2004-08
SPRUMONT, Yves
Aumann-Shapley Pricing : A Reconsideration of
the Discrete Case
C.P. 6128, succursale Centre-Ville Montréal (Québec) H3C 3J7 Canada http://www.sceco.umontreal.ca SCECO-information@UMontreal.CA 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 11-2004.
This working paper was also published by the Center for Interuniversity Research in Quantitative Economics (CIREQ), under number 11-2004.
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Wklv surshuw| frqqhfwv frvw vkduhv lq wzr sureohpv zlwk glhuhqw vhwv ri djhqwv= Q rewdlqv iurp Q e| vsolwwlqj djhqw l lqwr d vhw L ri djhqwv l zkrvh djjuhjdwh
ghpdqg htxdov l*v ruljlqdo ghpdqg1 Wkh rwkhu djhqwv lq Q dqg Q duh lghqwlfdo/
dqg wkhlu ghpdqgv gr qrw fkdqjh1 Wkh frvw ixqfwlrq F iru Q h{suhvvhv wkh vdph
whfkqrorj| dv F= wkh vxp ri wkh frqvxpswlrqv e| wkh djhqwv l lq L phuho| sod|v wkh
uroh ri djhqw l*v ruljlqdo frqvxpswlrq1
Wkh frqglwlrq suhyhqwv pdqlsxodwlrqv ri lghqwlw|= djhqw l lq Q grhv qrw jdlq e| vsolwwlqj khu ghpdqg lqwr vpdoohu slhfhv>wkh djhqwv l lq Q gr qrw jdlq e| phujlqj
wkhluv lqwr rqh odujhu eorfn1 Qrwh wkdw zh uhtxluh l 5 L G Qr Phujlqj ru Vsolwwlqj grhv qrw h{solflwo| uxoh rxw pruh udglfdo pdqlsxodwlrqv ri lghqwlw| zkhuhe| dq djhqw lv uhsodfhg zlwk d frpsohwho| glvmrlqw vhw ri ghpdqghuv1
Yduldqwv ri wkh Qr Phujlqj ru Vsolwwlqj frqglwlrq kdyh ehhq vwxglhg ehiruh/ qr0 wdeo| lq wkh vlpsohu prgho ri udwlrqlqj +ru edqnuxswf|,/ zkhuh d jlyhq dprxqw pxvw eh vsolw dprqj d vhw ri djhqwv kroglqj fodlpv rq lw1 Hduo| frqwulexwlrqv lqfoxgh R*Qhloo +4<;5,/ Prxolq +4<;:,/ dqg Fkxq +4<;;,1 Iru d uhfhqw wuhdwphqw frqwdlqlqj uhihuhqfhv dqg jhqhudol}dwlrqv/ vhh Mx/ Pl|djdzd/ dqg Vdndl +5336,1
4. An axiomatization Zh duh qrz uhdg| wr vwdwh dqg glvfxvv rxu pdlq uhvxow1
Wkhruhp1 Wkh Dxpdqq0Vkdsoh| phwkrg lv wkh rqo| frvw0vkdulqj phwkrg vdwlvi|lqj Dgglwlylw|/ Gxpp|/ dqg Qr Phujlqj ru Vsolwwlqj1
Rqh ihdwxuh ri wkh surri sursrvhg lq wkh Dsshqgl{ pd| eh ri lqghshqghqw lqwhuhvw1 Qr Phujlqj ru Vsolwwlqj lpsolhv d Zhdn V|pphwu| surshuw| vwdwlqj wkdw uhqdplqj h{dfwo| rqh djhqw grhv qrw dhfw khu rzq frvw vkduh1 Vwhs 7 ri wkh surri vkrzv wkdw rq wkh vxevhw ri sureohpv fruuhvsrqglqj wr vwdqgdug frrshudwlyh jdphv zlwk d yduldeoh srsxodwlrq 0qdpho|/ SEf0 Dgglwlylw|/ Gxpp| dqg Zhdn V|pphwu| fkdudfwhul}h wkh Vkdsoh| ydoxh1 Wkxv/ wkh ixoo irufh ri Vkdsoh|*v +4<86, v|pphwu| frqglwlrq lv qrw qhhghg wr slq grzq wkh ydoxh zkhq wkh vhw ri djhqwv lv doorzhg wr fkdqjh1
Zh pdnh wkuhh ixuwkhu frpphqwv1
4, Wkh d{lrpv lq wkh deryh wkhruhp duh lqghshqghqw1 Qr Phujlqj ru Vsolwwlqj lv reylrxvo| qhhghg= wkh +ulfk, fodvv ghqhg e| Dgglwlylw| dqg Gxpp|/ ghvfulehg lq Vwhs 4 ri wkh surri/ frqwdlqv/ iru lqvwdqfh/ wkh Vkdsoh|0Vkxeln phwkrg1 Wkh sursru0 wlrqdo phwkrg *s
lEQ> {> F ' E{l@
P
mMQ{mFE{ zkhqhyhu { 9' f +dqg *EQ> {> F ' f
zkhq { ' f vkrzv wkdw Gxpp| lv qrw uhgxqgdqw1 Wr vhh wkdw Dgglwlylw| fdqqrw eh gursshg/ frqvlghu wkh phwkrg *EQ> {> F ' *sEQ> {> F zkhqhyhu F lv vwulfwo| lq0
