Efficient Strategy-Proof Allocation Functions in Linear

9805
Efficient Strategy-Proof Allocation Functions in
Linear Production Economies
MANIQUET, François
SPRUMONT, Yves
Département de sciences économiques
Université de Montréal
Faculté des arts et des sciences
C.P. 6128, succursale Centre-Ville
Montréal (Québec) H3C 3J7
Canada
http://www.sceco.umontreal.ca
[email protected]
Téléphone : (514) 343-6539
Télécopieur : (514) 343-7221
Ce cahier a également été publié par le Centre interuniversitaire de recherche en
économie quantitative (CIREQ) sous le numéro 0698.
This working paper was also published by the Center for Interuniversity Research in
Quantitative Economics (CIREQ), under number 0698.
ISSN 0709-9231
CAHIER 9805
EFFICIENT STRATEGY-PROOF ALLOCATION FUNCTIONS IN
LINEAR PRODUCTION ECONOMIES
1
2
François MANIQUET and Yves SPRUMONT
1
Fonds National de la Recherche Scientifique and Facultés Universitaires NotreDame de la Paix, Belgium
2
Centre de recherche et développement en économique (C.R.D.E.) and
Département de sciences économiques, Université de Montréal
June 1998
_________________
The authors thank two anonymous referees for valuable comments. During the preparation
of this paper, the first author benefited from the excellent research atmosphere at the
Department of Economics and Economic History, Universitat Aut?noma de Barcelona,
Spain. Partial financial support from HCM grant ERBCHBGCT940699 is gratefully
acknowledged. The second author is grateful to the Fonds pour la formation de chercheurs
et l’aide à la recherche (FCAR) of Québec and the Social Sciences and Humanities
Research Council (SSHRC) of Canada for partial financial support.
RÉSUMÉ
Dans le cadre d'une économie de production à technologie linéaire, nous
caractérisons les règles d'allocation efficaces et non manipulables, tant au sens individuel
que coalitionnel. Différentes propriétés de symétries nous permettent ensuite d'isoler une
règle unique.
Mots clés : non-manipulabilité, efficacité, ensemble de production linéaire
ABSTRACT
In a linear production model, we characterize the class of efficient and strategy-proof
allocation functions, and the class of efficient and coalition strategy-proof allocation
functions. In the former class, requiring equal treatment of equals allows us to identify a
unique allocation function. This function is also the unique member of the latter class
which satisfies uniform treatment of uniforms.
Key words : strategy-proofness, efficiency, linear production set
W?|hL_U|L?
D jurxs ri djhqwv mrlqwo| rzq d exqgoh ri jrrgv dqg d olqhdu whfkqrorj|
zklfk hqdeohv wkhp wr wudqvirup wkhvh jrrgv lqwr rqh dqrwkhu dw frqvwdqw
udwhv1 Suhihuhqfhv duh vhovk dqg vwulfwo| prqrwrqlf1 Dv dq h{dpsoh/ zh
fdq wklqn ri dq duhd ri krprjhqhrxv odqg zklfk kdv wr eh doorfdwhg dprqj
djhqwv dv d ixqfwlrq ri wkh sxusrvhv +h1 j1 exloglqj/ iduplqj, iru zklfk wkh|
zrxog olnh wr xvh wkhlu sorw1 Lw lv uhdvrqdeoh wr dvvxph wkdw dq| xqlw ri odqg
xvhg iru d jlyhq sxusrvh fdq eh wudqviruphg lqwr d xqlw ri odqg xvhg iru dq
rwkhu sxusrvh1
Dq doorfdwlrq ixqfwlrq dvvljqv wr hdfk suhihuhqfh suroh d ihdvleoh exqgoh
ri lqglylgxdo frqvxpswlrq exqgohv1 Zh orrn iru hflhqw dqg qrq pdqlsx0
odeoh doorfdwlrq ixqfwlrqv1
Wkh qrq pdqlsxodelolw| uhtxluhphqwv zh duh lqwhuhvwhg lq duh vwudwhj|0
surriqhvv dqg frdolwlrq vwudwhj|0surriqhvv1 Dq doorfdwlrq ixqfwlrq lv vwudwhj|0
surri li wuxwkixoo| uhsruwlqj rqh*v suhihuhqfhv lv d grplqdqw vwudwhj| ri wkh
gluhfw uhyhodwlrq jdph dvvrfldwhg zlwk wkh doorfdwlrq ixqfwlrq +wkdw lv/ wkh
jdph zkhuh wkh vhw ri dgplvvleoh vwudwhjlhv lv wkh vhw ri dgplvvleoh suhi0
huhqfhv dqg wkh rxwfrph dvvrfldwhg wr d suroh ri vwudwhjlhv lv ghwhuplqhg
dffruglqj wr wkh doorfdwlrq ixqfwlrq,1 Dq doorfdwlrq uxoh lv frdolwlrq vwudwhj|0
surri li qr frdolwlrq ri djhqwv fdq hyhu ehqhw iurp mrlqwo| plvuhsruwlqj wkhlu
suhihuhqfhv1
Wkh uhvxowv duh= d fkdudfwhul}dwlrq ri wkh fodvv ri hflhqw dqg vwudwhj|0
surri doorfdwlrq ixqfwlrqv/ dqg wkh fodvv ri hflhqw dqg frdolwlrq vwudwhj|0
surri doorfdwlrq ixqfwlrqv1 Zh dovr lghqwli| wkh vxefodvvhv ri wkhvh wzr
fodvvhv zklfk zh rewdlq zkhq zh dovr lpsrvh krul}rqwdo htxlw| uhtxluhphqwv
+zklfk duh frqvhtxhqfhv ri wkh fodvvlfdo dqrq|plw| uhtxluhphqw,1
Lw lv zhoo nqrzq wkdw wkh uhtxluhphqwv ri hflhqf| dqg vwudwhj|0surriqhvv
ohdg wr glfwdwruvkls lq wzr djhqw sulydwh jrrg hfrqrplhv +fiu1 ]krx ^<`,/
dqg/ pruh jhqhudoo|/ frqlfw zlwk dqrq|plw| zkhq wkhuh duh pruh wkdq wzr
djhqwv +fiu1 Eduehud dqg Mdfnvrq ^4`,1Q Vwudwhj|0surri doorfdwlrq ixqfwlrqv
lq surgxfwlrq hfrqrplhv zlwkrxw wkh uhvwulfwlrq wr d olqhdu whfkqrorj| duh
dqdo|vhg lq Vkhqnhu ^:`1 Rq wklv odujhu grpdlq/ qr hflhqw/ vwudwhj|0surri/
dqg krul}rqwdoo| htxlwdeoh doorfdwlrq uxoh h{lvwv1
Q
Ai U*@tt Lu iUL?L4it ihi @ i_ @4L?| Lu @ Th@|i UL44L_|) @t |L Mi __i_
