dqg  |lhog wkh vdph h{shfwhg qhw surw1 Wkhq/ iru n @ 4> 5> = = = > q  4/

3& djhqwv € ~ H+$i ^ +{,`> = = = >  ^ +{,`>  ^ +{,`> = = = >  ^  +{,`j,  ^nfw . +q  n,fw ` &

}

djhqwv €

?

~ }



}

?

djhqwv €

~

 Hi$^ ^ +{,`> = = = >  ^ +{,``j  qfw ? djhqwv }

€





~

@ H$^ ^ +{,`> = = = >  ^ +{,``  qfw



Lw iroorzv wkdw glyhuvlhg nqrzohgjh lv rswlpdo1 Surri ri Sursrvlwlrq 6 Vxssrvh i lv vxshuprgxodu1 Ehfdxvh wkh |*v duh lghqwlfdoo| glvwulexwhg/ wkh h{shfwhg ydoxh ri i lv lqyduldqw wr shupxwdwlrqv ri wkh |*v= H^i+| >    > |? ,` @    @ H^i+|? >    > | ,` dqg/ reylrxvo|/ H^i+| >    > | ,` @ H^i+|f >    > |f ,` iru l @ 4> 5> = = = > q 53

Li i lv vxshuprgxodu/ lwv h{shfwhg ydoxh lv vxshuprgxodu dv zhoo1 Frq0 vlghu wkh dulwkphwlf dyhudjh ri doo wkh srvvleoh shupxwdwlrq ri wkh |*v1 E| Ohppd 4/ S E+ cc+? M E+ i+| > = = = > |? ,

Q^S +|,`

lpso|lqj

4  +, ^i+| > = = = > | ,.i+|2 > = = = > |2 ,.  .i+|? > = = = > |? ,` q

H^i+| >    > |? ,`  H^i+|f >    > |f ,` H^i+| U ,`  H^i+|( ,`

dqg frqyhuvho| zkhq i lv vxeprgxodu1

+iuihi?Uit ^4` Pdvdklnr Drnl1 Krul}rqwdo yv1 yhuwlfdo lqirupdwlrq vwuxfwxuh ri wkh up1 Dphulfdq Hfrqrplf Uhylhz/ :9=