The Proportional Random Allocation Of Indivisible Units - Core

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compositions of proportional methods. The proportional method is the only fair method in the family. Keywords: proportionality, fair division, random allocation, ...
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6

Wkh sursruwlrqdo phwkrg

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7

Orzhu dqg Xsshu Frpsrvlwlrq

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+9,

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+44,

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+45,

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Ri frxuvh/ wkh lqfuhphqwdo jdlq dqg orvv ri wkh sursruwlrqdo phwkrg duh = +Q> {,+hl , @ +Q> {,+hl , @

8

{l iru doo l {Q

Iluvw fkdudfwhul}dwlrq ri wkh sursruwlrqdo phwkrg

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46

+4:,

Zh xvh wkh v|vwhp +49, +4:, wr vkrz/ e| lqgxfwlrq rq {Q > wkdw  @  lv wkh sursruwlrqdo lqfuhphqwdo glvwulexwlrq1 Li { lv d xqlw yhfwru +{Q @ 4, wkhuh lv qrwklqj wr suryh/ qh{w frqvlghu wkh fdvh {Q @ 5= Li { @ 5hl > wkhuh lv qrwklqj wr suryh> zh duh ohiw zlwk wkh fdvh { @ h4 . h5 = Zh fdqqrw dsso| OF dv deryh ehfdxvh {Q  4 @ 4> exw HWH jlyhv l +{, @ 45 dqg  l +{, @ 45 iru l @ 4> 5> dv ghvluhg1 Qrwh wkdw wklv lv wkh rqo| sodfh lq wkh surri zkhuh zh xvh wkh htxlw| dvvxpswlrq1 Qh{w zh dvvxph wkh surshuw| krogv xs wr n  5>dqg slfn d yhfwru { zlwk {Q @ n . 4= Zh fdq dsso| +49, khqfh iru doo l 5 P+{, =  l +{, @ l +{,=

[ {l  4 {l m +{,= . {Q  4 {Q  4 P+{, l

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4 {l =l +{, @ {Q  4 {Q  4

Vlploduo|/ ghyhorslqj +4:, zlwk wkh khos ri wkh lqgxfwlyh dvvxpswlrq= l +{, .

4 {l = l +{, @ {Q  4 {Q  4

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Wkh frpsohwh dqvzhu wr wklv txhvwlrq iru dq| fduglqdolw| ri Q lv txlwh frp0 solfdwhg= vhh Prxolq dqg Vwrqj ^5333d`1 Lq wkh fxuuhqw sdshu/ zh dqvzhu wkh txhvwlrq rqo| lq wkh fdvh mQ m @ 5 = wklv wxuqv rxw wr eh hqrxjk iru rxu pdlq uhvxow/ Wkhruhp 5 lq wkh qh{w vhfwlrq1 Iru wkh uhvw ri wkh vhfwlrq/ zh dvvxph Q @ Q @ i4> 5j = Iru dq| uhdo qxpehu > zh zulwh  @ 4  > iru dq|  vxfk wkdw 3 ?  ? 4 dqg  9@ 45 / zh ghqh dq lqfuhphqwdo jdlq dqg dq lqfuhphqwdo orvv glvwulexwlrqv= 4 +{, @

 4 +{, @

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48

9

Frqvlvwhqf|/ dqg wkh pdlq fkdudfwhul}dwlrq uhvxow

Wkh Frqvlvwhqf| surshuw| +lq vkruw/ FV\,/ uhodwhv wkh vroxwlrqv ri udwlrqlqj sureohpv zlwk glhuhqw srsxodwlrqv/ l1h1/ glhuhqw vhwv ri djhqwv vkdulqj wkh uhvrxufhv1 Lq d qxwvkhoo/ wklv h{wuhpho| srsxodu d{lrpdwlf uhtxluhphqw uhtxluhv wkh iroorzlqj= li wkh phwkrg dssolhg wr srsxodwlrq Q vhohfwv d fhuwdlq doorfdwlrq ri wkh uhvrxufhv/ wkh phwkrg dssolhg wr d vxevhw Q 3 ri Q dqg wr wkh wrwdo uhvrxufhv uhfhlyhg e| Q 3 lq wkh Q  doorfdwlrq/ pxvw frqup wkh vkduhv ri wkhvh djhqwv1 Rqh zd| wr zulwh wkh ghqlwlrq ri wkh FV\ d{lrp iru d phwkrg lq U+Q , pdnhv lw irupdoo| lghqwlfdo wr wkh xvxdo ghqlwlrq iru ghwhuplqlvwlf phwkrgv= l +Q> w> {, @ 5 +Q l> w  4 l +Q> w> {,> {l , iru doo Q> w> { dqg l

