Preponderance of Evidence

Texte intégral

(1)2002s-61. Preponderance of Evidence Dominique Demougin et Claude Fluet. Série Scientifique Scientific Series. Montréal Juin 2002. © 2002 Dominique Demougin, Claude Fluet. Tous droits réservés. All rights reserved. Reproduction partielle permise avec citation du document source, incluant la notice ©. Short sections may be quoted without explicit permission, if full credit, including © notice, is given to the source..

(2) CIRANO Le CIRANO est un organisme sans but lucratif constitué en vertu de la Loi des compagnies du Québec. Le financement de son infrastructure et de ses activités de recherche provient des cotisations de ses organisationsmembres, d’une subvention d’infrastructure du ministère de la Recherche, de la Science et de la Technologie, de même que des subventions et mandats obtenus par ses équipes de recherche. CIRANO is a private non-profit organization incorporated under the Québec Companies Act. Its infrastructure and research activities are funded through fees paid by member organizations, an infrastructure grant from the Ministère de la Recherche, de la Science et de la Technologie, and grants and research mandates obtained by its research teams. Les organisations-partenaires / The Partner Organizations •École des Hautes Études Commerciales •École Polytechnique de Montréal •Université Concordia •Université de Montréal •Université du Québec à Montréal •Université Laval •Université McGill •Ministère des Finances du Québec •MRST •Alcan inc. •AXA Canada •Banque du Canada •Banque Laurentienne du Canada •Banque Nationale du Canada •Banque Royale du Canada •Bell Canada •Bombardier •Bourse de Montréal •Développement des ressources humaines Canada (DRHC) •Fédération des caisses Desjardins du Québec •Hydro-Québec •Industrie Canada •Pratt & Whitney Canada Inc. •Raymond Chabot Grant Thornton •Ville de Montréal. Les cahiers de la série scientifique (CS) visent à rendre accessibles des résultats de recherche effectuée au CIRANO afin de susciter échanges et commentaires. Ces cahiers sont écrits dans le style des publications scientifiques. Les idées et les opinions émises sont sous l’unique responsabilité des auteurs et ne représentent pas nécessairement les positions du CIRANO ou de ses partenaires. This paper presents research carried out at CIRANO and aims at encouraging discussion and comment. The observations and viewpoints expressed are the sole responsibility of the authors. They do not necessarily represent positions of CIRANO or its partners.. ISSN 1198-8177.

(3) Preponderance of Evidence * Dominique Demougin† et Claude Fluet‡ Résumé / Abstract Cet article analyse les propriétés incitatives du standard de preuve et de l’attribution de la charge de la preuve en responsabilité pour faute, lorsque l’information sur le comportement de l’auteur du dommage est imparfaite et relève des parties. Nous montrons que le standard de la « prépondérance de preuve » appliqué dans la common law, combiné aux règles d’exclusion définissant les faits admissibles en cour, fournit le maximum d’incitations à la prudence. Cette propriété est vérifiée même si, lorsque survient un dommage, les parties ont un accès inégal aux faits ou peuvent les présenter de manière partiale dans le cadre d’une procédure adversariale. Abstraction faite des coûts de litiges, l’attribution de la charge de la preuve s’explique également par le principe fondant le standard de preuve. Ces résultats sont obtenus comme propriétés d’un mécanisme optimal, mais nous analysons aussi la mise en oeuvre du mécanisme dans un équilibre séquentiel où le tribunal est modélisé comme un joueur. This paper analyzes the incentive properties of the standard and burden of proof for a finding of negligence, when evidence about injurers' behavior is imperfect and rests with the parties. We show that the `preponderance of evidence' standard used in common law, together with ordinary exclusion rules defining legally admissible evidence, provides maximal incentives for potential tort-feasors to exert care. This holds even though, following the occurrence of harm, litigants have unequal access to evidence and may distort information in adversarial procedures. Abstracting from litigation costs, the optimal assignment of the burden of proof is shown to follow from the principle underlying the standard of proof. Our main results are derived in a mechanism design framework, but we also consider implementation as a sequential equilibrium with the court as a player in the game. Mots clés : Faute, responsabilité, standard de preuve, charge de la preuve, prévention. Keywords: Negligence, tort rules, standard of proof, burden of proof, deterrence Codes JEL : D8, K4. * Nous remercions M. Boyer, C. Helm, C. Kirchner, E. Mackay, A. Miglo, F. Riedel, B. Schäfer et G. Thüsing pour leurs commentaires sur une première version. Cette recherche a bénéficié d’un financement FCAR (Québec). † Humboldt Universität zu Berlin. demougin@wiwi.hu-berlin.de ‡ Université du Québec à Montréal et CIRANO. fluet.claude-denys@uqam.ca.

(4) Suhsrqghudqfh ri Hylghqfh. 4. 4 Lqwurgxfwlrq D ixqgdphqwdo uxoh lq frpprq odz lv iru frxuwv wr uhdfk d ghflvlrq rq wkh edvlv ri d suhsrqghudqfh ri hylghqfh1 Iru lqvwdqfh/ lq wruw glvsxwhv xqghu wkh qhjoljhqfh uxoh/ wkh sodlqwl pxvw xvxdoo| shuvxdgh wkh frxuw wkdw wkh ghihqgdqw glg qrw h{huflvh gxh fduh1 Wkh frxuw ghflghv rq d vr0fdoohg cedodqfh ri suredelolwlhv* ru csuhsrqghudqfh ri hylghqfh*= Wr hvwdeolvk e| d suhsrqghudqfh ri hylghqfh phdqv wr suryh wkdw vrphwklqj lv pruh olnho| vr wkdq qrw vr1 +Ghylww dqg Eodfnpdq/ Ihghudo Mxu| Sudfwlfh dqg Lqvwuxfwlrqv/ 4<::, Wkxv/ wkh ghihqgdqw zloo eh khog oldeoh li/ xsrq wkh hylghqfh/ lw dsshduv pruh olnho| wkdq qrw wkdw kh glg qrw h{huflvh gxh fduh1 Wklv lv jhqhudoo| xqghuvwrrg dv lpso|lqj d wkuhvkrog ghjuhh ri fhuwdlqw| mxvw deryh 83 shufhqw iru d uxolqj lq idyru ri wkh sduw| zlwk wkh exughq ri surri=4 Wkh voljkwhvw suhsrqghudqfh ri wkh hylghqfh lq klv idyru hqwlwoh^v` wkh sodlqwl wr d yhuglfw1 Doo wkdw lv uhtxluhg lq d flylo fdvh ri rqh zkr kdv wkh exughq ri surri lv wkdw kh hvwdeolvk klv fodlp e| d suhsrqghudqfh ri wkh hylghqfh1 Zkhq wkh htxloleulxp ri surri lv ghvwur|hg/ dqg wkh ehdp lqfolqhv wrzdug klp zkr kdv wkh exughq/ krzhyhu voljkwo|/ kh kdv vdwlvhg wkh uhtxluhphqw ri wkh odz/ dqg lv hqwlwohg wr wkh yhuglfw1 D eduh suhsrqghudqfh lv vx!flhqw/ wkrxjk wkh vfdohv gurs exw d ihdwkhu*v zhljkw1 +Olydqrylwfk y1 Olydqrylwfk/ <<Yw1 65: 464D1 :<</ 4<59, Suhsrqghudqfh ri hylghqfh vwdqgv lq vkdus frqwudvw wr wkh rwkhu pdlq oh0 jdo vwdqgdug ri surri eh|rqg d uhdvrqdeoh grxew irxqg lq fulplqdo surfhhg0 lqjv1 Lw dovr glhuv iurp wkh lqwhuphgldwh vwdqgdug ri fohdu dqg frqylqflqj hylghqfh xvhg e| XV frxuwv rq vrph pdwwhuv1 Lq wkh frqwh{w ri ruglqdu| flylo glvsxwhv/ wkh prvw vwulnlqj glyhujhqfh e| idu lv ehwzhhq wkh frpprq odz dqg wkh flyloldq wudglwlrq1 Lq prvw flylo0odz frxqwulhv/ phuh suhsrqghudqfh lv qrw frqvlghuhg vx!flhqw15 Wr wdnh exw rqh h{dpsoh/ dffruglqj wr ]ùoohu/ dq dxwkrulw| rq Jhupdq flylo surfhgxuh=. Vhh Fduovrq hw do1 +4<<:, dqg Furvv dqg Wdsshu +4<;8, dprqj rwkhuv/ dqg wkh uhihu0 hqfhv wkhuhlq1 5 Wkhuh duh h{fhswlrqv/ iru h{dpsoh lq Txhehf= Hylghqfh lv vx!flhqw li lw uhqghuv wkh h{lvwhqfh ri d idfw pruh suredeoh wkdq lwv qrq0h{lvwhqfh/ xqohvv wkh odz uhtxluhv pruh frqylqflqj surri +Flylo Frgh ri Txhehf/ Errn YLL/ ¢ 5;37,1 4.

(5) Suhsrqghudqfh ri Hylghqfh. 5. Ohvv wkdq wkh frqylfwlrq ri wkh wuxwk lv qrw vx!flhqw iru d surri 111 Vwdqgdugv wkdw duh edvhg rq d phdvxuh ri olnholkrrg gr qrw qg dq| vxssruw lq wkh odz1 +]ùoohu/ ]lylosur}h rugqxqj/ 4<<<, Lq Iuhqfk odz/ wkh vdph phdqlqj lv frqyh|hg e| wkh uhtxluhphqw wkdw sduwlhv ehdulqj wkh exughq ri surri vdwlvi| wkh mxgjhv wr wkh srlqw ri lqwlph frqylf0 wlrq1 Lq vr idu dv lw kdv ehhq duwlfxodwhg/ wkh flyloldq vwdqgdug lv riwhq vdlg wr eh wdqwdprxqw wr uhdvrqdeoh0grxew=6 Lq frqwlqhqwdo Hxurshdq odz/ qr glvwlqfwlrq lv pdgh ehwzhhq flylo dqg fulplqdo fdvhv zlwk uhjdug wr wkh vwdqgdug ri surri1 Lq erwk/ vxfk d kljk ghjuhh ri suredelolw| lv uhtxluhg wkdw/ wr wkh ghjuhh wkdw wklv lv srvvleoh lq wkh ruglqdu| h{shulhqfh ri olih lwvhoi/ grxewv duh h{foxghg dqg suredelolw| dssurdfkhv fhuwlwxgh1 +Qdjho/ Hyl0 ghqfh/ lq Hqf|forshgld Eulwdqqlfd= Pdfursdhgld/ 4<:7, Duh vrph flylo vwdqgdugv ehwwhu wkdq rwkhuvB Lq zkdw vhqvh gr wkh| pdw0 whuB Zh dgguhvv wklv txhvwlrq zlwk uhvshfw wr wkh lvvxh ri hvwdeolvklqj qhj0 oljhqfh +rwkhu lvvxhv frxog eh h{dplqhg vxfk dv fdxvdwlrq ru lghqwlfdwlrq,1 Zh gr wklv iurp wkh xvxdo shuvshfwlyh lq wkh hfrqrplf dqdo|vlv ri oldelolw| uxohv/ frqvlghuhg dv lqvwlwxwlrqv iru ghwhuulqj vrfldoo| lqh!flhqw ehkdylru1 D fodvv ri vlwxdwlrqv lv h{dplqhg/ zlwklq wkh vr0fdoohg xqlodwhudo fduh sureohp/ zkhuh qhjoljhqfh kdv d xvhixo uroh dv rssrvhg wr d vlpsohu uxoh vxfk dv vwulfw oldelolw|1 Xqghu wkh odwwhu/ dq lqmxuhu pxvw sd| iru doo kdup fdxvhg wr wklug sduwlhv1 Dv lv zhoo nqrzq/ wklv surylghv lqdghtxdwh lqfhqwlyhv wr h{huflvh fduh zkhq srwhqwldo lqmxuhuv kdyh lqvx!flhqw zhdowk wr sd| gdpdjhv lq ixoo1 E| frqwudvw/ wkh qhjoljhqfh uxoh frqglwlrqv oldelolw| erwk rq wkh rffxuuhqfh ri gdpdjhv dqg rq dgglwlrqdo h{ srvw lqirupdwlrq derxw wkh wruw0ihdvru*v ohyho ri fduh1 Li fduh lv revhuyhg zlwkrxw huuru/ wkh uxoh fdq ghwhu fduhohvv ehkdylru hyhq wkrxjk srwhqwldo lqmxuhuv duh sduwldoo| mxgjphqw0surri +vhh iru lqvwdqfh Vkdyhoo 4<;9/ 4<;:,1 Zh h{whqg wklv dqdo|vlv wr wkh fdvh ri lpshuihfw h{ srvw lqirupdwlrq1 Lq0 gxflqj fduh iru dv pdq| srwhqwldo wruw0ihdvruv dv srvvleoh/ zkhq wkh| glhu lq zhdowk dqg vrph pd| eh mxgjphqw0surri/ udlvhv wkh lvvxh ri wkh h!flhqw xvh ri lqirupdwlrq1 Wkuhh vfhqdulrv zlwk lqfuhdvlqj frpsoh{lw| duh h{dplqhg1 Lq wkh uvw rqh/ rqfh dq dfflghqw kdv rffxuuhg/ vrflhw| +dv ehqhyrohqw sulqflsdo, gluhfwo| revhuyhv hylghqfh derxw wkh lqmxuhu*v dfwlrq1 Zh vkrz wkdw vrflhw|*v 6. Vhh wkh uhihuhqfhv lq Vkhuzlq dqg Fohuprqw +5334,1.

(6) Suhsrqghudqfh ri Hylghqfh. 6. phfkdqlvp ghvljq sureohp lv vroyhg e| dvvljqlqj oldelolw| iru gdpdjhv rq wkh edvlv ri d pruh olnho| wkdq qrw fulwhulrq1 Lq wkh vhfrqg vfhqdulr/ vrflhw| grhv qrw gluhfwo| revhuyh wkh hylghqfh dqg pxvw uho| rq ylfwlp dqg lqmxuhu/ zkr pd| wkhpvhoyhv kdyh xqhtxdo dffhvv wr hylghqfh1 Wdnlqj lqwr dffrxqw h{ srvw lqfhqwlyhv wr glvforvh +ru frqfhdo, hylghqfh lq dgyhuvduldo surfhgxuhv/ doorfdwlqj gdpdjhv rq wkh edvlv ri pruh olnho| wkdq qrw lv vwloo vkrzq wr sur0 ylgh h!flhqw h{ dqwh lqfhqwlyhv wr wdnh fduh1 Ilqdoo|/ wkh wklug vfhqdulr vkrzv wkdw lq hlwkhu fdvh wkh rswlpdo phfkdqlvp fdq eh lpsohphqwhg e| ghohjdwlqj ghflvlrqv wr d frxuw/ wkurxjk wkh xvh ri wkh jhqhudo uxoh ri suhsrqghudqfh ri hylghqfh1 Wkh ohjdo vwdqgdug ri surri lv lqwhusuhwhg khuh vtxduho| dv pruh olnho| wkdq qrw lq wkh xvxdo pdwkhpdwlfdo vhqvh1 D zhoo nqrzq dwwhpsw wr pdnh vhqvh ri suhsrqghudqfh kdv ehhq lq whupv ri Ed|hvldq xsgdwlqj17 Vxssrvh vwdqgv iru wkh fodlp wkdw wkh ghihqgdqw zdv fduhohvv dqg iru wkh frqwudu|1 Zkhq . lv wkh hylghqfh vxeplwwhg/ wkh srvwhulru rggv rq wkh ghihqgdqw ehlqj dw idxow duh hE . hE. hE ' hE . hE. hE zkhuh hE lv wkh frxuw*v sulru wkdw wkh ghihqgdqw zdv fduhohvv dqg hE. wkh suredelolw| ri wkh hylghqfh frqglwlrqdo rq lqdghtxdwh fduh +htxlydohqwo| lw lv wkh olnholkrrg ri lqdghtxdwh fduh jlyhq wkh hylghqfh,1 Dffruglqj wr wkh Ed|hvldq lqwhusuhwdwlrq/ wkhuh lv d suhsrqghudqfh ri hylghqfh djdlqvw wkh ghihqgdqw li hE . : hE . 1 E| frqwudvw/ zh lqwhusuhw suhsrqghudqfh dv hE. : hE. / zklfk phdqv wkdw lqdghtxdwh fduh dsshduv pruh olnho| xsrq wkh hylghqfh vxeplwwhg1 Zklfk lqwhusuhwdwlrq ehwwhu ghvfulehv frxuw ghflvlrqv lv ghedwdeoh/ vlqfh lq ohjdo glvfrxuvh pruh olnho| ru pruh suredeoh kdyh orqj ehhq xvhg lqwhu0 fkdqjhdeo|18 D frpprqo| hqfrxqwhuhg dujxphqw/ eruurzhg iurp vwdwlvwlfdo m. m. m. m. . m. m. m. m. m. H{whqvlyh uhihuhqfhv rq wkh ohjdo ru uhodwhg olwhudwxuh glvfxvvlqj wklv dssurdfk duh jlyhq lq Eurrn +4<;5, dqg Ilhqehuj +4<;;,1 8 Wklv iroorzv d uhvshfwdeoh wudglwlrq= Li ehvlghv wkh dujxphqwv wkdw frxqw lq idyrxu ri wkh wklqj/ rwkhu sxuh dujxphqwv suhvhqw wkhpvhoyhv/ zklfk lqglfdwh wkh rssrvlwh ri wkh wklqj/ wkh dujxphqwv ri erwk nlqgv pxvw eh zhljkwhg vhsdudwho| 111 lq rughu wkdw rqh pd| rewdlq d udwlr ehwzhhq wkh suredelolw| ri wkh wklqj dqg wkh suredelolw| ri wkh rssrvlwh ri wkh wklqj1 Khuh lw pxvw eh qrwhg wkdw li wkh dujxphqwv iru hdfk vlgh duh vwurqj hqrxjk/ lw fdq kdsshq wkdw wkh devroxwh suredelolw| ri hdfk vlgh qrwdeo| h{fhhgv kdoi ri fhuwdlqw|1 Wkxv hdfk ri wkh dowhuqdwlyh lv pdgh suredeoh/ dowkrxjk vshdnlqj uhodwlyho| rqh pd| eh ohvv wkdq wkh rwkhu1 Vr lw fdq kdsshq wkdw d wklqj srvvhvvhv 526 ri fhuwdlqw| dqg lwv rssrvlwh srvvhvvhv 627 ri fhuwdlqw|1 Lq wklv zd|/ hdfk ri wkh dowhuqdwlyh zloo eh suredeoh/ 7.