fuhdvlqj +wkdw lv/ FE} ? FE} li } ? }/ dqg *EQ> {> F ' *DVkEQ> {> F rwkhuzlvh1
5, Ehfdxvh Qr Phujlqj ru Vsolwwlqj frpsduhv sureohpv zlwk glhuhqw vhwv ri djhqwv/ lw lv d vrphzkdw frpsoh{ frqglwlrq1 Zh sursrvh d yduldqw ri rxu wkhruhp
Aumann-Shapley pricing: A reconsideration of the discrete case : edvhg rq d {hg0srsxodwlrq frqglwlrq zklfk lv d olwwoh vlpsohu wr judvs wkdq Qr Phujlqj ru Vsolwwlqj1 Jlyhq Q 5 Q dqg V Q> ghqh F1
VEQ ' iF 5 FEQ m
<f G FE} ' fE}Q.V>PlMV}lj= Wklv lv wkh vhw ri frvw ixqfwlrqv lq FEQ iru zklfk wkh
ghpdqgv iurp wkh djhqwv lq V duh shuihfw vxevwlwxwhv= wkhvh djhqwv ghpdqg hvvhqwldoo| wkh vdph jrrg1
Qr Ghpdqg Uhvkx"lqj1 Iru doo Q 5 Q > {> { 5 QQ> V Q> dqg F 5 F1 VEQ> iPlMV{l ' P lMV{l dqg {Q.V ' {Q.Vj , i P lMV*lEQ> {> F ' P lMV*lEQ> {> Fj=
Wklv frqglwlrq vd|v wkdw wkh djjuhjdwh frvw vkduh ri d jurxs ri djhqwv ghpdqglqj hvvhqwldoo| wkh vdph jrrg ghshqgv rqo| rq wkhlu djjuhjdwh ghpdqg1 Li wklv frqglwlrq zhuh ylrodwhg/ doo phpehuv ri V frxog ehqhw e| uhvkx"lqj lqglylgxdo ghpdqgv zlwklq V dqg shuiruplqj vxlwdeoh vlgh0wudqvihuv1 Wkh lghd zdv sursrvhg e| Prxolq +4<;:, lq wkh edqnuxswf| prgho1 Lw lv hdv| wr fkhfn wkdw Qr Ghpdqg Uhvkx"lqj lv lpsolhg e| Qr Phujlqj ru Vsolwwlqj1 Dvvxph wkh suhplvhv ri wkh iruphu/ fkrrvh dq djhqw v 5 V> dqg ghqh f 5 FEEQqV ^ v e| fE}Q.V>
P
lMV}l ' FE} iru doo } 5 QQ= Dsso|lqj Qr
Phujlqj ru Vsolwwlqj wzlfh/ zh jhwPlMV*lEQ> {> F ' *vEEQqV^v> E{Q.V>PlMV{l>
f ' *vEEQqV ^ v> E{Q.V>
P
lMV{l> f '
P
lMV*lEQ> {> F=
Wkh iroorzlqj h{dpsoh vkrzv wkdw Dgglwlylw|/ Gxpp|/ dqg Qr Ghpdqg Uhvkxi0 lqj duh qrw txlwh hqrxjk wr slq grzq wkh Dxpdqq0Vkdsoh| phwkrg1 Ohw
*1Ei> 2> j > ht1>2>3> F '
EC1FEf n C1FEht2 n C1FEht2>3>
*2Ei> 2> j > ht1>2>3> F '
EC2FEf n C2FEht3 n C2FEht1>3>
*3Ei> 2> j > ht1>2>3> F '
EC3FEf n C3FEht1 n C3FEht1>2
iru doo F 5 FEi> 2> j dqg ohw *EQ> {> F ' *DVkEQ> {> F iru hyhu| rwkhu frvw0vkdulqj
sureohp EQ> {> F= Dgglwlylw| dqg Gxpp| duh fohduo| phw1 Wr vkrz wkdw * vdwlvhv Qr Ghpdqg Uhvkx"lqj/ revhuyh wkdw wklv d{lrp frqqhfwv wkh frvw vkduhv iru wzr sureohpv EQ> {> F/ EQ> {> F rqo| zkhq wkh vhwv ri djhqwv duh wkh vdph +Q ' Q,
dqg wkh wrwdo ghpdqgv duh wkh vdph +PlMQ{l '
P
lMQ0{l,1 Vlqfh * frlqflghv zlwk
*DVk h{fhsw zkhq Q ' i> 2> j dqg { ' E> > > dqg vlqfh *DVk vdwlvhv Qr Ghpdqg
Uhvkx"lqj/ lw lv vx!flhqw wr fkhfn *1Ei> 2> j> E> > > F n *2Ei> 2> j> E> > > F
' *DVk
1 Ei> 2> j> E2> f> > F n *DVk2 Ei> 2> j> E2> f> > F iru doo F 5 Ft1>21 Ei> 2> j>
dqg wzr vlplodu frqglwlrqv rewdlqhg e| uhsodflqj i> 2j zlwk i> j dqg i2> j= Xvlqj exgjhw edodqfh/ wklv dprxqwv wr *3Ei> 2> j> E> > > F ' *DVk3 Ei> 2> j> E2> f> > F
iru doo F 5 F1
t1>2Ei> 2> j> dqg vlplodu frqglwlrqv iru djhqwv dqg 2= Wkhvh wkuhh
frqglwlrqv duh uhdglo| yhulhg1
Wkh phwkrg lq wklv h{dpsoh vdwlvhv wkh edvlf d{lrp ri ]hur Fkdujh iru ]hur Ghpdqg= iru doo EQ> {> F 5 S dqg l 5 Q/ i{l ' fj , i*lEQ> {> F ' fj= Exw
uhprylqj wkrvh zkr ghpdqg }hur iurp wkh vhw ri djhqwv dhfwv wkh frvw vkduhv ri wkh rwkhuv1 Iru lqvwdqfh/ frqvlghu wkh frvw ixqfwlrq F 5 FEi> 2> > ej jlyhq e| FE} ' f zkhqhyhu }1 ' }2 ' f> FE} ' rwkhuzlvh1 Wkhq *Ei> 2> > ej> E> > > f> F '