@4L?} @}i?|t @?} t?}*iTi@!i_ Thiuihi?Uitc Liihc t @? i @4T*i Lu @ 4L_i* ihi
t|h@|i})ThLLu?ittc iUi?U)c @?_ @?L?)4|) _L ?L| UL?U| Etii 5Th4L?| dHoc ?} deo
@?_ @hMih
@c a@U!tL? @?_ i4i d2o
5i?!ih hit|hU|t t @||i?|L? |L @**LU@|L? u?U|L?t U t@|tu) _gihi?|@M*|)
UL?_|L?tc |ihiM) i U*_?} @*4Lt| @** Lu |i h*it U i _i?|u)
4
Lq frqwudvw zlwk wkh uhvxowv rewdlqhg lq wkhvh sdshuv/ hflhqf| dqg
vwudwhj|0surriqhvv gr qrw frqlfw zlwk dqrq|plw| lq rxu vhwwlqj1 Dfwxdoo|/
zh lghqwli| d xqltxh dqrq|prxv doorfdwlrq ixqfwlrq wkdw lv hflhqw dqg
vwudwhj|0surri1
Wklv doorfdwlrq ixqfwlrq/ zklfk frxog eh fdoohg wkh htxdo
exgjhw iuhh fkrlfh ixqfwlrq/ dvvljqv wr hdfk djhqw khu suhihuuhg exqgoh lq d
exgjhw vhw zklfk lv wkh vdph iru doo djhqwv/ dqg zkrvh %vorsh% lv ghwhuplqhg
e| wkh {hg whfkqrorjlfdo wudqvirupdwlrq udwhv1
Rxu fkdudfwhul}dwlrq ri wkh htxdo exgjhw iuhh fkrlfh ixqfwlrq fdq eh uh0
odwhg wr wkh uhvxow rewdlqhg e| Eduehu
d dqg Mdfnvrq ^4`1 Wkh| vkrzhg wkdw/
lq h{fkdqjh hfrqrplhv/ vwudwhj|0surriqhvv dqg dqrq|plw| ohdg wr {hg sur0
sruwlrq wudglqj dzd| iurp htxdo vsolw +jlyhq wkh djhqwv* dqqrxqfhg suhi0
huhqfhv/ d {hg wudgh sursruwlrq lv fkrvhq rxw ri d vpdoo +lq sduwlfxodu
qlwh, vhw ri suhvshflhg sursruwlrqv> djhqwv duh wkhq dvvljqhg wkh exq0
goh uhvxowlqj iurp wudgh wdnlqj sodfh dzd| iurp wkh htxdo vsolw srlqw dqg
lq wkh fkrvhq sursruwlrqv/ jlyhq wkh xqlirup udwlrqlqj surfhvv +lqwurgxfhg
e| Ehqdvv| ^6`,,1
Wkh htxdo exgjhw iuhh fkrlfh ixqfwlrq lv yhu| pxfk lq
wkh vdph vslulw1 Djhqwv pd| wudgh dw {hg sulfhv zklfk duh ghwhuplqhg e|
wkh whfkqrorjlfdo wudqvirupdwlrq udwhv1 Ehfdxvh ri wkh olqhdu whfkqrorj| dv0
vxpswlrq/ wkhvh sulfhv duh htxloleulxp sulfhv iru doo suhihuhqfh surohv1 Wklv
lv zk| hflhqf| lv vdwlvhg1 Dfwxdoo|/ wkh Eduehu
d dqg Mdfnvrq ^4` prgho
dqg rxu prgho duh srodu fdvhv ri wkh jhqhudo prgho zkhuh wkh vhw ri ihdvleoh
doorfdwlrqv duh ghwhuplqhg e| d frqyh{ whfkqrorj|1
Wkh sdshu lv rujdql}hg dv iroorzv1 Lq vhfwlrq 5/ zh ghqh wkh prgho dqg
jlyh wkh edvlf ghqlwlrqv1
Wkhq/ zh vwxg| vwudwhj|0surriqhvv1
Lq vhfwlrq
6/ zh vwdwh wzr whfkqlfdo ohppdv/ zklfk suryh hvvhqwldo iru wkh uhvw ri wkh
lqtxlu|1 Lq sduwlfxodu/ Ohppd 5 vd|v wkdw dq| hflhqw dqg vwudwhj|0surri
doorfdwlrq ixqfwlrq dvvljqv wr dq djhqw rqh ri khu suhihuuhg exqgohv lq d
exgjhw vhw1 Wkh sulfhv dvvrfldwhg zlwk wklv exgjhw vhw duh ghwhuplqhg e|
wkh whfkqrorjlfdo wudqvirupdwlrq udwhv1 Wkh exgjhw lwvhoi lv lqghshqghqw ri
wkh djhqw*v suhihuhqfhv1
Lq vhfwlrq 7/ zh fkdudfwhul}h wkh fodvv ri hflhqw dqg vwudwhj|0surri
doorfdwlrq ixqfwlrqv/ dqg wkh rqo| ixqfwlrq zh rewdlq zkhq zh dgg wkh htxdo
wuhdwphqw ri htxdov uhtxluhphqw1
Htxdo wuhdwphqw ri htxdov vlpso| vd|v
wkdw wzr djhqwv kdylqj wkh vdph suhihuhqfhv pxvw hqg xs hqmr|lqj wkh vdph
zhoiduh1 Lw lv d fohdu frqvhtxhqfh ri wkh fodvvlfdo dqrq|plw| uhtxluhphqw1 Dv
dqqrxqfhg deryh/ wklv ixqfwlrq dvvljqv wr hdfk djhqw khu suhihuuhg exqgoh lq
d exgjhw vhw zklfk lv wkh vdph iru doo djhqwv/ dqg zkrvh vorsh lv ghwhuplqhg
e| wkh {hg whfkqrorjlfdo sdudphwhuv1
Lq vhfwlrq 8/ zh fkdudfwhul}h wkh fodvv ri hflhqw dqg frdolwlrq vwudwhj|0
surri doorfdwlrq ixqfwlrqv/ dqg zh vkrz wkdw zh dovr frph wr wkh htxdo
5
exgjhw iuhh fkrlfh ixqfwlrq li zh phuho| dgg wkh xqlirup wuhdwphqw ri xql0
irupv uhtxluhphqw1 Xqlirup wuhdwphqw ri xqlirupv vd|v wkdw li doo djhqwv
kdyh wkh vdph suhihuhqfhv/ wkhq wkh| doo pxvw hqg xs hqmr|lqj wkh vdph
zhoiduh1
2 Ai 4L_i*
Lq wklv vhfwlrq/ zh suhvhqw wkh prgho/ vrph whuplqrorj|/ dqg zh jlyh wkh
edvlf ghqlwlrqv1 Wkhuh duh o sulydwh jrrgv1 Wkhuh lv d vhw Q ri q djhqwv
zkrvh frqvxpswlrq vhw lv LUn 1 E| xvlqj d olqhdu whfkqrorj|/ wkh djhqwv
fdq wudqvirup jrrgv lqwr rqh dqrwkhu/ vr wkdw wkh surgxfwlrq vhw fdq eh
qrupdol}hg wr \ @ i| 5 LUn =
' | 4j1 D IKDEDKDGBD lv d suhrughulqj
U rq LUn wkdw lv frqwlqxrxv dqg vwulfwo| prqrwrqlf/ wkdw lv/ } A }
}S } 1
,
5
,
,
,
Wkh vhw ri doo suhihuhqfhv lv ghqrwhg e|
U
f1
suhihuhqfh suroh lv dq hohphqw U
D
ri phpehuv ri d frdolwlrq
U
V
3t Q
ri
lv ghqrwhg
U
f
U7 1
ri
3
f
1
Iru dq|
f
/ rqh shu djhqw1
1 Ohw
'
Dq
ghqrwh wkh vhw ri ihdvleoh doorfdwlrqv1 Ohw
zlwk d ihdvleoh doorfdwlrq1 Iru
wk frpsrqhqw ri
+
Ohw
p+U> u,
f/
eh jlyhq1 Wkhq/
, @
n
=
p
n/
f
U>
}
5
UU
f1
Dq
/ ohw
i +U ,
ghqrwh wkh
LU,
n
eh jlyhq1 Wkhq/ e| d voljkw dexvh ri qrwdwlrq/ zh xvh
wr ghqrwh
# +
U3
,1
+
n/
Ohw
/ ohw
dvvrfldwhv hdfk suhihuhqfh suroh lq lwv
=
grpdlq
l
V
Q / ohw U37 ghqrwh dq hohphqw
@ + } > = = = > } ? ,
LU,nf? lv d olvw ri exqgohv
doorfdwlrq } @ +} > = = = > }? , lv ihdvleoh li
1 Iru dq|
Dq
5Q
1 Wkh olvw ri suhihuhqfhv
U 7 doorfdwlrq }
5
} > l 5 Q
5? } 5 \ ]
doorfdwlrq ixqfwlrq i U $ ]
U 5 U U
l
i U
D LU, U 5 U
p U> D
i} 5 LU, ;} 5 LU, > } U } j=
u 5 LU U 5 U
ghqrwh dq hohphqw ri
,
n
=
,
[
'
}
u
,$
=
Zh zloo xvh wkh wzr iroorzlqj ulfkqhvv dvvxpswlrqv rq d vhw
e
hqfhv1
ViU|Lh ?i^@*|itG uLh @** %c % M U-n