+53,

Lq htxdwlrq +53,/ wkh ohiw kdqg whup lv wkh surmhfwlrq ri wkh udqgrp yduldeoh +Q> w> {, rqwr Ql +wkdw lv/ zh ljqruh wkh l  wk frruglqdwh,1 Wkh uljkw kdqg vlgh lv wkh frpsrvlwlrq ri wzr vwrfkdvwlfdoo| lqghshqghqw udqgrp yduldeohv= uvw 4 l ghwhuplqhv djhqwv l3 v +udqgrp, vkduh zkhq w lv glylghg dprqj Q > iru hyhu| uhdol}dwlrq }l ri wklv vkduh/ wkh udqgrp yduldeoh 5  ghwhuplqhv wkh doorfdwlrq ri w  }l dprqj Q l= Wkxv +53, lqfrusrudwhv wkh udqgrp fkdudfwhu ri wkh doorfdwlrq surfhvv lq wkh vwdqgdug irupxodwlrq ri Frqvlvwhqf|1 Lq rughu wr glylgh w xqlwv dprqj wkh djhqwv lq Q/ zh pd|/ iru dq| djhqw l lq Q / vhsdudwh wkh doorfdwlrq wr djhqw l iurp wkdw wr wkh djhqwv lq Ql = uvw/ zh udqgrpo| vhohfw djhqw l3 v vkduh }l lq wkh Q udwlrqlqj sureohp/ qh{w zh udqgrpo| doorfdwh wkh uhpdlqlqj w  }l xqlwv dprqj Ql= Hdfk ri wkh wzr vwhsv iroorzv wkh suredelolw| glvwulexwlrq vhohfwhg e| wkh udwlrqlqj phwkrg> wkh nh| ihdwxuh lv wkdw wkhvh wzr vwhsv duh vwrfkdvwlfdoo| lqghshqghqw1 Wkh FV\ d{lrp uhtxluhv wkdw wkh wzr vwhsv doorfdwlrq surfhvv mxvw ghvfulehg dqg wkh rqh vwhs doorfdwlrq ri w xqlwv dprqj Q eh htxlydohqw/ lq wkh vhqvh ri jhqhudwlqj suhflvho| wkh vdph wzr pdujlqdo suredelolw| glvwulexwlrqv iru wkh vkduh \l ri djhqw l> dqg wkh vdph wzr pdujlqdo suredelolw| glvwulexwlrqv iru wkh yhfwru ri vkduhv \^Q  l` ri wkh djhqwv rwkhu wkdq l= Qh{w zh jlyh wkh r!fldo ghqlwlrq ri FV\ lq whupv ri wkh suredelolw| glvwul0 exwlrq lqgxfhg e| = Zh lqwurgxfh d slhfh ri qrwdwlrq uvw1 Jlyhq Q> w> {> l 5 Q dqg }l > 3  }l  w> zh zulwh +Q> w> {,+\l @ }l , @ +Q> w> {,+i}l j  ]+Ql> w  }l > {l ,, Wklv lv wkh suredelolw| wkdw dw wkh udwlrqlqj sureohp +Q> w> {,/ djhqw l uhfhlyhv }l = wkh qrwdwlrq xvhv wkh udqgrp yduldeoh \l iru fodulw| rqo|1 Ghqlwlrq 6 Wkh suredelolvwlf udwlrqlqj phwkrg >  5 U+Q ,/ vdwlvhv Frqvlvwhqf| li iru dq| Q> w> { dqg l lq Q zh kdyh=

49

+Q> w> {,+}, @ +Ql> w  }l> {l ,+}l ,=+Q> w> {,+\l @ }l , iru doo } 5 ]+Q> w> {, +54, Zh qrwh wkdw surshuw| +54, lv ydfxrxvo| wuxh li mQm @ 5 wkhuhiruh wkh Frq0 vlvwhqf| d{lrp kdv qr elwh xqohvv mQ m  6= Dqrwkhu vwdqgdug ihdwxuh ri FV\ lv vhoi0gxdolw|= d phwkrg lq U+Q , phhwv FV\ li dqg rqo| li lwv gxdo phwkrg grhv +surri lv vwudljkwiruzdug,1 Ohppd 7 D sulrulw| phwkrg u  phhwv Frqvlvwhqf|> wkh udqgrp sulrulw| phwkrg grhv qrw1 Wkh sursruwlrqdo phwkrg phhwv Frqvlvwhqf| dv zhoo1 Surri Wkdw d sulrulw| phwkrg lv Frqvlvwhqw lv erwk fohdu dqg zhoo nqrzq +Prxolq ^5333`,1 Wr fkhfn wkdw udqgrp sulrulw| lv qrw Frqvlvwhqw/ zh frqvlghu irupxod +54, iru Q @ i4> 5> 6j > w @ 5> dqg { @ +5> 4> 4,= Frpsxwh uvw wkh suredelolw| glvwulexwlrq ri us +Q> 5> +5> 4> 4,, = +5> 3> 3, dqg +3> 4> 4, hdfk zlwk sured=4 @6 +4> 4> 3, dqg +4> 3> 4, hdfk zlwk sured=4 @9 Wkxv djhqw 6 jhwv 4 ru 3 xqlw/ hdfk zlwk sured 4 25 1 Vd| djhqw 6 jhwv 4 xqlw> wkh us phwkrg dprqj Q6 doorfdwhv wkh uhpdlqlqj xqlw wr hlwkhu djhqw 4 ru 5 zlwk htxdo suredelolw|1 Khqfh iru } @ +4> 3> 4, wkh uljkw0kdqg whup lq +54, lv +4 25 ,=+4 25 , @ 4 27 zkhuhdv wkh ohiw kdqg whup lv 4 29 = Htxdwlrq +54, idlov vlploduo| iru hyhu| } lq ] +Q> 5/+5> 4> 4,,= Zh fkhfn qh{w wkdw wkh sursruwlrqdo phwkrg phhwv Frqvlvwhqf|1 Il{ Q> w> {> l dqg } dv lq +54, dqg frpsxwh iurp +7,= sur+Q> w> {,+\l @ }l , @