(7) Suhsrqghudqfh ri Hylghqfh. 7. ghflvlrq wkhru| dqg riwhq xvhg wr ghprqvwudwh wkh vxshulrulw| ri wkh frpprq odz ryhu wkh flyloldq vwdqgdug/ lv wkdw suhsrqghudqfh lq whupv ri Ed|hvldq srvwhulruv zrxog plqlpl}h wkh ryhudoo iuhtxhqf| ri huuru1 Wklv dffrugv zlwk wkh lghd wkdw wulhuv ri idfwv vkrxog vhhn wuxwk lq wkh orqj uxq/ zklfk vhhpv dssursuldwh li frxuw ghflvlrqv lq flylo wuldov lpsrvh v|pphwulfdo jdlqv dqg orvvhv rq wkh sduwlhv19 Rqh zhoo nqrzq gl!fxow| zlwk wklv dujxphqw lv wkdw frxuwv duh w|slfdoo| uhoxfwdqw wr frqvlghu uhdo sulruv derxw idfwv lqyroylqj d vljqlfdqw hohphqw ri ghflvlrq ru yroxqwdulqhvv1 Hylghqwldu| uxohv ghwhuplqh zkdw hylghqfh lv ohjdoo| dgplvvleoh1 Lq d qhjoljhqfh fdvh/ wkh| zrxog h{foxgh frqvlghudwlrq wkdw vrph nqrzq sursruwlrq ri lqglylgxdov dfw fduhohvvo| lq vlwxdwlrqv vlplodu wr wkh fdvh dw kdqg1 Hylghqfh ri fkdudfwhu ru ri vlplodu idfwv/ dooxglqj wkdw wkh ghihqgdqw kdv ehhq qhjoljhqw +ru gloljhqw, rq rwkhu vlplodu rffdvlrqv/ zrxog dovr lq jhqhudo eh lqdgplvvleoh +h1j1 Ihghudo Uxohv ri Hylghqfh 737,1 Wklv kdv ohg vrph frpphqwdwruv xvlqj d Ed|hvldq whuplqrorj| wr vxjjhvw wkdw sulruv eh lqwhusuhwhg qrupdwlyho|1: Zlwk d qhxwudo svhxgr sulru hE ' *2/ Ed|hvldq suhsrqghudqfh lv ri frxuvh htxlydohqw wr pruh olnho| wkdq qrw1; Zkdw phdqlqj fdq eh dwwdfkhg wr huuru plqlpl}dwlrq zkhq qrupdwlyh sulruv duh xvhg lv xqfohdu1 Rq wkh rwkhu kdqg/ Ed|hvldq suhsrqghudqfh/ zlwk vlwxdwlrq frqvlvwhqw sulruv derxw ghihqgdqwv* ehkdylru/ lv lqfrpsdwleoh zlwk oldelolw| uxohv ehlqj vwuxfwxuhg wrzdugv h!flhqw ghwhuuhqfh1 Vhhnlqj wkh wuxwk lv qrw wkh vdph dv surylglqj lqfhqwlyhv1 Wklv lv zhoo nqrzq iurp wkh sulqflsdo0djhqw olwhudwxuh +h1j1 Kropvwuùp/ 4<:<,/ exw wkh srlqw hdvlo| h{whqgv wr oldelolw| uxohv1 Vxssrvh lqvroyhqf| lv qrw d sureohp/ vr wkdw vwulfw oldelolw| lpsohphqwv wkh uvw0ehvw/ dqg frqvlghu wkh qhjoljhqfh uxoh zkhq hylghqfh surylghv rqo| lpshuihfw lqirupdwlrq derxw fduh1 Zlwk Ed|hvldq suhsrqghudqfh/ udwlrqdo sulruv derxw d ghihqgdqw kdylqj h{huwhg gxh fduh fdq eh qhlwkhu }hur qru xqlw|= xqghu wkh iruphu/ lqmxuhuv duh dozd|v khog exw qhyhuwkhohvv wkh wklqj lv ohvv suredeoh wkdq lwv rssrvlwh/ lq wkh udwlr ri 526 wr 627/ ru ; wr <1 +M1 Ehuqrxool/ Duv frqmhfwdqgl/ 4:46/ txrwhg iurp Kdfnlqj/ 4<:8,1 9 Plqlpl}lqj wkh h{shfwhg frvwv ri huuru +srvvleo| wrjhwkhu zlwk olwljdwlrq ru survhfxwlrq frvwv, lv dovr frqvlghuhg lq pxfk ri wkh hfrqrplf olwhudwxuh rq vwdqgdugv ri surri/ dv lq Uxelqihog dqg Vdsslqjwrq +4<;:,/ Plfhool +4<<3,/ Gdylv +4<<7, dqg Vdqfklulfr +4<<:,1 : Iru lqvwdqfh Srvqhu +4<<<,= Lghdoo| zh zdqw wkh wulhu ri idfw wr zrun iurp sulru rggv ri 4 wr 4 wkdw wkh sodlqwl 111 kdv d phulwrulrxv fdvh1 ; Hyhq lq wkh hfrqrplfv olwhudwxuh/ sulruv glhulqj iurp rqh kdoi duh vrphwlphv ghvfulehg dv eldvhg +h1j1 Iurhe dqg Nred|dvkl/ 4<<9,1 Gdxjkhw| dqg Uhlqjdqxp +5333d/ 5333e, suhvhqw dq h{solflw prgholqj ri qrq0Ed|hvldq frxuwv/ prwlydwhg e| wkh frqvwudlqwv lpsrvhg e| hylghqwldu| uxohv1.

(8) Suhsrqghudqfh ri Hylghqfh. 8. oldeoh/ zklfk dprxqwv wr vwulfw oldelolw| dqg wkhuhiruh lqgxfhv gxh fduh> xqghu wkh odwwhu/ lqmxuhuv duh qhyhu khog oldeoh/ zklfk surylghv qr lqfhqwlyhv1 Lq d vhtxhqwldo htxloleulxp/ lqmxuhuv wkhuhiruh udqgrpl}h ehwzhhq gxh fduh dqg fduhohvvqhvv +ru vrph fkrrvh gxh fduh zkloh rwkhuv duh fduhohvv,/ zklfk lpsolhv ohvv ghwhuuhqfh wkdq xqghu vwulfw oldelolw|1 E| frqwudvw/ dv zloo eh vkrzq lq wkh sdshu/ suhsrqghudqfh dv pruh olnho| wkdq qrw lqgxfhv gxh fduh1 Xqghu d pruh olnho| wkdq qrw vwdqgdug/ wulhuv ri idfwv duh vlpso| du0 elwhuv lq d frqwhvw ri surri0vwuhqjwk= wkh| gr qrw vhhn wr hvwdeolvk wkh de0 vroxwh phulw ri d fodlp/ exw rqo| dvvhvv wkh frpsdudwlyh hylghqwldo vxssruw iru ru djdlqvw lw1< Hpskdvl}lqj ghwhuuhqfh wkhuhiruh surylghv d mxvwlfdwlrq/ edvhg rq h!flhqf| frqvlghudwlrqv/ iru uxohv ri h{foxvlrq frqqlqj wkh idfwv dqg dvvxpswlrqv zklfk frxuwv duh hqwlwohg wr wdnh lqwr dffrxqw1 Vxfk uxohv fdq eh vhhq dv d frpplwphqw ghylfh/ zkhuhe| wkh mxglfldo surfhvv suhyhqwv oldelolw| dvvhvvphqwv iurp ehlqj edvhg rq lqirupdwlrq wkdw zrxog plwljdwh ghwhuuhqfh1 Hylghqfh lv wkhq uhohydqw rqo| li lw uhodwhv wr wkh sduwlfxodu gh0 ihqgdqw*v ehkdylru lq wkh fdvh xqghu frqvlghudwlrq1 Wklv lv frqvlvwhqw zlwk prvw ri wkh ohjdo frpphqwdu|/ dv zhoo dv zlwk flylo mxu| lqvwuxfwlrqv ghqlqj ohjdoo| dgplvvleoh hylghqfh1 Frxuw huuru qhyhuwkhohvv dovr pdwwhuv iru ghwhuuhqfh vhh lq sduwlfxodu Srvqhu +4<:6/ 4<<<,/ Fudvzhoo dqg Fdoihh +4<;9,/ Srolqvn| dqg Vkdyhoo +4<;<,/ Ndsorz +4<<7,/ Ndsorz dqg Vkdyhoo +4<<7, dqg Ehuqdugr hw do1 +5333,1 Lq wklv olwhudwxuh/ wkh suredelolw| ri idovh srvlwlyhv ru idovh qhjdwlyhv dv wr zkhwkhu wkh ghihqgdqw h{huwhg gxh fduh lv hlwkhu h{rjhqrxv ru h{rjhqrxvo| uhodwhg wr frxuw h{shqglwxuh rq dffxudf| ru sduwlhv* olwljdwlrq frvwv/ zlwk qr sduwlfxodu uhihuhqfh wr wkh frpprq odz vwdqgdug ri surri ru wr wkh sureohp ri lqwhusuhwlqj hylghqfh1 Wkh srlqw pdgh lq wkh suhvhqw sdshu lv wkdw/ xqghu wkh suhsrqghudqfh vwdqgdug/ wkh uhvxowlqj w|sh L dqg w|sh LL huuruv duh vxfk dv wr surylgh pd{lpxp lqfhqwlyhv wr h{huflvh fduh/ jlyhq wkh lqirupdwlyhqhvv ri wkh hylghqfh1 Wkh iruhjrlqj glvfxvvlrq lv qrw phdqw wr lpso| wkdw doo sulruv duh luuhoh0 ydqw iurp d ghwhuuhqfh srlqw ri ylhz1 Iru lqvwdqfh/ zkhq wkh hylghqfh uhvwv zlwk wkh sduwlhv/ lw pd| eh wkdw/ edvhg rq vrph sulruv/ wkh exughq ri surri vkrxog eh dvvljqhg wr rqh sduw| udwkhu wkh rwkhu1 Lq vrph vlwxdwlrqv/ frxuwv pd| dovr glvfrxqw d olwljdqw*v vxeplvvlrq ehfdxvh lw lv frqfhlydeo| sduwldo dqg eldvhg1 Lq hlwkhu fdvhv wklv pd| orrn dv li wkh frxuw kdg sulruv zlwk Wkh mxgjh grhv qrw dvfhuwdlq wkh wuxwk lq dq| uhdo vhqvh1 Zkdw kh grhv lv wr jlyh d ghflvlrq rq wkh hylghqfh suhvhqwhg wr klp111zklfk lv lqfrpsohwh1 +Hjjohvwrq/ 4<;6, <.

(9) Suhsrqghudqfh ri Hylghqfh. 9. uhvshfw wr wkh lqmxuhu*v ehkdylru/ exw zkdw lv uhdoo| dw vwdnh lv wkh frxuw*v dwwhpsw wr lqwhusuhw wkh hylghqfh1 Rxu dqdo|vlv lq wklv uhvshfw eruurzv iurp wkh olwhudwxuh rq dgyhuvduldo shuvxdvlrq jdphv/ dv lq Vreho +4<;8,/ Plojurp dqg Urehuwv +4<;9,/ Vklq +4<<7/ 4<<;, dqg Ghzdwulsrqw dqg Wluroh +4<<<,1 Rqh glhuhqfh lv wkdw zh dgguhvv wkh glvforvxuh sureohp zlwk wkh remhfwlyh ri doljqlqj lqfhqwlyhv wr wdnh fduh1 Dq| vrsklvwlfdwhg vnhswlflvp rq wkh sduw ri wkh frxuw lv wkhq rq wkh edvlv ri wkh pruh olnho| wkdq qrw fulwhulrq zlwk uh0 vshfw wr wkh ghihqgdqw*v fduh/ jlyhq wkh frxuw*v eholhiv derxw wkh lqirupdwlrq dydlodeoh wr wkh sduwlhv1 Zh wdnh lw wkdw wkh sxusrvh ri oldelolw| uxohv lv wr plqlpl}h wkh vxp ri fduh frvwv dqg h{shfwhg dfflghqw orvvhv1 Rxu uhvxowv qhhg wr eh txdolhg li/ lq dgglwlrq wr wkhvh sulpdu| frvwv/ wkhuh duh qrq qhjoljleoh sulydwh dqg sxeolf frvwv ri suhvhqwlqj dqg khdulqj hylghqfh ru li wkh frvw lpsrvhg rq wkh lqmxuhu e| d uxolqj ri qhjoljhqfh lv qrw d sxuh wudqvihu wr wkh ylfwlp1 Vxfk vhfrqgdu| frvwv lqwurgxfh d wudgh0r ehwzhhq wkh ghwhuuhqfh surylghg e| oldelolw| uxohv dqg wkh frvwv ri wkh wruw phfkdqlvp lwvhoi1 D srvvleoh lpsolfdwlrq lv wkdw wuxh sulruv derxw wkh ghihqgdqwv* fduh pd| qrz pdwwhu1 Lq sduwlfxodu/ wkh| surylgh dq dgglwlrqdo ghwhuplqdqw iru wkh rswlpdo dvvljqphqw ri wkh exughq ri surri/ dorqj wkh olqhv ri Vreho +4<;8, ru Kd| dqg Vslhu +4<<:,1 Wkh sdshu lv rujdql}hg dv iroorzv1 Lq vhfwlrq 5 zh suhvhqw wkh edvlf iudphzrun dqg ghqh oldelolw| uxohv1 Vhfwlrq 6 dqdo|vhv wkh phfkdqlvp ghvljq sureohp zkhq vrflhw| kdv gluhfw dffhvv wr hylghqfh1 Vhfwlrq 7 glvfxvvhv wkh vwdqgdug ri surri1 Lq vhfwlrqv 8 dqg 9 zh ghdo zlwk wkh phfkdqlvp ghvljq sureohp zkhq rqo| wkh sduwlhv kdyh gluhfw dffhvv wr hylghqfh dqg ghulyh wkh lpsolfdwlrqv iru wkh doorfdwlrq ri wkh exughq ri surri1 Lq vhfwlrq : zh vkrz krz wkh rswlpdo phfkdqlvp lv lpsohphqwhg wkurxjk wkh suhsrqghudqfh uxoh/ zlwk frxuwv prghohg dv sod|huv lq wkh jdph1 Ilqdoo|/ h{whqvlrqv vxfk dv olwljdwlrq frvwv duh glvfxvvhg lq wkh frqfoxglqj vhfwlrq1. 5 Wkh prgho Lq wkh frxuvh ri wkhlu dfwlylwlhv/ lqglylgxdov pd| lpsrvh dq dfflghqwdo orvv ri dprxqw u rq d wklug sduw|1 Wkh suredelolw| ri fdxvlqj kdup ghshqgv rqo| rq wkh srwhqwldo lqmxuhuv* ohyho ri fduh/ zklfk fdq wdnh wzr ydoxhv uhihuuhg wr dv kljk dqg orz fduh zlwk rssruwxqlw| frvw S : S, dqg suredelolw| ri dfflghqw R R, uhvshfwlyho|1 Doo lqglylgxdov duh ulvn qhxwudo dqg kljk fduh lv wkh.