U
ri suhihu0
% $ % C % $ % c M tc c ,c % % C
d% $ % @?_ % ' % oc % J % C % % c M tc c ,
e hi}*@h 4L?L|L?U Uhi ? U-, t |i 4@}i Lu @ u?U|L? G U-n < U-, U
n
n
t UL?|?Ltc 4L?L|L?U E|@| tc o o i Eo Eo c ?ML?_i_c @?_ tU |@|
Ef ' f Thiuihi?Ui - M , t _gihi?|@M*i u |ihi i t|t @ _gihi?|@M*i u?U|L?
G U-,n < U-n tU |@| uLh @** 5c 5 M U-n, c 5 - 5 C E5 D E5 ,
c
$
F lq LUn / wkhuh lv vrph glhuhqwldeoh
U 5 U vxfk wkdw p+U> u, F iru hyhu| u 5 ^3> 4`/
+D5, iru doo U> U 5 U dqg u 5 LUn / wkhuh lv vrph U 5 U vxfk wkdw
+D4, iru hyhu| uhjxodu lqfuhdvlqj fxuyh
,
+p+U> u, ^ p+U > u,, p+U > u,1
Wkh vhw UB lwvhoi vdwlvhv Dvvxpswlrqv +D4, dqg +D5,1 Dq rwkhu h{dpsoh
lv wkh vhw ri frqyh{ suhihuhqfhv lq UB1 Wkh vhw ri txdvlolqhdu suhihuhqfhv
lq UB grhv qrw vdwlvi| Dvvxpswlrq +D4,/ zkhuhdv wkh vhw ri vwulfwo| frqyh{
suhihuhqfhv lq UB grhv qrw vdwlvi| Dvvxpswlrq +D5,1 Ohw xv qrwh wkdw +D5,
uhtxluhv hlwkhu dw sruwlrqv ri lqglhuhqfh fxuyhv ru qrq0frqyh{ suhihuhqfhv1
Qrz/ zh ghqh wkh d{lrpv1
Bij|N juuWjAW+
Iru hyhu| U @ +UQ> = = = > U? , 5 U / dqg } @ +}Q > = = = > }? , 5 ] /
^} U i +U ,>
; l 5 Q ` , ^} L i +U ,> ; l 5 Q `
Qh{w/ zh ghqh wzr krul}rqwdo htxlw| d{lrpv1
.`B, |ijB|6jA| Nu j`B,t
Iru hyhu| U @ +UQ> = = = > U? , 5 U / dqg l> m
Û @ U ` , î +U , L i +U ,`
5 Q/
PAuNi6 |ijB|6jA| Nu AuNi6t
Iru hyhu| U @ +UQ> = = = > U? , 5 U /
Û @ U > ;l> m
5 Q ` , î +U , L i +U ,> ;l> m 5 Q `
Revhuyh wkdw xqlirup wuhdwphqw ri xqlirupv lv orjlfdoo| zhdnhu wkdq
htxdo wuhdwphqw ri htxdov/ dqg wkdw erwk d{lrpv duh orjlfdoo| zhdnhu wkdq
wkh fodvvlfdo dqrq|plw| d{lrp1 Ilqdoo|/ zh ghqh wzr qrq0pdqlsxodelolw|
d{lrpv1 Zh glvwlqjxlvk ehwzhhq wkh fdvh zkhuh rqo| lqglylgxdo djhqwv fdq
wu| wr pdqlsxodwh wkh rxwfrph +vwudwhj|0surriqhvv, dqg wkh fdvh zkhuh dq|
frdolwlrq fdq wu| wr pdqlsxodwh wkh rxwfrph +frdolwlrq vwudwhj|0surriqhvv,1
7|iB|j}+ViNNuAjtt
Iru hyhu| U @ +UQ> = = = > U? , 5 U / l 5 Q dqg U
5 U/
i +U , U i +U > U3 ,
NB,|NA t|iB|j}+ViNNuAjtt
Iru hyhu| U @ +UQ> = = = > U? , 5 U / V
î +U7 > U
3
Q dqg U7 5 U7 /
7 , U i +U ,> ; l 5 V ` , î +U7 > U 7 , L i +U ,> ; l 5 V `
3
7
hi*4?@h) hit*|t
Lq wklv vhfwlrq/ zh vwdwh dqg suryh wzr suholplqdu|/ udwkhu whfkqlfdo/ ohppdv1
Wkh uvw ohppd lghqwlhv d jhqhudo surshuw| ri vwudwhj|0surri doorfdwlrq
ixqfwlrqv1
Wkh vhfrqg ohppd vwdwhv wkdw dq hflhqw dqg vwudwhj|0surri
5Q
doorfdwlrq ixqfwlrq dvvljqv wr hdfk djhqw l khu suhihuuhg exqgoh lq d exgjhw
vhw
i} 5
n
LU
,
,
=
} @ d
j zkhuh d
/ zklfk fdq eh lqwhusuhwhg dv djhqw
l*v lqfrph/ grhv qrw ghshqg rq khu suhihuhqfh/ iru hyhu| l
5 Q1
U Uf1 Dq doorfdwlrq ixqfwlrq i U $ ] vdwlvhv
li dqg rqo| li iru hyhu| l 5 Q / wkhuh h{lvwv d fruuhvsrq0
3t
ghqfh [ U
$ LU,n/ vxfk wkdw iru hyhu| U 5 U /
41 i U > U3 5 p U > [ U3 /
51 li {> { 5 [ U3 / wkhq { { , { { 1
Ohppd 4 Ohw
=
vwudwhj|0surriqhvv
=
+
,
+
+
Surri=
+
@
,
Wkh vwudljkwiruzdug %li % sduw lv rplwwhg1 Qrz/ ohw xv vxssrvh wkdw
lv vwudwhj|0surri1 Iru hyhu|
vhw
,,
l
5Q
/
U 3
5 U 3t
LU,n
R U3
i} 5 LU,n < U 5 U } i U> U3 j
i U > U3 5 p U > R U3
{ { , { {
{> { 5 R U3
R +U3 ,
+
e|
, @
=
=
E| vwudwhj|0surriqhvv/
+
,
@
+
+
,
+
,,1 E| vwulfw prqrwrqlf0
lw| ri wkh suhihuhqfhv dqg vwudwhj|0surriqhvv/ qr wzr hohphqwv ri
+
duh rughuhg= li
ghqh
[
i
/ zh fdq ghqh wkh rswlrq
@
,/ wkhq
e| dvvrfldwlqj hyhu|
U3
1
R +U3 ,
Wkhuhiruh/ zh fdq
zlwk wkh fruuhvsrqglqj
R +U3 ,1
T1H1G1
Ohppd 5 D Ohw U Uf vdwlvi| Dvvxpswlrq +D4,1 Dq doorfdwlrq ixqfwlrq
i
l
=
U $ ]
vdwlvhv
vwudwhj|0surriqhvv
dqg
hflhqf|
rqo| li iru hyhu|
5 Q / wkhuh h{lvwv d U 3t $ LUn / vxfk wkdw iru hyhu| U 5 U /
i U > U3 5 p U > d U3
Surri=
l5Q
U3 5 U 3t
[ U3
i >===> j
F
U5U
i U> U3 5 F
i U> U3 5
@F
ì @hi }h@|iu* |L aLti `_@h uLh t i*T | |i ThLLu Lu |t *i44@
=
+
,
+
+
Il{ dq duelwudu| djhqw
+
li
, @
+3
,,
1 Qrwh wkdw
/ dqg d suroh
3, / wkhq zh duh grqh1 Vr zh dvvxph wkdw lw lv qrw wkh
fdvh1 Zh ehjlq e| qrwlqj d vwudljkwiruzdug frqvhtxhqfh ri hflhqf|= iru hy0
hu| uhjxodu lqfuhdvlqj fxuyh
huwlhv ri Dvvxpswlrq +D4,/
dqg fruuhvsrqglqj
+
,
,
8
vdwlvi|lqj wkh surs0
1 Lqghhg/ vxssrvh
+
,
1
Ohw
K
@
5 i U> U3
+
5 p+U> u,1 Wkhq/
{
,1 Vlqfh
[ [
l,
{
@
F / i+U> U3, 5@ p+U> u,1
p+U> u,
i +U> U3 , @ u>
Slfn
dqg
{ S i +U> U3 ,=
ll,
Exw wkhq +{ > i3 +U> U3 ,, Sduhwr grplqdwhv
surri e| wkh iroorzlqj fodlp1
i +U> U3 ,1
Zh frqwlqxh wkh
, / F _ [ +U , 9@ >1 Vxs0
F lq LUn
3
srvh qrw1 E| Dvvxpswlrq +D4,/ wkhuh h{lvwv U 5 U vxfk wkdw iru doo u 5 ^3> 4`/
p +U> u, F 1 E| Ohppd 4 dqg vwudwhj|0surriqhvv/ i +U> U3 , 5 [ +U3 ,/
@ F / zklfk frqwudglfwv hflhqf|1
vr wkdw i +U> U3 , 5
Fodlp 4= iru hyhu| uhjxodu lqfuhdvlqj fxuyh
S
5U
9
/
U > U
,
, vxfk wkdw i +U > U , @ } / i +U > U , @ } / dqg
L
U
@
}
3
3
n
' ,
1 Ohw wkh sdudphwhuv } > } > U > U eh {hg iru wkh uhvw ri wkh surri1
}
' S
Qrz/ li wkh vwdwhphqw ri wkh Ohppd lv qrw wuxh/ wkhuh h{lvw
} > } 5
Zh {
d@
[
,
'
} =
+4,
E| fodlp 4/ wkhuh h{lvwv