[\ w$+{Q  w,$ {l $ {m $ = {Q $ }l $+{l  }l ,$ |m $+{m  |m ,$ Q l

zkhuh wkh vxp ehduv rq doo |l lq ]+Ql> w  }l > {l ,= Xvh +7, djdlq wr zulwh sur+Ql> w  }l > {l ,+]+Ql> w  }l > {l ,, @ 4 = +w  }l ,$+{Q  l . }l  w,$ [ \ {m $ @4 {Q  l $ |m $+{m  |m ,$ Ql

Dsso| +7, rqh pruh wlph= sur+Q l> w  }l> {l ,+}l , @

+w  }l ,$+{Q l . }l  w,$ \ {m $ {Q  l $ }m $+{m  }m ,$ Ql

4:

Pxowlso|lqj wkh wkuhh htxdwlrqv deryh +wkh surgxfw ri wkh wkuhh ohiw0kdqg whupv htxdov wkdw ri wkh wkuhh uljkw0kdqg whupv, dqg vlpsoli|lqj |lhogv wkh ghvluhg frqfoxvlrq1 Rxu pdlq uhvxow fkdudfwhul}hv doo wkh phwkrgv lq U+Q , phhwlqj wkh wkuhh surshuwlhv FV\/ OF dqg XF zkhq Q frqwdlqv wkuhh ru pruh djhqwv1 Iurp wkh h{dpsohv glvfxvvhg vr idu/ rqo| wkh sulrulw| phwkrgv u dqg wkh sursruwlrqdo phwkrg sur vxffhvvixoo| sdvv wkhvh wkuhh whvwv zkhq mQ m  6= Pruhryhu wkh phwkrgv  > 3 ?  ? 4/ lq Wkhruhp 4 sdvv wkh wkuhh whvwv zkhq mQ m @ 5 +FV\ ehlqj ydfxrxv,1 Wkh hqwluh idplo| gholqhdwhg e| rxu wkuhh d{lrpv lv pdgh ri vlpsoh frpelqdwlrqv ri wkhvh phwkrgv1 Ghqlwlrq 7 Jlyhq Q / d phwkrg  lq U+Q , dqg wzr djhqwv l > m lq Q / zh vd| wkdw  jlyhv sulrulw| wr l ryhu m li iru doo Q > w> { zh kdyh i+Q> w> {,+}, A 3 dqg }m A 3j @, }l @ {l

iru doo } 5 ]+Q> w> {,

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53

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  +} . g+n,,+g+n,  g+n  4,,

n@4

wdnlqj lqwr dffrxqw wkh uhsuhvhqwdwlrq ri g e| wkh vhtxhqfh +58,1 Diwhu uhduudqjlqj wkh whupv lq wkh qxphudwru dqg ghqrplqdwru/ zh jhw Tv4 Tv5 + n@4 +}4 . n,, = + n@4 +}5 . n,, v4 }5 . v } .* Tv45 = = 5 4 +{> }, @ n@4 +}45 . n,

+59,

zkhuh wkh lqwhjhuv  dqg * ghshqg rqo| xsrq g/ qrw rq } ru {1 Iru lqvwdqfh  @ d45 =d54 . d46 =+d54 . d55 , . === . d4n +d54 . === . d5n4 , . === Lw lv qrz d vlpsoh pdwwhu wr frpsxwh wkh fruuhvsrqglqj whupv lq wkh vxp0 pdwlrqv iru 3 +w  4> {  h4 ,+}  h4 , dqg 3 +w  4> {  h5 ,+}  h5 ,= +}45 . v45 , +}4 , 4 = = v5 +}45 , +}4 . v4 ,  +} +}5 , 4 . v , 45 45 +{  h5 > }  h5 , @ +{> },= = = +}45 , +}5 . v5 , v4

+{  h4 > }  h4 , @ +{> },=

Wkh yhulfdwlrq wkdw +{> }, @ 4 +{,=+{  h4 > }  h4 , . 5 +{,=+{  h5 > }  h5 , lv qrz lpphgldwh1

51 Surri ri Ohppd 6

Uhfdoo wkh iroorzlqj surshuw|/ iru dq| phwkrg  lq U zlwk lqfuhphqwdo orvv2jdlq >  4 +4> 4, @  5 +4> 4,> 5 +4> 4, @  4 +4> 4, 5;

+wkh surshuw| zdv xvhg lq wkh surri ri vwdwhphqw ll lq Wkhruhp 4> lw iroorzv dw rqfh iurp +43, +44,,1 Ohppd 5 vkrzv wkdw doo phwkrgv lq wkh idplo|  vdwlvi| OF dqg XF1 Qrz zh vwduw iurp d phwkrg  phhwlqj OF dqg XF/ dqg zlwk lqfuhphqwdo jdlq2orvv glvwulexwlrqv >  = zh pxvw vkrz wkdw  lv lq wkh idplo|  = Frqvlghu wkh v|vwhp +49, +4:,1 Zlwk wkh xvxdo frqyhqwlrq wkdw d qhjdwlyh fr0 ruglqdwh pdnhv wkh fruuhvsrqglqj suredelolw| }hur/ wklv v|vwhp zulwhv dv iroorzv iru dq| { 5 L‚+45, =  4 +{, @ 4 +{,= 4 +{4  4> {5 , . 5 +{,= 4 +{4 > {5  4,