(10) Suhsrqghudqfh ri Hylghqfh. :. vrfldoo| h!flhqw dfwlrq/ wkdw lv. Ru n S R, u n S, 1 +4, Xqghu wkh vwulfw oldelolw| uxoh/ lqglylgxdov duh khog oldeoh iru dq| kdup wkh| pd| fdxvh1 Dvvxplqj fdxvdolw| lv dozd|v hvwdeolvkhg zlwkrxw huuru/ lqmxuhuv wkhq sd| u ru xs wr wkhlu zhdowk ru oldelolw| olplw / li wkh odwwhu lv vpdoohu1 Wkxv/ dq djhqw zlwk zhdowk kdv klv sulydwh lqfhqwlyhv doljqhg zlwk wkrvh ri vrflhw| li +5, R 4?dc uo n S R, 4?dc uo n S, 1 Wkh zhdowk wkuhvkrog dw zklfk d srwhqwldo lqmxuhu lv mxvw lqglhuhqw ehwzhhq hlwkhu ohyho ri fduh lv 7 ' RS RS, +6, , Lqglylgxdov zlwk zhdowk vpdoohu wkdq 7 wkhuhiruh xqghuwdnh lqdghtxdwh fduh/ zkloh rwkhuv surgxfh uvw0ehvw fduh1 Xqghu wkh qhjoljhqfh uxoh/ lq dgglwlrq wr dvfhuwdlqlqj wkh rffxuuhqfh ri kdup dqg wkh fdxvdo uhodwlrq zlwk wkh ghihqgdqw*v dfwlrqv/ frxuwv frqvlghu dgglwlrqdo lqirupdwlrq derxw wkh vshflf flufxpvwdqfhv ri wkh dfflghqw1 Dq lqmxuhu lv wkhq khog oldeoh iru kdup li wkh frxuw lv vdwlvhg/ rq wkh edvlv ri wkh hylghqfh surgxfhg dw wuldo/ wkdw kh glg qrw h{huw gxh fduh +frxuwv duh dvvxphg wr htxdwh gxh fduh zlwk wkh vrfldoo| h!flhqw dfwlrq,1 Zkdw nlqg ri hylghqfh lv dgplvvleoh lq frxuw/ zklfk vlgh ehduv wkh exughq ri surgxflqj wkh hylghqfh/ krz wkh vxeplwwhg hylghqfh lv hydoxdwhg dqg rwkhu surfhgxudo lvvxhvduh doo fkdudfwhulvwlfv ri d qhjoljhqfh uxoh1 Xqghu d vshflf uxoh/ lqglylgxdov dqwlflsdwh wkh suredelolw| ri ehlqj irxqg qhjoljhqw iroorzlqj wkh rffxuuhqfh ri kdup dqg jlyhq wkhlu ohyho ri fduh1 Zh ghqrwh zlwk k dqg k, wkh uhvshfwlyh suredelolwlhv= k pd| eh lqwhusuhwhg dv wkh suredelolw| ri w|sh L huuru wkh lqmxuhu lv irxqg qhjoljhqw hyhq wkrxjk kh surgxfhg kljk fduh> vlploduo|/ k, lv wkh suredelolw| ri d w|sh LL huuru wkh lqmxuhu lv qrw khog oldeoh hyhq wkrxjk kh xqghusurgxfhg fduh1 Djhqwv zlwk zhdowk wkhuhiruh kdyh lqfhqwlyhv wr h{huflvh gxh fduh li Rk 4?dc uo n S R, k, 4?dc uo n S, / +7, lpso|lqj wkh zhdowk wkuhvkrog ' R kS RS,k +8, , , . . . . .

(11) Suhsrqghudqfh ri Hylghqfh. ;. Lq fkrrvlqj ehwzhhq oldelolw| uxohv/ vrflhw| zrxog zdqw wr wdnh lqwr df0 frxqw lqfhqwlyhv wr h{huflvh fduh/ wrjhwkhu zlwk wkh sulydwh dqg sxeolf frvwv dvvrfldwhg zlwk wkh uxohv wkhpvhoyhv1 Lq wkh suhvhqw frqwh{w/ frpsduhg wr vwulfw oldelolw|/ wkh qhjoljhqfh uxoh fdq lqgxfh pruh lqglylgxdov wr h{huw gxh fduh rqo| li 7 ru htxlydohqwo| R, k, Rk : R, R 1 Zkhwkhu wklv frqglwlrq krogv reylrxvo| ghshqgv rq wkh fkdudfwhulvwlfv ri wkh qhjoljhqfh uxoh xqghu frqvlghudwlrq dqg suhvxpdeo| dovr rq wkh lqirupdwlrqdo frqwhqw ri wkh hylghqfh1 Pruh jhqhudoo|/ vrflhw|*v sureohp lv wr fkrrvh dqg ghvljq uxohv zlwk uh0 vshfw wr d zkroh duud| ri srvvleoh vlwxdwlrqv ru hqylurqphqwv vxfk dv wkh rqh ghvfulehg deryh1 Hdfk vlwxdwlrq fruuhvsrqgv wr sduwlfxodu ydoxhv / u/ S/ R / S, / R, dqg wr d vshflfdwlrq ri wkh glvwulexwlrq ixqfwlrqv ri hylghqfh1 Ohw hdfk vxfk vlwxdwlrq eh lghqwlhg e| r 7 / zkhuh 7 lv vrph lqgh{ vhw zlwk phdvxuh S uh hfwlqj uhodwlyh iuhtxhqf|1 Dw wklv srlqw/ zh qhhg qrw ixuwkhu vshfli| wkh vwuxfwxuh ri 7 / exw dvvxph wkdw +4, krogv iru doo r 7 +l1h1 lq hyhu| vlwxdwlrq wkhuh lv wkh srvvlelolw| wkdw lqglylgxdov xqghuwdnh ohvv wkdq gxh fduh,1 Dovr/ zh wdnh lw wkdw wkhuh lv dw ohdvw d vxevhw ri vlwxdwlrqv lq zklfk lqglylgxdov zrxog qrw eh fduhixo xqghu vwulfw oldelolw|1 Iru hdfk vlwxdwlrq r/ jlyhq wkh ohyho ri fduh/ sulpdu| frvwv duh Ec r ' R EruEr n S Erc ' c , +9, Ghshqglqj rq lqfhqwlyhv/ lqglylgxdov lq hdfk sduwlfxodu hqylurqphqw xqghu0 wdnh vrph ohyho ri fduh1 Ghqrwlqj wkh fkrvhq ohyho e| Er/ wkh djjuhjdwh frvw wr vrflhw| lv ] ' E Erc r _S +:, . . 5. 5. 7. Lq zkdw iroorzv/ zh glvuhjdug wkh frvwv ri wkh wruw phfkdqlvp lwvhoi dqg frqvlghu zkdw uxohv plqlpl}h djjuhjdwh sulpdu| frvwv/ ghqhg dv wkh vxp ri fduh frvwv dqg dfflghqw orvvhv1. 6 Klgghq dfwlrq Zh lqlwldoo| ljqruh dq| ri wkh gl!fxowlhv dvvrfldwhg zlwk ohjdo surfhgxuhv dqg uhgxfh wkh lvvxh wr d vlpsoh sulqflsdo0djhqw sureohp zlwk klgghq dfwlrq1.

(12) Suhsrqghudqfh ri Hylghqfh. <. Vrflhw| +dv wkh sulqflsdo, lv dvvxphg wr revhuyh wkh rffxuuhqfh ri kdup/ xsrq zklfk lw lv deoh wr lghqwli| wkh lqmxuhu lqyroyhg dqg wkh vlwxdwlrq r lq zklfk kdup rffxuv1 Lq dgglwlrq/ wkh sulqflsdo wkhq dovr revhuyhv d vljqdo fruuhodwhg zlwk wkh lqmxuhu*v fduh ohyho1 Wklv lv uhsuhvhqwhg e| d udqgrp yhfwru f U? zlwk frqglwlrqdo ghqvlw| ixqfwlrqv sE%cr dqg s, E%c r/ ghshqglqj rq wkh ohyho ri fduh h{huwhg e| wkh lqmxuhu1 Wkh vljqdo pd| eh lqwhusuhwhg dv lqirupdwlrq derxw wkh flufxpvwdqfhv ri wkh sduwlfxodu dfflghqw/ zklfk pd| ehdu vrph uhodwlrq wr wkh lqmxuhu*v ehkdylru1 Wkh sulqflsdo*v sureohp lv wr ghvljq dq lqfhqwlyh vfkhph jlyhq lqirupdwlrq uhyhdohg rqo| xsrq wkh rffxuuhqfh ri kdup1 Wklv fdswxuhv rqh fkdudfwhulvwlf ri wkh wruw phfkdqlvp1 Dq dgglwlrqdo fkdudfwhulvwlf lv wkdw lqfhqwlyhv duh wr eh surylghg rqo| wkurxjk prqhwdu| shqdowlhv1 Wkhvh duh erxqghg e| wkh dprxqw ri wkh ylfwlp*v orvv ru wkh lqmxuhu*v oldelolw| olplw143 Jlyhq wkhvh uhvwulfwlrqv/ wkh lqfhqwlyh vfkhph xqghu frqvlghudwlrq lv d shqdow| ixqfwlrq |E%c r lpsrvhg rqo| lq wkh fdvh ri dfflghqw dqg vdwlvi|lqj 5. f |E%c r 4?dErc uEro . +;,. Wkhuh duh qr sduwlflsdwlrq frqvwudlqwv dv hyhu|rqh lv vxemhfw wr wkh odz1 Wkh sulqflsdo wkhuhiruh fkrrvhv |E%cr jlyhq wkh deryh frqvwudlqw vr dv wr plqlpl}h 1 Lq dq| vlwxdwlrq wkh uvw ehvw ohyho ri fduh lv xqghuwdnhq li +wr vlpsoli| qrwdwlrq zh rplw h{solflw uhihuhqfh wr r zkhq wkhuh lv qr dpeljxlw|, ]. ]. S n R |E%sE% _% S, n R, |E%s, E% _% . +<,. Idfwrul}lqj whupv/ wkh frqglwlrq lv htxlydohqw wr ]. |E%dR, s, E% Rs E%o _% S S, . . . +43,. Dq rswlpdo vfkhph pd{lpl}hv wkh qxpehu ri vlwxdwlrqv zkhuh gxh fduh lv xqghuwdnhq1 Vxfk d vfkhph lv qrw xqltxh/ exw rqh reylrxv vroxwlrq lv wr fkrrvh wkh shqdow| ixqfwlrq vr wkdw/ lq hyhu| vlwxdwlrq/ wkh lqwhjudo rq wkh ohiw kdqg vlgh ri +43, lv pd{lpl}hg1 Zkhq wkh vtxduh eudfnhw lq wkh lqwhjudo lv srvlwlyh/ wklv uhtxluhv vhwwlqj wkh shqdow| dv kljk dv srvvleoh> zkhq lw H{foxglqj prqhwdu| sxqlwlyh gdpdjhv lpsolhv qr uhdo uhvwulfwlrqv khuh/ vlqfh lqh!0 flhqflhv h{lvw rqo| ehfdxvh ri wkh olplwhg oldelolw| frqvwudlqwv1 43.

(13) Suhsrqghudqfh ri Hylghqfh. 43. lv qhjdwlyh/ wkh shqdow| vkrxog eh dv vpdoo dv srvvleoh1 Ilqdoo|/ zkhq wkh vtxduh eudfnhw mxvw ydqlvkhv wkh vl}h ri wkh shqdow| lv lqfrqvhtxhqwldo iru vhwwlqj lqfhqwlyhv1 Wkxv/ rqh ehvw vfkhph lv R, s, E% : R sE% W +44, | E% ' 4?dfc uo zkhq zkhq R, s, E% R sE% Wklv vfkhph pd| qrw eh xqltxh hyhq lq d phdvxuh wkhruhwlfdo vhqvh/ ehfdxvh wkh vhw ri uhdol}dwlrqv ri wkh vljqdo vdwlvi|lqj R, s, E% ' Rs E% frxog zhoo eh odujh1 Pruhryhu/ dv douhdg| qrwhg/ wkhuh pd| eh rwkhu shqdow| vfkhphv wkdw gr qrw vdwlvi| +44,/ exw lqgxfh wkh vdph ehkdylru ri doo lqglylgxdov144 Krzhyhu/ irfxvlqj rq |W doorzv d vwudljkwiruzdug lqwhusuhwdwlrq lq whupv ri d qhjoljhqfh uxoh dsso|lqj d pruh olnho| wkdq qrw vwdqgdug ri surri1 Revhuyh wkdw dq| shqdow| ixqfwlrq vdwlvi|lqj +;, fdq eh zulwwhq dv. |E%cr ' E%c r 4?dErc uEroc E%c r dfc o Wkh suhylrxv uhvxow fdq wkhq eh uhirupxodwhg dv iroorzv1 5. +45,. Sulpdu| frvwv duh plqlpl}hg e| dq| shqdow| vfkhph |W E%c r ' WE%c r 4?dErc uEro zlwk WE%c r 5 ifc j dqg. Sursrvlwlrq 4. . zkhq R, Ers, E%c r : R Ers E%c r E%c r ' f zkhq +46, R, Ers, E%c r RErsE%c r Dq hvvhqwldo fkdudfwhulvwlf ri wklv vroxwlrq lv wkdw wkh sulqflsdo*v ghflvlrq wr shqdol}h dqg wkh vl}h ri wkh shqdow| duh vhsdudeoh1 Lq wklv wzr0vwhs vfkhph/ W fdq eh xqghuvwrrg dv vrflhw|*v ghflvlrq fulwhulrq wr ghwhuplqh zkhwkhu dq lqmxuhu vkrxog eh khog oldeoh ru qrw1 Li khog oldeoh/ lqmxuhuv sd| gdpdjhv hlwkhu lq ixoo ru xs wr wkhlu zhdowk olplw1 Lq wkh uvw vwhs/ wkh |hv ru qr ghflvlrq uhjduglqj oldelolw| ghshqgv rq wkh vlwxdwlrq r rqo| wkurxjk wkh olnholkrrg ixqfwlrqv Rs dqg R, s, ri gxh fduh yhuvxv lqdghtxdwh fduh1 Oldelolw| lv wkhq ghwhuplqhg rq wkh edvlv ri d pruh0olnho|0wkdq0qrw fulwhulrq/ jlyhq wkh rffxu0 uhqfh ri kdup dqg wkh uhdol}dwlrq ri wkh dgglwlrqdo hylghqfh uhsuhvhqwhg e| f 1 Wkxv/ dq lqglylgxdo fdxvlqj kdup lv shqdol}hg li fduhohvvqhvv lv h{ srvw W. 44. Iru lqvwdqfh/ w+{> v,. @. . O+v, w. +{> v,. li O+v, z+v, rwkhuzlvh.