5 LUn vxfk wkdw +3> 5 5Q 5
5 [+U3,1 E| Ohppd 4 surylvr 5/
^3> ` $ LUn e|
j +, @ |
3[ 4
D
C
'
/ +|> > = = = > 5 5, 5 ,
Q
xqltxh1 Ghqh j =
S '2
}
}
wklv qxpehu lv
>===>
S '
},
}
,
5 [+U3,
vr wkdw
j
}
@ }
E| fodlp 4 dqg Ohppd 4 surylvr 5/
ghfuhdvlqj/ frqwlqxrxv ixqfwlrq1
+5,
j
Frqvhtxhqwo|/ e| Ohehvjxh*v wkhruhp/
lq ^3> `/ j
]
f
bQ
lv lqwhjudeoh/ dqg
j +,g j+ , j +3,
j
lv d zhoo0ghqhg/ erxqghg/ vwulfwo|
lv glhuhqwldeoh doprvw hyhu|zkhuh
+6,
9
E| hflhqf| dqg fodlp 4/ jQ +, @ f @ 4 zkhqhyhu jQ +, h{lvwv/ zkhuh 4
lv wkh ydoxh ri wkh surmhfwlrq rq wkdw k|shusodqh ri wkh gluhfwlrqdo ghulyd0
wlyh ri vrph glhuhqwldeoh suhihuhqfhv dvvrfldwhg wr d fxuyh F lqwhuvhfwlqj
[ +U , dw +jQ +,> 5
> = = = > 5
,1 Qrz frqvlghu wkh lqyhuvh ixqf0
Q
Q
Q
wlrq jQ 1 H{dfwo| dv deryh/ fodlp 4 dqg Ohppd 4 surylvr 5 lpso| wkdw jQ Q
lv glhuhqwldeoh doprvw hyhu|zkhuh lq ^3> jQ +3,`/ jQ Q lv lqwhjudeoh/ dqg
5
3
5 ,
5 5 3
3
3
]
}Q -B
j3 +|,g|
f
j3 +j +3,, j3 +3,