+5:,

5 +{, @  4 +{,=5 +{4  4> {5 , .  5 +{,=5 +{4 > {5  4,

+5;,

Rxu qh{w vwhs lv wr vkrz wkdw wkh v|vwhp +5:, +5;, ghwhuplqhv hqwluho|  dqg  iurp wkh vlqjoh qxpehu  @ 5 +4> 4, @  4 +4> 4,= Zh kdyh 3    4 dqg zh ghdo zlwk wkh fdvhv  @ 3> 4 uvw1 Dvvxph  @ 31 Iurp +5:, dssolhg wr +4> {5 , zh jhw  4 +4> {5 , @ 5 +4> {5 ,= 4 +4> {5  4, 5 +4> {5 , @  4 +4> {5 ,= 5 +4> {5 ,=5 +4> {5  4,`

+5 {5 , @ 5 +4> {5 , @ 3 iru doo {5 A 3= Rqh fkhfnv vlploduo|  4 +{4 > 4, @ 5 +{4 > 4, @ 3 iru doo {4 A 3= Ilqdoo|/ e| lqgxfwlrq rq {4 . {5 > v|vwhp +5:, +5;, |lhogv  4 +{, @ 5 +{, @ 3 iru doo { AA 3= Wkxv iru  @ 3 wkh phwkrg  lv wkh sulrulw| phwkrg zkhuh 4 suhfhghv 51 Vlploduo|/ iru  @ 4> zh qg  4 +{, @ 5 +{, @ 4 iru doo { v1w1 {l A 3> l @ 4> 5> qdpho|  lv wkh rwkhu sulrulw| phwkrg1 Iurp qrz rq zh dvvxph 3 ?  ? 4= Zh suryh vlpxowdqhrxvo| wkdw 5 +{, dqg  4 +{, duh xqltxho| ghwhuplqhg dqg wkdw iru doo { AA 3

3 ? 5 +{, ? 4>

3 ?  4 +{, ? 4

Zh surfhhg e| lqgxfwlrq/ dqg wdnh fduh ri wkh fdvhv +4> {5 , dqg +{4 > 4, uvw1 Iru { @ +4> {5 ,> {5  5/ frqvlghu v|vwhp +5 {5  4, dqg e @  4 +4> {5  4, duh erwk lq `3> 4^ > +5 {5 , @

de d > 5 +4> {5 , @ 4  e +4  d, 4  e +4  d, dqg erwk qxpehuv duh lq `3> 4^ dv ghvluhg=

D vlplodu dujxphqw wdnhv fduh ri wkh fdvh { @ +{4 > 4,= 5
e| lqgxfwlrq rq {4 . {5 = E| dvvxps0 wlrq/ wkh fodlp lv wuxh iru {4 . {5 @ 5= Frqvlghu qh{w dq duelwudu| { vxfk wkdw {  +5> 5, +li {l @ 4 iru l @ 4 ru 5/ wkh fodlp lv douhdg| hvwdeolvkhg,1 Zh vhw e4 @  4 +{4 4> {5 ,> e5 @  4 +{4 > {5 4,> d4 @ 5 +{4 4> {5 ,> d5 @ 5 +{4 > {5 4, = wkh lqgxfwlyh dvvxpswlrq jlyhv 3 ? el > dl ? 4 iru l @ 4> 5= Qh{w zh holplqdwh 4 +{, @ 4  5 +{, dqg  5 +{, @ 4   4 +{, iurp v|vwhp +5:, +5;, zklfk ehfrphv=  4 +{, . +e4  e5 ,=5 +{, @ e4 +d5  d4 , 4 +{, . 5 +{, @ d5 Wkh ghwhuplqdqw ri wklv v|vwhp lv vwulfwo| srvlwlyh= i4 ? e4  e5 ? 4> 4 ? d5  d4 ? 4j @, 4  +e4  e5 ,+d5  d4 , A 3

+63,

wkhuhiruh  4 +{, dqg 5 +{, duh xqltxho| ghwhuplqhg  4 +{, @

e4  d5 +e4  e5 , d5  e4 +d5  d4 , > 5 +{, @ 4  +e4  e5 ,=+d5  d4 , 4  +e4  e5 ,=+d5  d4 ,

Lw uhpdlqv wr fkhfn wkdw erwk qxpehuv duh lq `3> 4^ = Wklv uhvxowv dw rqfh iurp wkh idfw wkdw dl > el duh lq wkdw lqwhuydo/ wrr/ dqg iurp +63,  4 +{, 5 +{,  4 +{, 5 +{,

A A ? ?