E1 2> 1 2> f> f 9' *Ei> 2> j> E> > > Ft1>2>3 ' E 1 3> 2 3> f= Rxu qh{w d{lrp uxohv rxw wklv
sureohpdwlf ihdwxuh1 Iru dq| Q 5 Q dqg F 5 FEQ> zh ghqh FV 5 FEV e|
FVE}V ' FE}V> fQ.V iru doo } 5 QQ=
Lqghshqghqfh ri ]hur Ghpdqgv1 Iru doo EQ> { F 5 S dqg l 5 Q/ i{l ' fj ,
i*lEQ> {> F ' f dqg *Q.lEQ> {> F ' *EQql> {Q.l> FQ.lj=
Wklv yhu| qdwxudo yduldeoh0srsxodwlrq frqglwlrq lv zhoo nqrzq= vhh iru lqvwdqfh Prxolq dqg Vkhqnhu +4<<7, ru Vsuxprqw +4<<;,1 Lw lv fohduo| phw e| wkh Dxpdqq0 Vkdsoh| phwkrg dqg gholyhuv wkh iroorzlqj fruroodu| wr rxu wkhruhp1
Fruroodu|1 Wkh Dxpdqq0Vkdsoh| phwkrg lv wkh rqo| frvw0vkdulqj phwkrg vdwlvi|lqj Dgglwlylw|/ Gxpp|/ Lqghshqghqfh ri ]hur Ghpdqgv/ dqg Qr Ghpdqg Uhvkx"lqj1 Surri1 Lw vx!fhv wr vkrz wkdw Qr Ghpdqg Uhvkx"lqj dqg Lqghshqghqfh ri ]hur Ghpdqgv wrjhwkhu lpso| Qr Phujlqj ru Vsolwwlqj1 Ohw EQ> {> F> l> L> Q> F> dqg {
vdwlvi| wkh suhplvhv ri Qr Phujlqj ru Vsolwwlqj1 Qrwh wkdw F 5 F1
LEQ1 Ghqh
{ 5 QQ0
e| {
m ' {m iru doo m 5 Q dqg {l0 ' f iru doo l 5 L= Dsso| vxffhvvlyho|
Qr Ghpdqg Uhvkx"lqj dqg Lqghshqghqfh ri ]hur Ghpdqgv= Pl0ML*l0EQ> {> F '
P
l0MLEQ> {> F ' *lEQ> {> F=¥
6, Dv phqwlrqhg lq wkh Lqwurgxfwlrq/ Fdoyr dqg Vdqwrv +5333, rhu dq dowhuqdwlyh fkdudfwhul}dwlrq ri wkh glvfuhwh Dxpdqq0Vkdsoh| phwkrg1 Wkh nh| surshuw| wkh| xvh vwdwhv wkdw iru dq| sdlu ri srvlwlyh ghpdqghuv l> m> wkh lqfuhdvh lq l*v shu xqlw frvw vkduh jhqhudwhg e| d rqh0xqlw lqfuhdvh lq m*v ghpdqg lv htxdo wr wkh lqfuhdvh lq m*v shu xqlw frvw vkduh jhqhudwhg e| d rqh0xqlw lqfuhdvh lq l*v ghpdqg1 Zulwlqj Q+E{ ' il 5 Q m {l A fj/ wkh irupdo frqglwlrq uhdgv dv iroorzv1
Edodqfhg Frqwulexwlrqv1 Iru doo EQ> {> F 5 S dqg doo l> m 5 Q+E{>
{lE*lEQ> {> F *lEQ> { h
m> F '
{mE*mEQ> {> F *mEQ> { h
l> F= +4,
Fdoyr dqg Vdqwrv suryh wkdw wkh Dxpdqq0Vkdsoh| phwkrg lv wkh rqo| frvw0vkdulqj phwkrg vdwlvi|lqj Edodqfhg Frqwulexwlrqv dqg ]hur Fkdujh iru ]hur Ghpdqg1
Wr vhh zk| wklv lv wuxh/ { EQ> {> F 5 S dqg l 5 Q+E{= Vlqfh Q dqg F duh {hg
wkurxjkrxw wkh dujxphqw/ zh rplw wkhp lq rxu qrwdwlrq>zh dovr zulwh Q+ lqvwhdg ri
Q+E{1 Pxowlso|lqj erwk vlghv ri +4, e| {m dqg vxpplqj xs/
P mMQ+.l {m {lE*lE{ *lE{ h m ' P mMQ+.l E*mE{ *mE{ hl=
Aumann-Shapley pricing: A reconsideration of the discrete case < Dgglqj *lE{ rq erwk vlghv/ xvlqj exgjhw edodqfh dqg ]hur Fkdujh iru ]hur Ghpdqg/
dqg zulwlqj {EQ+ ' P mMQ+{m> zh jhw {EQ+ {l *lE{ P mMQ+.l {m {l*lE{ h m ' FE{ FE{ hl n * lE{ hl
ru/ diwhu uhduudqjlqj/ *lE{ ' {EQ{l +EFE{ FE{ h l n P mMQ+ {m {EQ+*lE{ h m= +5,
E| ]hur Fkdujh iru ]hur Ghpdqg/ *Ef ' f1 Vlqfh wkh uhfxuvlyh irupxod +5, krogv iru hyhu| l 5 Q+> lw ixoo| ghwhuplqhv d xqltxh frvw0vkdulqj phwkrg *1 Dv *DVk vdwlvhv
Edodqfhg Frqwulexwlrqv/ * ' *DVk=
Wkxv/ Edodqfhg Frqwulexwlrqv 0hvvhqwldoo| dorqh0 dprxqwv wr d uhfxuvlyh ghqlwlrq ri wkh Dxpdqq0Vkdsoh| frvw0vkdulqj phwkrg1 Lw lv d frpsoh{ uhtxluhphqw frpsrvhg ri vhyhudo lghdv wkdw duh qrw hdvlo| glvhqwdqjohg1 Rxu d{lrpdwl}dwlrq rhuv wkh dgydq0 wdjh ri d jhqxlqh ghfrpsrvlwlrq ri wkh Dxpdqq0Vkdsoh| phwkrg lqwr d vhw ri pruh hohphqwdu|/ mrlqwo| fkdudfwhulvwlf/ d{lrpv1
