Collusion via Signalling in Open Ascending Auctions with Multiple Objects and Complementarities by Sandro Brusco and Giuseppe Lopomo EC-99-05 L** t L? @ 5 }?@** ?} ? Ti? tUi?_ ?} U| L?t | G127 *| T*i M iU|t @?_ L4T*i4i?|@h | it 5@?_hL h tULc104 @?_ B tiTTi wLTL4Lc61 Th * .c bbb Devwudfw Frooxvlyh htxloleuld h{lvw lq rshq dvfhqglqj dxfwlrqv zlwk pxowlsoh remhfwv/ li wkh qxpehu ri elgghuv lv vx flhqwo| vpdoo uhodwlyh wr wkh qxpehu ri remhfwv/ hyhq zlwk odujh frpsohphwdulwlhv lq wkh ex|huv* xwlolw| ixqfwlrqv1 Wkh elgghuv frooxgh e| glylglqj wkh remhfwv dprqj wkhpvhoyhv/ zkloh nhhslqj wkh sulfhv orz1 Khqfh wkh frpsohphqwdulwlhv duh qrw uhdol}hg1 c87 `i | @?! aLt?i @4T@c 6@h ! B *c 6@Mh 3 L ihh c aLi* 5LMi*c ,?? L 5|@UU i|| c | i T@h| U T@?|t |L | i U|LMih bbH G127 _ it|ih? A iLh) L?uihi?Uic @?_ | i T@h| U T@?|t |L ti4 ?@ht @| | i N? iht |) Lu @* uLh? @ 5@? # i}Lc N UL**i}i ? wL?_L?c @?_ | i , hLTi@? N? iht |) W?t| | |i uLh i*Tu * UL44i?|t @?_ t }}it| L?t c104 #iT@h|@4i?|L _i ,UL?L4?>@ _i *@ ,4Thit@c N? iht _@_ @h*Lt WWW _i G127@_h _ , 4@ *G Mh tUL9i4T U 4 it c61 5|ih? 5U LL* Lu t ?ittc i vLh! N? iht |) , 4@ *G }*LTL4L9t|ih? ?) i_ 4 W?|hL_ U| L? Lw lv zhoo nqrzq wkdw wkh Hqjolvk dxfwlrq kdv pdq| ghvludeoh surshuwlhv zkhq d vlqjoh remhfw lv wr eh vrog1 Zlwk sulydwh ydoxhv/ wklv dxfwlrq lpsohphqwv wkh h flhqw doorfdwlrq xqltxho| lq zhdno| grplqdqw vwudwhjlhv/ dqg pd{lpl}hv wkh vhoohu*v h{shfwhg uhyhqxh zlwklq d odujh fodvv ri cvlpsoh* vhoolqj surfhgxuhv +Orsrpr ^43`,1 Krzhyhu/ wkh surshuwlhv ri jhqhudol}hg yhuvlrqv ri wkh Hqjolvk dxfwlrq lq vlwxdwlrqv lq zklfk pdq| remhfwv duh wr eh vrog/ dqg wkh ex|huv kdyh xvh iru pruh wkdq rqh remhfw/ duh |hw wr eh ixoo| xqghuvwrrg1 E| dqg odujh/ prvw ri wkh h{lvwlqj zrun rq vlpxowdqhrxv pxowlsoh remhfwv dxfwlrqv kdv irfxvhg rq wkh fdvh zlwk qrqlqfuhdvlqj pdujlqdo zloolqjqhvv wr sd| lq wkh elgghuv* xwlolw| ixqfwlrq1 Prvw forvho| uhodwhg wr wkh suhvhqw sdshu lq whupv ri wkh dxfwlrq uxohv/ Plojurp ^46` kdv dqdo|}hg wkh _vlpxowdqhrxv dvfhqglqj dxfwlrqv/% zklfk kdyh ehhq xvhg e| wkh XV jryhuqphqw wr vhoo olfhqvhv iru wkh xvh ri udglr iuhtxhqf| edqgv1 Prvwo| xqghu wkh k|srwkhvlv wkdw wkh elgghuv* xwlolw| ixqfwlrqv duh frpprq nqrzohgjh/ Plojurp glvfxvvhv lvvxhv vxuurxqglqj wkh dxfwlrq*v shuirupdqfh lq whupv ri lwv delolw| ri jhqhudwlqj h flhqw rxwfrphv dqg lwv srwhqwldo iru pd{lpl}lqj wkh vhoohu*v h{shfwhg uhyhqxh1 Lq sduwlfxodu/ kh ghvfulehv dq htxloleulxp iru wkh fdvh ri wzr elgghuv/ wzr remhfwv dqg qr sulydwh lqirupdwlrq/ wkdw lv vlplodu wr wkh rqh ghvfulehg lq Sursrvlwlrq 4 ri wklv sdshu= hdfk elgghu fdqex|rqhremhfwiruwkhplqlpxpsulfhdoorzhge|wkhuxohvriwkhdxfwlrq1 Sursrvlwlrq 4 doorzv iru sulydwh lqirupdwlrq exw uhvwulfwv wkh elgghuv* xwlolw| ixqfwlrqv wr eh dgglwlyh= l1h1 hdfk elgghu*v zloolqjqhvv wr sd| iru hdfk remhfw lv lqghshqghqw ri zkhwkhu vkh lv dovr ex|lqj wkh rwkhu remhfw1 Wklv uhvxow kdv dovr ehhq hvwdeolvkhg lqghshqghqwo| e| Hqjhoeuhfkw0 Zljjdqv dqg Ndkq ^9`1 Wkh| dovr hvwdeolvk wkh h{lvwhqfh ri rwkhu corz uhyhqxh* htxloleuld/ dozd|v iru wkh fdvh zlwk wzr elgghuv/ wzr remhfwv dqg dgglwlyh xwlolw| ixqfwlrqv1 Xqghu d frqglwlrq zklfk uxohv rxw frpsohphqwdulwlhv lq wkh ex|huv* xwlolw| ixqfwlrqv/ J? dqg Vwdffkhwwl ^;` kdyh vwxglhg d jhqhudol}hg yhuvlrq ri wkh Hqjolvk dxfwlrq dnlq wr d wdwrqqhphqw surfhvv/ zlwk hpskdvlv rq wkh uhodwlrq ehwzhhq lwv htxloleuld dqg wkh Zdoudvldq htxloleuld ri wkh xqghuo|lqj hfrqrp|1 Dxvxeho dqg Fudpwrq ^4` kdyh dovr vwxglhg hqylurqphqwv zlwk qrqlqfuhdv0 lqj pdujlqdo ydoxhv/ exw kdyh irfxvhg prvwo| rq vhdohg0elg dxfwlrqv1 Uhfhqwo|/ Nzdvqlfd ^<` kdv grqh h{shulphqwdo zrun rq frooxvlrq lq pxowlsoh remhfw vhdohg0elg dxfwlrqv/ zlwk dgglwlyh xwlolw| ixqfwlrqv1c129 Hqylurqphqwv lq zklfk wkh elgghuv kdyh lqfuhdvlqj pdujlqdo ydoxdwlrqv kdyh ehhq frqvlghuhg lq Fkdnuderuw| ^6`/ zkr kdv vwxglhg surshuwlhv ri ydulrxv vhdohg0elg dxfwlrqv irupdwv1 Klv sdshu dovr frqwdlqv d jrrg vxuyh| ri h{lvwlqj zrun rq pxowlsoh remhfw dxfwlrqv1 c129 A i TLtt M * |) Lu UL** t L? ? @ U| L?t @t @*tL Mii? t| _ i_ i |i?t i*) ? | i t ?}*i LM iU| U@ti E5ii @4TMi** d2oc Bh@ @4 @?_ G127@ht @** d.oc G127@ *@| @?_ ~i4t!) d oc G127U uii @?_ G127UG127 **@? d eo @?_ iti?_Lhuih d So 5 Lq wklv sdshu zh h{dplqh wkh fodlp wkdw jhqhudol}hg Hqjolvk dxfwlrqv fdq eh pruh yxoqhudeoh wr frooxvlrq lq wkh pxowlsoh remhfwv fdvh wkdq lq wkh vlqjoh remhfw fdvh1 Frqfhuqv derxw frooxvlyh ehkdylru ri elgghuv kdyh hphujhg/ iru h{dpsoh/ lq dq duwlfoh sxeolvkhg lq Wkh Hfrqrplvw +4<<:,1 Prvw uhfhqwo|/ Fudpwrq dqg Vfkzduw} ^7` kdyh lqglfdwhg hylghqfh ri frooxvlyh ehkdylru lq wkh IFF vshfwuxp dxfwlrqv/ dqg glvfxvvhg wkh h hfwlyhqhvv ri ydulrxv prgl fdwlrqv ri wkh dxfwlrq uxohv lq klqghulqj elgghuv* frooxvlrq1 Lq sduwlfxodu/ wkh iroorzlqj frqmhfwxuhv duh xvxdoo| khog derxw dxfwlrqv zlwk pxowlsoh remhfwv= Wkh suhvhqfh ri pxowlsoh remhfwv idflolwdwhv frooxvlrq e| doorzlqj wkh elgghuv wr vljqdo wkhlu zloolqjqhvv wr devwdlq iurp frpshwlqj ryhu fhuwdlq remhfwv/ surylghg wkh| duh qrw fkdoohqjhg rq rwkhuv1 Lq wklv zd|/ wkh djhqwv fdq doorfdwh wkh remhfwv dprqj wkhpvhoyhv zlwkrxw sd|lqj pxfk1 Dv wkh udwlr ri elgghuv wr remhfwv lqfuhdvhv/ wkh srvvlelolw| ri frooxvlyh vfkhphv dv wkh rqhv lqglfdwhg lq wkh suhylrxv frqmhfwxuh whqgv wr glvdsshdu1 Kljk frpsohphqwdulwlhv dprqj remhfwv klqghu frooxvlrq1 Wklv lv ehfdxvh hdfk elgghu lv ohvv vdwlv hg zlwk rzqlqj rqo| d vxevhw ri wkh remhfwv rq vdoh> vkh kdv wkhuhiruh dq lqfhqwlyh wr euhdn wkh frooxvlrq dqg frpshwh iru doo wkh remhfwv lq rughu wr ixoo| uhdol}h wkh v|qhujlhv1 Zh vwxg| krz wkh vljqdoolqj rssruwxqlwlhv surylghg e| wkh vhtxhqwldo qdwxuh ri rshq dvfhqglqj dxfwlrqv fdq eh h{sorlwhg e| wkh elgghuv lq wkh suhvhqfh ri pxowlsoh remhfwv wr frruglqdwh rq htxloleuld zklfk jhqhudwh orz uhyhqxh iru wkh vhoohu dqg lpsohphqw vrfldoo| lqh flhqw doorfdwlrqv ri wkh remhfwv1 Iru vlpsolflw|/ zh irfxv rq wkh fdvh ri wzr remhfwv/ dowkrxjk wkh uhvxowv fduu| ryhu wr wkhfdvhzlwkdq|qxpehuriremhfwv1 Wkh prgho lv ghvfulehg lq vhfwlrq 51 Lq vhfwlrq 6 zh ehjlq wkh dqdo|vlv zlwk wkh ehqfkpdun fdvh ri sxuho| dgglwlyh ydoxhv/ l1h1 zh dvvxph wkdw hdfk elgghu rewdlqv qr v|qhujlhv iurp rzqlqj pxowlsoh remhfwv/ khqfh khu zloolqjqhvv wr sd| iru rqh remhfw lv lqghshqghqw ri zkhwkhu vkh lv dovr ex|lqj rwkhu remhfwv1 Zh suhvhqw frqglwlrqv xqghu zklfk frooxvlrq0yld0vljqdolqj fdq eh vxvwdlqhg lq htxloleulxp1 Htxloleuld lq wklv fodvv fdq eh ghvfulehg iru wkh vlpsoh fdvh zlwk rqo| wzr elgghuv dv iroorzv1 Hdfk elgghu vwduwv e| sodflqj wkh vpdoohvw srvvleoh elg rq khu prvw ydoxhg remhfw/ dqg qr elg rq wkh rwkhu remhfw1 Li rqo| rqh elg lv sodfhg rq hdfk remhfw/ lw ehfrphv frpprq nqrzohgjh wkdw wkh elgghuv udqn wkh remhfwv gl huhqwo|/ dqg wkh elgghuv vlpso| frq up wkhlu elgv lq wkh qh{w urxqg wkxv iruflqj wkh dxfwlrq wr hqg zlwk hdfk ex|lqj rqh remhfw iru wkh plqlpxp sulfh1 Li/ lqvwhdg/ wkh lqlwldo elgv uhyhdo wkdw wkh wzr elgghuv kdyh d kljkhu ydoxh iru wkh vdph remhfw/ wkhq wkh elgglqj frqwlqxhv dffruglqj wr vrph htxloleulxp vwudwhj|/ zklfk fdq hqwdlo/ iru h{dpsoh/ d uhyhuvlrq wr _elgglqj vwudljkwiruzdugo|/% l1h1 udlvlqj wkh elg rq hdfk remhfw li wkh ydoxh 6 lv