Dovr/ e| hflhqf| dqg fodlp 4/
Iurp htxdwlrq +6,/ zh ghulyh
f

j3 +|, @

S
+7,
@
4 zkhqhyhu j3 +|, h{lvwv1

j +3,
+8,

Iurp htxdwlrq +7,/ zh ghulyh
4
f
j +3,
+9,

Sxwwlqj htxdwlrqv +8, dqg +9, wrjhwkhu/ zh jhw
htxdwlrqv +4, dqg +5,/ zh jhw
j +, @ d
f @
3} Ef 1
b
Qrz/ xvlqj
} ? d1 Vxssrvh qrw1 E| wkh suhfhglqj fodlp/
eh vxfk wkdw i +U
> U3 , @ +d> 3> = = = > 3,1
+d> 3> = = = > 3,
[ +U3 ,1 Ohw U
E| vwulfw prqrwrqlflw|/ } S +d> 3> = = = > 3,/ frqwudglfwlqj vwudwhj|0surriqhvv1
5
5
LUn eh ghqhg e| j2+, @ |
+} > |> 5 5 > = = = > 5 , 5 ,
Ohw j2 = LUn
Zh qrz fodlp wkdw
5
5U
$
/
' c2
'c 2
5 [+U3,1 Iroorzlqj wkh vdph dujxphqwv dv deryh/ zh jhw j +, @ +d } , 1 E| uhsolfdwlqj wkh dujxphqwv o 4 wlphv/ zh suryh wkdw +} / } / = = =/
5 , 5 [+U3, zklfk/ e| vwulfw prqrwrqlflw| ri wkh suhihu0
},3 / d ,3 }
hqfhv dqg vwudwhj|0surriqhvv ri i lv lqfrpsdwleoh zlwk } 5 [ +U3 ,1 T1H1G1
2