(14) Suhsrqghudqfh ri Hylghqfh. 44. pruh olnho|> frqyhuvho|/ kh lv qrw wr eh shqdol}hg zkhq dghtxdwh fduh dsshduv pruh olnho|1 Zkhq erwk dfwlrqv duh htxdoo| olnho| wkh doorfdwlrq ri gdpdjhv lv luuhohydqw1 Lq vrph vlwxdwlrqv wkhuh pljkw eh uhdol}dwlrqv ri wkh hylghqfh iru zklfk wkhuh lv shuihfw lqirupdwlrq derxw wkh lqmxuhu*v ehkdylru1 Iru lqvwdqfh/ zkhq sE% ' f dqg s, E% : fc d sulqflsdo vhhlqj % zrxog nqrz iru vxuh wkdw wkh lqmxuhu zdv fduhohvv1 Wr doljq lqfhqwlyhv/ lqmxuhuv vkrxog wkhq eh shqdol}hg1 Vlploduo|/ li sE% : f dqg s, E% ' f/ wkh sulqflsdo nqrzv wkdw wkh lqglylgxdo wrrn dghtxdwh fduh dqg/ dv d frqvhtxhqfh/ vkrxog qrw eh shqdol}hg1 Wkh sursrvlwlrq vkrzv wkdw lw lv rswlpdo wr xvh d pruh0olnho|0wkdq0qrw fulwhulrq wr h{whqg wkhvh frpprq vhqvh shqdow| dvvljqphqw uxohv wr flufxpvwdqfhv zkhuh srvw0dfflghqw lqirupdwlrq grhv qrw shuihfwo| uhyhdo wkh lqmxuhu*v eh0 kdylru1 Ri frxuvh/ dv lv vwdqgdug lq sulqflsdo0djhqw prghov/ wkhuh lv qr dfwxdo h{ srvw xqfhuwdlqw| vlqfh/ xqghu wkh vfkhph/ d sulqflsdo revhuylqj r zrxog xqghuvwdqg wkh lqmxuhu*v lqfhqwlyhv dqg zrxog wkhuhiruh nqrz zkhwkhu kh h{huwhg gxh fduh ru qrw1. 7 Wkh vwdqgdug ri surri Rwkhu lqfhqwlyh vfkhphv dovr plqlpl}h djjuhjdwh sulpdu| frvwv/ exw wkhuh lv d vhqvh lq zklfk wkh wzr0vwhs vfkhph ghvfulehg deryh lv xqltxh1 Frqvlghu wkh sureohp ri lpsohphqwlqj dq lqfhqwlyh vfkhph e| ghohjdwlqj ghflvlrqv wr frxuwv wkurxjk d oldelolw| uxoh1 Dq| vxfk uxoh zloo eh fkdudfwhul}hg e| lwv lqirupdwlrqdo uhtxluhphqwv dqg lwv ghflvlrq fulwhulrq1 Rxu wzr0vwhs vfkhph fruuhvsrqgv wr d qhjoljhqfh uxoh zkhuh oldelolw| lv ghwhuplqhg rqo| rq wkh edvlv ri lqirupdwlrq derxw wkh lqmxuhu*v ehkdylru dqg zkhuh wkh ghflvlrq fulwhulrq wdnhv wkh irup ri d vwdqgdug ri surri1 Vxfk d vfkhph lv eurdgo| frqvlvwhqw zlwk krz wruw odz rshudwhv1 Iru lqvwdqfh/ lq wkh XV uhvsrqvlelolw| iru dvvhvvlqj qhjoljhqfh +wkh |hv ru qr ghflvlrq, lv riwhq dwwulexwhg wr d od| mxu| zkr zloo eh lqvwuxfwhg rq zkdw hylghqfh lv dgplvvleoh dqg rq wkh uhtxluhg vwdqgdug ri surri1 Hyhq zkhq wkhuh lv qr mxu|/ wkh vdph wzr0vwhs dssurdfk jhqhudoo| krogv dv d pdwwhu ri sulqflsoh lq flylo wuldov145 Iurp wkh srlqw ri ylhz ri lpsohphqwdwlrq/ d qdwxudo txhvwlrq wr eh dg0 Xqghu Ihghudo Uxoh ri Flylo Surfhgxuh 75+e,/ lq rughu wr dyrlg suhmxglfh/ frxuwv vrphwlphv kdyh wkh rswlrq ri frqgxfwlqj d vhsdudwh wuldo wr uvw ghwhuplqh zkhwkhu wkh lqmxuhu lv oldeoh +lq wklv wuldo hylghqfh derxw wkh h{whqw ri gdpdjhv ru wkh lqmxuhu*v fdsdflw| wr sd| lv lqdgplvvleoh,1 45.

(15) Suhsrqghudqfh ri Hylghqfh. 45. guhvvhg lv zkdw plqlpdo lqirupdwlrq vkrxog dv d uxoh eh frppxqlfdwhg wr frxuwv iru wkhp wr eh deoh wr pdnh wkh dssursuldwh elqdu| ghflvlrq uhjduglqj oldelolw|11 Wklv txhvwlrq vkrxog eh dgguhvvhg iurp wkh vwdqgsrlqw ri ghvljqlqj d jhqhudo uxoh wkdw lv vlwxdwlrq lqghshqghqw1 Wkdw lv/ ehiruh nqrzlqj zkdw vlwxdwlrqv zloo rffxu/ lw pxvw eh ghflghg zkhwkhu dv d uxoh wkh zkroh ri r zloo eh frppxqlfdwhg ru rqo| sduw ri lw dqg zkhwkhu dv d uxoh wkh uhdol}dwlrq ri f zloo dovr eh frppxqlfdwhg1 Pruhryhu/ vrflhw| pxvw ehiruhkdqg lpsrvh rq frxuwv vrph jhqhudo ghflvlrq fulwhulrq/ whoolqj wkhp zkdw wr gr zlwk zkdwhyhu lqirupdwlrq zloo eh pdgh dydlodeoh wr wkhp146 Wkh suhfhglqj uhvxowv vkrz wkdw lw lv vx!flhqw wr frppxqlfdwh wkh olnhol0 krrg ixqfwlrqv Rs dqg R, s, / wrjhwkhu zlwk wkh uhdol}dwlrq ri f 1 Li wkh vhw 7 ri srvvleoh vlwxdwlrqv lv vx!flhqwo| ydulhg/ frppxqlfdwlqj wklv lqirupdwlrq lv dovr qhfhvvdu| iru lpsohphqwlqj dq rswlpdo vfkhph1 Lq rwkhu zrugv/ li dv d uxoh R s/ R, s, dqg f zhuh qrw frppxqlfdwhg/ frxuw ghflvlrqv zrxog rq dyhudjh ohdg wr vwulfwo| kljkhu sulpdu| frvwv/ luuhvshfwlyh ri zkdw rwkhu lqirupdwlrq lv pdgh dydlodeoh1 Plqlpdo lqirupdwlrqdo uhtxluhphqwv duh uh0 odwhg wr wkh qrwlrq ri wkh lqirupdwlrqdo h!flhqf| ri d uhvrxufh doorfdwlrq v|vwhp1 Dffruglqj wr wkh Kxuzlf} fulwhulrq +vhh iru lqvwdqfh Plojurp dqg Urehuwv/ 4<<5,/ lqirupdwlrqdo h!flhqf| ghshqgv rq krz pxfk lqirupdwlrq lw wdnhv wr ghwhuplqh zkhwkhu d sduwlfxodu uhvrxufh doorfdwlrq lv h!flhqw1 Lq wkh suhvhqw fdvh/ zh dvn zkdw plqlpdo lqirupdwlrq dq rxwvlgh revhuyhu zrxog uhtxluh lq rughu wr fkhfn zkhwkhu d sduwlfxodu doorfdwlrq ri |hv ru qr ghfl0 vlrqv xsrq wkh rffxuuhqfh ri kdup lv frqvlvwhqw zlwk plqlpl}lqj djjuhjdwh sulpdu| frvwv1 Li pruh lqirupdwlrq zhuh pdgh dydlodeoh lq hyhu| vlwxdwlrq/ pdq| ghflvlrq uxohv frxog lpsohphqw dq rswlpdo vfkhph1 D uhodwlyho| vlpsoh rqh/ zkhuh dvvhvvlqj oldelolw| dovr uhtxluhv nqrzohgjh ri dqg u/ lv wkh ghhs0srfnhw uxoh uEr Er E%c r ' WE%c r lirwkhuzlvh +47, Wkhuh duh rwkhu vwloo pruh frpsolfdwhg uxohv1 Zh nqrz wkdw W lv vx!flhqw iru rswlpdo lqfhqwlyhv xqghu zkdw kdv mxvw ehhq ghvfulehg dv wkh plqlpxp lqirupdwlrq vhw1 Surylghg 7 lv vx!flhqwo| ulfk/ wklv ghflvlrq fulwhulrq lv dovr qhfhvvdu|1 Wkdw lv/ li d jhqhudo ghflvlrq uxoh lv rqh zklfk pxvw eh xvhg lq doo frqfhlydeoh vlwxdwlrqv/ wkh pruh olnho| wkdq qrw fulwhulrq lv wkh rqo| rqh frq0 . Jhqhudo uxohv/ jhqxlqh odzv/ dv glvwlqjxlvkhg iurp vshflf rughuv/ pxvw 111 eh lqwhqghg wr rshudwh lq flufxpvwdqfhv zklfk fdqqrw eh iruhvhhq lq ghwdlo1 +Kd|hn/ 4<77, 46.

(16) Suhsrqghudqfh ri Hylghqfh. 46. vlvwhqw zlwk rswlpdo lqfhqwlyhv xqghu plqlpdo lqirupdwlrqdo uhtxluhphqwv1 Wkh qhjoljhqfh uxoh wrjhwkhu zlwk wkh pruh olnho| wkdq qrw vwdqgdug lv wkh rqo| jhqhudo uxoh iru dvvhvvlqj oldelolw| wkdw kdv plqlpxp lq0 irupdwlrqdo uhtxluhphqwv frqvlvwhqw zlwk plqlpl}lqj djjuhjdwh sulpdu| frvwv1 Sursrvlwlrq 5. Rswlpdolw| kdv douhdg| ehhq vkrzq1 Wr suryh qhfhvvlw| dv d jhqhudo uxoh/ rqh qhhg rqo| h{klelw d vhw 7 ri vlwxdwlrqv wkdw lv vx!flhqwo| ydulhg1 Jlyhq dq duelwudu| uxoh / wkh frqglwlrqdo suredelolw| ri ehlqj irxqg oldeoh xsrq wkh rffxuuhqfh ri kdup lv ]. k Er ' E%c rs E%c r _% ' c ,. +48,. Xqghu wkh pd{lpxp olnholkrrg uxoh W/ wkhvh frqglwlrqdo suredelolwlhv duh fkrvhq vr dv wr pd{lpl}h lqfhqwlyhv1 Vlqfh k ' k, ' lv dozd|v ihdvleoh +e| vhwwlqj ,/ lw qhfhvvdulo| iroorzv wkdw lq doo vlwxdwlrqv +49, ' R kSW RS, kW RS RS, ' 7 , , , Wkh zhdowk wkuhvkrog 7 rq wkh uljkw kdqg vlgh lv wkh rqh fkdudfwhul}lqj wkh vwulfw oldelolw| uxoh/ zklfk rqo| uhtxluhv nqrzohgjh ri wkh rffxuuhqfh ri kdup1 Wkh wkuhvkrog fkdudfwhul}hv wkh qhjoljhqfh uxoh zlwk wkh pruh olnho| wkdq qrw vwdqgdug1 Wkh fodlp lq wkh sursrvlwlrq lv wuxh li ryhu d vxevhw ri vlwxdwlrqv zlwk srvlwlyh phdvxuh wkh iroorzlqj krogv= +l, wkh lqhtxdolw| lq +49, lv vwulfw> +ll, wkhuh duh lqglylgxdov zlwk vx!flhqwo| vpdoo zhdowk> +lll, wkh pruh olnho| wkdq qrw uxoh lv qhfhvvdu| iru pd{lpl}lqj ghwhuuhqfh ghqhg dv B ' R, k, R k +4:, Frqglwlrq +l, krogv li f surylghv vx!flhqw dgglwlrqdo lqirupdwlrq/ frp0 sduhg wr vlpso| nqrzlqj wkdw dq dfflghqw rffxuuhg1 Wkh lqirupdwlrq frqwhqw ri f ghshqgv rq wkh yduldelolw| ri wkh olnholkrrg udwlr s, E%*sE%1 D wrwdoo| xqlqirupdwlyh f fruuhvsrqgv wr d frqvwdqw olnholkrrg udwlr htxdo wr xqlw|1 Hyhq li wklv udwlr lv qrw frqvwdqw/ wkh lqirupdwlrqdo frqwhqw ri f pd| eh wrr zhdn wr hyhu ryhuwxuq wkh olnholkrrg ri orz fduh uhvxowlqj iurp wkh rffxuuhqfh ri dq dfflghqw147 Zh lqwurgxfh wkh iroorzlqj ghqlwlrq= . . . . . . . Wklv frxog eh wkh fdvh iru lqvwdqfh zkhq wkh phuh rffxuuhqfh ri kdup vshdnv iru lwvhoi/ dv zkhq sk * 31 Dq duelwudulo| vpdoo io +{,@ik +{, lv wkhq uhtxluhg iru so io +{, sk ik +{, ? 31 47.

(17) Suhsrqghudqfh ri Hylghqfh Ghilqlwlrq 4= f. lv lqirupdwlyh li. U. R. ]. .. sE% _% R, . s E% _% : f. . . R, s, E% RsE% Dq lqirupdwlyh f lpsolhv uhdol}dwlrqv vxfk wkdw. 47. / zkhuh. .. lv wkh vhw ri. 1. ]. .. s, E% _% ' RE kW R, E kW, : fc . . . +4;,. zklfk dprxqwv wr R, kW, R kW : R, R1 Wkxv/ d vx!flhqw +dqg lqghhg qhfhvvdu|, frqglwlrq iru +l, wr krog lv wkdw f eh lqirupdwlyh dv ghqhg deryh1 Wr irupdol}h frqglwlrq +ll,/ vxssrvh wkhuh lv d vxevhw 7 ri vlwxdwlrqv wkdw glhu rqo| zlwk uhvshfw wr wkh lqglylgxdov* zhdowk1 Dvvxph dovr wkdw SE e 7 : f iru doo e deryh vrph vx!flhqwo| vpdoo ohyho dqg ohw wklv eh d frqwlqxrxv ixqfwlrq lq e1 Lqfuhdvlqj ghwhuuhqfh wkhq dozd|v lqfuhdvhv wkh qxpehu ri lqglylgxdov lq 7 h{huwlqj gxh fduh1 Li frqglwlrq +l, dovr krogv lq 7 / pd{lpl}lqj ghwhuuhqfh uhtxluhv oldelolw| ghflvlrqv wr eh frqglwlrqhg rq wkh lqirupdwlrq frqyh|hg e| f 1 Khqfh/ dv d uxoh/ frppxqlfdwlqj wkh dgglwlrqdo hylghqfh +wrjhwkhu zlwk R, s, dqg R s, lv qhfhvvdu| wr plqlpl}h djjuhjdwh sulpdu| frvwv1 Ilqdoo|/ frqvlghu wkh frxuwv* ghflvlrq uxoh1 Pd{lpl}lqj ghwhuuhqfh lv lq0 vxuhg e| W1 Krzhyhu/ lq dq| vlwxdwlrq/ vdwlvi|lqj W doprvw hyhu|zkhuh lv dovr qhfhvvdu| lq rughu wr pd{lpl}h ghwhuuhqfh1 Wkhuhiruh/ li wkhuh duh vxevhwv ri vlwxdwlrqv vxfk dv 7 zkhuh frqglwlrqv +l, dqg +ll, krog/ oldelolw| ghflvlrqv vkrxog dv d uxoh eh ghwhuplqhg dffruglqj wr wkh suhsrqghudqfh vwdqgdug1 . . . m. 8 Glvforvxuh Xqwlo qrz zh kdyh dvvxphg dq h{wuhph lqtxlvlwruldo zruog zkhuh vrflhw| kdv gluhfwo| revhuyhg wkh rffxuuhqfh ri dfflghqwv dqg wkh dvvrfldwhg hylghqfh1 Lq uhdolw|/ wkh lqflghqfh ri kdup dqg dq| dgglwlrqdo lqirupdwlrq uhvw zlwk wkh sduwlhv lqyroyhg1 Wklv lqwurgxfhv dq dx{loldu| sureohp ri klgghq lqirupdwlrq1 Lq rughu wr prwlydwh fduhixo ehkdylru rq wkh sduw ri srwhqwldo lqmxuhuv/ oldelolw| uxohv pxvw dovr surylgh lqfhqwlyhv iru wkh sduwlhv wr glvforvh hylghqfh1 Zh uhyhuw wr d sxuh phfkdqlvp ghvljq dssurdfk dqg surfhhg dv lq vhfwlrq 6/ wdnlqj lqwr dffrxqw wkh sduwlhv* lqfhqwlyhv wr glvforvh ru frqfhdo hylghqfh1 Lpsohphqwdwlrq/ zlwk ghflvlrqv ghohjdwhg wr frxuwv/ lv glvfxvvhg lq vhfwlrq :1 Wr { lghdv/ frqvlghu uvw wkh olplwlqj fdvh zkhuh lw lv frpprq nqrzohgjh wkdw erwk sduwlhv dozd|v revhuyh doo wkh srwhqwldo hylghqfh1 Dv lv zhoo nqrzq/.