kljkhu wkdw wkh fxuuhqw kljkhvw elg dqg wkh elgghu lv qrw dvvljqhg wkh remhfw1c50 Dowhuqdwlyho|/ wkh elgghuv pd| frqwlqxh elgglqj dffruglqj wr vrph rwkhu frqwlqxdwlrq vwudwhj| lq zklfk wkh| surfhhg wr vljqdo pruh ghwdlohg lqirupdwlrq derxw wkhlu ydoxhv lq rughu wr frruglqdwh zlwk hdfk rwkhu dqg ex| rqo| rqh remhfw hdfk iru d uhodwlyho| orz sulfh1 Lq doo htxloleuld ri wklv nlqg/ wkh rxwfrph lv vrfldoo| lqh flhqw G127 l1h1 wkh remhfwv duh qrw dozd|v dvvljqhg lq d zd| wkdw pd{lpl}hv wkh wrwdo elgghuv* zloolqjqhvv wr sd| G127 exw wkh elgghuv hqg xs sd|lqj ohvv wkdq wkh| zrxog e| elgglqj vwudljkwiruzdugo| wkurxjkrxw wkh hqwluh dxfwlrq1 Dqg lw wxuqv rxw wkdw wkh uhgxfhg sd|phqwv pdnh xs iru wkh orvv ri h flhqf| lq dvvljqlqj wkh remhfwv/ vr wkdw wkh hdfk elgghu*v lqwhulp h{shfwhg vxusoxv lv lqfuhdvhg1 Zh dovr vkrz krzhyhu wkdw/ iru wkhvh htxloleuld/ wkh suredelolw| wkdw wkh elgghuv fdq frooxgh yld vljqdolqj ghfuhdvhv dv wkhlu qxpehu lqfuhdvhv uhodwlyh wr wkh qxpehu ri remhfwv1 Wklv uhvxow fruurerudwhv wkh frqmhfwxuh wkdw frooxvlrq lv d corz qxpehuv* skhqrphqrq1 Lq vhfwlrq 7 zh frqvlghu wkh fdvh lq zklfk wkh elgghuv* xwlolw| ixqfwlrqv h{klelw odujh frpsohphq0 wdulwlhv/ l1h1 wkhlu zloolqjqhvv wr sd| iru wkh wzr remhfwv wrjhwkhu lv pxfk juhdwhu wkdq wkh vxp ri wkh wzr remhfwv*v _vwdqg dorqh% ydoxhv1 Zh vkrz wkdw wkh vroh suhvhqfh ri frpsohphqwdulwlhv grhv qrw klqghu frooxvlrq= wkh elgghuv fdq vwloo pdqdjh wr ex| rqh remhfw hdfk/ dw orz sulfhv1 Lq idfw/ lq wkh h{wuhph fdvh lq zklfk wkh v|qhujlhv duh frpprqo| nqrzq/ dqg qrw wrr gl huhqw dfurvv wkh elgghuv/ wkh lqfhqwlyh vwuxfwxuh iru wkh elgghuv lv hvvhqwldoo| lghqwlfdo wr wkh fdvh zlwk qr frpsohphqwdulwlhv1 Wkh h flhqf| orvv krzhyhu lv pxfk odujhu ehfdxvh lw lqfoxghv wkh xquhdol}hg frpsohphqwdulwlhv1 Zkhq frpsohphqwdulwlhv duh qrw rqo| odujh exw dovr yduldeoh krzhyhu/ wkh srvvlelolw| ri frooxvlrq lv vhulrxvo| uhgxfhg1 Wklv vxjjhvwv wkdw zkdw lv fuxfldo lq ghwhuplqlqj wkh olnholkrrg ri frooxvlrq lv qrw zkhwkhu wkh frpsohphqwdulwlhv duh +rq dyhudjh, codujh*/ exw pruh krz yduldeoh wkh| duh1 Vhfwlrq 8 frqwdlqv frqfoxglqj uhpdunv/ dqg dq dsshqgl{ froohfwv doo wkh surriv1 2 A i G127L_i* Wkhuh duh d vhw Q @ i4> ===> qj ri elgghuv dqg d vhw P @ i4>===>pj ri remhfwv/ zlwk p> q qlwh1 Wkh elgghuv kdyh txdvl0olqhdu xwlolw| ixqfwlrqv/ dqg wkh zloolqjqhvv wr sd| ri elgghu l 5 Q iru exqgoh M P lv jlyhq e| xc16 +M, = Elgghu l nqrzv khu ydoxhv ixc16 +M,jc97 c77 c50 c127 / zkloh wkh uhvw ri wkh zruog rqo| nqrzv wkdw vxfk ydoxhv duh gudzq dffruglqj wr d suredelolw| ixqfwlrq zlwk vxssruw rq d frpsdfw vxevhw ri ?c50 c54c110 1 Wkh p remhfwv duh vrog zlwk dq rshq dvfhqglqj dxfwlrq/ qdphg khuh wkh Jhqhudol}hg Hqjolvk Dxfwlrq/ ru JHD/ zklfk lv d qdwxudo h{whqvlrq ri wkh vwdqgdug rqh0remhfw Hqjolvk dxfwlrq wr c50 A i ti Lu | i i Thitt L? RM __ ?} t|h@ } |uLh @h_*) |L _i?L|i @ t|h@|i}) | @| t t 4 *@h |L | i t|@?_@h_ t|h@|i}) ? | i L?i LM iU| ,?}* t @ U| L? t _ i |L G127 *}hL4 d o 7 hqylurqphqwv zlwk pxowlsoh remhfwv1 Wkh dxfwlrq surfhhgv lq urxqgv1 Dw wkh lqlwldo urxqg hdfk elgghu l vxeplwv d yhfwru ri elgv ec129c16 +4,>===ec54c16 +4, /zkhuhec26c16 +4, ghqrwhv wkh dprxqw wkdw elgghu l ghfoduhv vkh lv suhsduhg wr sd| iru remhfw m dw urxqg 41 Wkh fdvh lq zklfk elgghu l sodfhv qr elg rq remhfw m lv wuhdwhg vhwwlqj ec26c16 +4, @ >/ dqg wkh plqlpxp h hfwlyh elg lv qrupdol}hg wr }hur1 Wkh dxfwlrq hqgv zlwk wkh vhoohu nhhslqj doo remhfwv li dqg rqo| li hdfk elgghu sodfhv qr elgrqdq|remhfw/l1h1 liec129c16 +4, @ > iru hdfk m 5 P> l 5 Q= Rwkhuzlvh/ erwk wkh kljkhvw elg ec26 +4, =@ pd{c16 ec26c16 +4, +> @ 4 e| frqyhqwlrq, dqg d srwhqwldo zlqqhu dprqj wkh elgghuv zkr kdyh r huhg ec26 +4, duh lghqwl hg iru hdfk remhfw m 5 P> dqgwkhdxfwlrqpryhvwrurxqg51 Dwurxqg 5/ hdfk elgghu l vxeplwv d qhz yhfwru ri elgv ec129c16 +5, >===ec54c16 +5, 1 Zh prgho wkh frqglwlrq wkdw suhylrxv elgv fdqqrw eh zlwkgudzq e| uhtxlulqj wkdw ec129c16 +5, >===ec54c16 +5, ec129c16 +4, >===ec54c16 +4, = Wkh dxfwlrq hqgv dw vwdjh 5 li qr elgghu uhylvhv khu elg rq dq| remhfw/ l1h1 li ec26c16 +5, @ ec26c16 +4, > doo m 5 P> l 5 Q= Rwkhuzlvh d srwhqwldo zlqqhu lv vhohfwhg djdlq iru hdfk remhfw m dprqj doo l vxfk wkdw ec26c16 +5, @ ec26 +5,=@pd{c16 c30 ec26c16 c30 +5,/ dqg wkh dxfwlrq pryhv wr wkh qh{w urxqg1 Surfhhglqj lq wklv idvklrq/ li urxqg w 5lvuhdfkhg/dqgliec26c16 +w,@ec26c16 +w 4, iru doo m 5 P dqg l 5 Q> wkhq wkh dxfwlrq hqgv/ dqghdfkremhfwm lv dvvljqhg wr wkh ex|hu vhohfwhg dw wkh hqg ri urxqg w 4dprqjdool vxfk wkdw ec26c16 +w 4, @ ec26 +w 4, =@ pd{c16 c30 ec26c16 c30 +w 4,= Wkh vhohfwhg ex|hu sd|v klv odvw elg ec26 +w 4, @ ec26 +w, = Wr nhhs wkh dqdo|vlv dqg wkh qrwdwlrq dv vlpsoh dv srvvleoh/ zh hvwdeolvk rxu pdlq uhvxowv iru wkh fdvh p @5> l1h1 rqo| wzr remhfwv rq vdoh1 Wkh pdlq lqvljkwv krzhyhu dsso| wr wkh pruh jhqhudo fdvh1 Zh gh qh= yc16 =@ xc16 +i4j,/ wkh ydoxh wr elgghu l ri kdylqj remhfw 4 rqo|> zc16 =@ xc16 +i5j, > wkh ydoxh wr elgghu l ri kdylqj remhfw 5 rqo|1 Zh zloo xvh lqwhufkdqjhdeo| wkh whupv cremhfw y*+remhfwz, dqg cremhfw 4* +remhfw 5,1 Zlwk rqo| wzrremhfwv/delge|djhqwl lq urxqg w lv mxvw dq rughuhg sdlu ec129c16 +w, >ec50c16 +w, 1 Ilqdoo|/ zh dvvxph wkdw wkh vl}h ri wkh frpsohphqwdulw| lv lqghshqghqw ri wkh wzr remhfwv* cvwdqg0dorqh* ydoxhv/ l1h1 wkh ydoxh wr elgghu l ri kdylqj erwk remhfwv lv xc16 +i4>5j,@yc16 . zc16 . nc16 > Iru hdfk l 5 Q> wkh ydoxhv +yc16 >zc16 >nc16 , duh gudzq iurp d mrlqw suredelolw| glvwulexwlrq zlwk ghqvlw| k +yc16 >zc16 >nc16 , dqg vxssruw ^3> 4`c50 N1 Zh dvvxph wkdw yc16 dqg zc16 duh lqghshqghqw dqg lghqwlfdoo| glvwulexwhg1 Wkxv/ yc16 dqg zc16 kdyh lghqwlfdo pdujlqdo glvwulexwlrq/ zkrvh ghqvlw| dqg f1g1i1 zh ghqrwh e| i dqg I uhvshfwlyho|1 Wkh pdujlqdo glvwulexwlrq ri nc16 lv hlwkhu ghjhqhudwh rq 3> ru lv uhsuhvhqwhg e| d ghqvlw| j +dqg f1g1i1 J, zlwk vxssruw ryhu wkh lqwhuydo n>n zlwk n A 3= Wkh ydoxhv +yc16 >zc16 >nc16 , duh gudzq lqghshqghqwo| ri +yc26 >zc26 >nc26 ,iruhdfkm 9@ l= 8 Lq wkh qh{w vhfwlrq zh h{dplqh wkh fdvh zlwk qr frpsohphqwdulwlhv/ l1h1 nc16 @3irudool 5 Q= Vhfwlrq 7 zloo eh ghyrwhg wr wkh fdvh zlwk srvlwlyh frpsohphqwdulwlhv1 L** t i ,^ * Mh @ | L L4T*i4i?