'
e 5|h@|i})ThLLu?itt
2
Zh duh qrz htxlsshg wr vwdwh rxu uvw wkhruhp/ zklfk surylghv d frpsohwh
fkdudfwhul}dwlrq ri hflhqw dqg vwudwhj|0surri doorfdwlrq ixqfwlrqv1 Wkh
ohvvrq lv wkdw zkhq djhqwv duh qrw deoh wr irup frdolwlrqv/ hflhqf| dqg
vwudwhj|0surriqhvv gr qrw suhyhqw d sodqqhu iurp wdnlqj doo suhihuhqfhv lqwr
dffrxqw zkhq doorfdwlqj wkh jrrgv1
:
Ohw U Uf vdwlvi| Dvvxpswlrq +D4,1 Dq doorfdwlrq ixqfwlrq
dqg
li dqg rqo| li iru
U $ ] vdwlvhv
hyhu| l 5 Q / hyhu| frdolwlrq V Q ri fduglqdolw| q qrw frqwdlqlqj l/
wkhuh lv d pdsslqj e7 U7 $ LUn vxfk wkdw iru hyhu| U 5 U /
57G*M7c7 '?32 e7 U7 > ; l 5 Q
5M 57G*M7c7'?32 e7 U7 > dqg
i U 5 p U> 57G*M7c7'?32 e7 U7 > ; l 5 Q 1
Wkhruhp 4
i
=
vwudwhj|0surriqhvv
hflhqf|
5
=
+
,
3
+
+
,
, @ 4
+
+
,,
Ehiruh surylqj Wkhruhp 4/ lw pd| eh xvhixo wr looxvwudwh lw lq wkh wkuhh0
iV
j ii j> i jj
djhqw fdvh1
mm
V> V
@
Wkxv/ li
e2 / e /
q
i lv
dqg
Lq wkdw fdvh/
5
@
4
= 4
5@ V> mVm q j ii j> i jj iV 5@
iV 5@ V> mVm q j ii j> i jj
@
/ dqg
5
= 6
@
5
@
6
5
vxfk wkdw iru dq| suroh +
5
5
5
S
U > U2> U ,/
p+U > e2 +U2 , . e +U ,,
p+U2 > e2 +U , . e2 +U ,,
p+U > e +U , . e2 +U2 ,,
/
@
vwudwhj|0surri dqg hflhqw/ wkhuh h{lvw pdsslqjv
e2 /
i +U> U2 > U,
i2 +U> U2 > U,
i +U> U2 > U,
Lqhtxdolwlhv
6
= 5
4
5
1
e2 / e / e2 /
<
@
>
+:,
frqvxpswlrq olhv
' e +U , 3 jxdudqwhh wkdw
S hdfk
S djhqw*v
e
+
U
,
' @ 4 hqvxuhv wkdw
lq wkh frqvxpswlrq vhw dqg wkh frqvwudlqw
wkh djjuhjdwh frqvxpswlrq lv lq wkh surgxfwlrq vhw1
Htxdwlrq +:, phdqv
wkdw hdfk djhqw lv pd{lpl}lqj khu suhihuhqfh lq d exgjhw vhw wkdw lv ghwhu0
plqhg e| wkh rwkhu djhqwv* suhihuhqfhv1 Qrwlfh wkdw wkhvh suhihuhqfhv dhfw
wkh exgjhw lq dq dgglwlyho| vhsdudeoh zd|1 Dffruglqj wr Wkhruhp 4/ d zhdn
irup ri dgglwlyh vhsdudelolw| shuvlvwv zkhq wkhuh duh pruh wkdq wkuhh djhqwv1
S
Surri ri Wkhruhp 4= Wkh surri ri wkh vwudljkwiruzdug %li % sduw lv rplwwhg1
$ ] eh dq vwudwhj|0surri dqg hflhqw doorfdwlrq ixqfwlrq1 Iru
Ohw i = U
.t $ LUn vxfk
hdfk l 5 Q / wkhuh h{lvwv/ e| Ohppd 5/ d pdsslqj d = U
[
wkdw iru hyhu| U
5U
/ i +U ,
5 p U >d
+
U3 ,, iru doo l
+
d +U3 , @ 4
5Q
dqg
+;,
M
5
Il{ qrz dq duelwudu| m
Q / vd|/ m @ 41 Zh uvw fodlp wkdw wkh pdsslqj
dQ lv dgglwlyho| vhsdudeoh lq wkh vhqvh wkdw
dQ +U3Q , @
[
dQ7 +U7 ,>
7 /QM
* 7c7 $?3
3Q
.tQ
,^@|L? E. U@? Mi uh|ih t4T*i_ M) ?L|?} |@| K2 ' !K2 c K ' !K2 @?_
' !K 5U LMLt t4T*U@|L?t @hi ?L| @@*@M*i uLh 4Lhi |@? |hii @}i?|t
S
K2
;U 5 U
;
d Q% /
iru surshuo| fkrvhq pdsslqjv
zkhuh zh lqwhusuhw
Wr suryh wklv fodlp/ { dq duelwudu| suroh
dQ
dv d frqvwdqw1
5 U .tQ1
U3Q
U zklfk lv glvwlqfw iurp U
UB
eh
ri surohv U
5
Ohw
5 Q q i4j1
B
iru
Frqvlghu qrz wkh vhw U+U > U3Q , ri doo suhihuhqfh surohv U 5 U
B
B
zklfk UQ @ U dqg U
5 iU
> U j/ iru doo l 5 Q q i4j1 Iru hdfk l 5 Q q i4j
B
dqg U 5 U+U > U3Q ,/ ghqh
dq duelwudu| suhihuhqfh lq
w
+U , @
,
w+U
U+Uf > U3 , iru
dqg ohw
4
U
U
li
5M
5
@
li
iru doo
U
B
@ U
@
.t w +U ,1 Vlqfh wkh qxpehu
w+U , lv rgg htxdov wkh qxpehu ri
zklfk
[
lv hyhq/ zh nqrz iurp +;, wkdw
#
+ 4,
- M,E-B c- 3Q |E-
l
wkrvh iru zklfk lw
$
n?3 [
d +U3 ,
M @3
S
M,E-B c-3Q +4,|E- n?3d +U3 , @ 3 +vlqfh
, dqg +Uf > U , dqg wkxv dsshduv
hdfk d +U3 , lv jhqhudwhg e| erwk +U > U3
3
Exw iru hdfk l
5 Q q i4j/
-
rqfh zlwk d .4 frhflhqw/ dqg rqfh zlwk d
[
-
M,E-B c-3Q |E-
+ 4,
zklfk fdq eh uhzulwwhq dv
d +U3 , @
-
4 frhflhqw,1
Wkhuhiruh/
n?3
d +U3 , @ 3
[
M,E-B c-3Q .tE-B c-3Q + 4,
|E-
n?
d +U3 ,
Qrz/ zh fdq ghqh d rqh0wr0rqh fruuhvsrqghqfh ehwzhhq wkh surohv lq
zlwk wkh frdolwlrq
+Uf > U3 , dqg wkh vxevhwv ri Q
4 e| dvvrfldwlqj U
U
qi j
V ri wkrvh djhqwv l iru zkrp U @ U 1 Revhuyh wkdw w+U , @ 5+q
zklfk phdqv wkdw
rewdlq
d +U3 , @
w+U
, . q lv rgg li dqg rqo| li q
[
+
M $?32
4,
?
37 d
4, mV m/
mV m lv rgg1
Wkxv zh
f
f
+U 7 > U > = = = > U ,
7 G * 7c 7
q
^i j
zkhuh wkh suhihuhqfh Uf lv khog e| doo phpehuv ri Q +V
l ,1 Ghqlqj
?37
f
f
d7 +U7 , @ + 4,
d +U7 > U > = = = > U , frpsohwhv wkh surri ri wkh fodlp1
Zh kdyh qrz wr ghulyh wkh e7 pdsslqjv iurp wkh deryh d7 pdsslqjv1
Lqgh{ dqg udqn wkh frdolwlrqv ri vl}h q 5 qrw frqwdlqlqj l +wkhuh duh q 4
<
ri wkhp, dffruglqj wr wkh oh{lfrjudsklf rughulqj ghulyhg iurp wkh qdwxudo
rughulqj ri wkh djhqwv=.
V ! V 2 ! = = = ! V ?3
Iru hdfk n/ 4 n
q 4/ dqg hdfk U7 5 U7 / ohw
e7 +U7 , =@
[
7 G7 a7 dqg 7 a7 & &
d7 +U7 ,
Wkh surri lv qrz frpsohwh vlqfh wkh fkrlfh ri djhqw 4 zdv duelwudu|1 T1H1G1
Qrwh wkdw Wkhruhp 4 grhv qrw dvvhuw ~ dqg lw lv jhqhudoo| idovh ~ wkdw
wkh v|vwhp ri e7 pdsslqjv lv xqltxh1 Rq wkh rwkhu kdqg/ lw pd| vhhp
fxulrxv wkdw wkh e7 pdsslqjv pxvw eh ghqhg iru frdolwlrqv frqwdlqlqj dv
pdq| dv q 5 djhqwv1 Lw lv lpsrvvleoh/ krzhyhu/ wr uhvwulfw rxuvhoyhv wr
e7 ghqhg zlwk uhvshfw wr vwulfwo| vpdoohu frdolwlrqv/ dv h{dpsolhg e| wkh
iroorzlqj ixqfwlrq1 Ohw L +=, ghqrwh wkh lqglfdwru ixqfwlrq1 Ohw UW Uf 1
Ohw i = Uf $ ] eh ghqhg e|=
ec .tc2 @
e2c .tc2 @
4
q
4
.
4
q
L +U 5 UW > = = = > U? 5 UW,
4
L +U 5 UW > = = = > U? 5 UW,
q q
; V Q> 4 5@ V> mVm @ q 5 dqg V 9@ Q q i4> 5j = e7 3
; V Q> 5 5@ V> mVm @ q 5 dqg V 9@ Q q i4> 5j = e27 3
57 e7 1 Ilqdoo|/ qrwh wkdw li wkhuh h{lvw
dqg iru doo m 5 Q q i4> 5j/
?
, / dqg suhihuhqfhv U> U 5 Uf vxfk wkdw
d uhjxodu lqfuhdvlqj fxuyh F lq LUn
p+U> u, F dqg p+U > u, F iru hyhu| u 5 ^3> 4`/ dqg U 5 UW zkhuhdv
U 5@ UW/ wkhq i lv ervv|1H
Rxu qh{w uhvxow lv wkdw li htxdo wuhdwphqw ri htxdov lv dgghg wr hflhqf|
dqg vwudwhj|0surriqhvv/ wkhq doo exgjhwv pxvw eh frqvwdqw dqg htxdo1