3 +, +4  d5 ,=e4 . d5 =e5 3 +, +4  e4 ,=d5 . e4 =d4 4 +, +4  d4 ,=e4 . d4 =e5 4 +, +4  e5 ,=d5 . e5 =d4

A3 A3 ?4 ?4

Vxpplqj xs rxu qglqjv= iru dq| qxpehu > 3    4> zh kdyh vkrzq wkdw wkhuh lv dw prvw rqh phwkrg  lq U/ phhwlqj OF/ XF dqg vxfk wkdw 5 +4> 4, @  4 +4> 4, @ = Rq wkh rwkhu kdqg/ wkh phwkrg  ghqhg iru doo > 3    4> +vhh Ohppd 6 dqg wkh glvfxvvlrq wkhuhdiwhu, phhwv OF dqg XF1 Lw lv lpphgldwh wr fkhfn iurp +4;, +4 4,+5, @   +4> 4,+4, @ > dqg wkh fdvhv  @ 3> 45 > 4 duh htxdoo| fohdu1 Wkh frqvflhqwlrxv uhdghu pd| dovr fkhfn wkdw wkh irupxodv +4;, +4 5 eh wzr phwkrgv lq U phhwlqj OF/ dqg  @ 4 . 5 eh d surshu frqyh{ frpelqdwlrq/ iru vrph > 3 ?  ? 4> zlwk wkh xvxdo  @ 4  = qrwdwlrq Wkh phwkrg  5 U fdq eh lqwhusuhwhg dv d orwwhu| ryhu 4 > 5 zlwk suredelo0 lwlhv  dqg  uhvshfwlyho|1 Zh dvvxph wkdw  phhwv OF dv zhoo dqg frqvlghu/ iru dq| l dqg dq| { v1w1 {l  5/ wkh htxdolw| +5> {,+5hl , @ 4 +5> {,+5hl , . =5 +5> {,+5hl , E| OF/ hdfk whup fdq eh frpsxwhg iurp wkh lqfuhphqwdo jdlq glvwulexwlrq/ ghqrwhg uhvshfwlyho| > 4 > 5 = l +{,=l +{  hl , @ =4l +{,=4l +{  hl , . =5l +{,=5l +{  hl , Xvlqj l @ =4l . =5l dqg uhduudqjlqj/ zh jhw +4l +{,  5l +{,,=+4l +{  hl ,  5l +{  hl ,, @ 3 r q Ohw Dl @ { 5 L‚Q m 4l +{, @ 5l +{, = Zh kdyh mxvw vkrzq wkdw iru doo {> {l  4>rqh ri { ru {  hl lv lq Dl +li {l @ 4 wklv lv reylrxv,1 D vlplodu dujxphqw iru +6> {,+6hl , vkrzv wkdw iru doo {> {l  5> dw prvw rqh ri {> {  hl > {  5hl l lv dujxphqw wkhq jlyhv wkdw lq dq| olqh E+{l , @ r q qrw lq D 1 Dq lqgxfwlrq Q } 5 L‚ m }l @ {l > wkhuh lv dw prvw rqh srlqw rxwvlgh Dl = Lq rwkhu zrugv/ wkh lqfuhphqwdo glvwulexwlrqv ri 4 dqg 5 fdqqrw eh wrr glhuhqw1 Iru vlpsolflw|/ frqvlghu wkh wzr djhqwv fdvh/ Q @ i4> 5j1 Khuh D4 @ D5 > khqfh 4 dqg 5 fdq glhu dw prvw rq wkh judsk d elmhfwlrq iurp L‚ lqwr lwvhoi1 Xsrq h{dplqlqj wkh lpsdfw ri  @ 4 . 5 lq irupxod +46,/ zh qg wkdw wkh vhw rq zklfk 4 dqg 5 glhu pxvw lqwhuvhfw dq| lqfuhdvlqj sdwk lq L‚ 5 dw prvw rqfh1 Wkhuhiruh lw lv wkh judsk ri d ghfuhdvlqj pds iurp L‚ ^ i3j lqwr lwvhoi1 Iru lqvwdqfh zh fdq kdyh= 9 5 +{, iru doo { v=w= {4 . {5 @ 43 4 +{, @ 4  +{, @ 5 +{, hovhzkhuh

61 Surri ri Wkhruhp 5 dqg Fruroodu|

Vwdwhphqw l iroorzv dw rqfh iurp Ohppdv 5/ 7 dqg 91 Wkh surri ri vwdwhphqw ll surfhhgv lq irxu vwhsv1 Vwhs 4 Dq dx{loldu| uhvxow Dvvxph mQ m  6 dqg ohw  5 U+Q , eh d phwkrg vdwlvi|lqj Frqvlvwhqf| dqg zlwk dq lqfuhphqwdo jdlq  vxfk wkdw iru doo l> m 5 Q / doo { 5 L‚il>mj = {l A 3 @, l +il> mj > {, A 3 64

Wkhq wkhuh h{lvwv iru doo l d uhdoydoxhg ixqfwlrq il ghqhg rq ‚L > zlwk il +3, @ 3> il +{l , A 3 iru {l A 3> dqg vxfk wkdw= l +Q> {, @ S

il +{l , Q im +{m ,

iru doo Q  Q >

doo l 5 Q

dqg doo { 9@ 3

+64,

Surri= iru dq| qlwh vxevhw Q ri Q / Frqvlvwhqf| +54, dssolhg wr w @ 4/ dq| {> } @ hm dqg dq| l 9@ m |lhogv m +Q> {, @ m +Ql> {l ,=+4  l +Q> {,,