|@h | it Lq wklv vhfwlrq zh frqvlghu wkh fdvh ri sxuho| dgglwlyh ydoxhv xc16 +i4>5j,@yc16 . zc16 >l5 Q> ru qr frpsohphqwdulwlhv/ nc16 @3iruhdfkl1 Lqwklvfdvhwkhuhlvuhdoo|qrsrlqwlqdxfwlrqlqjwkh remhfwv vlpxowdqhrxvo|/ vlqfh wkh h flhqw doorfdwlrq ri hdfk remhfw lv lqghshqghqw ri wkh doorfdwlrq ri wkh rwkhu remhfwv1 D vhtxhqfh ri vlqjoh0remhfw Hqjolvk dxfwlrqv zrxog doorfdwh hdfk remhfw wr wkh elgghu zlwk wkh kljkhvw ydoxh/ dqg/ zlwk vxlwdeo| fkrvhq uhvhuyh sulfhv/ zrxog dovr pd{lpl}h wkh vhoohu*v h{shfwhg uhyhqxh1 Krzhyhu/ wkh dqdo|vlv ri wkh elgghuv* htxloleulxp ehkdylru lq wkh JHD zloo surylgh d xvhixo ehqfkpdun iru wkh pruh uhdolvwlf fdvh lq zklfk frpsohphqwdulwlhv duh suhvhqw1 Lq sduwlfxodu/ wkh sxuho| dgglwlyh fdvh zloo vkhg oljkw rq wkh uroh sod|hg e| pxowlsoh remhfwv lq idflolwdwlqj frooxvlrq dprqj wkh elgghuv1 Zh ehjlq zlwk wkh iroorzlqj vlpsoh exw lpsruwdqw revhuydwlrq= zlwk qr frpsohphqwdulwlhv/ lw lv d shuihfw Ed|hvldq htxloleulxp iru wkh JHD wkdw hdfk elgghu l iroorz d cVhsdudwhg Hqjolvk Dxfwlrqv* vwudwhj| +VHD,/ elgglqj rq hdfk remhfw m xqwlo wkh sulfh uhdfkhv wkh ydoxh xc16 +imj,= Lw lv fohdu wkdw/ li doo rwkhu djhqwv duh iroorzlqj wklv vwudwhj|/ wkhq sod|hu l*v ehvw uhso| lv wr iroorz wkh vdph vwudwhj|c12 1 Qrwh wkdw wkh VHD vwudwhjlhv duh zhdno| grplqdqw/ khqfh rswlpdo iru hdfk elgghu lqghshqghqw ri khu eholhiv derxw khu rssrqhqw*v ydoxhv1 Zh vwdwh wkh uhvxow iru dq duelwudu| qxpehu ri remhfwv lq wkh iroorzlqj Sursrvlwlrq1 Sursrvlwlrq 3 Zlwk qr frpsohphqwdulwlhv/ iru dq| q dqg p> wkh vhsdudwhg Hqjolvk dxfwlrqv vwudw0 hj| +VHD, sur oh irup d shuihfw Ed|hvldq htxloleulxp +zlwk vrph frqvlvwhqw eholhi v|vwhp, diwhu dq| klvwru| lq wkh Jhqhudol}hg Hqjolvk dxfwlrq1 Wkh VHD vwudwhj| sur oh fdq eh xvhg lq wkh vdph zd| dv Sduhwr lqihulru htxloleuld duh xvhg lq uhshdwhg jdphv wr vxssruw frooxvlyh rxwfrphv1 Wkh| frqvwlwxwh wkh wkuhdw xvhg wr ghwhu wkh elgghuv iurp xvlqj djjuhvvlyh elgglqj vwudwhjlhv1 Wkh qh{w revhuydwlrq lv dq lpphgldwh lpsolfdwlrq ri wkh zhoo0nqrzq Uhyhqxh Htxlydohqfh Wkhr0 uhp +P|huvrq ^48`,1 Vlqfh wkh elgghu*v w|shv duh gudzq iurp lqghshqghqw suredelolw| glvwulexwlrqv/ e| lqfhqwlyh frpsdwlelolw| wkh VHD htxloleulxp lv wkh xqltxh SEH zklfk lpsohphqwv wkh vrfldoo| c12 W| t Lh| TL ?| ?} L | ihi | @| | t @h} 4i?| L *_ u@ * ? | i Thiti?Ui Lu UL4T*i4i?|@h | it 9 h flhqw doorfdwlrq1 Wkh lqfhqwlyh frpsdwlelolw| frqvwudlqwv slq grzq wkh lqwhulp h{shfwhg sd|0 phqw ixqfwlrq ri hdfk elgghu iru dq| jlyhq remhfwv* doorfdwlrq uxoh1 Wkhuhiruh/ lq dq| htxloleulxp ri dq| wudglqj jdph zlwk qr frpsohphqwdulwlhv zklfk lpsohphqwv wkh vrfldoo| h flhqw doorfdwlrq wkh ex|huv* lqwhulp h{shfwhg sd|phqwv pxvw eh htxdo wr wkh vxp ri wkh h{shfwhg sd|phqwv lq p vhsdudwh vwdqgdug Hqjolvk dxfwlrqv1 Wklv dovr lpsolhv wkdw lq dq| shuihfw Ed|hvldq htxloleulxp ri wkh JHD lq zklfk doo elgghuv duh ehwwhu r wkdq lq wkh VHD htxloleulxp wkh remhfwv fdqqrw eh doorfdwhg dffruglqj wr wkh vrfldoo| h flhqw uxoh1 A L __iht Zh ehjlq zlwk wkh fdvh lq zklfk wkhuh duh rqo| wzr elgghuv1 Sursrvlwlrq 4 hvwdeolvkhv wkh h{lvwhqfh ri d v|pphwulf Shuihfw Ed|hvldq htxloleulxp zklfk grplqdwhv wkh VHD lq whupv ri ex|huv*v lqwhulp h{shfwhg vxusoxvc101 1UhfdoowkdwI lv wkh frpprq pdujlqdo f1g1i1 ri yc16 dqg zc16 1 Sursrvlwlrq 4 Vxssrvh wkdw H +{, U c129c102 {gI+{, c129c50 = Wkhq wkh iroorzlqj vwudwhj|/ wrjhwkhu zlwk vrph frqvlvwhqw eholhi v|vwhp/ irup d v|pphwulf shuihfw Ed|hvldq htxloleulxp= W|shv +yc16 >zc16 , vxfk wkdw yc16 zc16 rshq zlwk ec129c16 +4, >ec50c16 +4, @+3> >,> W|sh +yc16 >zc16 , vxfk wkdw yc16 ?zc16 rshq zlwk ec129c16 +4, >ec50c16 +4, @+>> 3, > Li wkh lqlwldo elgv duh gl huhqw/ doo w|shv frq up wkhlu elg lq urxqg 51 Li dw dq| urxqg wkh elgv gl hu iurp wkh rqhv jlyhq deryh/ doo w|shv uhyhuw wr wkh VHD vwudwhj| ghvfulehg lq Sursrvlwlrq 31 Wklv htxloleulxp fdq eh ghvfulehg dv iroorzv1 Hdfk elgghu rshqv e| pdnlqj wkh plqlpxp elg +}hur, rqo| rq khu prvw suhihuuhg remhfw1 Li/ dw wkh hqg ri wkh uvw urxqg/ wkh elgghuv glvfryhu wkdw wkh| udqn wkh remhfwv gl huhqwo|/ wkhq wkh| vwrs elgglqj/ dqg hdfk lv deoh wr ex| khu suhihuuhg remhfw dw wkh orzhvw srvvleoh sulfh1 Li lqvwhdg wkh| glvfryhu wkdw wkh| udqn wkh wzr remhfwv lq wkh vdph zd|/ wkhq wkh| uhyhuw wr wkh Vhsdudwhg Hqjolvk Dxfwlrqv vwudwhjlhv1 Wkh frqglwlrq H +{, c129c50 jxdudqwhhv wkdw/ iru hdfk w|sh ri hdfk elgghu/ wkh h{shfwhg vxusoxv iurp wuljjhulqj wkh VHD vwudwhjlhv diwhu dq rshqlqj zlwk gl huhqw elgv lv orzhu wkdq wkh vxusoxv rewdlqhg e| ex|lqj mxvw khu prvw suhihuuhg remhfw iru wkh plqlpxp sulfh1 Lq wkh qh{w vxevhfwlrq/ zh vkrz wkdw wkh vhw ri shuihfw Ed|hvldq htxloleuld ri wkh JHD frqwdlqv rwkhu/ _pruh frooxvlyh% htxloleuld/ l1h1 htxloleuld lq zklfk wkh elgghuv hqg xs zlwk d kljkhu lqwhulp h{shfwhg vxusoxv1 c101 A i i t|i?Ui Lu | t i^ * Mh 4 @t Mii? it|@M* t i_ ?_iTi?_i?|*) M) ,?}*i*MhiU | ` }}@?t @?_ k@ ? dSo : 2 Bi|| ?} G127Lhi L | Lu L** t L? Wkh htxloleulxp ghvfulehg lq Sursrvlwlrq 4 suhvfuleh wkdw wkh elgghuv uhyhuw wr wkh VHD vwudwhjlhv zkhq wkh| rshq zlwk wkh vdph elgv/ l1h1 zkhq lw ehfrphv frpprq nqrzohgjh wkdw wkhlu suhihuuhg remhfw lv wkh vdph1 Lw lv wkxv qdwxudo wr dvn zkhwkhu lq wklv fdvh wkh elgghuv fdq gr ehwwhu e| wu|lqj djdlq wr frruglqdwh wkhpvhoyhv dqg ex| rqh remhfw hdfk dw uhodwlyho| orz sulfhv1 Wkh d updwlyh dqvzhu wr wklv txhvwlrq lv surylghg e| wkh qh{w sursrvlwlrq1 Sursrvlwlrq 5 Ohw {/ | eh wzr lqghshqghqw udqgrp yduldeohv zlwk f1g1i1 I1 Dvvxph wkdw/ iru hdfk d 5 ^3>4`> wkh iroorzlqj lqhtxdolwlhv krog= H ^{ m 3 { 4 d`.H ^{m d { 4` 4> +4, H ^{m { d . |>4 d |`.H ^{m | d { 3>| d` 4= +5, Wkhq wkh iroorzlqj vwudwhj|/ wrjhwkhu zlwk vrph frqvlvwhqw eholhi v|vwhp/ irup d v|pphwulf shuihfw Ed|hvldq htxloleulxp= Iluvw urxqg= W|shv +yc16 >zc16 , vxfk wkdw yc16 zc16 rshq zlwk ec129c16 +4, >ec50c16 +4, @+3> >,> W|shv +yc16 >zc16 , vxfk wkdw yc16 ?zc16 rshq zlwk ec129c16 +4, >ec50c16 +4, @+>> 3, = Vxevhtxhqw urxqgv= Li wkh lqlwldo elgv duh hlwkhu ++3> >, >+>>3,, ru ++>>3,> +3> >,, wkhq doo w|shv frq up wkhlu elgv> Li wkh lqlwldo elgv duh ++3>>,> +3>>,,> wkhq w|shv +yc16 >zc16 , vxfk wkdw yc16 zc16 @ dc16 nhhs udlvlqj wkhlu elg rq remhfw y zkloh uhiudlqlqj iurp elgglqj rq z xqwlo hlwkhu l, wkh rssrqhqw vwrsv/ ru ll, wkh elg uhdfk wkh ydoxh dc16 1 Lq fdvh l,/ wkh| gr qrw uhylvh dq| elg iru wkh qh{w wzr urxqgv>dqg lq fdvh ll, wkh| elg +dc16 >3,> iru wzr frqvhfxwlyh urxqgv/ wkxv prylqj wkh rxwvwdqglqj elg rq z iurp > wr 3= Li wkh lqlwldo elgv duh ++>> 3, >+>>3,, > wkh vwudwhj| lv v|pphwulf/ zlwk wkh urohv ri y dqg z vzlwfkhg1 Rxw~ri~htxloleulxp sdwkv= Li dw dq| urxqg d elg qrw lq dffrugdqfh wr wkh deryh ghvfulehg vwudwhj| lv revhuyhg/ wkhq hdfk w|sh uhyhuwv wr wkh VHD vwudwhj|1 ; Wkh htxloleulxp irupdoo| vwdwhg lq Sursrvlwlrq 5 fdq eh ghvfulehg dv iroorzv1 Wkh elgghuv rshq e| vljqdoolqj zklfk remhfw wkh| suhihu1 Li wkh| suhihu gl huhqw remhfwv/ wkhq wkh jdph hqgv/ dv lq wkh htxloleulxp ri Sursrvlwlrq 41 Li wkh| suhihu wkh vdph remhfw/ vd| y/ wkhq wkh| nhhs udlvlqj wkh sulfh rq y zkloh devwdlqlqj iurp frpshwlqj rq z/ zlwk elgghu l suhsduhg wr elg xs wr wkh gl huhqfh ehwzhhq khu wzr ydoxhv dc16 @ yc16 zc16 1 Wkxv/ li dc16 Adc12 c51 c16 /elgghul hqgv xs ex|lqj remhfw y iru wkh gl huhqfh ehwzhhq khu rssrqhqw*v ydoxhv dc12 c51 c16 = Wkh rssrqhqw vwrsv frpshwlqj rq y zkhq wkh sulfh uhdfkhv wkh gl huhqfh ehwzhhq khu ydoxhv/ dqg ex|v z iru wkh plqlpxp elg1 Lq wklv htxloleulxp/ hdfk elgghu*v w|sh vhw ^3>4`c50 lv sduwlwlrqhg lqwr olqhv zlwk vorsh 4= w|shv rq wkh vdph olqh G127 l1h1 zlwk wkh vdph gl huhqfh ehwzhhq wkh wzr remhfwv* ydoxhv G127 ehkdyh lghqwlfdoo| khqfh uhpdlq lqglvwlqjxlvkdeoh xqwlo wkh hqg ri wkh dxfwlrq1 Lw lv ehfdxvh ri wklv srrolqj dprqj orz dqg kljk w|shv wkdw wkh elgghuv devwdlq iurp wuljjhulqj wkh VHD vwudwhjlhv dw wkh hqg ri wkh elgglqj rq rqh remhfw1 Lw lv zruwk qrwlqj wkdw wkh htxloleulxp vwudwhj| lv vlpsoh= rqo| dw wkh qdo urxqg wkh elgghuv* eholhiv sod| d uroh lq vxvwdlqlqj wkh htxloleulxp1 Lw lv fuxfldo krzhyhu wkdw qr remhfw lv dvvljqhg ehiruh wkh hqg ri wkh dxfwlrq/ l1h1 dq| remhfw fdq eh vwloo erxjkw hyhq diwhu pdq| urxqgv lq zklfk lwv rxwvwdqglqj elg kdv qrw pryhg1 Lw