5. Conclusion
Xqghu Dgglwlylw|/ vwuhqjwkhqlqj wkh Sursruwlrqdolw| d{lrp dqg uhlqwurgxflqj wkh id0 ploldu Gxpp| frqglwlrq |lhogv d fkdudfwhul}dwlrq ri wkh glvfuhwh Dxpdqq0Vkdsoh| frvw0vkdulqj phwkrg1 Wkh h{huflvh lq wklv sdshu zdv prwlydwhg e| wkh revhuydwlrq wkdw Vfdoh Lqyduldqfh/ d fuxfldo lqjuhglhqw lq wkh fodvvlfdo d{lrpdwl}dwlrqv/ lv phdq0 lqjohvv lq wkh glvfuhwh frqwh{w1 Exw rxu frpelqdwlrq ri d{lrpv pd| eh ri lqghshqghqw lqwhuhvw= gr wkh| fkdudfwhul}h wkh Dxpdqq0Vkdsoh| phwkrg lq wkh vwdqgdug prgho zkhuh ghpdqgv duh uhdo qxpehuvB
6. Appendix
Surri ri wkh wkhruhp1 Lw lv fohdu wkdw *DVk vdwlvhv Dgglwlylw|/ Gxpp|/ dqg Qr
Phujlqj ru Vsolwwlqj1 Frqyhuvho|/ ohw * eh d frvw0vkdulqj phwkrg vdwlvi|lqj wkhvh d{lrpv1
Vwhs 4= * dgplwv d rz uhsuhvhqwdwlrq
Jlyhq Q 5 Q dqg { 5 QQ/ zh zulwh yhfwru lqhtxdolwlhv > ?> dqg ghqh
df> {d ' i} 5 QQ m f } ? {j dqg of> {d ' i} 5 QQ m f ? } ? {j= D rz wr {
lv d pdsslqj iEQ> {> = G df> {d$ UQ
+ vdwlvi|lqj wkh frqyhqwlrq wkdw ilEQ> {> } ' f
zkhqhyhu }l ' {l> dqg wkh vr0fdoohg rz frqvhuydwlrq frqvwudlqwv
P lMQilEQ> {> f ' > P lMQilEQ> {> } ' P lMQ+(})
zkhuh Q+E} ' il 5 Q m }l A fj=
Exloglqj rq Zdqj +4<<<,/ Prxolq dqg Yrkud +5336, suryh wkdw d frvw0vkdulqj phwkrg * vdwlvhv Dgglwlylw| dqg Gxpp| li dqg rqo| li/ iru doo Q 5 Q dqg { 5 QQ>
wkhuh lv d rz iEQ> {> = wr { vxfk wkdw *lEQ> {> F '
P
}M[0>{[ilEQ> {> }ClFE} +6,
iru doo l 5 Q dqg F 5 FEQ= Iru hdfk Q dqg {> wkh rz iEQ> {> = dvvrfldwhg zlwk *EQ> {> = lv xqltxh>zh fdoo i wkh rz uhsuhvhqwdwlrq ri *= Dq lpphgldwh fruroodu| wr Prxolq dqg Yrkud*v uhvxow lv wkdw Dgglwlylw| dqg Gxpp| lpso| ]hur Fkdujh iru ]hur Ghpdqg1 Iurp qrz rq/ zh ghqrwh e| i wkh rz uhsuhvhqwdwlrq ri rxu phwkrg *=
Vwhs 5= Wkh rz uhsuhvhqwdwlrq ri wkh Dxpdqq0Vkdsoh| phwkrg1
Vlqfh wkh Dxpdqq0Vkdsoh| phwkrg *DVk vdwlvhv Dgglwlylw| dqg Gxpp|/ lw wrr
dgplwv d rz uhsuhvhqwdwlrq= zh ghqrwh lw e| iDVk1 E| ghqlwlrq/ wkh Dxpdqq0
Vkdsoh| phwkrg dssolhv wkh Vkdsoh| ydoxh wr wkh jdph zkhuh hdfk xqlw ghpdqghg e| hyhu| djhqw lq Q frqvwlwxwhv d vhsdudwh sod|hu dqg frpsxwhv dq djhqw*v frvw vkduh e| vxpplqj xs wkh frruglqdwhv ri wkh Vkdsoh| ydoxh zklfk fruuhvsrqg wr wkh xqlwv kh ghpdqgv1 Wklv dprxqwv wr wdnlqj wkh xqlirup dyhudjh ri wkh vr0fdoohg lqfuhphqwdo frvw0vkduh yhfwruv jhqhudwhg e| doo srvvleoh rughulqjv ri doo xqlwv ghpdqghg e| doo djhqwv1 Vwudljkwiruzdug frpsxwdwlrqv vkrz wkdw wkh fruuhvsrqglqj rz uhsuhvhqwd0 wlrq lv
iDVk
l EQ> {> } ' E}E{ } h l
E{ iru doo EQ> { 5 G/ l 5 Q> dqg } 5 df> { hlo> zkhuh E} ' EPmMQ}m-@
Q
mMQ}m-1 Vhh/ iru lqvwdqfh/ Prxolq +4<<8, ru Prxolq dqg
Vsuxprqw +5336,1 Dv revhuyhg lq Vsuxprqw dqg Zdqj +4<<;,/ wkh ghqlqj ihdwxuh ri wklv irupxod lv wkdw wkh rz dw } lv vsolw lq sursruwlrq wr wkh vl}hv ri wkh uhpdlqlqj ghpdqgv= ilEQ> {> }@ P mMQ imEQ> {> } ' E{l }l@ P mMQE{m }m iru doo l 5 Q= Vwhs 6= * lv zhdno| v|pphwulf1 Li G Q $ Q lv d elmhfwlrq/ Q 5 Q / | 5 UQ