Ai Lh_ih?} A t uLh4@**) _i?i_ @t uL**Lt +ih|i i@U UL@*|L? 7 '
tQ E7c c ?3 E7 tL |@| Q E7 ?3 E7 Ai? 7 A A u |ihi i t|t &c
$ & $ ? 3 2c tU |@| & E7 ' & EA @?_ & E7 & EA i?iih $ & &
0
+iU@** |@| @ u?U|L? t MLtt) u |ihi i t| c M c - M , @?_ - M , tU |@|
.
!
s E- ' s E- c -3 ihi@t s E- ' s E- c -3 "!
!
Ohw 7 7B vdwlvi| Dvvxpswlrq +D4,1 Dq doorfdwlrq ixqfwlrq
/
/ dqg
U $ ] vdwlvhv
li dqg rqo| li iru hyhu| U 5 U / l 5 Q /
Wkhruhp 5
i
=
vwudwhj|0surriqhvv
htxdov
i +U ,
Surri=
5p
U >
4
q
hflhqf|
htxdo wuhdwphqw ri
=
Wkh surri ri wkh vwudljkwiruzdug %li % sduw lv rplwwhg1
Ohw wkh doorfdwlrq ixqfwlrq
i
vdwlvi| wkh wkuhh d{lrpv1 Ohw djdlq
d > = = = > d ?
eh wkh pdsslqjv lghqwlhg lq Ohppd 51 Zh pxvw vkrz wkdw
d +U3 , @
4
q
>
; l 5 Q> ; U 5 U
+<,
Uf1 Vlqfh htxdo wuhdwphqw ri htxdov uhtxluhv
f
f
f
f
f
Q dqg vlqfh Uf lv vwulfwo|
wkdw i +U > = = = > U , L i +U > = = = > U , iru doo l> m
f
f
f
f
prqrwrqlf/ d +U > = = = > U , @ d +U > = = = > U ,/ iru doo l> m
Q 1 Xvlqj +;,/ zh
Il{ dq duelwudu| suhihuhqfh
5
rewdlq
4
d +Uf > = = = > Uf , @
q
5
; 5Q
> l
Zh frpsohwh wkh surri e| dq lqgxfwlrq dujxphqw rq wkh qxpehu ri suhihu0
hqfhv glhulqj iurp
Uf
j
/ dqg
V +U ,
odwwhu vhw1
qrz {
/
@
Uf1 Iru hdfk U
Q V f +U ,/ dqg
ohw
V f +U ,
/ ohw
v+U ,
@
4
q
U
l 5 V U
Iru hdfk
+
dqg revhuyh wkdw
v+U , @ /
3 ,
@
|lhogv wkdw
d +U3 , @
4
41
M7 B E- E| htxdo wuhdwphqw ri htxdov/
d + U 3 , @
4
; l 5 V U
,
d +U3 , @ 4
q
q
4
q
dqg wkh surri lv frpsohwh1 T1H1G1
44
@
41
zlwk
Wkh lqgxfwlrq k|srwkhvlv wkxv
Wdnlqj +;, lqwr dffrxqw/ zh jhw
[
@
Zh
Uf> U3 , e| uhsodflqj U
+
q
U
dqg suryh +<, iru wkdw suroh1
,/ frqvwuxfw wkh suroh +
v+Uf > U
=
v+U , @ 31
zkhqhyhu v+U ,
/ dqg dvvxph wkdw +<, krogv
iru zklfk
il 5 Q
eh wkh fduglqdolw| ri wkh
Zh kdyh mxvw vkrzq wkdw htxdwlrq +<, krogv li
Zh slfn d suroh
Uf
q
5 U
4
q
; 5 Vf U
> l
+
,
D L@*|L?
t|h@|i})ThLLu?itt
Lq wklv vhfwlrq/ zh dvvxph wkdw wkh djhqwv kdyh wkh rssruwxqlw| wr frruglqdwh
wkhlu plvuhsruwv1 Li wkh suhihuhqfh grpdlq lv ulfk hqrxjk/ frdolwlrq vwudwhj|0
surriqhvv dqg hflhqf| irufh xv wr jlyh hdfk djhqw klv ehvw fkrlfh lq d
{hg
exgjhw vhw1
Ohw U UB vdwlvi| Dvvxpswlrqv +D4, dqg +D5,1 Dq doorfdwlrq
ixqfwlrq i = U $ ] vdwlvhv frdolwlrq vwudwhj|0surriqhvv dqg hflhqf|
? vxfk wkdw iru hyhu|
li dqg rqo| li wkhuh h{lvwv @ + > = = = > ? , 5 LUn
U 5 U /
Wkhruhp 6
5?' @ 4/ dqg
i+U , 5 p+U> ,> ; l 5 Q 1
Surri=
Khuh djdlq/ zh ohdyh wkh hdv| surri ri wkh %li% sduw wr wkh uhdghu1
U vdwlvi|lqj Dvvxpswlrqv +D4, dqg +D5, dqg dq
i rq U 1 Iru hdfk
l 5 Q / wkhuh lv d pdsslqj d vxfk wkdw i +U , 5 p+U > d +U3 ,, iru hdfk
U / dqg htxdwlrq +;, krogv iru hdfk U 1
f
f
f
Fkrrvh d suhihuhqfh U 5 U dqg vhw @ d +U > = = = > U , iru doo l 5 Q 1
Zh fodlp wkdw d +U3 , @ iru doo U 5 U dqg doo l 5 Q 1 Vlqfh U3 fdq
Il{ d suhihuhqfh grpdlq
hflhqw dqg frdolwlrq vwudwhj|0surri doorfdwlrq ixqfwlrq
eh wudqviruphg lqwr +U
f> = = = > Uf , e| vxffhvvlyho| uhsodflqj hdfk frpsrqhqw/
zh qhhg rqo| suryh wkdw
d +U3 , @ d +U3 > U ,>
Il{
m / U
U iru
vrph U
dqg
wkhuh h{lvwv
p+U > d +U3 ,,1
; l 5 Q qimj> ; m 5 Q> ; U 5U > ; U 5U +43,
wkh uhvw ri wkh surri1 Wkdqnv wr Dvvxpswlrq +D5,/
5 U vxfk wkdw +p+U > d +U3 ,, ^ p+U > d +U3 ,,, E| ghqlwlrq/
i +U > U3 , L i +U , L i +U > U3 ,1
Exw
wkhq/ frdolwlrq vwudwhj|0surriqhvv uhtxluhv
i +U > U3 , U i +U ,>
; l 5 Q q im j
+44,
dqg
+45,
; l 5 Q q im j
Lqghhg/ li +44, lv ylrodwhg iru vrph l @
9 m/ wkhq frdolwlrq il> mj pdqlsxodwhv
dw suroh +U > U3 , e| dqqrxqflqj +U > U ,1 Olnhzlvh/ li +45, lv ylrodwhg
iru vrph l 9@ m / wkhq frdolwlrq il> m j pdqlsxodwhv dw suroh +U > U3 , e|
i +U > U3 , U i +U > U3 ,>
dqqrxqflqj +U
> U ,1
45
Vlqfh i +U > U , 5 p+U > d +U
lw iroorzv iurp +44, wkdw
d +U
3 3 3
> U , d +U ,>
3
5 p+U > d +U ,,/
> U ,, dqg i +U ,
; l 5 Q q im j
3
+46,
ehfdxvh U lv vwulfwo| prqrwrqlf1 Olnhzlvh/ +45, |lhogv
d +U
3 > U , d +U
5
3
> U ,>
; l 5 Q q im j
5
+47,
Vlqfh d +U , . ' d +U > U , @ 4 @ d +U , . ' d +U ,/ doo lq0
htxdolwlhv lq +46, dqg +47, pxvw eh htxdolwlhv1 Vlqfh zh kdyh fkrvhq m / U /
dqg U duelwudulo|/ htxdwlrq +43, iroorzv1 T1H1G1
3
3
3
3
Dvvxpswlrq +D5, lq Wkhruhp 6 fdqqrw eh gursshg1 Wr vhh wklv/ frqvlghu
wkh iroorzlqj 60djhqw/ 50jrrg h{dpsoh1 Wkh grpdlq U frqvlvwv ri wkh frq0
wlqxrxv/ vwulfwo| frqyh{/ vwulfwo| prqrwrqlf suhihuhqfhv1 Iru hdfk U 5 U/ ohw
d+U, 5 ^3> ` eh wkh uvw frruglqdwh ri wkh xqltxh hohphqw ri p+U> ,1 Iru
hdfk U 5 U / ohw
4
i +U , @ p U >
64
i2 +U , @ p U2 > . d+U ,
6
4
i +U , @ p U > d+U ,
6
Wklv uxoh lv hflhqw dqg frdolwlrq vwudwhj|0surri1 b
Dq lpphgldwh frqvhtxhqfh ri Wkhruhp 6 lv
Wkhruhp 7 Ohw U Uf vdwlvi| Dvvxpswlrqv +D4, dqg +D5,1 Wkh doorfd0
wlrq ixqfwlrq i = U $ ] vdwlvhv frdolwlrq vwudwhj|0surriqhvv/ hflhqf|/
dqg xqlirup wuhdwphqw ri xqlirupv li dqg rqo| li i +U , 5 p+U > ? , iru doo
l 5 Q dqg U 5 U 1
Dq dowhuqdwlyh fkdudfwhul}dwlrq ri wkh htxdo exgjhw iuhh fkrlfh ixqfwlrq
zdv jlyhq lq Wkhruhp 51 D vlpsoh wkuhh0djhqw h{dpsoh vkrzv wkdw frdolwlrq
vwudwhj|0surriqhvv fdqqrw eh uhsodfhg zlwk vwudwhj|0surriqhvv lq Wkhruhp
t i 4i?|L?i_ @MLic |i ti| Lu t|hU|*) UL?i Thiuihi?Uit _Lit ?L| t@|tu) @tt4T
|L? E2 `i @i ?L| Mii? @M*i |L U@h@U|ih3i |i U*@tt Lu UL@*|L? t|h@|i})ThLLu @?_
iUi?| u?U|L?t _i?i_ Lih |i _L4@? Lu t|hU|*) UL?i Thiuihi?Uit
>
"$
7
7
71 Sduwlwlrq
lq wzr duelwudu| vxevhwv n dqg
pd{lpl}h khu suhihuhqfh lq wkh exgjhw vhwv
;
?
d+U2> U , @
=
;
?
d2+U> U , @
=
;
?
d+U> U2 , @
=