lv wkxv whpswlqj wr frqfoxgh wkdw vlpso| lpsrvlqj dq _dfwlylw| uxoh% G127 l1h1 d frqglwlrq vshfli|lqj wkdw dq| remhfw zkrvh sulfh grhv qrw pryh iru d fhuwdlq qxpehu ri urxqgv eh dvvljqhg wr d elgghu zkr kdv pdgh wkh kljkhvw elg G127 fdq eh h hfwlyh lq suhyhqwlqj wkh elgghuv iurp dfklhylqj wklv ohyho ri frooxvlrq1 Lq oljkw ri Sursrvlwlrq 5 krzhyhu/ lw lv srvvleoh wr revhuyh wkdw +xqghu vrph frqglwlrqv rq wkh f1g1i1 I, wkhuh h{lvwv dqrwkhu v|pphwulf shuihfw Ed|hvldq htxloleulxp |lhoglqj wkh rxwfrph ri Sursrvlwlrq 5 lq dw prvw wkuhh urxqgv= li wkh elgghuv rshq vljqdolqj wkdw wkh| erwk suhihu wkh vdph remhfw/ vd| y/lqwkhvhfrqgurxqghdfksodfhvdelg rq y htxdo wr wkh h{shfwhg gl huhqfh ehwzhhq khu rssrqhqw*v ydoxhv> dqg lq wkh wklug urxqg/ wkh orzhvw elgghu fdq ex| z iru wkh plqlpxp sulfh1 Wkh frpshwlwlrq skdvh rq y lv wkxv frpsuhvvhg lq mxvw rqh urxqg/ khqfh wklv htxloleulxp fdqqrw eh ghvwur|hg e| dq| uxoh zklfk doorzv dw ohdvw wzr urxqgv ri lqdfwlylw| ehiruh forvlqj wkh elgglqj rq dq remhfw1 Li dq| srvlwlyh zhljkw lv jlyhq wr wkh vhoohu*v vxusoxv/ wkh rxwfrph ri wklv htxloleulxp lv Sduhwr lqihulru hyhq wr wkh rqh jhqhudwhg e| wkh htxloleulxp ri Sursrvlwlrq 41 Vrfldo h flhqf| uhtxluhv wkdw hdfk remhfw eh dvvljqhg wr wkh djhqw zkr ydoxhv lw prvw/ dqg wklv lv zkdw kdsshqv zkhq wkh VHD vwudwhjlhv duh wuljjhuhg1 Lq wkh htxloleulxp ri Sursrvlwlrq 5 wkh VHD vwudwhjlhv duh qhyhu wuljjhuhg1 Frqglwlrqv +4, dqg +5, lpso| H +{, c129c50 > vlqfh wkh odwwhu lv rewdlqhg vhwwlqj d @ 3 lq +4,1 Wkh wzr frqglwlrqv duh vdwlv hg/ iru h{dpsoh/ e| wkh xqlirup glvwulexwlrq ryhu ^3>4` = < G127Lhi | @? A L __iht Wkh htxloleuld ghvfulehg lq Sursrvlwlrqv 4 dqg 5 pd| vhhp wr uho| khdylo| rq wkh idfw wkdw wkh qxpehu ri elgghuv lv htxdo wr wkh qxpehu ri remhfwv1 Krzhyhu/ vrph ghjuhh ri frooxvlrq lv vwloo srvvleoh zkhq wkhuh duh pruh elgghuv wkdq remhfwv1 Wkh edvlf lghd lv wkdw wkh elgghuv fdq iroorz wkh VHD vwudwhj| xqwlo rqo| 5 sod|huv duh ohiw/ dqg wkhq dgrsw wkh vwudwhjlhv ghvfulehg lq Sursrvlwlrqv 4 ru 5 wr glylgh wkh remhfwv1 Sursrvlwlrq 6 Li wkhuh duh qA5 elgghuv dqg wkh f1g1i1 I +{, vdwlv hv H ^{ m{ }` c129c110 c53c50 iru hdfk } 5 ^3>4`> wkhq wkh iroorzlqj vwudwhj|/ wrjhwkhu zlwk vrph frqvlvwhqw eholhi v|vwhp/ irup d v|pphwulf shuihfw Ed|hvldq htxloleulxp= Urxqg 4= Li yc16 zc16 > rshq zlwk +3> >,/ rwkhuzlvh rshq zlwk +>> 3,> Urxqg w= li pruh wkdq wzr elgghuv zhuh dfwlyh dw urxqg w 4> doo w|shv xvh wkh VHD vwudwhj|/ l1h1 wkh| lqfuhdvh wkhlu elg rq hdfk remhfw li wkhlu ydoxh lv kljkhu wkdq wkh fxuuhqw kljkhvw elg dqg wkh| duh qrw dvvljqhg wkh remhfw1 Li dw urxqg w 4 rqo| l dqg m 9@ l zhuh dfwlyh/ dqg elgghu m rshqhg zlwk +>> 3, > wkhq w|shv +yc16 >zc16 , vxfk wkdw yc16 zc16 ulvh wkh elg rq y rqo| e| d vpdoo dprxqw1 W|shv +yc16 >zc16 , vxfk wkdw yc16 zc16 xvh v|pphwulf vwudwhj| li m rshqhg zlwk +3>>,1 Li wkh revhuyhg klvwru| ri elgv lv qrw rewdlqhg dffruglqj wr wkh vwudwhjlhv suhylrxvo| ghvfulehg wkhq dgrsw wkh VHD vwudwhj|1 D idplo| ri f1g1i1*v zklfk vdwlv hv wkh frqglwlrq H ^{m{ }` c129c50 +4 . },iruhdfk} 5 ^3>4` lv I +{,@{c107 /zlwk 41 Lq wklv fdvh zh kdyh= H +{m{ },@ .44 } c107 c110c129 4 }c107 dqg lw fdq eh fkhfnhg wkdw wkh lqhtxdolw| lv vdwlv hg iru } 5 ^3>4`= Wkh htxloleulxp ri Sursrvlwlrq 5 fdq dovr eh h{whqghg wr wkh fdvh ri qA5elgghuv1 Sursrvlwlrq 7 Vxssrvh wkdw wkhuh duh qA5 elgghuv dqg wkh f1g1i1 I lv vxfk wkdw iru hdfk sdlu +d> }, vxfk wkdw } 5 ^3>4` dqg d 5 ^3>4 }` wkh wzr iroorzlqj frqglwlrqv duh vdwlv hg= H +{ m} { 4 d,.H +{m} . d { 4, 4.} +6, H +{m { d . |>4 d | },.H +{m | d { }>| d . }, 4.}= +7, Wkhq wkh iroorzlqj vwudwhj| lv sduw ri d v|pphwulf shuihfw Ed|hvldq htxloleulxp= Ehkdyh dv lq Sursrvlwlrq 6 h{fhsw dw wkh iroorzlqj srlqw= 43 Li dw urxqg w 4 rqo| |rx dqg dqrwkhu elgghu zhuh dfwlyh wkhq= Li yc16 zc16 dqg |rx rshqhg zlwk +3> >, zkloh wkh rwkhu elgghu rshqhg zlwk +>>3, wkhq lqfuhdvh wkh elg rq y dqg qrw rq z/ wkhq vwrs1 Li yc16 ?zc16 dqg |rx rshqhg zlwk +>> 3, zkloh wkh rwkhu elgghu rshqhg zlwk +3>>, wkhq lqfuhdvh wkh elg rq y dqg qrw rq z/ wkhq vwrs1 Li erwk sod|huv rshqhg zlwk +3> >, dqg } zdv wkh odvw r hu iru erwk remhfwv wkhq lqfuhdvh wkh elg rq y xs wr } .dc16 / zkloh nhhslqj wkh r hu iru z dw }1 Li wkh rwkhu elgghuv r huv pruh wkdq } . dc16 wkhq jhw z iru }1 Rwkhuzlvh/ jhw y dw wkh sulfh dw zklfk frpshwlwlrq hqgv/ dqg ohdyh z wr wkh rwkhu elgghu1 Wklv htxloleulxp zrunv dv wkh rqh ri Sursrvlwlrq +6,= wkh elgghuv vwduw vljqdoolqj zklfk remhfw wkh| suhihu dqg wkhq sxvk xs erwk sulfhv xqwlo rqo| wzr sod|huv duh ohiw1 Wkh gl huhqfh lv wkdw dw wkdw srlqw wkh vdph vwudwhjlhv dv lq Sursrvlwlrq +5, duh xvhg= li elgghuv kdyh rshqhg vkrzlqj wkdw wkh| udqn wkh wzr remhfwv lq wkh vdph zd|/ wkhq wkh| frpshwh rqo| rq wkh wrs udqnhg remhfw1 Wkh vwrsslqj srlqw iru hdfk sod|huv lv } . dc16 / wkdw lv wkh odvw elg soxv wkh gl huhqfh ehwzhhq wkh wzr ydoxhv1 Dq lpsruwdqw revhuydwlrq lv wkdw lq wkh htxloleuld h{klelwhg lq Sursrvlwlrqv +4, dqg +6,/ ru lq Sursrvlwlrqv +5, dqg +7,/ wkh suredelolw| ri frooxvlrq ghfuhdvhv dv wkh qxpehu ri elgghuv lqfuhdvhv1 Wr eh pruh suhflvh/ wkh suredelolw| ri dvvljqlqj hdfk remhfw wr wkh elgghu zlwk wkh kljkhvw ydoxh lqfuhdvhv dv wkh qxpehu ri elgghuv lqfuhdvhv1 Frqglwlrqv +6, dqg +7, duh dovr vdwlv hg e| wkh xqlirup glvwulexwlrq1 e L** t i ,^ * Mh @ | w@h}i L4T*i4i?|@h | it Lq wklv vhfwlrq zh frqvlghu wkh fdvh ri frpsohphqwdulwlhv/ l1h1 xc16 +D ^ E, Axc16 +D,.xc16 +E,1 Dv vwdwhg lq Vhfwlrq 5/ zh gh qh xc16 +4, @ yc16 >xc16 +5, @ zc16 dqg xc16 +i4>5j,@yc16 . zc16 . nc16 >l@4>5> dqg zh dvvxph wkdw yc16 dqg zc16 duh gudzq iurp d v|pphwulf glvwulexwlrq zlwk vxssruw ^3> 4`c50 > pdujlqdo ghqvlw| i/ dqg pdujlqdo f1g1i1 I1 Zh dovr dvvxph wkdw iru hdfk sod|hu l @4> 5> wkh ydoxh ri wkh frpsohphqwdulw| nc16 lv gudzq iurp d glvwulexwlrq zlwk frqwlqxrxv ghqvlw| j dqg vxssruw ryhu dq lqwhuydo n>n 1 Hdfk udqgrp yduldeoh nc16 lv lqghshqghqw ri +yc26 >zc26 >nc26 ,iruhdfkm 9@ l1 Ilqglqj htxloleuld iru wkh fdvh lq zklfk frpsohphqwdulwlhv duh suhvhqw lv frpsolfdwhg e| wkh idfw wkdw/ dw dq| jlyhq urxqg ri wkh dxfwlrq/ d elgghu*v zloolqjqhvv wr sd| iru d jlyhq remhfw ghshqgv rq wkh suredelolw| ri zlqqlqj wkh rwkhu remhfw1 Wklv ghvwur|v wkh ceholhi0iuhh* qdwxuh ri wkh VHD htxloleulxp ghvfulehg lq Sursrvlwlrq 31 Zh fdq vkrz krzhyhu wkdw/ li wkh frpsohphqwdulwlhv duh 44 frpprqo| nqrzq wr eh codujh*/ lq d vhqvh wr eh pdgh suhflvh/ wkhq d uhvxow vlplodu wr wkh rqh irxqg lq Sursrvlwlrq 3 fdq eh rewdlqhg1 Gh qh c16 =@ yc16 . zc16 . nc16 / wkh wrwdo ydoxh ri wkh exqgoh iru djhqw l1 Sursrvlwlrq 8 Vxssrvh wkdw wkhuh duh q sod|huv/ 5 remhfwv/ dqg n A 41 Wkhq wkhuh lv d shuihfw Ed|hvldq htxloleulxp zlwk wkh iroorzlqj rxwfrph= Wkh wzr remhfwv duh doorfdwhg wr wkh djhqw zlwk wkh kljkhvw c16 > dw d sulfh htxdo wr wkh vhfrqg kljkhvw ydoxdwlrq +l1h1 pd{c26 c11c39 c16 c26 ,1 Wkh edvlf lqwxlwlrq khuh lv dv iroorzv1 Xqghu wkh dvvxpswlrqv ri Sursrvlwlrq 8/ li wkh ex|huv frpshwh rq erwk remhfwv/ wkh dxfwlrq fdqqrw hqg zlwk hdfk elgghu ex|lqj mxvw rqh remhfw ehfdxvh wkh ydoxh iru hdfk elgghu ri rzqlqj d vhfrqg remhfw lv kljkhu1 Wkhuhiruh/ erwk sod|huv ehkdyh dv li elgglqj iru d vlqjoh remhfw/ wkh exqgoh iy>zj1 Wkh htxloleulxp ri Sursrvlwlrq 8 fdq eh xvhg dv d wkuhdw wr vxvwdlq frooxvlyh htxloleuld zkhq odujh frpsohphqwdulwlhv duh suhvhqw1 Wkh qh{w sursrvlwlrq hvwdeolvkhv wkh h{lvwhqfh ri dq htxloleulxp zklfk |lhogv d vxshulru h{shfwhg vxusoxv iru erwk sod|huv1 Gh qh c15 =@ q +y>z>n, 5 ^3>4`c50 n> n m yAz r dqg c25 =@ q +y>z>n, 5 ^3> 4`c50 n>n m y z r = Sursrvlwlrq 9 Wkhuh h{lvw wzr vhwv Dc15 c15 dqg Dc25 c25 vxfk wkdw wkh iroorzlqj vwudwhj| irup d SEH= W|shv +yc16 >zc16 >n, 5 ^3>4`c50 n>n qDc15 ^ Dc25 rshq zlwk +3>3, dqg frpshwh iru erwk remhfwv> W|shv +yc16 >zc16 , 5 Dc25 rshq zlwk +>> 3, W|shv +yc16 >zc16 , 5 Dc15 rshq zlwk +3> >, = Li wkh lqlwldo elgv duh i+3> >, >+>>3,j ru i+>>3, >+3>>,j wkhq elgghuv gr qrw sodfh dq| ixuwkhu elg1 Lq doo rwkhu fdvhv/ wkh elgghuv sod| wkh VHD htxloleulxp ghvfulehg lq Sursrvlwlrq 81 Li/ dw dq| vwdjh/ wkh elgv gl hu iurp wkh rqhv jlyhq deryh/ wkh elgghuv sod| wkh VHD htxloleulxp ghvfulehg lq Sursrvlwlrq 81 Wkh vhw Dc15 dqg Dc25 kdyh wkh surshuw| wkdw li +y>z>n, 5 Dc15 lpsolhv +z>y>n, 5 Dc25 dqg ylfh0yhuvd1 45 Wkh htxloleulxp ri Sursrvlwlrq 9 lv d qdwxudo jhqhudol}dwlrq ri wkh htxloleulxp ghvfulehg lq Sursrvlwlrq +4,1 Wkh vhw ri w|shv ri hdfk elgghu lv glylghg lqwr wkuhh vxevhwv1 Wkh uvw vxevhw frqvlvwv ri wkrvh w|shv zkr fdqqrw eh lqgxfhg wr frooxgh1 Wkhvh duh wkh w|shv zkr kdyh yhu| orz vwdqg0dorqh ydoxhv iru hdfk remhfw/ dqg zkr wkhuhiruh rqo| ydoxh wkh wzr remhfwv wrjhwkhu1 Wr looxvwudwh/ vxssrvh wkdw djhqw 4 lv ri w|sh +3>3>nc129 ,> dqg gh qh vc50 =@ yc50 .zc50 .nc50 1 Li 4 dffhswv wr ex| rqo| rqh remhfw dw sulfh }hur/ wkh xwlolw| lv }hur1 Rq wkh rwkhu kdqg/ frpshwlqj iru erwk remhfwv |lhogv d xwlolw| htxdo wr Su +vc50 nc129 ,+nc129 H +vc50 mvc50 nc129 ,,/ zklfk lv srvlwlyh/ dowkrxjk srvvleo| vpdoo1 Lw lv fohdu wkdw w|shv olnh +%c129 >%c50 >n,/ iru %c129 dqg %c50 vx flhqwo| vpdoo/ zloo dovr eh xqzloolqj wr frooxgh1 Krzhyhu/ w|shv zlwk d vwdqg0dorqh ydoxh iru y vx flhqwo| kljk duh lq idfw zloolqj wr frooxgh1 Lq sduwlfxodu dvvxph wkdw elgghu 4 kdv w|sh +yc129 >zc129 >nc129 ,/ zlwk yc129 Azc129 1Gh qhvc129 =@ yc129 . zc129 . nc129 1 Vxssrvh wkdw dw wkh uvw urxqg elgghu 4 ohduqv wkdw khu rssrqhqw*v w|sh olhv lq vrph vxevhw Dc25 c25 1 Wkhq frooxvlrq lv ehwwhu wkdq frpshwlwlrq li= yc129 ] c114 c129 c38 +vc129 vc50 ,gK +vc50 m+yc50 >zc50 >nc50 , 5 Dc25 ,+8, zkhuh K ghqrwhv wkh frqglwlrqdo f1g1i1 ri vc50 1 Lq htxloleulxp/ wkh vhw Dc15 zloo eh h{dfwo| wkh vhw ri wkrvh w|shv iru zkrp lqhtxdolw| +8, lv vdwlv hg1 D vlplodu lqhtxdolw| zloo gh qh Dc25 1 Lq htxloleulxp wkh wzr vhwv Dc15 dqg Dc25 kdyh wr eh gh qhg vlpxowdqhrxvo|1 Lw lv lqwxlwlyh iurp lqhtxdolw| +8,wkdw wkh wzr vhwv zloo eh v|pphwulf1 Wkh vkdsh ri wkh vhw Dc15 lv urxjko| dv iroorzv1 Vxssrvh wkdw elgghu 4 kdv yc129 zc129 1Ohwxv vxppdul}h wkh w|sh ri djhqw 4 e| wkh sdlu +yc129 >vc129 ,/ zlwk yc129 5 ^3>4` dqg vc129 5 n> 5.n 1Wkhqlw lvfohduwkdwliwkhsdlu+yc87c129 >vc87c129 , vdwlv hv lqhtxdolw| +8, wkhq doo sdluv +yc87c129 >vc129 ,zlwkvc129 ?vc87c129 zloo dovr vdwlvi| wkh lqhtxdolw|1 Wkh lqhtxdolw| lv dovr vdwlv hg e| wkh w|shv fkdudfwhul}hg e| wkh sdlu +3>n,1 Wklv w|sh kdv qr xvh iru d vlqjoh remhfw/ exw lv dovr vxuh wr orvh wkh frpshwlwlrq iru wkh wzr remhfwv1 Wkxv/ +8, krogv zlwk htxdolw|1 Lw lv dovr fohdu wkdw doo w|shv fkdudfwhul}hg e| sdluv olnh +yc129 >yc129 . n, duh zloolqj wr frooxgh1 Wkhvh duh w|shv iru zkrp zc129 @ 3 dqg kdyh wkh orzhvw srvvleoh ydoxh iru wkh v|qhuj|1 Li wkh| frpshwh iru erwk remhfwv wkh| sd| dw ohdvw n +wkh orzhvw srvvleoh ydoxh iru vc50 ,/ dqg uhfhlyh ohvv xwlolw| wkdq yc129 / zklfk lv zkdw wkh| zrxog jhw dffhswlqj frooxvlrq1 Lq jhqhudo/ iru djlyhqyc129 wkhuh zloo eh d fruuhvsrqglqj ydoxh vc129 +yc129 ,vxfkwkdww|shvzlwkvc129 ?vc129 +yc129 , duh zloolqj wr dffhswv frooxvlrq dqg w|shv zlwk vc129 Avc129 +yc129 , suhihu wr frpshwh iru erwk remhfwv udwkhu wkdq wr dffhsw frooxvlrq1 Wkh vkdsh ri wkh vhw Dc15 lv wkxv vlplodu wr wkh rqh vkrzhg lq jxuh 41 LQVHUW ILJXUH 4 KHUH Rqh fdvh zklfk lv sduwlfxoduo| vlpsoh dqg vwulnlqj lv wkh rqh lq zklfk wkh h{whqw ri wkh frp0 sohphqwdulw| lv nqrzq dqg lghqwlfdo dfurvv elgghuv/ l1h1 wkh glvwulexwlrq ri nc16 lv ghjhqhudwh rq 46 vrph ydoxh nc87 1 Lq wklv fdvh/ surylghg wkdw wkh frqglwlrq H +{, c129c50 krogv/ wkh vwudwhjlhv sursrvhg lq sursrvlwlrq 4 duh vwloo htxloleulxp vwudwhjlhv1 Lq rwkhu zrugv/ wkh vhw Dc15 dqg Dc25 ghvfulehg lq sursr0 vlwlrq+9,fdqehwdnhqwreh c15 dqg c25 uhvshfwlyho|/ zkhq wkh frpsohphqwdulwlhv nc16 duh nqrzq dqg lghqwlfdo dfurvv djhqwv1 Wkh lqwxlwlrq lv vwudljkwiruzdug1 Li nc16 lv wkh vdph iru hdfk elgghu/ wkhq lw zloo eh hqwluho| frpshwhg dzd| zkhqhyhu wkh htxloleulxp ri sursrvlwlrq 8 lv wuljjhuhg1 Wklv pdnhv dq| dwwhpsw wr jhw erwk remhfwv xqdwwudfwlyh/ khqfh hyhq w|shv zlwk yhu| orz cvwdqg0dorqh* ydoxhv fdq eh lqgxfhg wr frooxgh1 Zh frqfoxgh wklv vhfwlrq e| uhfrqvlghulqj wkh frqmhfwxuh dffruglqj wr zklfk frooxvlrq ghfuhdvhv zkhq frpsohphqwdulwlhv duh suhvhqw1 Zh kdyh vkrzq wkdw wkh suhvhqfh ri frpsohphqwdulwlhv grhv qrw ghvwur| frooxvlrq1 Lq idfw/ zh kdyh vhhq wkdw odujh frpsohphqwdulwlhv zklfk duh nqrzq dqg frpprq dprqj wkh sod|huv gr qrw uhgxfh wkh srvvlelolw| ri frooxvlrq dw doo1 Zkdw uhdoo| pdwwhuv lq klqghulqj frooxvlrq lv wkh yduldelolw| ri wkh h{whqw ri frpsohphqwdulwlhv/ udwkhu wkdq wkhlu devroxwh ydoxhv1 D L?U* t L?t Zkhq vhtxhqwldo surfhgxuhv duh xvhg wr vhoo pxowlsoh remhfwv/ wkh ex|huv fdq frooxgh lq rughu wr uhgxfh wkhlu sd|phqwv wr wkh vhoohu1 Wkh jhqhudo ihdwxuh ri frooxvlyh htxloleuld lq rshq dvfhqglqj dxfwlrqv lv wkdw hdfk elgghu vljqdov wr wkh rwkhuv zklfk remhfw kdv wkh kljkhvw ydoxh wr khu1 Diwhu wkh vljqdolqj urxqg/ wkh elgghuv lpsolflwo| surplvh hdfk rwkhu qrw wr frpshwh rq wkh remhfwv wkdw wkh| ydoxh ohvv/ surylghg wkh| duh qrw fkdoohqjhg rq wkh remhfwv wkh| ydoxh pruh1 Zh kdyh surylghg frqglwlrqv xqghu zklfk wklv ehkdylru fdq eh pdgh d shuihfw Ed|hvldq htxloleulxp1 Zh kdyh dovr vkrzq wkdw dw ohdvw vrph ghjuhh ri frooxvlrq pd| vwloo eh suhvhqw zkhq wkh udwlr ri elgghuv wr remhfwv lv kljk/ dqg zkhq wkhuh duh frpsohphqwdulwlhv lq wkh elgghuv xwlolw| ixqfwlrqv1 Dv d pruh jhqhudo srlqw/ wkh vhw ri htxloleuld lq dxfwlrqv zlwk pxowlsoh remhfwv dsshduv wr eh pxfk ulfkhu wkdq lq wkh vlqjoh remhfw fdvh1 Lq wklv sdshu/ zh kdyh vkrzq vrph ri wkhvh htxloleuld1 Lw lv zruwk srlqwlqj rxw wkdw lq doo htxloleuld lq zklfk frooxvlrq0yld0vljqdoolqj rffxuv lw pxvw eh wkh fdvh wkdw qrw wrr pxfk lqirupdwlrq lv uhyhdohg e| wkh htxloleulxp elgglqj vwudwhj|1 Wr vhh wklv/ vxssrvh/ iru h{dpsoh/ wkdw wkh elgglqj vwudwhj| zhuh wr uhyhdo wkdw rqh elgghu kdv yhu| orz ydoxhv iru erwk remhfwv1 Wkhq wkh rwkhu elgghu zloo ghflgh wr frpshwh iru erwk remhfwv/ l1h1 wr uhyhuw wr wkh VHD vwudwhjlhv/ vlqfh khu h{shfwhg sd|phqwv rq erwk remhfwv zloo eh orz1 D elgghu zlwk kljk ydoxhv zloo dffhsw d frooxvlyh rxwfrph rqo| li wkh lqirupdwlrq uhyhdohg lv vxfk wkdw khu h{shfwhg sd|phqw lq rshq frpshwlwlrq lv vx flhqwo| kljk1 Exw wklv pxvw lpso| wkdw wkhuh lv dozd|v vrph srrolqj dprqj orz dqg kljk ydoxhv1 Wklv lq wxuq lpsolhv wkdw lq jhqhudo frooxvlrq0yld0vljqdoolqj qrw rqo| uhgxfhv wkh uhyhqxh wr wkh vhoohu/ exw dovr uhgxfhv wkh h flhqf| ri wkh qdo doorfdwlrq1 47 TTi?