+/ dqg F 5 FEQ/ zulwh Q ' iEl m
l 5 Qj= Iru dq| } 5 UQ
+> ghqh } 5 UQ+ e| E}(l) ' }l iru doo l 5 Q= Ilqdoo|/
ghqh F G QQ $ U
+ e| FE} ' FE} iru doo } 5 QQ G qrwh wkdw F 5 FEQ1
Zh fodlp wkdw * vdwlvhv wkh iroorzlqj surshuw|1
Zhdn V|pphwu|1 Ohw EQ> {> F 5 S/ l 5 Q> dqg ohw G Q $ Q eh d elmhfwlrq1 Li wkhuh h{lvwv vrph l 5 Q vxfk wkdw Em ' m iru doo m 5 Qql> wkhq *(l)EQ> {> F '
Aumann-Shapley pricing: A reconsideration of the discrete case 44 Wklv frqglwlrq vd|v wkdw uhqdplqj dq djhqw vkrxog qrw dhfw khu frvw vkduh1 D sulrul/ lw grhv qrw uxoh rxw wkh srvvlelolw| wkdw wkh frvw vkduhv ri wkh rwkhuv pljkw eh dhfwhg e| wkdw uhqdplqj=
Zhdn V|pphwu| lv d frqvhtxhqfh ri Qr Phujlqj ru Vsolwwlqj1 Wr vhh zk|/ ohw EQ> {> F> l> dqg vdwlvi| wkh suhplvhv ri Zhdn V|pphwu|1 Wkh fdvh El ' l ehlqj wulyldo/ dvvxph El 9' l= Ohw Q ' Q ^El> ghqh F 5 FEQ e| FE} ' FE}
Q.l> }ln
}(l)> dqg ghqh { 5 QQ0 e| {(l) ' f> {m ' {m iru doo m 5 Q1
Ohw L ' il> Elj= Qrwh wkdw Q ' EQql ^ L dqg {
l n {(l) ' {l= Dsso|lqj Qr
Phujlqj ru Vsolwwlqj wr EQ> {> F dqg djhqw l jlyhv
*lEQ> {> F ' *lEQ> {> F n *(l)EQ> {> F=
Qh{w/ revhuyh wkdw Q ' EQqEl^L/ FE} ' FE}
Q.l> }ln}(l)> dqg {ln{(l) '
E{(l)= Dsso|lqj Qr Phujlqj ru Vsolwwlqj wr EQ> {> F dqg djhqw El jlyhv
*(l)EQ> {> F ' *lEQ> {> F n *(l)EQ> {> F=
Khqfh/ *lEQ> {> F ' *(l)EQ> {> F> dv fodlphg1
Vwhs 7= * h{whqgv wkh Vkdsoh| ydoxh1
Iru dq| Q 5 Q dqg dq| qrqhpsw| N Q/ ghqh wkh vr0fdoohg xqdqlplw| jdph EQ> KN e| KNEW ' li N W> dqg f rwkhuzlvh1 Zh fodlp wkdw
*lEQ> KN ' @n li l 5 N> dqg f li l 5 QqN> +7,
zkhuh n ' mNm1
Wkh surri glhuv iurp wkh xvxdo dujxphqw edvhg rq wkh ixoo v|pphwu| d{lrp= lw uholhv fuxfldoo| rq wkh dvvxpswlrq wkdw wkh vhw ri djhqwv/ Q/ lv doorzhg wr fkdqjh51 Il{
Q 5 Q dqg d qrqhpsw| N Q= Li n ' > wkh fodlp iroorzv gluhfwo| iurp Gxpp|1 Iurp qrz rq/ dvvxph n 2= Ohw P 5 Q eh vxfk wkdw N P> EQqN _ P ' B> dqg mPm ' 2n =6 Ghqh PEn ' iV P m mVm ' nj dqg zulwh En ' mPEnm =
E| Gxpp| dqg exgjhw edodqfh/ P
lMV*lEEQqN ^ V> KV ' iru doo V 5 PEn= +8,
5Li Q zhuh {hg/ Zhdn V|pphwu| zrxog eh ydfxrxv ehfdxvh qr elmhfwlrq iurp Q wr Q uhqdplqj h{dfwo| rqh phpehu ri Q ohdyhv Q xqfkdqjhg1 Lq wklv fdvh/ Dgglwlylw|/ Gxpp|/ dqg Zhdn V|pphwu| zrxog qrw |lhog wkh Vkdsoh| ydoxh/ dv wkh phwkrg ghqhg iru wkh {hg srsxodwlrq Q = i1> 2> 3j lq wkh h{dpsoh lq frpphqw 5/ Vhfwlrq 7/ looxvwudwhv1
6Revhuyh wkdw wklv frqvwuxfwlrq lv srvvleoh iru hdfk Q ehfdxvh wkh vhw ri srwhqwldo djhqwv/ Q > lv lqqlwh1 Lq idfw/ wkh fodlp lq Vwhs 7 zrxog eh idovh li wkh vhw ri srwhqwldo djhqwv zhuh qlwh1
Ghqh PEn ' iV P m mV m ' n j= Vlqfh V 5 PEn li dqg rqo| li PqV 5 PEn> zh kdyh mPEn m ' mPEnm ' En= Iru dq| V 5 PEn > Zhdn
V|pphwu| hqvxuhv wkdw wkhuh h{lvwv d qxpehu |EV vxfk wkdw
*lEEQqN ^ V^ l> KV0l ' |EV iru doo l 5 PqV=
Zh pd| wkhuhiruh uhzulwh +8, dv iroorzv1 P
lMV|EVql ' iru doo V 5 PEn= +9,
Wklv lv d v|vwhp ri En olqhdu htxdwlrqv lq wkh En yduldeohv |EV> V 5 PEn 1
Fohduo|/ |EV ' @n iru doo V lv d vroxwlrq1 Li lw lv wkh rqo| vroxwlrq/ wkhq iru doo l 5 N>