(18) Suhsrqghudqfh ri Hylghqfh. 48. zkhq sduwlhv kdyh gldphwulfdoo| rssrvhg lqwhuhvwv lw lv vwudljkwiruzdug wr lq0 gxfh uhyhodwlrq ri doo uhohydqw lqirupdwlrq148 Wr vhh wklv/ dvvxph wkdw lq rughu wr hvwdeolvk rffxuuhqfh dqg fdxvdolw| wkh ylfwlp pxvw frppxqlfdwh wkh fkdu0 dfwhulvwlfv ri wkh vlwxdwlrq/ dv fdswxuhg e| r 7 / wrjhwkhu zlwk d vxeyhfwru f ri wkh dgglwlrqdo hylghqfh/ zkhuh f ghvfulehv vrph ri wkh flufxpvwdqfhv ri wkh dfflghqw dqg zkhuh wkh frpsohwh hylghqfh lv f ' Efc f21 Wkh lq0 whusuhwdwlrq lv wkdw wkh ylfwlp fdqqrw exw suhvhqw erwk r dqg wkh uhdol}dwlrq ri f li kh zlvkhv wr hvwdeolvk wkh rffxuuhqfh ri kdup/ exw qhhg qrw glvforvh wkh frpsohwh srwhqwldo hylghqfh1 Rswlpdo phfkdqlvpv duh wkhq d vlpsoh h{whqvlrq ri wkh rqhv h{dplqhg lq wkh suhfhglqj vhfwlrqv dqg wkh| lqgxfh wkh vdph sulpdu| frvwv1 Rqh ehvw vfkhph lv iru wkh ylfwlp wr eh lqghpqlhg rqo| li wkh hqwluh hylghqfh f ' Ef c f2 lv vxeplwwhg wr wkh sulqflsdo dqg lwv uhdol}dwlrq vdwlvhv R, s, E% : R sE%1 Zkhq wkh hylghqfh lv idyrudeoh wr wkh ylfwlp/ kh zloo vxeplw lw dqg wkh lqmxuhu zloo eh irxqg oldeoh1 Lq frqwudvw/ zkhq lw lv xqidyrudeoh/ wkh ylfwlp grhv qrw oh vxlw dqg wkh lqmxuhu lv qrw khog oldeoh1 Exw ri frxuvh/ hyhq li wkh hylghqfh kdg ehhq uhyhdohg/ wkh lqmxuhu zrxog vwloo qrw kdyh ehhq khog oldeoh rq wkh edvlv ri wkh pruh olnho| wkdq qrw fulwhulrq1 Dq dowhuqdwlyh vfkhph lv rqh zkhuhe| wkh ylfwlp lv lqghpqlhg li kh vxeplwv f/ h{fhsw zkhq f2 lv dovr glvforvhg dqg R, s, E% RsE%1 Frqvh0 txhqwo|/ zkhq R, s, E% R sE% wkh ylfwlp ohv vxlw dqg wkh lqmxuhu lv khog oldeoh1 Lq doo rwkhu fdvhv wkh ylfwlp grhv qrw oh vxlw +li kh glg/ wkh lqmxuhu zrxog mxvw vxeplw %2 dqg hvfdsh oldelolw|,1 Lq wkh frqwh{w ri oldelolw| uxohv zlwklq dq dgyhuvduldo surfhgxuh/ wkh glvwlqfwlrq ehwzhhq wkh wzr vfkhphv lv vlpso| zklfk sduw| fduulhv wkh exughq ri surri zlwk uhvshfw wr wkh lvvxh ri qhjoljhqfh1 Xqghu hlwkhu doorfdwlrq ri wkh exughq/ doo uhohydqw lqirupdwlrq lv uhyhdohg dqg lqfhqwlyhv wr wdnh fduh duh wkh vdph dv ehiruh1 Wkh iruhjrlqj dujxphqw uholhv rq wkh dvvxpswlrq wkdw f lv nqrzq wr eh revhuydeoh e| erwk sduwlhv1 Lw grhv qrw jhqhudol}h wr vlwxdwlrqv zkhuh sduwlhv pd| eh lpshuihfwo| lqiruphg dqg wkh sulqflsdo grhv qrw nqrz wkh h{whqw ri wkh lqirupdwlrq dydlodeoh wr wkhp1 Iru lqvwdqfh/ vxssrvh wkh exughq lv rq wkh lqmxuhu zkr grhv qrw dozd|v revhuyh f21 Qrw frppxqlfdwlqj wkh uhdol}dwlrq ri f2 wkhq grhv qrw qhfhvvdulo| lpso| wkdw wkh hylghqfh zrxog kdyh ehhq idyrudeoh wr wkh ylfwlp/ vlqfh wkh lqmxuhu pd| vlpso| eh xqlqiruphg1 Lq zkdw iroorzv zh h{dplqh wkh phfkdqlvp ghvljq sureohp iru wkh fdvh zkhuh 5. . Vhh Plojurp dqg Urehuwv +4<;9,/ Olspdq dqg Vhssl +4<<8, dqg Vhlgpdq dqg Zlqwhu +4<<:,1 48.

(19) Suhsrqghudqfh ri Hylghqfh. 49. vrflhw|/ lq dgglwlrq wr ehlqj xqdeoh wr revhuyh wkh hylghqfh gluhfwo|/ grhv qrw nqrz krz zhoo lqiruphg wkh sduwlhv duh1 Zh dvvxph lw lv frpprq nqrzohgjh wkdw erwk sduwlhv nqrz wkh vlwxdwlrq r/ wkdw wkh| revhuyh f xsrq wkh rffxuuhqfh ri kdup exw rqo| revhuyh f2 zlwk vrph suredelolw|= iru wkh ylfwlp dqg iru wkh lqmxuhu + dqg duh qrz sduw ri wkh vshflfdwlrq ri r,1 Dv sduwlhv pd| eh xqlqiruphg/ zkhq rqo| f lv glvforvhg wkh sulqflsdo grhv qrw nqrz zkhwkhu wklv lv ehfdxvh wkh sduwlhv glg qrw revhuyh doo wkh srwhqwldo hylghqfh ru zkhwkhu dq lqiruphg sduw| fkrvh qrw wr glvforvh f21 Qrwh wkdw zh pdnh wkh frpprq dvvxpswlrq wkdw hylghqfh lv yhuldeoh/ l1h1 idovh hylghqfh fdqqrw eh ideulfdwhg149 Wkh sulqflsdo*v sureohp lv wr ghvljq d wudqvihu vfkhph rq wkh edvlv ri wkh hylghqfh 5 %c E%c %2 vxeplwwhg wr klp1 Dv ehiruh/ zh irfxv rq phfkdqlvpv vdwlvi|lqj wkh frqvwudlqw 5 i. j. f |E5c r 4?dErc uEroc. +4<,. zkhuh |E5c r lv qrz wkh wudqvihu iurp wkh lqmxuhu wr wkh ylfwlp1 Wkh sulqfl0 sdo*v remhfwlyh lv wr fkrrvh wkh wudqvihu vfkhph zklfk plqlpl}hv djjuhjdwh sulpdu| frvwv/ vxemhfw wr +4<, dqg wr dgglwlrqdo glvforvxuh frqvwudlqwv1 Wkh glvforvxuh jdph lv dv iroorzv1 Zkhq dq dfflghqw rffxuv/ erwk sduwlhv revhuyh f dqg srvvleo| dovr f2> wkh ylfwlp wkhq ghflghv zkhwkhu ru qrw wr oh vxlw1 Lq rughu wr hvwdeolvk wkh rffxuuhqfh ri kdup dqg wkh lghqwlw| ri wkh lqmxuhu/ d vxlw hqwdlov wkh vxeplvvlrq ri wkh uhdol}dwlrq ri f wrjhwkhu zlwk wkh fkdudfwhulvwlfv ri wkh vlwxdwlrq r1 Rqfh d vxlw lv ohg/ erwk sduwlhv ghflgh zkhwkhu wr vxeplw dgglwlrqdo hylghqfh1 Wkh| dfw vlpxowdqhrxvo| dqg qhlwkhu nqrzv zkhwkhu wkh rwkhu kdv vhhq wkh frpsohwh srwhqwldo hylghqfh1 Wkh iroorzlqj vwudwhj| sdlu iru lqmxuhu dqg ylfwlp lv hdvlo| vhhq wr frqvwlwxwh dq htxloleulxp iru wklv jdph +zh rplw h{solflw uhihuhqfh wr r,= +l, Li wkh ylfwlp revhuyhv wkh frpsohwh hylghqfh dqg li |E% |E%c %2/ wkh ylfwlp ohv vxlw dqg glvforvhv E%c %2 > zkhq f2 lv qrw revhuyhg ru zkhq |E%c %2 |E%/ wkh ylfwlp ohv vxlw e| vxeplwwlqj rqo| % surylghg |E% : f> lq doo rwkhu fdvhv wkh ylfwlp grhv qrw oh vxlw1 +ll, Li d vxlw kdv ehhq ohg dqg |E%c %2 |E%/ wkh lqmxuhu glvforvhv E%c%2 li kh fdq> rwkhuzlvh kh uhyhdov qrwklqj1 . Lq wkh fxuuhqw frqwh{w/ wkh dvvxpswlrq fdq eh mxvwlhg e| wkh vhyhuh shqdowlhv iru shumxu|1 49.

(20) Suhsrqghudqfh ri Hylghqfh. 4:. Lq htxloleulxp hdfk sduw| glvforvhv lqirupdwlrq rqo| li wklv vwulfwo| ehq0 hwv klp xqghu wkh wudqvihu vfkhph1 Revhuyh wkdw zh kdyh irfxvhg rq dq htxloleulxp zkhuh sduwlhv gr qrw glvforvh zkhq lqglhuhqw1 Wklv fdq eh lq0 whusuhwhg lq whupv ri duelwudulo| vpdoo frvwv ri vxeplwwlqj hylghqfh/ zlwk wkh glvforvxuh ri E%c%2 ehlqj lwvhoi 00pruh frvwo| wkdq vxeplwwlqj % rqo|1 Zkloh wklv kdv qr vxevwdqwldo hhfw rq wkh sulqflsdo*v sureohp/ wkh dvvxps0 wlrq kdv wkh dgydqwdjh wkdw wkh deryh vwudwhjlhv +qrz grplqdqw vwudwhjlhv, fdq eh wudqvsrvhg zlwkrxw prglfdwlrq wr vlwxdwlrqv zkhuh suhvhqwlqj hyl0 ghqfh lv frvwo|/ surylghg ri frxuvh frvwv duh qrw wrr odujh1 Zh eulh | glvfxvv wklv lvvxh lq wkh frqfoxglqj vhfwlrq1 Iru dq| wudqvihu vfkhph/ wkh h{shfwhg oldelolw| frvwv lpsrvhg rq srwhq0 wldo lqmxuhuv ghshqg rq wkh sduwlhv* delolw| wr glvforvh lqirupdwlrq dqg rq wkhlu lqfhqwlyhv wr gr vr/ zkhuh wkhvh iroorz iurp wkh wudqvihu vfkhph lwvhoi1 Krzhyhu/ wkh glvforvxuh sureohp iru f fdq eh glvuhjdughg/ dv wkh rxwfrph lv dozd|v wkh vdph dv li wkh sulqflsdo gluhfwo| revhuyhg wkdw sduw ri wkh srwhqwldo hylghqfh1 Wkh uhdvrq lv wkdw/ lq htxloleulxp/ f uhpdlqv xqglvforvhg rqo| zkhq wkh lqmxuhu*v sd|phqw lv wkh vdph zkhwkhu ru qrw wkhuh lv glvforvxuh14: Uhjduglqj f/ wkh vlwxdwlrq lv wkhuhiruh vlplodu wr wkh fdvh ri shuihfwo| lq0 iruphg sduwlhv glvfxvvhg hduolhu dqg zh fdq surfhhg dv li wkh sulqflsdo qhhghg rqo| eh frqfhuqhg derxw wkh sduwlhv*v delolw| ru lqfhqwlyhv wr glvforvh f21 Lq rughu wr ghulyh wkh h{shfwhg oldelolw| frvwv ri srwhqwldo lqmxuhuv/ lw lv xvhixo wr lqwurgxfh wkh iroorzlqj qrwdwlrq1 Ohw |E% ghqrwh wkh wudqvihu zkhq rqo| % lv glvforvhg1 Dffruglqjo|/ dv d ixqfwlrq ri wkh srwhqwldo hylghqfh/ |E% lv frqvwudlqhg wr eh frqvwdqw lq %21 Ohw |2E% ghqrwh wkh wudqvihu zkhq wkh zkroh hylghqfh lv glvforvhg1 Iru d jlyhq wudqvihu vfkhph/ wkh vhw ri srvvleoh uhdol}dwlrqv % lv sduwlwlrqhg lqwr wkuhh uhjlrqv= rqh zkhuh wkh ylfwlp zrxog vwulfwo| ehqhw iurp glvforvxuh ri wkh frpsohwh hylghqfh +dv rssrvhg wr wkh glvforvxuh ri % rqo|,/ rqh zkhuh wkh lqmxuhu zrxog vwulfwo| ehqhw/ dqg rqh zkhuh erwk sduwlhv zrxog eh lqglhuhqw1 Zh fkdudfwhul}h wkhvh uhjlrqv e| wzr lqglfdwru ixqfwlrqv1 Dv lqglfdwru iru wkh uvw uhjlrq/ ohw ) E% ' li |E% |2E% dqg }hur rwkhuzlvh1 Vlploduo|/ iru wkh vhfrqg uhjlrq ohw )E% ' li |E% : |2E% dqg }hur rwkhuzlvh1 Wkh uhpdlqlqj uhjlrq ghqhg e| |E% ' |2E% kdv wkh lqglfdwru ixqfwlrq E ) E%E )E%1 Wdnlqj lqwr dffrxqw wkh sduwlhv* delolw| dqg lqfhqwlyhv wr glvforvh f2/ wkh . . 4: Zkhq w+{ , A 3/ wkh ylfwlp vxeplwv { dqg wkh lqmxuhu pd| ru pd| qrw dovr uhyhdo { 1 4 4 5 Hlwkhu zd| wkh rxwfrph lv dv li wkh sulqflsdo kdg gluhfwo| revhuyhg {4 1 Zkhq w+{4 , @ 3/ hlwkhu wkh ylfwlp glvforvhv erwk {4 dqg {5 ru qr lqirupdwlrq lv uhyhdohg dqg wkh lqmxuhu sd|v qrwklqj1 Djdlq/ wkh rxwfrph lv wkh vdph dv xqghu gluhfw revhuydwlrq ri {4 1.

(21) Suhsrqghudqfh ri Hylghqfh. 4;. h{shfwhg oldelolw| frvwv wr d srwhqwldo lqmxuhu h{huwlqj wkh fduh ohyho ' c , lv wkhq. R. ]. ) d|2 n E |o n ) d|2 n E |o n E ) E ) | s _% . i. . . . . j. Wkh lqwhusuhwdwlrq lv vwudljkwiruzdug1 Zkhq ) E% ' wkh ylfwlp zloo uhyhdo zkhqhyhu kh revhuyhv %2> wklv rffxuv zlwk suredelolw| dqg lqgxfhv wkh wudqvihu |21 Zkhq wkh ylfwlp grhv qrw revhuyh %2 wkh wudqvihu lv |1 Wkh vdph orjlf dssolhv iru wkh lqmxuhu zkhq )E% ' 1 Lq wkh wklug uhjlrq/ %2 lv qrw uhyhdohg dqg wudqvihuv duh |1 H{shfwhg oldelolw| frvwv fdq eh uhzulwwhq dv ] ] ] ) E|2 |R s _% n ) E|2 |R s _% n |R s _% Wkh rswlpdo vfkhph pd{lpl}hv wkh glhuhqfh lq wkh h{shfwhg oldelolw| frvwv iurp h{huwlqj fduh ohyho , dv rssrvhg wr 1 Wklv lv qrz . ]. . ]. { ' ) E|2 | ER, s, R s _% n ) E|2 | ER, s, R s _% . ]. n | ER, s, R s _%. +53,. Zulwlqj s E%c %2 ' } E%s E%2 %/ zkhuh } E% dqg s E%2 % duh uhvshf0 wlyho| wkh pdujlqdo dqg frqglwlrqdo glvwulexwlrqv jlyhq fduh ohyho / zh kdyh wkh iroorzlqj uhvxow +wkh surri lv lq wkh dsshqgl{,1 m. m. Zkhq rqo| wkh sduwlhv kdyh gluhfw dffhvv wr wkh hylghqfh dqg pd| eh lpshuihfwo| lqiruphg/ vrflhw|*v sulpdu| frvwv duh plqlpl}hg e| dq| vfkhph |E5 ' wE5 4?dc uo/ 5 5 i% c %j/ zlwk wE% ' WE% dv ghqhg e| +46, lq sursrvlwlrq 4 dqg wE% 5 ifc j vdwlvi|lqj Sursrvlwlrq 6. . zkhq R, }, E% , E% : R } E% E% wE% ' f zkhq R, }, E%, E% R } E% E% zkhuh. ]. +54,. ]. E% ' E%s E%2 % _%2 d WE%os E%2 % _%2 +55, . W. m. . . m. Wkh sursrvlwlrq h{whqgv wkh pruh olnho| wkdq qrw fulwhulrq wr vlwxdwlrqv zkhuh glvforvxuh lv dq lvvxh1 E% uhsuhvhqwv wkh frqglwlrqdo suredelolw|.