_ Sursrvlwlrqv 4 dqg 5 duh vshfldo fdvhv/ zlwk } @3> ri Sursrvlwlrqv 6 dqg 7 uhvshfwlyho|1 SurririSursrvlwlrq61 Jlyhq wkh v|pphwu| ri wkh sureohp/ lw lv hqrxjk wr fkhfn wkh rswlpdolw| ri wkh vwudwhj| iru w|shv kdylqj yc16 zc16 1 Zh zloo gr wklv surfhhglqj edfnzdug1 Frqvlghu wkh uvw urxqg dw zklfk rqo| wzr djhqwv uhpdlq/ vd| 4 dqg 51 Vxssrvh wkdw elgghu 4 kdv yc129 zc129 dqg rshqhg dw urxqg }hur zlwk +3>>, > zkloh elgghu 5 rshqhg zlwk +>>3,1 Vxssrvh dovr wkdw wkh rxwvwdqglqj sdlu ri elgv dw urxqg w 4 zdv +}>}, = Ohw Ic84 +yc50 m Oc53 ,dqgIc96 +yc50 m Oc53 ,ghqrwhwkh f1g1i1 ri yc50 dqg zc50 uhvshfwlyho|/ erwk frqglwlrqdo rq wkh vhw Oc53 =@ q +yc50 >zc50 , 5 ^3>4`c50 m } yc50 zc50 r = Li elgghu 4 fkdqjhv khu elgv/ wkhq wkh VHD vwudwhjlhv duh wuljjhuhg1 Wkh h{shfwhg xwlolw| lq wklv fdvh lv= V +yc129 >zc129 m Oc53 ,@ ] c15 c129 c53 +yc129 yc50 , gIc84 +yc50 m Oc53 ,. ] c25 c129 c53 +zc129 zc50 , gIc96 +zc50 m Oc53 , dqg zh kdyh wr fkhfn wkdw wkh ghyldwlrq lv xqsur wdeoh/ wkdw lv= yc129 } V +yc129 >zc129 m Oc53 , iru hdfk sdlu +yc129 >zc129 ,vxfkwkdwyc129 zc129 1VlqfhV +yc129 >zc129 m Oc53 , lv lqfuhdvlqj lq zc129 / lw lv hqrxjk wr fkhfn wkh lqhtxdolw| iru wkh w|shv rq wkh gldjrqdo/ l1h1 w|shv vxfk wkdw yc129 @ zc129 1 Gh qlqj= c53 +yc129 , V +yc129 >yc129 m Oc53 ,>yc129 5 ^}>4` > wkh lqhtxdolwlhv wr eh fkhfnhg duh= yc129 } c53 +yc129 , > iru hdfk yc129 5 ^}>4` = Zh vwduw e| qrwlqj wkdw wklv krogv dw yc129 @ }/ vlqfh erwk vlghv duh }hur> dqg wkhq revhuyh wkdw wkh ghulydwlyh ri wkh OKV zlwk uhvshfw wr yc129 lv 4/ zkloh wkh UKV ghulydwlyh c30c53 +yc129 ,@ ] c15 c129 c53 gIc84 +yc50 m Oc53 ,. ] c15 c129 c53 gIc96 +zc50 m Oc53 , > lv }hur dw yc129 @ }/ dqg lqfuhdvlqj/ khqfh srvlwlyh iru hdfk yc129 5 +}>4` = Wkxv wkh ixqfwlrq c53 +yc129 ,lv frqyh{/dqgzhduhgrqhlizhfdqsuryhwkdw 4 } c53 +4, = Wklv fdq eh uhzulwwhq dv= 4 } H ^4 yc50 m Oc53 `.H ^4 zc50 m Oc53 `@5 H ^yc50 m Oc53 ` H ^zc50 m Oc53 `> 48 ru/ xvlqj wkh v|pphwu| ri wkh mrlqw glvwulexwlrq ri yc50 dqg zc50 /dv H ^yc50 m Oc53 `.H ^yc50 m Xc53 ` 4.}> +9, zkhuh Xc53 =@ q +yc50 >zc50 , 5 ^3>4`c50 m } zc50 yc50 r = E| v|pphwu|/ zh kdyh c129c50 @Su+Oc53 m } yc50 >} zc50 ,@ Su +Xc53 m } yc50 >} zc50 ,> khqfh H ^yc50 m Oc53 `.H ^yc50 m Xc53 `@5H ^yc50 m } yc50 >} zc50 ` = Lqghshqghqfh ri yc50 dqg zc50 lpsolhv H ^yc50 m } yc50 >} zc50 `@H ^yc50 m } yc50 `/ vr wkdw frqglwlrq 9 fdq eh zulwwhq dv= H +yc50 m} yc50 >} zc50 , 45 +4 . },= Wklv lv wkh frqglwlrq vwdwhg lq wkh wkhruhp/ dqg zh fdq wkhuhiruh frqfoxgh wkdw wkh djhqwv zloo frooxgh zkhq wkh rssruwxqlw| dulvhv1 Wkh rswlpdolw| ri wkh vwudwhjlhv zkhq pruh wkdq wzr djhqwv duh ohiw iroorzv iurp wkh idfw wkdw dq| rwkhu vwudwhj| vlpso| ghvwur|v wkh rssruwxqlw| ri frooxvlrq vkrxog lw dulvh/ dqg grhv qrw lpsuryh wkh rxwfrph rwkhuzlvh1 Wkh rqo| wklqj zklfk lv ohiw wr vkrz lv wkdw dw wkh rshqlqj d sod|hu lv zloolqj wr vljqdo wuxwkixoo| wkh wuldqjoh lq zklfk khu w|sh lv1 Wklv lv jrlqj wr pdwwhu rqo| zkhq wkh sod|hu hqgv xs ehlqj rqh ri wkh wzr odvw sod|huv dqg erwk sod|huv duh frpshwlqj iru erwk remhfwv1 Zh vkrz wkdw iru dq| jlyhq } dw zklfk wklv pd| kdsshq lw lv ehwwhu wr kdyh dqqrxqfhg wkh fruuhfw wuldqjoh dw gdwh 31 Li elgghu 4 dqqrxqfhv wkh fruuhfw wuldqjoh/ wkhq wkh h{shfwhg sd|r frqglwlrqdo rq ehlqj rqh ri wkh wzr odvw elgghuv/ dqg rq } ehlqj wkh odvw elg iru erwk elgghuv/ lv= 4 5 +yc129 },. 4 5V +yc129 >zc129 m Oc53 ,+:, Wklv lv ehfdxvh/ jlyhq wkh v|pphwu| lq wkh glvwulexwlrqv ri yc16 dqg zc16 iru hdfk l/ zlwk suredelolw| c129c50 wkhrssrqhqwlvriw|shzc50 yc50 / vr wkdw khu lqlwldo elg lv +>>3,/ dqg zlwk suredelolw| c129c50 wkh rssrqhqw lv ri w|sh yc50 zc50 1 Lq wkh uvw fdvh wkh dxfwlrq hqgv lpphgldwho|/ |lhoglqj d sd|r yc129 }/zklohlq wkh vhfrqg fdvh elgghuv jr rq sod|lqj wkh VHD htxloleulxp1 Li wkh elgghu rshqv zlwk +>>3, wkhq wkh h{shfwhg sd|r frqglwlrqdo rq ehlqj rqh ri wkh wzr odvw sod|huv dqg erwk kdylqj ydoxdwlrq dw ohdvw } iru erwk remhfwv lv= 4 5 +zc129 },. 4 5V +yc129 >zc129 m Xc53 ,+;, +qrwlfh wkdw qrz V lv frqglwlrqdo wr yc50 zc50 udwkhu wkdq wr yc50 zc50 ,1 Wkh h{suhvvlrq lq +;, grhv qrw h{fhhg wkh rqh lq +:, li yc129 . V +yc129 >zc129 m Oc53 , zc129 . V +yc129 >zc129 m Xc53 ,> 49 zklfk krogv zlwk htxdolw| li yc129 @ zc129 1 Pruhryhu/ wkh ghulydwlyhv zlwk uhvshfw wr yc129 duh 4.Ic84 +yc129 m Oc53 , iru wkh OKV/ dqg I +yc129 m Xc53 , iruwkhUKV1KhqfhwkhOKVjurzvidvwhuwkdqwkhUKVdvyc129 lv lqfuhdvhg/ wkxv lpso|lqj wkdw wkh lqhtxdolw| krogv iru hdfk yc129 Azc129 1 SurririSursrvlwlrq71 Djdlq/ ehfdxvh ri v|pphwu| lw vx fhv wr fkhfn wkh rswlpdolw| ri wkh vwudwhj| dorqj wkh htxloleulxp sdwk iru d elgghu zkrvh w|sh lv lq wkh corzhu wuldqjoh1* Zh surfhhg edfnzdug1 Vxssrvh uvw wkdw rqo| wzr sod|huv duh ohiw/ vd| 4 dqg 51 Li 4 rshqhg zlwk +3>>, dqg 5 rshqhg zlwk +>>3,> wkhq wkh dqdo|vlv ri Sursrvlwlrq +6, dssolhv/ vlqfh frqglwlrq +6, lpsolhv H +{m} { 4, +4 . }, @5irud @ 3/ khqfh ghyldwlqj wr wkh VHD vwudwhj| lv qrw sur wdeoh1 Li erwk elgghuv kdyh rshqhg zlwk +3> >,> wkhq zh kdyh wr vkrz wkdw elgghu 4 zlwk w|sh yc129 zc129 @ dc129 lv zloolqj wr udlvh wkh elg rq wkh uvw remhfw rqo| li vkh lv qrw dvvljqhg remhfw y dqg wkh rxwvwdqglqj elg lv +s>},zlwk s?}. dc129 1 Wkhuh duh wzr srvvleoh ghyldwlrqv iurp wkh htxloleulxp sdwk= 4, Vwrs elgglqj rq y/ dqg udlvh wkh elg rq z e| d vpdoo dprxqw li qhfhvvdu|/ l1h1 li 4 lv qrw fxuuhqwo| dvvljqhg z1 Wklv ghyldwlrq |lhogv dw prvw zc129 }= Gh qh Wc53 +s,@ q +yc50 >zc50 , 5 ^}>4`c50 m s . } yc50 zc50 r = Wkh vhw Wc53 +s, lv wkh vxssruw ri elgghu 4*v eholhiv derxw 5*v ydoxhv frqglwlrqdo rq wkh odvw urxqg*v elgv ehlqj +s> }, iru hdfk elgghu1 Wkh h{shfwhg xwlolw| iurp iroorzlqj wkh htxloleulxp vwudwhj| lv= Xc87 +yc129 >zc129 mWc53 +s,, @ Su idc50 dc129 m Wc53 +s,j +yc129 } H +dc50 m dc50 dc129 >Wc53 +s,,, .Suidc50 dc129 m Wc53 +s,j +zc129 },> zklfk fdq eh zulwwhq dv= Xc87 +yc129 >zc129 mWc53 +s,, @ zc129 } .Suidc50 dc129 m Wc53 +s,j +yc129 zc129 H +dc50 m dc50 dc129 >Wc53 +s,,,= Lw lv fohdu wkdw wkh odvw h{suhvvlrq kljkhu wkdq zc129 }1 5, Udlvh wkh elg rq z> zlwkrxw vwrsslqj wkh elgglqj rq y1 Lq wklv fdvh/ wkh VHD htxloleulxp lv wuljjhuhg dqg zh kdyh wr yhuli| wkdw= Xc87 +yc129 >zc129 mWc53 +s,, V +yc129 >zc129 m Wc53 +s,, 4: Lw lv hqrxjk wr fkhfn wkh lqhtxdolw| dw s @ } . dc129 = Wuljjhulqj wkh VHD htxloleulxp ehiruh s uhdfkhv wkdw ohyho fdq rqo| gr zruvh1 Xvlqj yc129 @ zc129 . dc129 / wkh uhohydqw lqhtxdolw| wr eh fkhfnhg lv wkhuhiruh= zc129 } ] c25 c129 c110 c64 c129 c64 c129 c110 c53 +zc129 . dc129 yc50 , gIc84 +yc50 m Wc53 +dc129 . },, . ] c25 c129 c53 +zc129 zc50 , gIc96 +zc50 m Wc53 +dc129 . },, Wkh lqhtxdolw| lv vdwlv hg dw zc129 @ } dqg wkh UKV lv lqfuhdvlqj dqg frqyh{1 Dsso|lqj wkh vdph uhdvrqlqj dv lq Sursrvlwlrq +6, zh frqfoxgh wkdw lw lv hqrxjk wr fkhfn wkh lqhtxdolw|= 4 dc129 } ] c129 c64 c129 c110 c53 +4 yc50 , gIc15 +yc50 m Wc53 +dc129 . },, . ] c129 c51 c64 c129 c53 +4 dc129 zc50 , gIc25 +zc50 m Wc53 +dc129 . },, zkhuh xvh lv pdgh ri wkh idfw wkdw wkh kljkhvw srvvleoh ydoxh iru zc129 zkhq yc129 zc129 dc129 lv 4 dc129 1 Wkh lqhtxdolw| lv htxlydohqw wr= H +yc50 m yc50 dc129 . zc50 >4 dc129 zc50 },.H +zc50 m yc50 dc129 zc50 }>yc50 dc129 . }, 4.