fkrrvlqj V ' Nql |lhogv *
lEQ> KN ' *lEEQqN ^ ENql ^ l> K(N.l)l ' |ENql '
@n/ dqg zh duh grqh1
Wr suryh wkdw +9, kdv d xqltxh vroxwlrq/ lw vx!fhv wr vkrz wkdw wkh En htxdwlrqv duh olqhduo| lqghshqghqw1 Uhzulwh wkh v|vwhp dv
xV | ' iru doo V 5 PEn>
zkhuh | ' E|EV 5 U}(n31)dqg xV 5 U}(n31)lv ghqhg e| xVEV ' li V V> dqg
f iru doo rwkhu V 5 PEn1 Zh fodlp wkdw ixV m V 5 PEnj lv d edvlv ri U}(n31)= Lw
lv hqrxjk wr vkrz wkdw dq| yhfwru lq wkh fdqrqlfdo edvlv ihV0
m V 5 PEn j +zkhuh
hV0
EW ' li W ' V dqg f rwkhuzlvh, lv d olqhdu frpelqdwlrq ri wkh yhfwruv xV= Il{
V 5 PEn= Iru hdfk u ' f> > ===> n> ghqh P
uEn ' iV 5 PEn m mV _ Vm ' uj=
Rqh fkhfnv wkdw hV0 'n31P u=0 En3u+1En u -n===Eu n P VM}u(n) xV>
surylqj rxu fodlp1 Zh kdyh vkrzq +7,1
Ehfdxvh wkh xqdqlplw| jdphv EQ> KN irup d edvlv ri SEf> Dgglwlylw| lpsolhv
wkdw *EQ> K ' *VkEQ> K iru doo EQ> K 5 SEf=
Vwhs 8= * lv wkh Dxpdqq0Vkdsoh| phwkrg1
Zh ehjlq zlwk wkh revhuydwlrq wkdw vlqfh * vdwlvhv Zhdn V|pphwu|/ lwv rz uhsuhvhqwdwlrq i lv zhdno| v|pphwulf= iru dq| EQ> { 5 G/ l 5 Q/ dqg dq| elmhfwlrq G Q $ Q vxfk wkdw Em ' m iru doo m 5 Qql>
i(l)EQ> {> } ' ilEQ> {> } iru doo } 5 df> {d= +:,
Iru d surri/ { Q> {> l> dqg dv uhtxluhg/ dqg } 5 df> {d1 Frqvlghu d frvw ixqfwlrq F vxfk wkdw ClFE} htxdov li } ' } dqg f rwkhuzlvh1 E| +6, dqg Zhdn V|pphwu|/
Aumann-Shapley pricing: A reconsideration of the discrete case 46 Wkh uhvw ri wkh surri vkrzv wkdw i ' iDVk1 Iru dq| n ' f> > 2> ===> ohw GEn '
iEQ> { 5 G m mil 5 Q m {l A jm nj1 Wklv lv wkh vhw ri ghpdqg surohv zkhuh qr
pruh wkdq n djhqwv ghpdqg vhyhudo xqlwv1 E| Vwhs 7/ iEQ> { ' iDVkEQ> { iru doo
EQ> { 5 GEf= Zh qrz { n f> pdnh wkh lqgxfwlrq k|srwkhvlv
iEP> { ' iDVkEP> { iru doo EP> { 5 GEn> +;,
dqg suryh wkdw
iEQ> { ' iDVkEQ> { iru doo EQ> { 5 GEn n = +<,
+d, Iluvw zh vkrz wkdw
ilEQ> { ' ilDVkEQ> { iru doo EQ> { 5 GEn n dqg l 5 Q vxfk wkdw {l A = +43,
Il{ EQ> { 5 GEn n = E| wkh lqgxfwlrq k|srwkhvlv +;,/ zh pd| dvvxph wkdw EQ> { 5 GEn n qGEn G h{dfwo| n n djhqwv ghpdqg vhyhudo xqlwv1 Frqvlghu dq duelwudu| djhqw l 5 Q vxfk wkdw {l A dqg dq duelwudu| F 5 FEQ= Ohw L 5 Q eh d
vhw ri fduglqdolw| mLm ' {l/ glvmrlqw iurp Qql> dqg frqwdlqlqj l= Ohw Q ' EQql ^ L=
Ghqh FE} ' FE} Q.l>Pl0ML}l0 iru doo } 5 QQ 0 dqg ghqh { 5 QQ0 e| { l0 ' iru doo l 5 L dqg {
m ' {m iru doo m 5 Qql= Dsso| vxffhvvlyho| Qr Phujlqj ru Vsolwwlqj/
wkh lqgxfwlrq k|srwkhvlv +;,/ dqg wkh ghqlwlrq ri wkh Dxpdqq0Vkdsoh| phwkrg wr rewdlq *lEQ> {> F ' P l0ML*l 0EQ> {> F ' P l0ML* DVk l0 EQ> {> F ' *DVk l EQ> {> F=
Vlqfh wklv krogv iru dq| frvw ixqfwlrq F 5 FEQ> lw iroorzv wkdw ilEQ> { ' ilDVkEQ> {>
dv ghvluhg1
+e, Qh{w zh vkrz wkdw
iEQ> {> f ' iDVkEQ> {> f iru doo EQ> { 5 GEn n = +44,
Djdlq/ ohw EQ> { 5 GEnn dqg dvvxph zlwkrxw orvv wkdw EQ> { 5 GEnnqGEn= Vxssrvh ixuwkhupruh wkdw {l A f iru doo l 5 Q= Wklv dvvxpswlrq lv iru qrwdwlrqdo
frqyhqlhqfh= li lw lv qrw phw/ wkh dujxphqw ehorz fduulhv ryhu surylghg wkdw Q lv uhsodfhg zlwk QE{> wkh vhw ri srvlwlyh ghpdqghuv1