(22) Suhsrqghudqfh ri Hylghqfh. 4<. ri f2 qrw ehlqj glvforvhg ghshqglqj rq wkh lqmxuhu*v fduh ohyho/ jlyhq wkh rffxuuhqfh ri kdup dqg f ' %1 Wkh udwlrqdoh lv wkdw f2 uhpdlqv xqglv0 forvhg/ xqohvv sduwlhv duh lqiruphg dqg fdqqrw eh kduphg iurp glvforvlqj1 Dowhuqdwlyho|/ wkh h{suhvvlrq fdq eh uhzulwwhq dv ]. E% ' E E n E d WE%os E%2 % _%2 . . n E . ]. . . WE%s E%2 % _%2 m. m. +56,. Wkdw lv/ f2 uhpdlqv xqglvforvhg hlwkhu ehfdxvh erwk sduwlhv duh xqlqiruphg ru rqo| rqh lv lqiruphg exw zrxog qrw glvforvh xqidyrudeoh hylghqfh1 Wkxv/ R } E% E% lv wkh suredelolw| ri wkh hyhqw % dqg f2 qrw glvforvhg jlyhq wkh lqmxuhu*v fduh ohyho1 Htxlydohqwo|/ lw lv wkh olnholkrrg ri fduh ohyho jlyhq wkh revhuydwlrq ri % dqg f2 qrw glvforvhg1 Zlwk uhvshfw wr dvvhvvlqj qhjoljhqfh/ wkh fulwhulrq ghqhg e| +54, lv wkhuhiruh pruh olnho| wkdq qrw wdnlqj lqwr dffrxqw lqfhqwlyhv wr glvforvh hylghqfh14;. 9 Wkh exughq ri surri Wkh phfkdqlvp ghvfulehg lq sursrvlwlrq 6 kdv ehhq lqwhusuhwhg dv ghqlqj wkh vwdqgdug ri surri iru dvvhvvlqj qhjoljhqfh1 Lw fdq dovr eh lqwhusuhwhg lq whupv ri wkh doorfdwlrq ri wkh exughq ri surri1 Lq ohjdo sudfwlfh/ wkh ylfwlp kdv wkh exughq ri surri li kh ehduv wkh orvv xqohvv hylghqfh lv surgxfhg vkrzlqj zlwk d suhsrqghudqfh wkdw wkh lqmxuhu zdv fduhohvv1 Frqyhuvho|/ wkh exughq uhvwv rq wkh lqmxuhu li kh lv khog oldeoh xqohvv hylghqfh lv vxeplwwhg vkrzlqj zlwk d suhsrqghudqfh wkdw kh h{huwhg gxh fduh1 Qhyhuwkhohvv/ lq doo fdvhv wkh surfhvv lv lqlwldwhg e| wkh ylfwlp zkr ehduv wkh sulpdu| exughq ri hvwdeolvklqj wkh rffxuuhqfh ri kdup dqg fdxvdolw| zlwk uhvshfw wr wkh lqmxuhu*v dfwlylw|1 Lq wkh frqwh{w ri wkh prgho/ d qhfhvvdu| wkrxjk jhqhudoo| qrw vx!flhqw frqglwlrq iru wkh ylfwlp wr rewdlq gdpdjhv lv iru wkh uhdol}dwlrq ri f wr eh vxeplwwhg1 Qrwh wkdw rqo| wkh ylfwlp frxog srvvleo| kdyh dq lqwhuhvw lq glvforvlqj f dorqh1 Dffruglqjo|/ wkh lqmxuhu pd| eh vdlg wr ehdu wkh Wkh olnholkrrg ri k yhuvxv o lv frpsxwhg khuh dv li sduwlhv glvforvhg hylghqfh zkhq lqglhuhqw ehwzhhq glvforvlqj ru qrw1 Lq vhfwlrq :/ zkhuh lpsohphqwdwlrq lv glvfxvvhg/ wkh frxuw lv prghohg dv d sod|hu lq wkh jdph1 Pruh olnho| wkdq qrw lv wkhq lq whupv ri wkh dfwxdo htxloleulxp suredelolwlhv ri glvforvxuh1 4;.

(23) Suhsrqghudqfh ri Hylghqfh. 53. exughq ri surri zlwk uhvshfw wr qhjoljhqfh +ru gloljhqfh, li dq| uhdol}dwlrq % vxeplwwhg e| wkh ylfwlp zrxog eh vx!flhqw e| lwvhoi iru wkh lqmxuhu wr eh khog oldeoh1 Wkh ylfwlp wkhq qhyhu kdv dq lqfhqwlyh wr vxeplw f2 dqg wkh lqmxuhu fdq hvfdsh oldelolw| rqo| e| frplqj iruzdug zlwk vx!flhqw frxqwhu0hylghqfh/ l1h1 vxeplwwlqj %2 vxfk wkdw WE%c %2 ' f1 Lq doo rwkhu fdvhv zkhuh f lv qrw dozd|v vx!flhqw e| lwvhoi iru d ghflvlrq djdlqvw wkh lqmxuhu/ wkh ylfwlp pd| eh vdlg wr ehdu wkh exughq ri surri1 Wkh ylfwlp zloo wkhq vrphwlphv qrw oh vxlw ru/ zkhq lqiruphg ri wkh frpsohwh hylghqfh/ zloo dovr vrphwlphv zdqw wr vxeplw f2 zkhq wkh uhdol}dwlrq ri f lv qrw vx!flhqw e| lwvhoi wr zlq wkh vxlw1 Wklv fdswxuhv sduw ri wkh phdqlqj ri wkh exughq ri surri/ dw ohdvw uhjduglqj wkh vr0fdoohg exughq ri surgxfwlrq/ vlqfh lw dssruwlrqv ehwzhhq sduwlhv wkh wdvn ri surgxflqj hylghqfh14<. Ghilqlwlrq 5= wr wkh lqmxuhu li. 1. Lq d jlyhq vlwxdwlrq/ d vfkhph dvvljqv wkh exughq ri surri. wEf ' . zlwk suredelolw| rqh1 Rwkhuzlvh wkh exughq lv rq. wkh ylfwlp. Wkh doorfdwlrq ri wkh exughq ghshqgv rq wkh sduwlhv* dffhvv wr lqirupdwlrq dqg rq wkh lqirupdwlrqdo frqwhqw ri wkh srwhqwldo hylghqfh1 Lw pd| dovr eh d pdwwhu ri lqglhuhqfh/ vlqfh wkh vfkhphv ghqhg lq sursrvlwlrq 6 gr qrw suhvfuleh d ghflvlrq zkhq wkh hylghqfh lv lq htxlsrlvh1 Wkh frqglwlrqv ghwhuplqlqj wE% duh lq whupv ri wkh vljq ri wkh olnholkrrg glhuhqfh. #E%. R, }, E%, E% R }E% E% ' E dR,]}, E% R } E% o n E W E%c %2 dR, s, E%c %2 R s E%c %2 o _%2 +57, . . . . . . Zkhq #E% : f iru doo %/ dq| vfkhph vdwlvi|lqj sursrvlwlrq 6 dvvljqv wkh exughq wr wkh lqmxuhu1 Zkhq #E% f iru doo % exw #E% ' f rffxuv zlwk srvlwlyh suredelolw|/ hlwkhu dvvljqphqw ri wkh exughq lv frqvlvwhqw zlwk wkh sursrvlwlrq1 Ilqdoo|/ zkhq #E% f ryhu d vhw ri srvlwlyh phdvxuh wkh exughq pxvw eh rq wkh ylfwlp1 Zh qrz h{dplqh krz wkhvh srvvlelolwlhv duh uhodwhg wr wkh lqirupdwlrqdo frqwhqw ri wkh hylghqfh1 Uhfdoo iurp ghqlwlrq 4 wkdw f lv lqirupdwlyh li lw surylghv xvhixo dgglwlrqdo lqirupdwlrq iru ghwhuuhqfh sxusrvhv/ frpsduhg wr vlpso| revhuylqj wkh rffxuuhqfh ri kdup1 Wklv dulvhv zkhq wkh pruh olnho| . 4<. Vhh Kd| dqg Vslhu +4<<:, iru d uhodwhg dqdo|vlv1.

(24) Suhsrqghudqfh ri Hylghqfh. 54. wkdq qrw fulwhulrq uhtxluhv lqmxuhuv h{huwlqj gxh fduh wr idfh d vwulfwo| srvlwlyh suredelolw| ri hvfdslqj oldelolw|1 Wr dqdo|}h wkh dvvljqphqw ri wkh exughq ri surri/ zh qhhg wr h{whqg wklv ghqlwlrq1. Ghilqlwlrq 6= f. lv wkh vhw ri uhdol}dwlrqv vxfk wkdw. U. 8 s E% _% ' R, s, E% : R sE% f. lv vwulfwo| xqlqirupdwlyh li. / zkhuh. 1. Zkhq. 8. lv qhlwkhu. lqirupdwlyh lq wkh vhqvh ri ghqlwlrq 4 qru vwulfwo| xqlqirupdwlyh/ lw lv vdlg wr eh zhdno| lqirupdwlyh1. Wkh prwlydwlrq iru wkhvh ghqlwlrqv lv dv iroorzv1 Li f lv lqirupdwlyh/ wkhuh duh uhdol}dwlrqv ri wkh hylghqfh zkhuh xqghu wkh pruh olnho| wkdq qrw fulwhulrq wkh lqmxuhu vkrxog qrw eh khog oldeoh1 Li f lv vwulfwo| xqlqirupdwlyh/ kh pxvw eh khog oldeoh xqghu dq| uhdol}dwlrq ri wkh hylghqfh1 Ilqdoo|/ d zhdno| lqirupdwlyh f phdqv wkdw R, s, Ef Rs Ef zlwk suredelolw| rqh/ exw R, s, Ef ' Rs Ef dovr rffxuv zlwk vwulfwo| srvlwlyh suredelolw|1 Ryhu wkh odwwhu uhdol}dwlrqv ri wkh hylghqfh/ lw lv luuhohydqw iru ghwhuuhqfh sxusrvhv zkhwkhu wkh lqmxuhu lv khog oldeoh ru qrw1 Wkh hylghqfh f lv dw ohdvw zhdno| lqirupdwlyh lq doo vlwxdwlrqv zkhuh kdup/ dowkrxjk uhodwhg wr wkh lqmxuhu*v dfwlylw|/ fdq dovr eh fdxvhg e| hyhqwv xquhodwhg wr klv ohyho ri fduh1 Iru lqvwdqfh/ vxssrvh kdup fdq eh fdxvhg hlwkhu e| wkh hyhqw rffxuulqj zlwk suredelolw| Z ru e| d plvwdnh rq wkh sduw ri wkh lqmxuhu/ zklfk rffxuv zlwk suredelolw| ^ zkhuh ^, : ^1 Dvvxplqj lqghshqghqfh/ wkh suredelolw| ri kdup lv wkhq R ' Z n E Z^ 1 Wkh srvvlelolw| ri revhuylqj zkhwkhu rffxuuhg ru qrw lv zhdno| lqirupdwlyh= zkhq lv revhuyhg/ fduh ohyhov dqg , duh htxdoo| olnho|> zkhq lv nqrzq qrw wr kdyh rffxuuhg/ , lv vwulfwo| pruh olnho|1 Li erwk sduwlhv duh nqrzq wr revhuyh zkhwkhu kdv rffxuuhg dqg li wklv lv wkh rqo| srwhqwldo hylghqfh/ doorfdwlqj wkh exughq ri surri rq wkh lqmxuhu ru wkh ylfwlp grhv qrw pdwwhu iurp d ghwhuuhqfh srlqw ri ylhz +wkh exughq zrxog eh rq wkh ylfwlp li wr zlq wkh fdvh kh zhuh uhtxluhg wr vkrz wkdw glg qrw rffxu,1 Wkh vdph ghqlwlrqv dsso| wr f iru vlwxdwlrqv zkhuh rqo| wkdw sduw ri U wkh hylghqfh lv revhuydeoh1 Wkxv/ zh vd| wkdw f lv lqirupdwlyh li . } E% _% : f/ zkhuh . lv wkh vhw ri uhdol}dwlrqv vxfk wkdw R, }, E% R } E%/ hwf1 Iru lqvwdqfh/ lq wkh iruhjrlqj h{dpsoh f lv zhdno| lqiru0 pdwlyh li lw fruuhvsrqgv wr wkh srvvlelolw| ri revhuylqj zkhwkhu wkh hyhqw rffxuuhg ru qrw1 Frqvlghu uvw wkh fdvh zkhuh f lv vwulfwo| xqlqirupdwlyh/ zklfk reylrxvo| lpsolhv wkdw f lv dovr vwulfwo| xqlqirupdwlyh1 Lw lv hdvlo| yhulhg wkdw zh wkhq . .

(25) Suhsrqghudqfh ri Hylghqfh. 55. kdyh #E% : f iru doo uhdol}dwlrqv +lpso|lqj wE% ' iru doo %,1 Fruroodu| 4 ^Vwulfw oldelolw|` Lq vlwxdwlrqv zkhuh f lv vwulfwo| xqlqiru0 pdwlyh/ wE% ' iru doo % 1 Wkh ylfwlp dozd|v ohv vxlw/ vxeplwwlqj f / dqg wkh lqmxuhu lv dozd|v khog oldeoh +f2 lv qhyhu vxeplwwhg,1. Wkh lqwhusuhwdwlrq lv wkdw wkh rswlpdo vfkhph lpsolhv wkh htxlydohqw ri wkh vwulfw oldelolw| fulwhulrq/ zlwkrxw wkh qhfhvvlw| ri hvwdeolvklqj qhjoljhqfh dv vxfk1 Dq dowhuqdwlyh lqwhusuhwdwlrq lv wkdw d vwulfwo| xqlqirupdwlyh f uhihuv wr vlwxdwlrqv zkhuh wkh phuh rffxuuhqfh ri kdup lv vx!flhqw wr suryh qhjoljhqfh1 Lq wkh vhtxho zh frqvlghu vlwxdwlrqv zkhuh f lv lqirupdwlyh1 Wkh iroorzlqj uhvxow lghqwlhv vlwxdwlrqv iru zklfk sursrvlwlrq 6 suhvfulehv dq xqdpeljxrxv dvvljqphqw ri wkh exughq ri surri153 Fruroodu| 5. ^Qhjoljhqfh` Zkhq f lv lqirupdwlyh/ wkh doorfdwlrq ri wkh. exughq ri surri lv lqfrqvhtxhqwldo li ' ' 1 Rwkhuzlvh/ lw lv dv iroorzv= +l, Zkhq f lv lqirupdwlyh/ wkh exughq lv rq wkh ylfwlp xqohvv lv vxi0 flhqwo| odujhu wkdq 1 +ll, Zkhq f lv rqo| zhdno| lqirupdwlyh/ wkh exughq lv rq wkh ylfwlp li : +dqg lv lqfrqvhtxhqwldo li ' ,1 +lll, Zkhq f lv vwulfwo| xqlqirupdwlyh/ wkh exughq lv rq wkh lqmxuhu xqohvv lv vx!flhqwo| odujhu wkdq 1. Wkh surri lv lq wkh dsshqgl{/ exw zh zloo qrz glvfxvv d ihz vlpsoh fdvhv1 Frqvlghu uvw vlwxdwlrqv zkhuh ' 1 Lqwxlwlyho|/ vlqfh kh lv shuihfwo| lqiruphg ri wkh hylghqfh/ wkh lqmxuhu vkrxog eh khog oldeoh zkhq rqo| f lv uhyhdohg lq rughu wr lqgxfh glvforvxuh ri wkh frpsohwh hylghqfh1 Lq rwkhu zrugv/ uhfdoolqj ghqlwlrq 5/ wkh exughq ri surri vkrxog eh rq wkh lqmxuhu1 Wklv dovr iroorzv iurp +57, vlqfh wkh lqwhjudo lv qrq qhjdwlyh e| ghqlwlrq ri W/ vr wkdw #E% f iru doo uhdol}dwlrqv dqg lv frqvlvwhqw zlwk dvvljqlqj wkh exughq wr wkh lqmxuhu1 Frqyhuvho|/ zkhq ' / lw fdq eh vkrzq iurp +57, wkdw #E% f iru doo uhdol}dwlrq/ zklfk lv frqvlvwhqw zlwk wkh exughq uhvwlqj rq wkh ylfwlp1 Ilqdoo|/ zkhq erwk sduwlhv duh shuihfwo| lqiruphg/ #E% ' f iru doo uhdol}dwlrqv1 Zklfk sduw| kdv wkh exughq ri surri lv wkhq luuhohydqw/ dv glvfxvvhg dw wkh ehjlqqlqj ri vhfwlrq 81 . . Lq fdvhv wkdw duh qrw h{solflwo| fkdudfwhul}hg/ wkh lqirupdwlrqdo surshuwlhv ri wkh hyl0 ghqfh zrxog qhhg wr eh ixuwkhu vshflhg iru dq xqdpeljxrxv dvvljqphqw1 53.