+65,

Qrz wdnh Q3 @ i4> 5> 6j > { { vxfk wkdw {l A 3 iru l @ 4> 5> 6> dqg zulwh iru vlpsolflw| l @ l +Q3 > {, dqg l +l> m, @ l +il> mj > +{l > {m ,,= Dsso| +65, wr doo vl{ rughuhg sdluv +l> m, lq Q3 = 4 4

@ +4  5 ,=4 +46, > 5 @ +4  6 ,=5 +45, > 6 @ +4  4 ,=6 +56, @ +4  6 ,=4 +45, > 5 @ +4  4 ,=5 +56, > 6 @ +4  5 ,=6 +46,

Wkh dvvxpswlrq l +l> m, A 3 lpsolhv l ? 4 iru doo l +h1j1/ 4 @ 4 eulqjv d frqwudglfwlrq iurp 5 @ +4  6 ,=5 +45,,= Wkxv zh fdq pxowlso| wkh wkuhh wrs htxdwlrqv/ wkh wkuhh erwwrp rqhv/ dqg vlpsoli|= 4 +46,=5 +45,=6 +56, @ 4 +45,=5 +56,=6 +46, Htxlydohqwo| 5 6 4 +45> +{4 > {5 ,,= +56> +{5 > {6 ,,= +46> +{4 > {6 ,, @ 4 5 6 4 Ohwwlqj {4> {5 ydu| dqg nhhslqj {6 {hg/ wkh deryh htxdwlrq lpsolhv wkdw 4 @5 wdnhv wkh irup 4 i+{4 , +45> +{4 > {5 ,, @ 5 j+{5 ,

iru doo {4> {5 @ 4> 5===

zkhuh wkh ixqfwlrqv i> j duh uhdoydoxhg dqg srvlwlyh iru doo {l A 3= Zh fdq dsso| wkh vdph dujxphqw wr 5 @6 +56, dqg 6 @4 +46,> pruhryhu d vwudljkwiru0 zdug qrupdol}dwlrq |lhogv wkuhh uhdoydoxhg ixqfwlrqv il / vxfk wkdw iru doo l> m lq i4> 5> 6j > dqg doo {l > {m A 3= il +{l , il +{l , l +, l +lm> +{l > {m ,, @ +l> m> +{l > {m ,, @ m im +{m , il +{l , . im +{m , 65

Li {l @ 3>zh nqrz wkdw l +lm> +{l > {m ,, @ 3 wkhuhiruh e| h{whqglqj il wr il +3, @ 3> zh kdyh frqvwuxfwhg wkuhh ixqfwlrqv i4 > i5 > i6 dv lq wkh vwdwhphqw ri Ohppd 9 dqg vxfk wkdw +64, krogv iru doo wkuhh vxevhwv ilmj ri Q3 @ i4> 5> 6j = Lw lv qrz d vlpsoh pdwwhu wr h{whqg wkh ghqlwlrq ri il wr doo l 5 Q 1 Iru dq| l 5 Q Q3 >dsso| wkh dujxphqw deryh wr i4> 5> lj >nhhslqj wkh vdph ixqfwlrqv i4 > i5 mxvw ghqhg dqg ghqlqj d qhz il = Iru dq| sdlu l> m glvmrlqw iurp i4> 5j > zh dsso| wkh suhylrxv dujxphqw wr wkh vhw i4> l> mj wr jhw surshuw| +64, iru wkh sdlu il> mj = Lw uhpdlqv wr h{whqg wkh surshuw| +64, iurp wkh vhwv Q ri vl}h wzr wr doo qlwh vxevhwv ri Q 1 Wklv iroorzv iurp +65, dq hdv| lqgxfwlrq dujxphqw rq mQm = Dvvxph +64, krogv xs wr mQ m  4> wkhq slfn { AA 3 dqg dq duelwudu| sdlu l> m= htxdwlrq +65, zulwhv dv iroorzv S m +Q> {, 4 Q  l n +Q> {, @ S @S im +{m , i +{ , i Ql n n Q n +{n , dv ghvluhg1 Vwhs 51 Dqrwkhu dx{loldu| uhvxow Ohw  5 U+Q , eh d phwkrg vdwlvi|lqj Frqvlvwhqf|1 Zh zulwh vxs+> Q> {, iru wkh vxssruw ri wkh suredelolw| glvwulexwlrq +> Q> {, zkhq w ydulhv iurp 3 wr {Q = vxs+> Q> {, @ i} 5 ^3> {` m +Q> }Q > {, A 3j Iru doo qlwh vxevhwv P > Q ri Q zlwk P  Q dqg doo { lq ‚L Q / zh kdyh= vxs+> P> surmP +{,,  surmP ^vxs+> Q> {,`