Ohw N ' il 5 Q m {l ' j dqg n ' mNm = Li n ' > rxu fodlp iroorzv gluhfwo|
dqg PEn dv lq Vwhs 71 Iru doo V 5 PEn ghqh { 5 U e| {l ' {l
iru l 5 QqN dqg {(V)
l ' iru l 5 V= Wkxv/ lq sduwlfxodu/ {(N) ' {= E| +43,/ rz
frqvhuydwlrq/ dqg wkh ghqlwlrq ri wkh Dxpdqq0Vkdsoh| rz uhsuhvhqwdwlrq iDVk lq
Vwhs 5/ P lMVilEEQqN ^ V> { (V)> f ' P lMQ.Ni DVk
l EEQqN ^ V> {(V)> f ' {EN{EQ
iru doo V 5 PEn> zkhuh zh xvh wkh qrwdwlrq {EW ' PlMW {l= Wkdqnv wr wkh zhdn
v|pphwu| ri i +vhh +:,,/ zh fdq uhzulwh wklv v|vwhp P
lMV|EVql '
{EN {EQ iru doo V 5 PEn/ zkhuh |EV ' i
lEEQqN ^ EV^ l> {(V0l)> f iru doo V 5 PEn
dqg doo l 5 PqV= Wkh vdph dujxphqw dv lq Vwhs 7 vkrzv wkdw wkh htxdwlrqv lq wklv
v|vwhp duh olqhduo| lqghshqghqw/ vr wkdw wkh vroxwlrq |EV ' 1
n{(N){(Q) ' {(Q)1 iru doo V
lv xqltxh1 Wkhuhiruh
ilEQ> {> f ' |EN ' {EQ ' ilDVkEQ> {> f
iru doo l 5 N/ zklfk wrjhwkhu zlwk +43, suryhv +44,1
+f, Zh suryh +<,1 Ohw EQ> { 5 GEn n qGEn dqg dvvxph zlwkrxw orvv wkdw {l A f iru doo l 5 Q= Iru o ' f> > ===> {EQ > ohw ]Eo ' i} 5 df> {dm PlMQ}l ' oj=
E| +44,/ iEQ> {> } ' iDVkEQ> {> } iru doo } 5 ]Ef= Surfhhglqj e| lqgxfwlrq rq o> {
o f dqg dvvxph wkdw
iEQ> {> } ' iDVkEQ> {> } iru doo } 5 ]Eo= +45,
Zh qrz { } 5 ]Eo n dqg vkrz wkdw iEQ> {> } ' iDVkEQ> {> }= Vhw N '
il 5 Q m {l ' dqg }l ' fj/ n ' mNm = E| wkh rz frqvhuydwlrq frqvwudlqwv/ +45, dqg +43,/ P lMNilEQ> {> } ' P lMQ+(}) ilEQ> {> } hl P lMQ.NilEQ> {> } ' P lMQ+(}) iDVk l EQ> {> } hl P lMQ.Ni DVk l EQ> {> }=
Ehfdxvh i lv zhdno| v|pphwulf/ plplfnlqj wkh dujxphqw lq +e, jlyhv ilEQ> {> } ' n " P lMQ+(}) iDVk l EQ> {> } hl P lMQ.Ni DVk l EQ> {> } # iru doo l 5 N= E| ghqlwlrq ri iDVk/ wklv phdqv wkdw i
lEQ> {> } ' ilDVkEQ> {> } iru
Aumann-Shapley pricing: A reconsideration of the discrete case 48
References
^4` Dxpdqq/ U1M1/ dqg Vkdsoh|/ O1 +4<:7,1 Ydoxhv ri Qrqdwrplf Jdphv1 Sulqfhwrq= Sulqfhwrq Xqly1 Suhvv1
^5` Eloohud/ O1/ dqg Khdwk/ G1 +4<;5,1 Doorfdwlrq ri Vkduhg Frvwv= D Vhw ri D{lrpv \lhoglqj d Xqltxh Surfhgxuh/ Pdwk1 Rshu1 Uhv1 :/ 6506<1
^6` Eloohud/ O1G1/ Khdwk/ G1/ dqg Uddqdq/ M1 +4<:;,1 Lqwhuqdo Whohskrqh Eloolqj Udwhv= D Qryho Dssolfdwlrq ri Qrqdwrplf Jdph Wkhru|/ Rshu1 Uhv1 59/ <890 <981
^7` Fdoyr/ H1/ dqg Vdqwrv/ M1F1 +5333,1 D Ydoxh iru Pxowlfkrlfh Jdphv/ Pdwk Vrf1 Vfl1 73/ 67406871
^8` Fdvwdqr0Sdugr/ D1/ dqg Jdufld0Gld}/ D1 +4<<8,1 Kljkzd| Frvw Doorfdwlrq= Dq Dssolfdwlrq ri wkh Wkhru| ri Qrq0Dwrplf Jdphv/ Wudqvsruw1 Uhv1 5</ 4;:05361 ^9` Fkxq/ \1 +4<;;,1 Wkh Sursruwlrqdo Vroxwlrq iru Uljkwv Sureohpv/ Pdwk1 Vrf1
Vfl1 48/ 56405791
^:` Kdlpdqnr/ R1/ dqg Wdxpdq/ \1 +5335,1 Uhfhqw Ghyhorsphqwv lq Dxpdqq0 Vkdsoh| Sulflqj/ Sduw 4= Wkh Qrqglhuhqwldeoh Fdvh/ iruwkfrplqj lq Jdph Wkhru| dqg Uhvrxufh Doorfdwlrq= Wkh D{lrpdwlf Dssurdfk +Wkrpvrq/ Z1/ hg1, QDWR DVL Vhulhv1
^;` Kdyly/ P1 +5334,1 Wkh Dxpdqq0Vkdsoh| Sulfh Phfkdqlvp iru Doorfdwlqj Frq0 jhvwlrq Frvwv/ Rshu1 Uhv1 Ohwwhuv 5</ 54405481
^<` Mx/ E1J1/ Pl|djdzd/ H1/ dqg Vdndl/ W1 +5336,1 Qrq0Pdqlsxodeoh Glylvlrq Uxohv lq Fodlp Sureohpv dqg Jhqhudol}dwlrqv/ plphr/ Froxpeld Xqlyhuvlw|1