(26) Suhsrqghudqfh ri Hylghqfh. 56. Lq doo wkhvh vlwxdwlrqv dw ohdvw rqh ri wkh sduwlhv lv nqrzq wr revhuyh wkh frpsohwh srwhqwldo hylghqfh1 Rqh wkhq h{shfwv wkh ohyho ri ghwhuuhqfh dfklhyhg wr eh wkh vdph dv li vrflhw| gluhfwo| revhuyhg wkh zkroh hylghqfh1 Wr vhh wkdw wklv iroorzv iurp dq rswlpdo dvvljqphqw ri wkh exughq/ frqvlghu wkh frqglwlrqdo suredelolw| ri ehlqj irxqg oldeoh xsrq wkh rffxuuhqfh ri kdup1 Wklv fdq eh vkrzq wr htxdo ]. ]. k ' wE%} E% E% _% n WE%s E% _%c ' c ,. +58,. Wkh uvw lqwhjudo lv wkh suredelolw| ri d qglqj ri qhjoljhqfh rq wkh edvlv ri % dqg f2 qrw glvforvhg1 Wkh vhfrqg whup lv wkh dgghg suredelolw| iru d qglqj qhjoljhqfh rq wkh edvlv ri wkh frpsohwh hylghqfh1 Revhuyh wkdw wkh frpsohwh hylghqfh lv rqo| vxeplwwhg e| wkh ylfwlp zkhq ixoo| lqiruphg/ zklfk rffxuv zlwk suredelolw| / dqg zkhq WE% ' 1 Ghwhuuhqfh lv wkhuhiruh. B ' R], k, Rk ' wE% # E% _% n BW . +59,. zkhuh BW lv wkh ohyho ri ghwhuuhqfh wkdw zrxog eh dfklhyhg li vrflhw| gluhfwo| revhuyhg wkh zkroh srwhqwldo hylghqfh/ dv lq vhfwlrq 71 Vlqfh wkh lqwhjudo lv qrq qhjdwlyh/ B ' BW li wkh ylfwlp lv shuihfwo| lqiruphg1 Zkhq wkh exughq lv rq wkh lqmxuhu/ wE% 1 Vxevwlwxwlqj iru #E% iurp +57,/ lw fdq eh fkhfnhg wkdw B ' E ER, R n BW +5:, Li wkh lqmxuhu lv shuihfwo| lqiruphg/ wklv dovr lpsolhv wkh vdph ohyho ri ghwhu0 uhqfh dv xqghu gluhfw revhuydwlrq ri wkh frpsohwh hylghqfh1 Zkhq qhlwkhu sduw| lv shuihfwo| lqiruphg/ wkh dvvljqphqw ri wkh exughq ri surri ghshqgv rq wkh lqirupdwlrq frqwhqw ri f e| lwvhoi/ rq wkh h{whqw ri wkh sduwlhv* dffhvv wr doo wkh srwhqwldo hylghqfh/ dv zhoo dv rq wkh glhuhqfh ehwzhhq wkh lqirupdwlrqdo frqwhqw ri f dqg f 1 Rqh sduwlfxoduo| vlpsoh fdvh lv zkhq erwk sduwlhv duh htxdoo| zhoo lqiruphg rq dyhudjh1 Iurp +57,/ zlwk ' wkh olnholkrrg glhuhqfh uhgxfhv wr . . . #E% ' E dR, }, E% R}E%o Lq wklv fdvh/ vwudwhjlf lqfhqwlyhv wr frqfhdo hylghqfh fdqfho rxw dqg wkh sulqfl0 . .

(27) Suhsrqghudqfh ri Hylghqfh. 57. sdo pd| wdnh vxeplvvlrqv dw wkhlu idfh ydoxh154 Wkh exughq lv wkhq qhfhvvdulo| rq wkh ylfwlp zkhq f lv lqirupdwlyh dqg pxvw eh rq wkh lqmxuhu zkhq f lv vwulfwo| xqlqirupdwlyh1. : Uxoh0edvhg lpsohphqwdwlrq Wr frpsohwh wkh dqdo|vlv/ zh uhh{dplqh wkh lvvxh ri lpsohphqwdwlrq zkhq ghflvlrqv duh ghohjdwhg wr frxuwv xqghu wkh suhsrqghudqfh ri hylghqfh vwdq0 gdug1 Rqh glhuhqfh ehwzhhq pruh0olnho|0wkdq0qrw lq d phfkdqlvp ghvljq iudphzrun dqg wkh suhsrqghudqfh ri hylghqfh uxoh lv wkdw wkh odwwhu uhtxluhv d sulru dvvljqphqw ri wkh exughq ri shuvxdvlrq1 Wkh exughq ri shuvxdvlrq lv rq wkh ylfwlp li wkh frxuw pxvw eh frqylqfhg wkdw qhjoljhqfh lv vwulfwo| pruh olnho| wkdq gloljhqfh1 Lw lv rq wkh lqmxuhu li kh pxvw shuvxdgh wkh frxuw wkdw gloljhqfh lv vwulfwo| pruh olnho|1 D vhfrqg glhuhqfh lv wkdw suhsrqghu0 dqfh ri hylghqfh zloo eh dssolhg rq wkh edvlv ri wkh htxloleulxp suredelolwlhv ri glvforvxuh/ zlwk wkh frxuw dv d sod|hu lq wkh jdph1 Lq sudfwlfh/ wkh exughq ri shuvxdvlrq lv xvxdoo| dvvljqhg wr wkh sduw| ehdulqj wkh exughq ri surgxfwlrq rq wkh pdlq frqwhvwhg lvvxh1 Iurp wkh iruhjrlqj vhfwlrq zh nqrz wkdw lq vrph vlwxdwlrqv wkh exughq ri surgxfwlrq pxvw uhvw rq d vshflf sduw|/ zkloh lq rwkhuv wkh dvvljqphqw lv duelwudu|1 Ohw KEr iru r 7 ghqrwh wkh exughq dvvljqphqw iru doo vfkhphv vdwlvi|lqj sursr0 vlwlrq 6/ zlwk KEr ' zkhq wkh dvvljqphqw lv xqdpeljxrxvo| rq wkh lqmxuhu dqg KEr ' f zkhq lw lv xqdpeljxrxvo| rq wkh ylfwlp ru lqglhuhqw1 Dprqj doo srvvleoh vfkhphv/ ohw xv vhohfw wkh vfkhph wE5c r dv iroorzv1 Zkhq KEr ' / dqg qrwlqj wkdw wklv fruuhvsrqgv wr vlwxdwlrqv zkhuh #E%cr : f iru doo %/ wE%c r ' iru doo % dqg wE%c r ' xqohvv R, Ers, E%c r RErs E%c r1 Zkhq KEr ' f/ wE%c r ' f xqohvv #E%c r : f dqg wE%c r ' f xqohvv R, Ers, E%c r : R ErsE%c r1 Xqghu wkh vfkhph wE5cr/ zklfk reylrxvo| vdwlvhv sursrvlwlrq 6/ erwk wkh exughq ri surgxfwlrq dqg wkh exughq ri shu0 vxdvlrq duh rq wkh vdph sduw|1 Pruhryhu/ wkh ylfwlp lv dvvljqhg wkh exughq ri surgxfwlrq lq doo vlwxdwlrqv zkhuh KEr ' f1 Lw lv hdvlo| fkhfnhg wkdw wkh fkrvhq vfkhph plqlpl}hv wkh iuhtxhqf| ri vxlwv155 5. Wklv h{whqgv wr lpshuihfwo| lqiruphg sduwlhv wkh lghd wkdw frpshwlwlrq ehwzhhq sdu0 wlhv doorzv wkh sulqflsdo wr eh qdlyh/ dv lq Plojurp dqg Urehuwv +4<;9, ru Iurhe dqg Nred|dvkl +4<<9,1 55 Dq dowhuqdwlyh exughq dvvljqphqw sduwlwlrq lv e3 +v, @ 4 li wkh exughq ri surgxfwlrq lv xqdpeljxrxvo| rq wkh lqmxuhu ru lqglhuhqw dqg e3 +v, @ 3 rwkhuzlvh1 Xqghu wkh vfkhph 54.

(28) Suhsrqghudqfh ri Hylghqfh. 58. Zlwk 00glvforvxuh frvwv/ wE5cr lpsolhv d xqltxh htxloleulxp lq wkh glv0 forvxuh vxejdph dv ghvfulehg lq vhfwlrq 81 Zh qrz prgli| wklv vxejdph e| dsshqglqj d whuplqdo vwdjh zkhuh d frxuw ghwhuplqhv oldelolw| rq wkh edvlv ri wkh vxeplwwhg hylghqfh1 Wklv vwdjh vxevwlwxwhv iru wkh phfkdqlvp wE5c r lp0 srvhg e| wkh sulqflsdo1 Vshflfdoo|/ zh dvvxph dv ehiruh wkdw d ylfwlp olqj vxlw qhfhvvdulo| vxeplwv wkh uhdol}dwlrq ri f1 Lq dgglwlrq wr frppxqlfdwlqj f dv vxfk/ olqj vxlw vkrzv wkh rffxuuhqfh ri kdup dqg wkh fdxvdo uhodwlrq wr wkh lqmxuhu1 Lw dovr frppxqlfdwhv vrph ri wkh fkdudfwhulvwlfv ri wkh vlwxdwlrq r/ lqfoxglqj Rs / R, s, / / dqg wkh exughq ri surri dvvljqphqw KEr1 Rqfh d vxlw lv ohg/ erwk sduwlhv ghflgh vlpxowdqhrxvo| zkhwkhu wr vxeplw f21 Dw wkh forvh ri wkh hylghqfh/ wkh frxuw ghflghv zkhwkhu wkh lqmxuhu lv wr eh khog oldeoh1 Dq rswlpdo vfkhph fdq eh lpsohphqwhg dv d vhtxhqwldo htxl0 oleulxp zlwk frxuwv dsso|lqj suhsrqghudqfh ri wkh hylghqfh/ jlyhq wkh exughq ri surri dvvljqphqw1 Ohw _E5c r 5 ifc j eh wkh frxuw*v vwudwhj|/ zkhuh _ ' ghqrwhv wkh gh0 flvlrq wr krog wkh lqmxuhu oldeoh1 Fohduo|/ wkh vfkhph wE5c r lv lpsohphqwhg li _E5cr ' wE5c r lv dq htxloleulxp vwudwhj| xqghu wkh suhsrqghudqfh vwdq0 gdug/ jlyhq wkh exughq ri surri dvvljqphqw KEr1 Lw lv vwudljkwiruzdug wr Sursrvlwlrq 7. vkrz wkdw wklv vwudwhj| fdq lqghhg eh sduw ri dq htxloleulxp dqg zh rqo| jlyh wkh rxwolqh ri wkh dujxphqw iru vlwxdwlrqv zkhuh wkh exughq ri surri lv rq wkh ylfwlp1 Vxssrvh wkh sursrvlwlrq lv wuxh1 Sduwlhv wkhq h{shfw wkh frxuw wr sod| _E5 ' wE5/ zkhuh uhihuhqfh wr r lv rplwwhg iru vlpsolflw|1 Lq htxloleulxp/ 5 ' % rffxuv rqo| zkhq #E% : f dqg zkhq wkh lqmxuhu lv hlwkhu xqlqiruphg ru lv lqiruphg exw R, s, E% : Rs E%1 Wkh htxloleulxp frqglwlrqdo suredelolw| ri % dqg qrw f2 lv wkhuhiruh. e E% ' n ] ' . ]. wE%c %2s E%2 % _%2 m. d wE% c %2 o s E%2 m % _%2. ' E% n . ]. wE%c %2s E%2 % _%2 c m. +5;,. ghqhg dv deryh/ exw qrz zlwk uhvshfw wr e3 +v,/ wkh ylfwlp ohv vxlw zkhq kh kdv hylghqfh wkdw qhjoljhqfh lv dw ohdvw dv olnho| dv gloljhqfh/ udwkhu wkdq vwulfwo| pruh olnho| dv xqghu +}> v,1 3. +}> v,.

(29) Suhsrqghudqfh ri Hylghqfh. 59. zkhuh E% lv wkh vdph dv lq sursrvlwlrq 61 Wkh uhvxowlqj htxloleulxp olnholkrrg glhuhqfh lv e# E% '. R, }, E%e, E]% R} E%eE% ' # E% n w E% c %2 dR, s, E% c %2 R s E% c %2 o _%2 +5<, . . Wkh lqwhjudo lv qrqqhjdwlyh e| ghqlwlrq ri wE%1 Khqfh #E% : f lpsolhv e# E% : f/ zklfk phdqv wkdw wkh frxuw zrxog zdqw wr krog wkh lqmxuhu oldeoh xqghu wkh suhsrqghudqfh vwdqgdug1 Vxssrvh qrz wkdw d vxlw lv ohg/ exw wkdw dw wkh forvh ri wkh hylghqfh 5 ' % zkhuh #E% f1 Wklv fdq rqo| rffxu dw dq lqirupdwlrq vhw r wkh htxloleulxp sdwk1 Wkh frxuw*v eholhiv dw wklv lqirupdwlrq vhw fdq eh udwlrqdol}hg e| frqvlghulqj wkdw ylfwlpv vrphwlphv plvwdnhqo| oh vxlw1 Rqfh d vxlw lv ohg/ wkh| qhyhuwkhohvv ehkdyh udwlrqdoo| zlwk uhvshfw wr glvforvlqj f2/ jlyhq wkh frxuw*v vwudwhj| _E% ' wE%1 Vlqfh wkh lqmxuhu*v htxloleulxp vwudwhj| lv qrw wr glvforvh f2 zkhq wkh ylfwlp vxhv zlwk #E% f/ wkh frqglwlrqdo suredelolw| ri % dqg qrw f2 lv qrz eholhyhg wr eh . . e E% ' n ] ' . ]. d wE%c %2 o s E%2 m % _%2. wE%c %2s E%2 % _%2. ' E% n . ]. m. d wE% c %2o s E%2 m % _%2 . +63,. Wkh olnholkrrg glhuhqfh lv. R, }, E%e], E% R } E%e E% ' # E% n d wE% c %2 o dR, s, E%c %2 R s E% c %2o _%2 +64,. e# E% '. . . . Wkh lqwhjudo lv qrz qrq srvlwlyh e| ghqlwlrq ri wE%/ vr wkdw #E% f lpsolhv e#E% f1 Wkh exughq ri shuvxdvlrq ehlqj rq wkh ylfwlp/ suhsrq0 ghudqfh ri wkh hylghqfh phdqv wkdw wkh lqmxuhu vkrxog qrw eh khog oldeoh1 Qrwh wkdw wkh frxuw*v ghflvlrq uhjduglqj wkh ghihqgdqw*v oldelolw| pd| eh lqwhusuhwhg dv uho|lqj rqo| rq ohjdoo| dgplvvleoh hylghqfh1 Dq rxwvlgh revhuyhu vhhnlqj wkh wuxwk rq wklv pdwwhu zrxog iru lqvwdqfh zdqw wr nqrz S/ S, / dqg u vr dv wr lqihu wkh lqmxuhu*v ehkdylru iurp wkh lqfhqwlyhv kh . .