+66,

zkhuh surm P lv wkh surmhfwlrq rshudwru iurp ‚L Q rqwr ‚L P = Slfn dq duelwudu| hohphqw } 5 vxs+> Q> {, dqg dq djhqw l5 Q P= Lq wkh Frqvlvwhqf| htxdwlrq +54,/ wkh ohiw0kdqg whup lv srvlwlyh/ khqfh wkh uljkw0kdqg whupv duh srvlwlyh dv zhoo/ lpso|lqj }l 5 vxs+> Ql> {l ,= Uhshdw wkh du0 jxphqw e| vxffhvvlyho| ghohwlqj doo djhqwv lq Q P> wr rewdlq surmP +}, 5 vxs+> P> surmP +{,,> dqg wkh lqfoxvlrq +66, lv hvwdeolvkhg1 Qrwh wkdw wkh frqyhuvh lqfoxvlrq krogv wuxh dv zhoo/ iru dq| phwkrg phhwlqj Frqvlvwhqf| dqg Orzhu dqg2ru Xsshu Frpsrvlwlrq1 Zh rplw wkh surri1 Vwhs 61 Frqvwuxfwlqj wkh sulrulw| suhrughulqj Zh vwduw qrz wkh surri ri vwdwhphqw ll= zh { d phwkrg  5 U+Q , phhwlqj FV\/ OF dqg XF1 Lq wklv vwhs zh dqdo|}h wkh sulrulw| uhodwlrq +Ghqlwlrq 7, ri  = zh vkrz wkdw wklv uhodwlrq lv d suhrughulqj/ dqg dsso| wkhq Ohppd 8 wr ghfrpsrvh  lqwr d idplo| ri sulrulw|0iuhh phwkrgv1 Iru dq| sdlu l> m frqvlghu wkh surshuw|=  jlyhv sulrulw| wr l ryhu m> Ghql0 wlrq 71 E| vwhs 5/ zh kdyh iru doo Q> {= 66

vxs+> il> mj > i{l > {m j,  surmil>mj ^vxs+> Q> {,` Wkhuhiruh li surshuw| +55, krogv iru Q @ il> mj > lw krogv iru dq| Q1 Iru Q @ il> mj > surshuw| +55, vd|v wkdw wkh phwkrg +lm, 5 U+lm, lv wkh sulrulw| phwkrg zlwk l uvw1 Rq wkh rwkhu kdqg/ +lm, phhwv OF dqg XF/ vr e| Ohppd 4/ +lm, lv wkh sulrulw| zlwk l uvw li dqg rqo| li m +lm> +4> 4,, @ 3 +uhfdoo wkdw wkh phwkrg  lv hqwluho| ghwhuplqhg e|  @ l +4> 4,,= Zh ghqh wkh sulrulw| uhodwlrq ( ri > lwv vwulfw frpsrqhqw dqg lqglhuhqfh uhodwlrq dv iroorzv= " y +, ( l

l l

m +,  jlyhv sulrulw| wr l ryhu m +, m +lm> +4> 4,, @ 3 +, 3 ? l +lm> +4> 4,, ? 4 m +, +lm, lv sulrulw|0iuhh 3 ? m +lm> +4> 4,, ? 4 +67, m +, m +lm> +4> 4,, A 3

Zh vkrz qh{w wkdw ( lv d suhrughulqj1 Lw lv reylrxvo| frpsohwh> iru wudqvl0 wlylw|/ zh slfn wkuhh duelwudu| djhqwv 4/5/6 dqg suryh wkh iroorzlqj= i4 +45> +4> 4,, A 3 dqg 5 +56> +4> 4,, A 3j @, i4 +46> +4> 4,, A 3j

+68,

Zh xvh uhshdwhgo| surshuw| +65, iru Q @ 4> 5> 6/ xvlqj wkh vkruwkdqg= l @ l +Q> +4> 4> 4,, dqg l +lm, @ l +lm> +4> 4,,= Zh suryh +68, e| frqwudglfwlrq1 Dvvxph wkh lqhtxdolwlhv lq wkh suhplvhv dqg 4 +46, @ 3= Dsso| +65, wkuhh wlphv lq vhtxhqfh= 4 +46, @ 3 dqg 4 @ +4  5 ,=4 +46, @, 4 @ 3 4 +45, A 3 dqg 4 @ +4  6 ,=4 +45, @, 6 @ 4 @, 5 @ 3 5 +56, A 3 dqg 5 @ +4  4 ,=5 +56, @, frqwudglfwlrq Li wkh sulrulw| uhodwlrq ri d phwkrg  ghqhv d suhrughulqj ( +wkh vwulfw frpsrqhqw ri zklfk lv wkh uhodwlrq lq Ghqlwlrq 7,/ wkh phwkrg lwvhoi lv wkh ( sulrulw| frpsrvlwlrq ri lwv surmhfwlrqv +P, ryhu doo lqglhuhqfh fodvvhv ri ( = Wklv idfw lv mxvw d uhzruglqj ri Ohppd 81 Lq wkh fdvh ri rxu phwkrg  phhwlqj FV\/ OF dqg XF/ zh qrwh wkdw wkh phwkrgv +P, phhw wkhvh wkuhh d{lrpv dv zhoo1 Wkh dgglwlrqdo surshuw| ri +P, lv wr eh sulrulw|0iuhh1 Lq wkh qh{w dqg odvw vwhs ri wkh surri/ zh vkrz wkdw wklv irufhv wkh sursruwlrqdo phwkrg zkhqhyhu mPm  6= Vwhs 71 Wkh fdvh ri sulrulw|0iuhh phwkrgv1 67