^43` Ohh/ G1 +5335,1 Jdph0Wkhruhwlf Surfhgxuhv iru Ghwhuplqlqj Sdyhphqw Wklfn0 qhvv dqg Wud!f Odqh Frvwv lq Kljkzd| Frvw Doorfdwlrq/ Sk1G1 wkhvlv/ Wh{dv D)P Xqlyhuvlw|1
^44` Phuwhqv/ M1I1 +4<;;,1 Wkh Vkdsoh| Ydoxh lq wkh Qrq0Glhuhqwldeoh Fdvh/ Lqw1 M1 Jdph Wkhru| 4:/ 40981
^45` Plupdq/ O1/ dqg Wdxpdq/ \1 +4<;5,1 Ghpdqg Frpsdwleoh Htxlwdeoh Frvw Vkdu0 lqj Sulfhv/ Pdwk1 Rshu1 Uhv1 :/ 730891
^46` Prxolq/ K1 +4<;:,1 Htxdo ru Sursruwlrqdo Glylvlrq ri d Vxusoxv dqg Rwkhu Phwk0 rgv/ Lqw1 M1 Jdph Wkhru| 49/ 49404;91
^47` Prxolq/ K1 +4<<8,1 Rq Dgglwlyh Phwkrgv wr Vkduh Mrlqw Frvwv/ Mdsdq Hfrq1 Uhy1 79/ 63606651
^48` Prxolq/ K1 +5335,1 D{lrpdwlf Frvw dqg Vxusoxv Vkdulqj/ lq Kdqgerrn ri Vrfldo Fkrlfh dqg Zhoiduh +Duurz/ N1M1/ Vhq/ D1/ dqg Vx}xpxud/ N1/ hgv1,/ Dpvwhugdp= Hovhylhu/ 5;<068:1
^49` Prxolq/ K1/ dqg Vkhqnhu/ V1 +4<<7,1 Dyhudjh Frvw Sulflqj yhuvxv Vhuldo Frvw Vkdulqj= Dq D{lrpdwlf Frpsdulvrq/ M1 Hfrq1 Wkhru| 97/ 4:;05341
^4:` Prxolq/ K1/ dqg Vsuxprqw/ \1 +5336,1 Rq Ghpdqg Uhvsrqvlyhqhvv lq Dgglwlyh Frvw Vkdulqj/ plphr/ Ulfh Xqlyhuvlw|/ iruwkfrplqj lq M1 Hfrq1 Wkhru|1
^4;` Prxolq/ K1/ dqg Yrkud/ U1 +5336,1 D Uhsuhvhqwdwlrq ri Dgglwlyh Frvw Vkdulqj Phwkrgv/ Hfrq1 Ohwwhuv ;3/ 6<<073:1
^4<` P|huvrq/ U1 +4<;3,1 Frqihuhqfh Vwuxfwxuhv dqg Idlu Doorfdwlrq Uxohv/ Lqw1 M1 Jdph Wkhru| </ 49<04;51
^53` R*Qhloo/ E1 +4<;5,1 D Sureohp ri Uljkwv Duelwudwlrq iurp wkh Wdopxg/ Pdwk1 Vrf1 Vfl1 5/ 67806:41
^54` Vdphw/ G1/ dqg Wdxpdq/ \1 +4<;5,1 Wkh Ghwhuplqdwlrq ri Pdujlqdo Frvw Sulfhv Xqghu d Vhw ri D{lrpv/ Hfrqrphwulfd 83/ ;<80<3<1
^55` Vdphw/ G1/ Wdxpdq/ \1/ dqg ]dqj/ L1 +4<;7,1 Dq Dssolfdwlrq ri wkh Dxpdqq0 Vkdsoh| Sulfhv wr Wudqvsruwdwlrq Sureohpv/ Pdwk1 Rshu1 Uhv1 43/ 580751 ^56` Vkdsoh|/ O1 +4<86,1 D Ydoxh iru q0Shuvrq Jdphv/ lq Frqwulexwlrqv wr wkh Wkh0
ru| ri Jdphv LL +Nxkq/ K1Z1/ dqg Wxfnhu/ Z1/ hgv1,/ Dqq1 Pdwk1 Vwxg1 Yro1 5;/ QM= Sulqfhwrq Xqly1 Suhvv1
^57` Vkxeln/ P1 +4<95,1 Lqfhqwlyhv/ Ghfhqwudol}hg Frqwuro/ wkh Dvvljqphqw ri Mrlqw Frvwv/ dqg Lqwhuqdo Sulflqj/ Pdqdjh1 Vfl1 ;/ 65806761
^58` Vsuxprqw/ \1 +4<<;,1 Ruglqdo Frvw Vkdulqj/ M1 Hfrq1 Wkhru| ;4/ 45904951 ^59` Vsuxprqw/ \1 +5333,1 Frkhuhqw Frvw0Vkdulqj Uxohv/ Jdphv Hfrq1 Ehkdy1 66/
45904771
^5:` Vsuxprqw/ \1/ dqg Zdqj/ \1W1 +4<<;,1 D Fkdudfwhul}dwlrq ri wkh Dxpdqq0 Vkdsoh| Phwkrg lq wkh Glvfuhwh Frvw Vkdulqj Prgho/ plphr/ Xqlyhuvlw| ri Prqwuhdo1
Aumann-Shapley pricing: A reconsideration of the discrete case 4: ^5;` Wdxpdq/ \1 +4<;;,1 Wkh Dxpdqq0Vkdsoh| Sulfhv= D Vxuyh|/ lq Wkh Vkdsoh| Ydoxh= Hvvd|v lq Krqru ri Oor|g Vkdsoh| +Urwk/ D1/ hg1, Fdpeulgjh= Fdpeulgjh Xqly1 Suhvv/ 5:<06371
^5<` ydq ghq Qrxzhodqg/ D1/ Srwwhuv/ M1/ Wlmv/ V1/ dqg ]du}xhor/ M1 +4<<8,1 Fruhv dqg Uhodwhg Vroxwlrq Frqfhswv iru Pxowl0Fkrlfh Jdphv/ ]RU0Pdwk1 Phwk1 Rshu1 Uhv1 74/ 5;<06441
^63` Zdqj/ \1 +4<<<,1 Wkh Dgglwlylw| dqg Gxpp| D{lrpv lq wkh Glvfuhwh Frvw Vkdulqj Prgho/ Hfrq1 Ohwwhuv 97/ 4;:04<51
^64` \rxqj/ K1S1 +4<;8,1 Surgxfhu Lqfhqwlyhv lq Frvw Doorfdwlrq/ Hfrqrphwulfd 86/ :8:0:961