(30) Suhsrqghudqfh ri Hylghqfh. 5:. idfhg1 E| frqwudvw/ wkh frxuw ghdov zlwk d frqwhvw ehwzhhq wzr sduwlhv/ rqh ri zkrp kdv wkh exughq ri surri zlwk uhvshfw wr wkh fodlp ri qhjoljhqfh +ru gloljhqfh,1 Wr uhdfk lwv ghflvlrq wkh frxuw frqvlghuv rqo| d uhvwulfwhg vhw ri hylghqfh1 Pruhryhu/ wkh rqo| vwudwhjlf frqvlghudwlrqv wdnhq lqwr dffrxqw duh wkrvh shuwdlqlqj wr wkh glvforvxuh ri hylghqfh dqg wkh lqwhusuhwdwlrq ri wkh vxeplwwhg hylghqfh/ jlyhq wkh lqfhqwlyhv idfhg e| wkh olwljdqwv1. ; Frqfoxglqj frpphqwv Wklv sdshu kdv dqdo|}hg wkh h!flhqf| surshuwlhv ri wkh vwdqgdug ri surri lq frpprq odz iru d qglqj ri qhjoljhqfh dqg ri exughq ri surri dvvljqphqwv zkhq hylghqfh lv lpshuihfw dqg uhvwv zlwk wkh sduwlhv1 Rxu uhvxowv fdq eh h{whqghg ru txdolhg lq qxphurxv zd|v1 Iluvw/ rxu dqdo|vlv kdv ghdow rqo| zlwk wkh vr0fdoohg xqlodwhudo fduh sureohp zkhuh wkh suredelolw| ri kdup grhv qrw ghshqg rq wkh dfwlrqv ri srwhqwldo ylfwlpv1 Frqvlghudwlrq ri d qhjoljhqfh uxoh lv wkhq prwlydwhg e| wkh idfw wkdw srwhqwldo wruw0ihdvruv pd| eh zhdowk0 frqvwudlqhg1 Zh kdyh dovr dvvxphg wkdw olwljdwlrq frvwv zhuh qhjoljleoh/ zklfk mxvwlhg plqlpl}lqj djjuhjdwh sulpdu| frvwv ghqhg dv wkh vxp ri fduh frvwv dqg h{shfwhg kdup1 Wkdw dvvxpswlrq kdv doorzhg xv wr devwudfw iurp rxw ri frxuw vhwwohphqwv/ vlqfh wkhvh surylgh qr edujdlqlqj vxusoxv lq vxfk d frqwh{w1 Zh kdyh dovr wdnhq iru judqwhg wkdw dw wkh vwduw ri surfhgxuhv sduwlhv nqhz h{dfwo| zkdw hylghqfh wkh| zrxog eh deoh wr suhvhqw lq frxuw1 Wklv djdlq pd| eh mxvwlhg e| wkh dvvxpswlrq ri qhjoljleoh olwljdwlrq frvwv/ lq wkh vhqvh wkdw zh frxog glvuhjdug wkh sduwlhv* ghflvlrq sureohp dv wr zkhwkhu wkh| vkrxog lqyhvw lq xqfryhulqj hylghqfh1 Zh eulh | glvfxvv wkh lpsolfdwlrqv ru lvvxhv udlvhg zkhq vrph ri wkhvh dvvxpswlrqv duh uhod{hg1 Lq rxu iudphzrun/ wkh sxusrvh ri d exughq ri surri dvvljqphqw lv wr lqgxfh sduwlhv wr glvforvh hylghqfh/ jlyhq wkdw wkh h{whqw ri wkh hylghqfh dydlodeoh wr wkh sduwlhv lv xqnqrzq dqg wkdw sduwlhv pd| eh dv|pphwulfdoo| lqiruphg1 E| frqwudvw/ Kd| dqg Vslhu +4<<:, dqdo|}h exughq ri surri uxohv iru wkh fdvh zkhuh erwk sduwlhv duh nqrzq wr eh shuihfwo| lqiruphg/ exw zkhuh wkhuh duh qrq qhjoljleoh frvwv ri vxeplwwlqj hylghqfh1 Dvvxplqj vxfk frvwv duh qrw odujh hqrxjk wr suhyhqw glvforvxuh dowrjhwkhu/ wkh| vkrz wkdw exughq ri surri dvvljqphqwv fdq eh xvhg wr hfrqrpl}h rq wkh frvwv ri wudqvplwwlqj lqirupdwlrq wr frxuwv1 Wkh dssursuldwh doorfdwlrq ri wkh exughq wkhq ghshqgv rq vrflhw|*v sulruv derxw zkhwkhu wkh hylghqfh lv olnho| wr idyru wkh sodlqwl ru wkh ghihqgdqw dqg rq wkh sduwlhv*v frvwv ri surgxflqj hylghqfh1 Vlplodu.

(31) Suhsrqghudqfh ri Hylghqfh. 5;. uhvxowv fdq dovr eh ghulyhg lq rxu iudphzrun1 Wr jlyh d vlpsoh h{dpsoh/ frqvlghu d vxevhw ri vlwxdwlrqv glhulqj rqo| e| wkh lqmxuhuv* zhdowk1 Vxssrvh wkdw olqj vxlw uhyhdov rqo| wkh rffxuuhqfh ri kdup dqg wkh fdxvdo uhodwlrq wr wkh lqmxuhu/ zkloh wkh frpsohwh hylghqfh f lv lqirupdwlyh dv ghqhg lq wkh suhylrxv vhfwlrqv1 Vxssrvh dovr wkdw erwk sduwlhv duh dssur{lpdwho| htxdoo| lqiruphg rq dyhudjh/ zlwk suredelolwlhv rqo| voljkwo| vpdoohu wkdq xqlw|1 Iurp wkh srlqw ri ylhz ri lqgxflqj glvforvxuh dqg ri wkh uhdol}hg ohyho ri ghwhuuhqfh/ lw grhv qrw pdwwhu lq vxfk d fdvh zklfk sduw| ehduv wkh exughq ri surri1 Ilqdoo|/ vxssrvh wkdw olqj vxlw lqyroyhv qhjoljleoh frvwv/ zkloh vxeplwwlqj wkh frpsohwh hylghqfh kdv frvwv iru wkh sodlqwl dqg ( iru wkh ghihqgdqw1 Li wkh sodlqwl kdv wkh exughq ri surri/ kh ohv vxlw rqo| zkhq kh fdq vxeplw wkh frpsohwh hylghqfh dqg zkhq R, s, E% : Rs E%1 Li q lv wkh sursruwlrq ri fduhohvv lqmxuhuv +wklv lv hqgrjhqrxv dv lw ghshqgv rq wkh ohyho ri ghwhuuhqfh,/ dvvljqlqj wkh exughq ri surri wr wkh ylfwlp ohdgv wr dyhudjh olwljdwlrq frvwv *. dE qRkW n qR, kW, o c zkhuh kW lv wkh suredelolw| wkdw dq lqmxuhu zlwk fduh ohyho zloo eh irxqg oldeoh xqghu wkh suhsrqghudqfh vwdqgdug1 Rq wkh rwkhu kdqg/ li wkh exughq lv rq wkh lqmxuhu/ wkh sodlqwl dozd|v ohv vxlwv ehfdxvh wklv lqyroyhv qhjoljleoh frvwv dqg kh pd| h{shfw wr zlq zlwk vrph suredelolw| +wkh ghihqgdqw lv qrw dozd|v deoh wr surgxfh frxqwhu0 hylghqfh,1 Zkhq lqiruphg/ wkh ghihqgdqw surgxfhv wkh frpsohwh hylghqfh surylghg R, s, E% RsE%1 Dyhudjh olwljdwlrq frvwv duh wkhq dE qR E kW n qR, E kW, o ( Xqghu wkh deryh dvvxpswlrqv/ wkh exughq dvvljqphqw plqlpl}lqj olwljdwlrq frvwv ghshqgv rq wkh sursruwlrq ri fduhohvv lqmxuhuv dqg rq wkh sduwlhv*v frvwv ri suhvhqwlqj hylghqfh156 Krzhyhu/ zkhq sduwlhv duh xqhtxdoo| lqiruphg ru zkhq dqg glhu vljqlfdqwo| iurp xqlw|/ d exughq dvvljqphqw dovr dhfwv ghwhuuhqfh1 D frpsohwh dqdo|vlv wkhq uhtxluhv wudglqj0r sulpdu| frvwv dqg olwljdwlrq frvwv1 Ixuwkhupruh/ zlwk qrq qhjoljleoh olwljdwlrq frvwv/ wkh srvvlelolw| ri rxw ri frxuw vhwwohphqwv vkrxog dovr eh wdnhq lqwr dffrxqw1 . . . . Iru lqvwdqfh/ xqohvv kh idfhv pxfk juhdwhu frvwv ri vxeplwwlqj hylghqfh/ wkh ylfwlp vkrxog ehdu wkh exughq li lv vpdoo dqg wkh hylghqfh kdv vwurqj lqirupdwlrq frqwhqw +l1h1 li k lv vpdoo,1 56.

(32) Suhsrqghudqfh ri Hylghqfh. 5<. Gr olwljdwlrq frvwv dhfw wkh rswlpdo vwdqgdug ri surriB Vxssrvh lq wkh iruhjrlqj h{dpsoh wkdw vxfk frvwv mxvwli| dvvljqlqj wkh exughq ri surri wr wkh ylfwlp1 Zlwk suhsrqghudqfh ri hylghqfh dv wkh vwdqgdug ri surri/ wkh uhdol}hg ohyho ri ghwhuuhqfh lv B ' BW/ zkhuh BW ' R, kW, R kW lv wkh ghwhuuhqfh wkdw zrxog eh rewdlqhg li wkh frpsohwh hylghqfh zhuh gluhfwo| revhuydeoh1 D vwdqgdug ri surri vwurqjhu wkdq suhsrqghudqfh zrxog uhtxluh wkh sodlqwl wr vkrz wkdw qhjoljhqfh lv pruh olnho| wkdq gloljhqfh e| vrph pdujlq1 Wkdw lv/ wkh ylfwlp zlqv wkh vxlw rqo| li wkh hylghqfh vxeplwwhg vdwlvhv R, s, E% : &R sE%/ iru vrph & : 1 Vxfk d vwdqgdug ohdgv wr ohvv ghwhuuhqfh/ exw lw dovr uhgxfhv wkh qxpehu ri vxlwv dqg wkh uhvxowlqj olwljdwlrq frvwv1 Wklv iroorzv iurp wkh idfw wkdw & : lpsolhv vpdoohu ydoxhv iru erwk k dqg k, 1 Krzhyhu/ devwudfwlqj iurp mxpsv lq wkh olnholkrrg udwlr/ lw fdq eh vkrzq wkdw & voljkwo| odujhu wkdq xqlw| lqyroyhv rqo| d vhfrqg0 rughu hhfw rq ghwhuuhqfh/ hyhq wkrxjk lw kdv uvw rughu hhfwv rq k dqg k, 1 Khqfh/ xvlqj d vwurqjhu vwdqgdug wkdq suhsrqghudqfh zrxog vhhp wr eh zduudqwhg1 Ghvslwh wklv revhuydwlrq/ lw frxog qhyhuwkhohvv eh uhdvrqdeoh iru vrflhw| wr uhtxluh frxuwv wr xvh wkh suhsrqghudqfh vwdqgdug +xqohvv wkh rswlpdo & lv yhu| glhuhqw iurp xqlw|,1 Wkh uhdvrq lv vlpso| wkdw wkh phdqlqj ri suhsrqghudqfh lv hdvlo| frppxqlfdwhg/ zkhuhdv rswlpdo ghyldwlrqv iurp wklv vwdqgdug zrxog eh kljko| vlwxdwlrq vshflf dqg wkhuhiruh gl!fxow wr lqfoxgh lq d vhw ri lqvwuxfwlrqv wr d frxuw ru mxu|157 Dgglwlrqdo vwudwhjlf frqvlghudwlrqv pxvw eh wdnhq lqwr dffrxqw zkhq olw0 ljdwlrq frvwv dovr lqfoxgh wkh frvw ri xqfryhulqj hylghqfh1 Li wkh sodlqwl kdv wkh exughq ri surri/ kh pxvw dvvhvv wkh suredelolw| ri dftxlulqj idyrudeoh hylghqfh ehiruh olqj vxlw1 Wkh uhwxuq rq wklv lqyhvwphqw ghshqgv rq wkh suredelolw| wkdw wkh ghihqgdqw kdv ehhq fduhohvv/ rq wkh lqirupdwlrq ydoxh ri srwhqwldo hylghqfh dqg rq wkh vwdqgdug ri surri158 Ghshqglqj rq zkhwkhu kh kdv ehhq qhjoljhqw ru gloljhqw/ wkh lqmxuhu zloo dovr idfh glhuhqw lqfhqwlyhv wr lqyhvw lq wkh surgxfwlrq ri hylghqfh1 Krz wkhvh ghflvlrqv dhfw wkh rswlpdo vwdqgdug ri surri dqg exughq ri surri dvvljqphqw lv xqfohdu1 Dqrwkhu h{whqvlrq zrxog eh wr h{dplqh elodwhudo fduh sureohpv/ zkhuh . Li d qglqj ri qhjoljhqfh lpsrvhg uhdo vrfldo frvwv +vwljpd/ hwf1,/ dv rssrvhg wr ehlqj d sxuh wudqvihu iurp lqmxuhu wr ylfwlp/ d pruh ghpdqglqj vwdqgdug ri surri zrxog dovr eh zduudqwhg1 Lq wkh olplw/ li vxfk frvwv duh yhu| odujh/ rqh frxog rewdlq wkh uhdvrqdeoh grxew vwdqgdug dv lq fulplqdo surfhhglqjv1 58 Iru lqvwdqfh/ lw lv qrw zruwkzkloh olqj vxlw li wkh hylghqfh lv olnho| wr eh shuihfwo| lqirupdwlyh/ exw wkhuh lv olwwoh fkdqfh wkdw wkh lqmxuhu kdv dfwxdoo| ehhq qhjoljhqw +h1j1 K|owrq/ 4<<3,1 57.

(33) Suhsrqghudqfh ri Hylghqfh. 63. wkh suredelolw| ri kdup ghshqgv rq wkh ehkdylru ri erwk sduwlhv1 Lq vxfk d frqwh{w/ lw lv zhoo nqrzq wkdw wkh qhjoljhqfh uxoh lv xvhixo hyhq zkhq djhqwv duh qhyhu mxgjphqw surri1 Xqghu wkh sxuh qhjoljhqfh uxoh wkh lqmxuhu lv khog oldeoh li kh kdv ehhq irxqg qhjoljhqw1 Xqghu wkh wudglwlrqdo uxoh ri qhjoljhqfh zlwk frqwulexwru| qhjoljhqfh/ wkh lqmxuhu lv khog oldeoh li irxqg qhjoljhqw dqg li wkh ylfwlp lv qrw irxqg qhjoljhqw1 Erwk uxohv duh nqrzq wr eh htxlydohqw zkhq wkh hylghqfh surylghv shuihfw lqirupdwlrq derxw fduh1 Grhv wkh htxlydohqfh vwloo krog zkhq wkh hylghqfh lv lpshuihfwB Lq sduwlfxodu/ vkrxog wkh vdph vwdqgdug ri surri eh dssolhg wr dvvhvv wkh sodlqwl*v dqg wkh ghihqgdqw*v qhjoljhqfhB Dqg krz vkrxog wkh exughqv ri surri eh dvvljqhg zlwk uhvshfw wr wkhvh lvvxhvB Zh ohdyh wkhvh txhvwlrqv iru ixuwkhu uhvhdufk1. D Dsshqgl{ Iru dq| vfkhph |E% dqg |2E%/ ghqh |2E% : |E% |2E% ' ||2EE%% lirwkhuzlvh. Surri ri sursrvlwlrq 6=. dqg. . |2E% |E% |2 E% ' ||2EE%% lirwkhuzlvh Htxdwlrq +53, fdq wkhq eh uhzulwwhq dv . { ' . ]. +66,. E|2 E% | E% ER, s, E% R s E% _%. n n. +65,. ]. ]. E|2 E% | E% ER, s, E% R s E% _%. |E% ER, s, E% Rs E% _% . +67,. Lq wkh rswlpdo vfkhph/ |E%/ |2 E% dqg |2 E% pxvw eh vxfk dv wr pd{lpl}h { vxemhfw wr |2 E% | E%/ |2 E% | E%/ wkh orzhu dqg xsshu erxqgv rq wudqvihuv dqg wr |E% ehlqj frqvwdqw lq %21 Wdnlqj |E% dv jlyhq/ wkh uvw lqwhjudo lv fohduo| pd{lpl}hg e| . . . c uo li R, s, E% : Rs E% |2 E% ' 4?d |E% li R, s, E% Rs E% . +68,.