Doo wkdw uhpdlqv wr eh vkrzq lv wklv= ohw  5 U+Q , eh d sulrulw|0iuhh phwkrg phhwlqj FV\/ OF dqg XF> li mQ m  6>  pxvw eh wkh sursruwlrqdo phwkrg1 Iru hyhu| sdlu ri djhqwv l> m/ wkh phwkrg +lm, +wkh surmhfwlrq ri  rq il> mj > vhh Ohppd 8, phhwv OF dqg XF/ wkhuhiruh lw lv d phwkrg  iurp Ohppd 51 Sulrulw|0iuhhqhvv lpsolhv 3 ?  ? 4= Rq wkh rwkhu kdqg/ zh fdq dsso| wkh dx{loldu| uhvxow ri Vwhs 4/ dqg l +lm> {, wdnhv wkh irup +64, iru vrph uhdo0ydoxhg ixqfwlrq il =Zh qrz frpelqh wkhvh wzr uhsuhvhqwdwlrqv ri l +lm> +{l > {m ,,= E| ghqlwlrq ri wkh phwkrgv  > wkh udwlr l +lm> {, @ m +lm> {, fdq rqo| wdnh rqh ri wzr irupv= l +lm> {, @ m l +lm> {, @ m

{l li +lm, lv wkh sursruwlrqdo phwkrg {m {l  {l {m rwkhuzlvh1 = {l {m  {m

Ghqh +l> m, @ 45 lq wkh iruphu fdvh dqg +l> m, @  lq wkh odwwhu1 Frpsdulqj wkhvh irupxodv zlwk +64, jlyhv il +{l , @ dl ={l iru vrph frqvwdqw dl A 3> li+lm,lv sursruwlrqdo {l  {l iru vrph frqvwdqw dl A 3> rwkhuzlvh= il +{l , @ dl = {l Ghqh +l> m, @ 45 li +l> m, lv wkh sursruwlrqdo phwkrg dqg +l> m, @  rwkhuzlvh1 Wkh deryh v|vwhp vkrzv wkdw +l> m, lv lqghshqghqw ri m +ehfdxvh il lv,= +l> m, @ +l> n, iru doo wulsoh l> m> n1 Lw dovr lpsolhv +l> m, . +m> l, @ 4= Wkhvh wzr surshuwlhv/ wrjhwkhu zlwk mQ m  6> |lhog +l> m, @ 45 iru doo l> m1 Wkxv +lm, lv wkh sursruwlrqdo phwkrg iru doo l> m= D qdo lqgxfwlrq dujxphqw edvhg rq surshuw| +65, +dqg vlplodu wr wkh du0 jxphqw dw wkh hqg ri Vwhs 4, vkrzv wkdw l +Q> {, lv sursruwlrqdo wr {l iru doo l> Q> {/ dqg wkh surri ri Wkhruhp 5 lv frpsohwh1 Vwhs 81 Surri ri wkh Fruroodu|1 Lw lv suhflvho| wkh remhfw ri Vwhs 7 deryh1

71 Surri ri Wkhruhp 6

Ohw  5 U+Q , phhw FV\/ OF dqg Srvlwlyh Vkduh1 E| wkh uhvxow lq Vwhs 4 ri wkh suhylrxv surri/ wkh lqfuhphqwdo jdlq glvwulexwlrq ri  wdnhv wkh irup +64, iru vrph uhdoydoxhg ixqfwlrqv il dv ghvfulehg lq wkh uhvxow1 Zh dsso| Frqvlvwhqf| +54, wr dq duelwudu| wulsoh Q @ i4> 5> 6j lq Q dqg wr l @ 5> w @ 5> { v=w= {m  4 iru m @ 4> 5> 6> dqg } @ +4> 4> 3, = +Q> 5> {,+4> 4> 3, @ +46> 4> +{4 > {6 ,,+4> 3,=+Q> 5> {,+\5 @ 4, +:,

+69,

Zh frpsxwh wkh ohiw0kdqg whup lq +69, zlwk wkh khos ri Orzhu Frpsrvlwlrq

68

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4 v



4 4 . v4 v5



Vlploduo|/ +Q> 5> {,+\5

@ 4, @ +Q> 5> {,+4> 4> 3, . +Q> 5> {,+3> 4> 4, @     4 4 4 4 4 4 . i5 =i6 = = i4 =i5 = = . . v v4 v5 v v5 v6

Qrz +69, ghyhorsv dv i4 i5 v



4 4 . v4 v5



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Lq wklv frpsxwdwlrq/ wkh djhqwv 4/6 zhuh duelwudu|/ dqg vr zhuh {4  4 dqg {6  4= Wkhuhiruh il +{l ,  il +{l  4, @ d lv lqghshqghqw erwk ri l dqg ri {l  4= Dv il +3, @ 3 dqg il +4, A 3 iru doo l/ zh ghgxfh d A 3 dqg il +{l , @ d={l iru doo l dqg doo {l  3= Wkhuhiruh +Q> {, lv sdudooho wr wkh yhfwru {/ dv ghvluhg1 Wkh surri ri vwdwhphqw l lv frpsohwh1 Wkdw ri vwdwhphqw ll iroorzv dw rqfh e| gxdolw|1

69