} ru= H +{m { d . |>4 d | },.H +{m | d { }>| d . }, 4.} zklfk lv wkh frqglwlrq vwdwhg lq wkh Sursrvlwlrq1 Dw odvw/ zh fkhfn wkdw d elgghu zdqwv wr vwrs diwhu wkh rwkhu elgghu kdv vwrsshg wkh elgglqj/ udwkhu wkdq frpshwlqj iru erwk remhfwv1 Vxssrvh wkdw wkh elgghu kdv yc129 zc129 @ d dqg wkh rwkhu elgghu vwrsshg dw } . dc30 zlwk dc30 d1 Lq wklv fdvh gh qh= G13c64 c30 c99c53 @ q +yc50 >zc50 , 5 ^}>4`c50 m yc50 zc50 @ dc30 r = Wkhq wkh lqhtxdolw| ehfrphv= yc129 dc30 } ] c15 c129 c64 c30 c110 c53 +yc129 yc50 , gIc15 yc50 m G13c64 c30 c99c53 . ] c25 c129 c53 +zc129 zc50 ,gIc25 zc50 m G13c64 c30 c99c53 Xvlqj zc129 @ yc129 d zh fdq uhzulwh wkh lqhtxdolw| dv= yc129 dc30 } ] c15 c129 c64 c30 c110 c53 +yc129 yc50 , gIc15 yc50 m G13c64 c30 c99c53 . ] c15 c129 c51 c64 c53 +yc129 d zc50 ,gIc25 zc50 m G13c64 c30 c99c53 Djdlq/ wkh lqhtxdolw| lv vdwlv hg dw yc129 @ } . d/ wkh UKV lv lqfuhdvlqj dqg frqyh{ dqg zh kdyh rqo| wr fkhfn= 4 dc30 } ] c129 c64 c30 c110 c53 +4 yc50 ,gIc15 yc50 m G13c64 c30 c99c53 . ] c129 c51 c64 c53 +yc129 d zc50 , gIc25 zc50 m G13c64 c30 c99c53 4; Lq rughu wr frpsxwh wkh lqwhjudov revhuyh= Su yc50 { yc50 @ zc50 . dc30 >yc50 }>zc50 } @Su zc50 { dc30 4 dc30 zc50 } @ Su +{ d c30 zc50 }, Su +4 dc30 zc50 }, @ + c56 c69 c37 c51 c64 c30 c28 c51 c56 c69 c53 c28 c56 c69c129 c51 c64 c30 c28 c51 c56 c69 c53 c28 li { } . d c30 3 rwkhuzlvh Wkhuhiruh= i yc50 yc50 @ zc50 . dc30 >yc50 }>zc50 } @ + c115 c69 c15 c50 c51 c64 c30 c28 c56 c69c129 c51 c64 c30 c28 c51 c56 c69 c53 c28 li yc50 } . d c30 3 rwkhuzlvh Vlplodu frpsxwdwlrqv ohdg wr= i zc50 yc50 @ zc50 . dc30 >yc50 }>zc50 } @ + c115 c69 c25 c50 c110 c64 c30 c28 c129 c51 c56 c69 c53 c110 c64 c30 c28 li 4 d c30 zc50 } 3 rwkhuzlvh Zh wkhuhiruh kdyh= ] c129 c64 c30 c110 c53 +4 yc50 ,gIc15 yc50 m G13c64 c30 c99c53 @4 U c129 c64 c30 c110 c53 yc50 i +yc50 d c30 ,gyc50 I +4 dc30 , I +}, dqg U c129 c64 c30 c110 c53 yc50 i +yc50 d c30 ,gyc50 I +4 dc30 , I +}, @ U c129 c51 c64 c30 c53 +| . d c30 ,i +|,g| I +4 dc30 , I +}, @ H { } { 4 dc30 . dc30 Vlploduo|/ zh kdyh= ] c129 c51 c64 c53 +4 d zc50 ,gIc25 zc50 m G13c64 c30 c99c53 @ U c129 c51 c64 c53 +4 d zc50 , i +zc50 . d c30 , gzc50 4 I +} . dc30 , @+4 d, I +4 +d d c30 ,, I +dc30 . }, 4 I +dc30 . }, U c129 c51 c64 c53 zc50 i +zc50 . d c30 , gzc50 4 I +} . dc30 , dqg U c129 c51 c64 c53 zc50 i +zc50 . d c30 , gzc50 4 I +} . dc30 , @ U c129 c51 c69 c64 c51 c64 c30 c28 c53 c110 c64 c30 +| d c30 ,i +|,g| 4 I +} . dc30 , U c129 c51 c69 c64 c51 c64 c30 c28 c53 c110 c64 c30 +| d c30 ,i +|,g| 4 I +} . dc30 , @ U c129 c53 c110 c64 c30 |i +|,g| 4 I +} . d, d Frpelqlqj wkhvh uhvxowv zh rewdlq wkh iroorzlqj frqglwlrq= 4 dc30 } 4 H { } { 4 dc30 dc30 .+4 d, I +4 +d d c30 ,, I +dc30 . }, 4 I +dc30 . }, U c129 c51 c64 c53 zc50 i +zc50 . d c30 , gzc50 4 I +} . dc30 , 4< Wkh lqhtxdolw| kdv wr krog iru hdfk d dc30 = Qrwlflqj wkdw wkh UKV lv ghfuhdvlqj lq d/wkhuhohydqw frqglwlrq lv rewdlqhg vhwwlqj d @ dc30 1 Wklv |lhogv= H +{ m} { 4 d,.H +{m} . d { 4, 4.} zklfk lv wkh frqglwlrq vwdwhg lq wkh Sursrvlwlrq1 Wkh dujxphqw iru rswlpdolw| zkhq pruh wkdq wkuhh elgghuv duh dfwlyh lv lghqwlfdo wr wkh rqh ri Sursrvlwlrq +6,= wkhuh lv qr srlqw lq wuljjhulqj wkh VHD vwudwhjlhv dw wkh rshqlqj/ vlqfh wkh ghflvlrq fdq dozd|v eh wdnhq odwhu1 Wkh rqo| wklqj wkdw uhpdlq wr eh suryhg lv wkdw lw lv frqyhqlhqw wr rshq lq wkh cwuxh* wuldqjoh1 Srvvleoh ghyldwlrqv lq wklv fdvh duh rshqlqj lq wkh czurqj* wuldqjoh ru rshqlqj elgglqj rq erwk remhfwv/ wkxv wuljjhulqj wkh VHD htxloleulxp1 Wkh lqlwldo elg lv rqo| uhohydqw li wkh elgghu hqgv xs dprqj wkh wzr odvw elgghuv1 Zh zloo vkrz wkdw iru hyhu| }/ dqg frqglwlrqdo rq ehlqj rqh ri wkh wzr odvw elgghuv/ rshqlqj lq wkh cwuxh* wuldqjoh jlyhv d kljkhu h{shfwhg xwlolw| wkdq dq| ghyldwlrq1 Wkh h{shfwhg xwlolw| frqglwlrqdo rq ehlqj rqh ri wkh wzr uhpdlqlqj elgghuv dw } iru d w|sh +yc129 >zc129 ,vxfkwkdwyc129 zc129 @ dc129 3lv= Xc69c102 c99 c18 c28 @ 45yc129 . 45 +zc129 .Su+dc50 dc129 ,+dc129 H +dc50 mdc50 dc129 ,,, } +<, zkhuh dc50 @ yc50 zc50 dqg wkh suredelolw| glvwulexwlrq lv frqglwlrqdo wr yc50 }>zc50 }1Wklvlvehfdxvh zlwk suredelolw| c129c50 wkh rwkhu elgghu kdv rshqhg lq wkh xsshu wuldqjoh/ vr wkdw wkh dxfwlrq hqgv dqg 4 rewdlqv yc129 dw sulfh }/ zkloh zlwk suredelolw| c129c50 wkh rwkhu elgghu rshqv lq wkh orzhu wuldqjoh1 Lq wkh odwwhu fdvh wkh elgghu sd|v dw ohdvw } dqg rewdlqv dw ohdvw zc129 Lw dgglwlrqdoo| rewdlqv dc129 plqxv wkh sulfh zkhq wkh dxfwlrq lv zrq1 Wuljjhulqj wkh VHD htxloleulxp zlwk dq rshqlqj rwkhu wkdq +>>3, ru +3> >, lv reylrxvo| grplqdwhg/ vlqfh wkh VHD htxloleulxp fdq eh wuljjhuhg odwhu dw qr frvw1 Zh kdyh wkhuhiruh rqo| wr fkhfn wkdw lw lv qrw frqyhqlhqw wr rshq lq wkh zurqj wuldqjoh1 Vxssrvh 4 rshqv elgglqj +>>3,/ l1h1 vljqdolqj wkh czurqj* wuldqjoh1 Li wkh rwkhu elgghu dovr rshqv zlwk +>>3, wkhq wkh ehvw vwudwhj| lv wr suhwhqg wr kdyh dc129 @3 dqgjhwy iru }1 Wklv lv fohduo| ehwwhu wkdq jhwwlqj z iru d sulfh juhdwhu wkdq }= Wkh rwkhu srvvlelolw| lv wr wuljjhu wkh VHD vwudwhjlhv= Wr vkrz wkdw wklv fdqqrw eh rswlpdo zh kdyh wr fkhfn wkh lqhtxdolw|= yc129 } V +yc129 >zc129 m Oc53 , Xqghu wkh dvvxpswlrqv vwdwhg lq wkh Sursrvlwlrq wkh lqhtxdolw| lv vdwlv hg +Wkh dqdo|vlv lv wkh vdph dv ehiruh,1 Li wkh rwkhu elgghu rshqv zlwk +3>>,wkhqdq|dwwhpswwrfrpshwhrqjrrgy wuljjhuv wkh VHD htxloleulxp1 Wkh sd|r lq wklv fdvh lv wkhuhiruh zkdwhyhu lv ehvw ehwzhhq rewdlqlqj zc129 dw } dqg wuljjhulqj wkh VHD htxloleulxp/ wkdw lv pd{ izc129 }>V +yc129 >zc129 m Xc53 ,j1 Zh wkhuhiruh frqfoxgh wkdw 53 wkhh{shfwhgsd|r /frqglwlrqdorqehlqjrqhriwkhwzrsod|huvohiwdw}/ zkhq wkh rshqlqj lv lq wkh zurqj wuldqjoh lv= Xc69 c18 c99 c102c28 @ 45 +yc129 },.45 pd{ izc129 }>V +yc129 >zc129 m Xc53 ,j +43, Li zc129 } V +yc129 >zc129 m Xc53 , wkhq wklv lv fohduo| ohvv wkdw wkh xwlolw| rewdlqhg lq htxloleulxp1 Li zc129 }?V+yc129 >zc129 m Xc53 , wkh frqglwlrq wkdw wkh ghyldwlrq eh qrw sur wdeoh/ wkdw lv +<, lv juhdwhu wkdq +43,/ fdq eh zulwwhq dv= zc129 .Su+dc50 dc129 ,+dc129 H +dc50 mdc50 dc129 ,, } V +yc129 >zc129 m Xc53 , zklfk lv vdwlv hg xqghu wkh frqglwlrqv vwdwhg lq wkh Sursrvlwlrq ehfdxvh lw lv htxlydohqw wr wkh frqglwlrq wkdw lw lv rswlpdo wr iroorz wkh htxloleulxp vwudwhj| diwhu rshqlqj lq wkh cwuxh* wuldqjoh1 Surri ri Sursrvlwlrq 8 1 Wkh iroorzlqj lv d v|pphwulf shuihfw Ed|hvldq htxloleulxp |lhoglqj wkh ghvluhg rxwfrph1 Wkh frqyhqwlrq lv wkdw elgv duh d sdlu/ zlwk wkh uvw hohphqw uhihuulqj wr y1 Rshq elgglqj +4> 3, = Li wkh rxwvwdqglqj elgv ec15 > ec25 duh vxfk wkdw ec15 . ec25 ?yc26 . zc26 . nc26 wkhq nhhs wkh elg rq y {hg dqg nhhs udlvlqj wkh elg rq z1 Li wkh rxwvwdqglqj elgv ec15 > ec25 duh vxfk wkdw ec15 . ec25 yc26 . zc26 . nc26 wkhq= ~ Elg rqo| rq y li yc16 A ec15 >z ec25 = ~ Elg rqo| rq z li yc16 ec15 >zA ec25 = ~ Vwrs elgglqj rwkhuzlvh1 Li dw dq| srlqw dq djhqw pdnhv dq rxw ri htxloleulxp elg wkhq doo wkh rwkhu djhqwv eholhyh wkdw vkh lv w|sh 4>4> n dqg wkh| elg yhu| kljk iru erwk remhfwv/ frq ghqw wkdw wkh| grq*w kdyh wr sd|1 Lw lv fohdu wkdw wkh rxw ri htxloleulxp vwudwhjlhv duh rswlpdo jlyhq wkh eholhiv1 Zh