li +U2 > U,
li +U2 > U,
rwkhuzlvh1

li +U > U,
li +U > U,
rwkhuzlvh1

li +U > U2,
li +U > U2,
rwkhuzlvh1
7! / dqg ohw hdfk djhqw
# +7! 7n ,>
# +7n 7! ,>
# +7n 7! ,>
# +7! 7n ,>
# +7! 7n ,>
# +7n 7! ,>
Wklv ixqfwlrq lv hflhqw/ vwudwhj|0surri/ dqg vdwlvhv xqlirup wuhdwphqw ri
xqlirupv1
+iuihi?Uit
^4`
7 BiOjiB BAa cBW"tNA/ *Vwudwhj|0Surri H{fkdqjh */ Hfrqr0
phwulfd
/ 96 +4<<8,/ ss 840;:1
^5`
7 BiOjiB e cBW"tNAe BAa j6j/
*Vwudwhj|0Surri
Do0
orwphqw Uxohv */ Zrunlqj sdshu/ Xqlyhuvlwdw Dxw
rqrpd gh Edufhorqd/
4<<81
^6`
c jABtt+/ *Wkh Hfrqrplfv ri Pdunhw Glvhtxloleulxp/ Dfd0
ghplf Suhvv= Qhz0\run1
^7`
^8`
7 A}/ *Dq Dowhuqdwlyh Fkdudfwhul}dwlrq ri wkh Xqlirup Uxoh */ Vr0
fldo Fkrlfh dqg Zhoiduh
/ 44 yro 5 +4<<7,/ ss 46404691
7B||ji|B|j BAa Q 7NAAjAtWjA/ *Vwudwhj|surri Doorfdwlrq
Phfkdqlvpv dw Glhuhqwldeoh Srlqwv */
Uhy1 Hfrq1 Vwxglhv
/ 7; +4<;4,/
ss 8;:08<:1
^9`
7 7ji5BB/
*Vwudwhj|0Surri
dqg Lqglylgxdoo| Udwlrqdo Vrfldo Fkrlfh
Ixqfwlrqv iru Sxeolf Jrrg Hfrqrplhv */
Hfrq1 Wkhru|
/ : +4<<9,/ ss
83408451
^:`
7 7jA"ji/ *Rq wkh Vwudwhj|0Surri dqg Vprrwk Doorfdwlrq ri Sulydwh
Jrrgv lq d Surgxfwlrq Hfrqrp| / plphr/ [hur{ SDUF/ 4<<51
47
^;`
v 7Vi6NA| Wkh Glylvlrq Sureohp zlwk Vlqjoh0Shdnhg Suhihuhqfhv=
d Fkdudfwhul}dwlrq ri wkh Xqlirup Doorfdwlrq Uxoh Hfrqrphwulfd
/ *
*/
/
8< +4<<4,/ ss 83<08531
^<`
w ~N Lqhflhqf| ri Vwudwhj|0Surri Doorfdwlrq Phfkdqlvpv lq Sxuh
/ *
H{fkdqjh Hfrqrplhv Vrf1 Fkrlfh Zhoiduh
*/
48
/ ; +4<<4,/ ss 57:05871

Download Report

Efficient Strategy-Proof Allocation Functions in Linear

Paperzz.com

Your Paperzz