kdyh wr fkhfn rswlpdolw| dorqj wkh htxloleulxp sdwk1 Jlyhq wkh rshqlqj elg/ lw lv fohdu wkdw lw lv lpsrvvleoh wr ex| y iru ohvv wkdq 41 Dovr/ yc16 ec15 iurp wkdw srlqw rq Wkhuhiruh qr djhqw zloo udlvh wkh elg rq y> dqg hdfk djhqw l zloo udlvh wkh elg rq z xs wr yc16 . zc16 . nc16 4= Dq| ghyldwlrq jlyhv }hur xwlolw|/ vlqfh wkh rwkhu djhqwv zloo elg ec15 >5.n ec15 1 54 Surri ri Sursrvlwlrq 91 Xvlqj wkh dujxphqwv ri Sursrvlwlrq 8 zh kdyh wkdw wkh vwudwhjlhv ghvfulehg lq wkh odvw srlqw ri wkh Sursrvlwlrq frqvwlwxwh d shuihfw Ed|hvldq htxloleulxp dw dq| jlyhq vwdjh1 Zh duh ohiw zlwk wkh wdvn ri qglqj wkh dssursuldwh vhwv Dc15 / Dc25 / vkrz wkdw wkh suhvfulehg vwudwhj| lv rswlpdo iru doo w|shv dw vwdjh 3/ dqg wkdw iru w|shv lq Dc15 / Dc25 lw lv rswlpdo wr vwrs elgglqj zkhq wkh lqlwldo elgv duh ++3>>,>+>> 3,, ru ++3>>,>+>> 3,,1 Ohw c15 @ +y>z>n,m y 5 ^3> 4` >z5 ^3>y`>n5 n> n dqg c25 @ +y>z>n,m y 5 ^3>z`>z5 ^3>4`>n5 n>n = Gh qh v y . z . n/dqgohwK +v, eh wkh f1g1i1 rq v> wkdw lv= K +{,@Suiy . z . n {j Jlyhqrxudvvxpswlrqrqwkhvxssruwriy>z dqg n lw lv fohdu wkdw K +n,@3dqgK 5.n @41 Ixuwkhupruh/ jlyhq wkh v|pphwu| ri +y>z, dqg wkh lqghshqghqfh ri wkh glvwulexwlrqv ri y>z>n zh kdyh wkdw K +vm c15 ,@K +vm c25 ,1 Gh qh wkh vhwv Dc102c15 @ c15 / Dc102c25 @ c25 / dqg gh qh= Dc129c15 @ +y>z>n, 5 c15 m y ] c15 c110 c25 c110 c38 c38 +y . z . n v,gK +vm c25 , Dc129c25 @ +y>z>n, 5 c25 m z ] c15 c110 c25 c110 c38 c38 +y . z . n v, gK +vm c15 , Wkxv/ Dc129c15 lv wkh vhw ri w|shv lq c15 zkrsuhihuwrkdyhy iru iuhh udwkhu wkdq frpshwlqj iru wkh exqgoh zkhq lw lv nqrzq wkdw wkh w|sh ri wkh rwkhu djhqw olhv lq c25 D v|pphwulf lqwhusuhwdwlrq krogv iru Dc129c25 1 Revhuyh wkdw wkh vhwv Dc129c15 dqg Dc129c25 duh frpsdfw dqg frqqhfwhg1 Lw lv fohdu wkdw wkh wzr vhwv duh v|pphwulf/ phdqlqj wkdw li +d>e> f, 5 Dc129c15 wkhq +e>f>d, 5 Dc129c25 Ixuwkhupruh/ lw lv dovr fohdu wkdw K vm Dc129c25 @ K vm Dc129c15 1 Qrz/ jlyhq wzr v|pphwulf vhwv Dc63c15 dqg Dc63c25 zlwk wkh surshuw| wkdw K +vm Dc63c25 ,@K +vm Dc63c15 ,gh qhwkhvhwv= Dc63 c110c129c15 @ +y>z>n, 5 c15 m y ] c15 c110 c25 c110 c38 c38 +y . z . n v, gK +vm Dc63c25 , Dc63 c110c129c25 @ +y>z>n, 5 c25 m z ] c15 c110 c25 c110 c38 c38 +y . z . n v,gK +vm Dc63c15 , 55 Li Dc63c15 dqg Dc63c25 duh frpsdfw dqg frqqhfwhg wkhq Dc63 c110c129c15 dqg Dc63 c110c129c25 duh dovr frpsdfw dqg frqqhfwhg1 Zh fodlp wkdw wkh vhtxhqfh iDc63c15 j kdv d frqyhujlqj vxevhtxhqfh/ dqg wkdw wkh vhw Dc15 wr zklfk wkh vxevhtxhqfh frqyhujhv lv wkh vhw zh duh orrnlqj iru1 Ohw K + c15 , eh wkh vhw ri qrq0hpsw| frpsdfw vxevhwv ri c15 1 Iru d jlyhq vhw I 5 K + c15 ,gh qh wkh vhw= Ec48 +I,@i| 5 c15 mm| {m ?%iru vrph { 5 I j Wkh vsdfh K + c15 , lv d phwulf vsdfh zkhq hqgrzhg zlwk wkh Kdxvgru glvwdqfh= +I>J,@plqi%A3 mI Ec48 +J,dqgJ Ec48 +I,j Vlqfh wkh vhw c15 lv frpsdfw/ wkh vhw K + c15 , lv dovr frpsdfw +vhh h1j1 Pdv Frohoo +4<;8,/ Sursrvlwlrq D1814,1 Wkh vhtxhqfh iDc63c15 j lvdvhtxhqfhrihohphqwvlqK + c15 ,/ dqg vlqfh wkh vhw lv frpsdfw wkhuh h{lvwv d frqyhujlqj vxevhtxhqfh1 Ohw Dc15 eh wkh hohphqw wr zklfk wkh vxevhtxhqfh frqyhujh/ dqg revhuyh wkdw vlqfh doo hohphqwv lq iDc63c15 j duh frqqhfwhg wkhq D lv frqqhfwhg wrr +vhh h1j1 Pdv Frohoo +4<;8,/ Sursrvlwlrq D1814,1 Wkh vhw Dc25 fdq eh rewdlqhg xvlqj h{dfwo| wkh vdph surfhgxuh1 Wkh vhwv Dc15 dqg Dc25 vdwlvi| wkh htxloleulxp frqglwlrqv1 Revhuyh uvw wkdw iru hdfk v dqg q zh kdyh K +vm Dc63c15 , K +vm Dc63c25 , @ 3 Wklv lpsolhv wkdw iru hdfk v= olpc63 c60c34 K +vm Dc63c15 , K +vm Dc63c25 ,@K +vm Dc15 , K +vm Dc25 ,@3 +44, Frqvlghu qrz wkdw d w|sh +y>z>n, 5 Dc15 1 Wkh htxloleulxp vwudwhj| suhvfulehv= 41 Rshq zlwk +3> >,1 51 Li wkh rwkhu elgghu rshqv zlwk +>>3, wkhq vwrs elgglqj1 Lq doo rwkhu fdvhv/ xvh wkh VHD vwudwhj|1 Ohw xv uvw fkhfn wkdw wkh vwudwhj| diwhu rshqlqj zlwk +3>>, dqg revhuylqj +>>3,1 Wkh rqo| srvvleoh ghyldwlrq lv wr wuljjhu wkh VHD htxloleulxp/ zklfk |lhogv= V +yc129 >zc129 >nm Dc25 ,@ ] c15 c129 c110 c25 c129 c110 c38 c38 +yc129 . zc129 . n v, gK +vm Dc25 , Xvlqj +44, dqg wkh idfw wkdw +yc129 >zc129 >n, 5 Dc15 zh rewdlq= y V +yc129 >zc129 >nm Dc25 , Zh qrz fkhfn rswlpdolw| dw vwdjh 31 Lw fohduo| pdnhv qr vhqvh wr wuljjhu wkh VHD vwudwhj|1 Wkh rqo| rwkhu srvvleoh ghyldwlrq lv wr elg +>>3,/ wkxv vljqdoolqj wkdw wkh w|sh ehorqjv wr Dc25 1Lwlvqrw sur wdeoh wr xvh wkh VHD htxloleulxp diwhu wkh rwkhu w|sh vljqdov Dc15 / vlqfh wklv lv htxlydohqw wr 56 wuljjhulqj gluhfwo| wkh VHD htxloleulxp zlwk suredelolw| 4/ zklfk zh nqrz qrw wr eh sur wdeoh1 Vxssrvh qrz wkdw frooxvlrq lv dffhswhg1 Wkhq zh frpsduh wkh h{shfwhg xwlolw| ri wkh ghyldwlrq= Su +Dc15 ,zc129 .+4 Su +Dc15 ,, V +yc129 >zc129 >nc129 mtDc15 , zlwk wkh h{shfwhg xwlolw| ri wkh htxloleulxp vwudwhj|= Su +Dc25 ,yc129 .+4 Su +Dc25 ,,V +yc129 >zc129 >nc129 mtDc25 , Exw qrz revhuyh wkdw wkh v|pphwu| ri Dc15 dqg Dc25 lpsolhv Su+Dc15 ,@Su+Dc25 ,dqg= V +yc129 >zc129 >nc129 mtDc15 ,@V +yc129 >zc129 >nc129 mtDc25 , Vlqfh yc129 zc129 zh frqfoxgh wkdw wkh ghyldwlrq lv qrw sur wdeoh1 D v|pphwulf uhdvrqlqj vkrzv wkdw w|shv +yc129 >zc129 >nc129 , @ Dc15 ^ Dc25 duh qrw ehwwhu r dqqrxqflqj +3>>,ru+>> 3,1 LqwklvfdvhwkhdjhqwlvjrlqjwrwuljjhuwkhVHDvwudwhj|qrpdwwhuzkdwwkh dqqrxqfhphqw ri wkh rwkhu djhqw lv/ vr wkdw dqqrxqflqj +3> 3, dqg wuljjhulqj wkh VHD htxloleulxp iurp wkh yhu| ehjlqqlqj lv rswlpdo1 57 +iuihi?Uit ^4` Dxvxeho/ O1 dqg S1 Fudpwrq +4<<9, cGhpdqg Uhgxfwlrq dqg Lqh flhqf| lq Pxowl0Xqlw Dxf0 wlrqv% plphr/ Xqlyhuvlw| ri Pdu|odqg1 ^5` Fdpsehoo/ F1 +4<<;, cFrruglqdwlrq lq Dxfwlrq zlwk Hqwu|/* plphr/ Rklr Vwdwh Xqlyhuvlw|1 ^6` Fkdnuderuw|/ L1 +4<<;, cHtxloleulxp lq Vhdohg0Elg Dxfwlrqv iru Pxowlsoh Xqlwv zlwk Lqfuhdvlqj Uhwxuqv/* plphr1 ^7` Fudpwrq/ S1 dqg M1 Vfkzduw} +4<<<, cFrooxvlyh Elgglqj= Ohvvrqv iurp wkh IFF Vshfwuxp Dxfwlrqv* plphr/ Xqlyhuvlw| ri Pdu|odqg1 ^8` Wkh Hfrqrplvw +4<<:, cOhduqlqj wr Sod| wkh Jdph*/ Pd| 4:/ sdjh <71 ^9` Hqjhoeuhfkw0Zljjdqv/ U1 dqg F1 Ndkq +4<<;, cOrz Uhyhqxh Htxloleuld lq Dvfhqglqj Sulfh/ Pxowl0Remhfw Dxfwlrqv*/ plphr/ Xqlyhuvlw| ri Loolqrlv1 ^:` Judkdp/ G1 dqg U1 Pduvkdoo +4<;:, cFrooxvlyh Elgghu Ehkdylru dw Vlqjoh0Remhfw Vhfrqg0Sulfh dqg Hqjolvk Dxfwlrqv*/ Mrxuqdo ri Srolwlfdo Hfrqrp|> <8= 454:06<1 ^;` J? I1 dqg H1 Vwdffkhwwl +4<<:, cHqjolvk dqg Grxeoh Dxfwlrqv zlwk Gl huhqwldwhg Frpprgl0 wlhv*/ Xqlyhuvlw| ri Plfkljdq Zrunlqj Sdshu <:0351 ^<` Nzdvqlfd/ D1 +4<<;, cFrooxvlrq lq Pxowlsoh Remhfw Vlpxowdqhrxv Dxfwlrqv= Wkhru| dqg H{0 shulphqwv*/ zrunlqj sdshu/ Fdowhfk VVZS43431 ^43` Orsrpr/ J1 +4<<;, cWkh Hqjolvk Dxfwlrq lv Rswlpdo dprqj Vlpsoh Vhtxhqwldo Dxfwlrqv*/ / Mrxuqdo ri Hfrqrplf Wkhru|> ;5= 4770991 ^44` Pdlodwk/ J1 dqg S1 ]hpvn| +4<<4, cFrooxvlrq lq Vhfrqg Sulfh Dxfwlrqv zlwk Khwhurjhqhrxv Elgghuv* Jdphv dqg Hfrqrplf Ehkdylru 6= 79:0;91 ^45` Pdv Frohoo/ D1 +4<;8, Wkh Wkhru| ri Jhqhudo Hfrqrplf Htxloleulxp= D Gl huhqwldeoh Ds0 surdfk/ Hfrqrphwulf Vrflhw| Prqrjudskv qc74 <6 +6,= 8:<0<<1 ^48` P|huvrq/ U1 +4<;4,= _Rswlpdo Dxfwlrq Ghvljq/% Pdwkhpdwlfv ri Rshudwlrqv Uhvhdufk 9= 8;~ :61 58 ^49` Shvhqgruihu/ P1/ +4<<9, cD Vwxg| ri Frooxvlrq lq Iluvw Sulfh Dxfwlrqv= Fduwho Djuhhphqwv* Sk1 G1 wkhvlv/ Qruwkzhvwhuq Xqlyhuvlw|1 59 01v1 2+k s1 Av k Figure 1 5: