• Aucun résultat trouvé

Aumann-Shapley Pricing : A Reconsideration of the Discrete Case

N/A
N/A
Protected

Academic year: 2021

Partager "Aumann-Shapley Pricing : A Reconsideration of the Discrete Case"

Copied!
19
0
0

Texte intégral

(1)

2004-08

SPRUMONT, Yves

Aumann-Shapley Pricing : A Reconsideration of

the Discrete Case

(2)

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.

(3)

Dxpdqq0Vkdsoh| sulflqj= D uhfrqvlghudwlrq ri wkh glvfuhwh fdvh Yves SprumontW

Mxqh 4</ 5337

Abstract. Zh uhfrqvlghu wkh iroorzlqj frvw0vkdulqj sureohp= djhqw l = 1> ===> q ghpdqgv d txdqwlw| {l ri jrrg l>wkh fruuhvsrqglqj wrwdo frvw

F({1> ===> {q) pxvw eh vkduhg dprqj wkh q djhqwv1 Wkh Dxpdqq0Vkdsoh| sulfhv

(s1> ===> sq) duh jlyhq e| wkh Vkdsoh| ydoxh ri wkh jdph zkhuh hdfk xqlw ri hdfk

jrrg lv uhjdughg dv d glvwlqfw sod|hu1 Wkh Dxpdqq0Vkdsoh| frvw0vkdulqj phwkrg dvvljqv wkh frvw vkduh sl{l wr djhqw l1

Zkhq jrrgv frph lq lqglylvleoh xqlwv/ zh vkrz wkdw wklv phwkrg lv fkdudf0 whul}hg e| wkh wzr vwdqgdug d{lrpv ri Dgglwlylw| dqg Gxpp|/ dqg wkh surshuw| ri Qr Phujlqj ru Vsolwwlqj= djhqwv qhyhu qg lw surwdeoh wr vsolw ru phujh wkhlu ghpdqgv1

MHO fodvvlfdwlrq qxpehuv= F:4/ G961

Nh|zrugv= frvw vkdulqj/ Dxpdqq0Vkdsoh| sulflqj/ phujlqj/ vsolwwlqj1 1. Introduction

Dxpdqq0Vkdsoh| sulflqj lv d phwkrg iru vroylqj frvw0vkdulqj sureohpv lq zklfk djhqwv ghpdqg duelwudu| txdqwlwlhv ri srvvleo| glhuhqw jrrgv= jlyhq wkdw hdfk djhqw l lq d qlwh vhw Q ' i> ===> qj uhtxhvwv d txdqwlw| {l ri jrrg l> zh zlvk wr vsolw wkh

frvw FE{ ri surgxflqj wkh ghpdqg suroh { xvlqj qr rwkhu lqirupdwlrq wkdq wkh frvw ixqfwlrq F=

Wkh vlpsohvw fdvh rffxuv zkhq jrrgv duh lqglylvleoh dqg hdfk djhqw ghpdqgv qr pruh wkdq rqh xqlw/ wkdw lv/ { ehorqjv wr if> jQ1 Doo wkh uhohydqw lqirupdwlrq lv wkhq fdswxuhg lq wkh fruuhvsrqglqj vwdqg0dorqh frvw frrshudwlyh jdph EQ> K zkhuh wkh frvw dvvrfldwhg zlwk d frdolwlrq V lv mxvw wkh frvw ri phhwlqj wkh ghpdqgv ri lwv phpehuv= lq vwudljkwiruzdug qrwdwlrq/ KEV ' FE{V> fQ.V= Wkh doorfdwlrq

uhfrpphqghg e| wkh Dxpdqq0Vkdsoh| phwkrg lq wklv fdvh lv mxvw wkh Vkdsoh| ydoxh ri wkdw jdph1

Wkh vwdqg0dorqh frvw jdph EQ> K uhpdlqv shuihfwo| zhoo ghqhg zkhq vrph djhqwv ghpdqg vhyhudo xqlwv1 Lq idfw/ Vkxeln +4<95, uhfrpphqghg wkh Vkdsoh| ydoxh ri wkdw

Gìsduwhphqw gh Vflhqfhv Hfrqrpltxhv dqg FLUHT/ Xqlyhuvlwì gh Prqwuìdo/ F1S1 945;/ vxffxu0 vdoh fhqwuh0ylooh/ Prqwuìdo K6F 6M:/ Fdqdgd +yves.sprumont@umontreal.ca)1 L zurwh wklv sdshu zkloh ylvlwlqj FRGH/ Xqlyhuvlwdw Dxwõqrpd gh Edufhorqd1 Vxssruw iurp wkh Vsdqlvk Plqlvwu| ri Hgxfdwlrq dqg wkh VVKUF ri Fdqdgd lv judwhixoo| dfnqrzohgjhg1 L wkdqn E1 Nodxv/ K1 Prxolq/ Z1 Wkrpvrq/ dqg \1 Zdqj iru khosixo frqyhuvdwlrqv1

(4)

jdph dv d vroxwlrq wr wkh xqghuo|lqj frvw0vkdulqj sureohp EQ> {> F= Wklv yhu| vlpsoh vroxwlrq/ nqrzq dv wkh Vkdsoh|0Vkxeln phwkrg/ srvvhvvhv dwwudfwlyh surshuwlhv= vhh Prxolq +4<<8, dqg Vsuxprqw +5333, iru d glvfxvvlrq dqg ydulrxv d{lrpdwl}dwlrqv1 Lq jhqhudo/ krzhyhu/ wkh vwdqg0dorqh frvw jdph ljqruhv pxfk ri wkh ulfk lqirupdwlrq frqwdlqhg lq wkh frvw ixqfwlrq F= Wkh Dxpdqq0Vkdsoh| phwkrg h{sorlwv wklv lqiru0 pdwlrq e| frqvwuxfwlqj d jdph EQ{> K{ zkhuh hdfk xqlw ghpdqghg e| hdfk djhqw

lv uhjdughg dv d vhsdudwh sod|hu= wkh sod|hu vhw lv Q{ ' ^lMQQl> zkhuh hdfk ri wkh

glvmrlqw vhwv Ql frqwdlqv {l hohphqwv/ dqg wkh frvw dvvrfldwhg zlwk d frdolwlrq V lv

K{EV ' FEmV _ Q1m > ===> mV _ Qqm1 Wkh Vkdsoh| ydoxh ri wkdw jdph ghwhuplqhv d

sulfh iru hyhu| xqlw ri ghpdqg1 Wkh Dxpdqq0Vkdsoh| phwkrg fkdujhv wr djhqw l wkh vxp ri wkh sulfhv dwwdfkhg wr wkh xqlwv vkh ghpdqgv1

Wkh odujh olwhudwxuh rq Dxpdqq0Vkdsoh| sulflqj/ urrwhg lq Dxpdqq dqg Vkdsoh|*v +4<:7, wkhru| ri ydoxh iru qrqdwrplf jdphv/ irfxvhv rq wkh fdvh ri shuihfwo| glylvleoh jrrgv= ghpdqgv duh uhdo qxpehuv1 Lq wkdw frqwh{w/ wkh Dxpdqq0Vkdsoh| frvw vkduhv rewdlq dv wkh olplw ri wkh frvw vkduhv lq wkh glvfuhwh prgho dv jrrgv ehfrph pruh dqg pruh glylvleoh1 Xqghu vxlwdeoh glhuhqwldelolw| dvvxpswlrqv/ rqh fdq vkrz wkdw dq djhqw sd|v wkh lqwhjudo ri khu pdujlqdo frvwv dorqj wkh ud| wr {= wklv lv wkh vr0 fdoohg gldjrqdo irupxod1 D vxuyh| ri wkh olwhudwxuh xs wr wkh plg0hljkwlhv lv rhuhg lq Wdxpdq +4<;;,1 Pruh uhfhqw vwxglhv h{whqg wkh wkhru| wr wkh qrq0glhuhqwldeoh fdvh +vhh Phuwhqv +4<;;, dqg wkh vxuyh| e| Kdlpdqnr dqg Wdxpdq +5335,, dqg sursrvh dv|pphwulf yduldqwv +vhh PfOhdq dqg Vkdunh| +4<<;,,1

Dssolfdwlrqv ri wkh Dxpdqq0Vkdsoh| sulflqj wkhru| duh qxphurxv= iurp wkh hduo| zrun ri Eloohud/ Khdwk/ dqg Uddqdq +4<:;, dqg Vdphw/ Wdxpdq/ dqg ]dqj +4<;7, wr wkh uhfhqw frqwulexwlrqv ri Fdvwdqr0Sdugr dqg D1 Jdufld0Gld} +4<<8,/ Kdyly +5334, ru Ohh +5335,/ wkh| udqjh iurp wkh sulflqj ri xwlolwlhv vxfk dv zdwhu/ skrqh ru hohfwulflw| wr wkh doorfdwlrq ri kljkzd| frqvwuxfwlrq frvwv/ dqg wkh vkdulqj ri zdlwlqj wlph dw d frqjhvwhg vhuyhu1

Pruh gluhfwo| frqqhfwhg wr wkh suhvhqw sdshu duh wkh d{lrpdwl}dwlrqv ri wkh gldj0 rqdo irupxod rhuhg e| Eloohud dqg Khdwk +4<;5,/ Plupdq dqg Wdxpdq +4<;5,/ dqg Vdphw dqg Wdxpdq +4<;5,1 Qrwh wkdw zkhq doo jrrgv duh shuihfwo| vxevwlwxwdeoh/ wkdw lv/ FE} ' fEPlMQ}l iru vrph ixqfwlrq f/ frvw vkduhv duh sursruwlrqdo wr gh0

pdqgv= zh vd| wkdw wkh phwkrg vdwlvhv wkh Sursruwlrqdolw| d{lrp41 Lw wxuqv rxw

wkdw wkh frqwlqxrxv Dxpdqq0Vkdsoh| frvw0vkdulqj phwkrg lv wkh rqo| phwkrg vdwlvi|0 lqj Sursruwlrqdolw|/ Dgglwlylw| +dvnlqj wkdw frvw vkduhv ghshqg dgglwlyho| rq wkh frvw ixqfwlrq,/ dqg Vfdoh Lqyduldqfh +uhtxlulqj wkdw wkh vfdohv xvhg wr phdvxuh ghpdqgv eh luuhohydqw,1 D txlwh glhuhqw d{lrpdwl}dwlrq edvhg rq d srzhuixo frvw prqrwrqlflw|

4Wklv surshuw| lv qrw vwdwhg dv dq h{solflw d{lrp lq wkh sdshuv flwhg deryh1 Udwkhu/ lw lv lpsolflwho| dvvxphg e| uhvwulfwlqj dwwhqwlrq wr sulflqj phwkrgv/ xqghu zklfk dq djhqw*v frvw vkduh lv qhfhvvdulo| wkh surgxfw ri khu ghpdqg e| d shu xqlw sulfh1

(5)

Aumann-Shapley pricing: A reconsideration of the discrete case 6 frqglwlrq lv gxh wr \rxqj +4<;8,1

Vrphzkdw vxusulvlqjo|/ wkh glvfuhwh yhuvlrq ri wkh Dxpdqq0Vkdsoh| frvw0vkdulqj phwkrg qhyhu uhfhlyhg d surshu d{lrpdwlf ghihqvh1 Fdoyr dqg Vdqwrv +5333, sursrvh dq lqwhuhvwlqj fkdudfwhul}dwlrq edvhg rq P|huvrq*v +4<;3, lghd ri edodqfhg frqwul0 exwlrqv1 Xqghu wkh yhu| plog uhvwulfwlrq wkdw djhqwv ghpdqglqj qrwklqj sd| }hur/ Fdoyr dqg Vdqwrv* yhuvlrq ri wkh edodqfhg frqwulexwlrqv frqglwlrq vx!fhv wr fkdudf0 whul}h wkh Dxpdqq0Vkdsoh| phwkrg1 Wkhlu fkdudfwhul}dwlrq lv wkhuhiruh ehvw vhhq dv dq dowhuqdwlyh ghqlwlrq ri wkh phwkrg udwkhu wkdq d ghfrpsrvlwlrq ri lw lqwr glvwlqfw hohphqwdu| d{lrpv= vhh Vhfwlrq 7 iru d glvfxvvlrq1

Rxu d{lrpdwl}dwlrq exlogv rq wkh iroorzlqj wkuhh revhuydwlrqv1

4, Vfdoh Lqyduldqfh kdv qr frxqwhusduw lq wkh glvfuhwh frqwh{w1 Zkhq jrrgv frph lq lqglylvleoh xqlwv/ wkhuh lv qr lvvxh ri phdvxuhphqw ehfdxvh wkhuh lv qr phdqlqjixo uhvfdolqj rshudwlrq= wkh rqo| lqfuhdvlqj elmhfwlrq iurp wkh vhw ri qrqqhjdwlyh lqwhjhuv lqwr lwvhoi lv wkh lghqwlw|1 \hw/ Vfdoh Lqyduldqfh lv fuxfldo iru wkh fodvvlfdo fkdudf0 whul}dwlrq1 Sursruwlrqdolw| dqg Dgglwlylw| gr qrw fkdudfwhul}h wkh Dxpdqq0Vkdsoh| phwkrg= mxvw frqvlghu wkh sodlq sursruwlrqdo irupxod fkdujlqj E{l@

P

lMQ{mFE{ wr

djhqw l iru hyhu| frvw ixqfwlrq F=

5, Wkh h{lvwlqj d{lrpdwl}dwlrqv ri wkh Dxpdqq0Vkdsoh| phwkrg gr qrw uho| rq wkh Gxpp| d{lrp/ d nh| hohphqw lq Vkdsoh|*v +4<86, fkdudfwhul}dwlrq ri wkh ydoxh1 Lq wkh frvw0vkdulqj frqwh{w zlwk qr {hg frvw/ Gxpp| uhtxluhv wkdw li wkh wrwdo frvw grhv qrw ydu| zlwk djhqw l*v ghpdqg/ wkdw djhqw vkrxog sd| qrwklqj1 Wkh d{lrp kdv elwh= iru lqvwdqfh/ lw uxohv rxw wkh sodlq sursruwlrqdo phwkrg1 \hw/ Sursruwlrqdolw|/ Dgglwlylw|/ dqg Gxpp| duh qrw hqrxjk wr fkdudfwhul}h wkh Dxpdqq0Vkdsoh| phwkrg1 Wklv lv hdvlo| vhhq zlwk mxvw wzr djhqwv= dsso| wkh Dxpdqq0Vkdsoh| phwkrg zkhqhyhu {1 9' {2 exw vsolw wkh edodqfh eh|rqg vwdqg0dorqh frvwv htxdoo| li {1 ' {2= Wkh

gl!fxow| lv fohdu= qrqh ri wkh wkuhh d{lrpv frqqhfwv frvw0vkdulqj sureohpv zlwk glhuhqw ghpdqg surohv1

6, Sursruwlrqdolw| lv xvxdoo| prwlydwhg e| wkh iroorzlqj lqfhqwlyh frqvlghudwlrqv= li wkh sulqflsoh zhuh ylrodwhg/ vrph djhqwv zrxog qg lw surwdeoh wr hlwkhu phujh dqg suhwhqg wr eh d vlqjoh odujh ghpdqghu/ ru vsolw dqg dfw dv d froohfwlrq ri vpdoohu ghpdqghuv1 Lw lv zhoo nqrzq wkdw zkhq doo jrrgv duh shuihfw vxevwlwxwhv/ wkh rqo| frvw0vkdulqj phwkrg lppxqh wr vxfk phujlqj ru vsolwwlqj wdfwlfv lv wkh sursruwlrqdo phwkrg= vhh iru lqvwdqfh Prxolq +5335, ru/ iru d pruh frpsuhkhqvlyh wuhdwphqw/ Mx/ Pl|djdzd/ dqg Vdndl +5336,1

Wkh lghd ri suhyhqwlqj phujlqj ru vsolwwlqj pdqhxyhuv/ krzhyhu/ lv phdqlqjixo eh|rqg wkh fdvh zkhuh doo ghpdqgv duh shuihfwo| vxevwlwxwdeoh1 Vxssrvh wkdw d srv0 vleo| vwulfw vxevhw ri wkh jrrgv/ V/ duh shuihfw vxevwlwxwhv= wkh frvw ixqfwlrq wdnhv wkh irup FE} ' fE}Q.V>

P

lMV}l= Lq wklv fdvh rqh zrxog olnh wr hqvxuh wkdw wkh djhqwv

(6)

ru Vsolwwlqj d{lrp lpsrvhv wklv uhtxluhphqw iru dq| vxevhw ri jrrgv= lw lv lv wkxv d vwuhqjwkhqlqj ri Sursruwlrqdolw|1

Zh zloo vkrz wkdw Dgglwlylw|/ Gxpp|/ dqg Qr Phujlqj ru Vsolwwlqj fkdudfwhul}h wkh Dxpdqq0Vkdsoh| phwkrg lq wkh glvfuhwh frqwh{w1 Vhfwlrq 5 ghqhv rxu yhuvlrq ri wkh frvw0vkdulqj prgho dqg wkh Dxpdqq0Vkdsoh| phwkrg/ Vhfwlrq 6 suhvhqwv wkh d{lrpv/ Vhfwlrq 7 vwdwhv dqg glvfxvvhv rxu wkhruhp/ dqg Vhfwlrq 8 frqfoxghv1 Wkh Dsshqgl{ frqwdlqv d surri ri wkh wkhruhp1

2. Aumann-Shapley pricing in the discrete context

Zh frqvlghu d yduldeoh0srsxodwlrq yhuvlrq ri wkh glvfuhwh frvw0vkdulqj prgho rhuhg lq Prxolq +4<<8, dqg ydq ghq Qrxzhodqg hw do1 +4<<8, dqg ixuwkhu vwxglhg e| Zdqj +4<<<,/ Vsuxprqw +5333,/ Prxolq dqg Yrkud +5336,/ dqg Prxolq dqg Vsuxprqw +5336,1 Ohw Q ghqrwh wkh vhw ri qrqqhjdwlyh lqwhjhuv dqg Q wkh vhw ri qrqhpsw| qlwh vxevhwv ri Q1 D ghpdqg suroh lv d sdlu EQ> {/ zkhuh Q 5 Q lv wkh vhw ri djhqwv dqg { 5 QQ lv wkh olvw ri wkhlu ghpdqgv1 Zh ghqrwh e| G wkh vhw ri doo ghpdqg

surohv1 Iru hdfk Q 5 Q / zh ghqrwh e| FEQ wkh vhw ri frvw ixqfwlrqv iru Q G wklv lv wkh vhw ri doo qrqghfuhdvlqj pdsslqjv F G QQ $ U

+ vxfk wkdw FEf ' f1 D

+frvw0vkdulqj, sureohp lv d olvw EQ> {> F> zkhuh EQ> { 5 G dqg F 5 FEQ1 Wkh vhw ri doo sureohpv lv ghqrwhg S= D +frvw0vkdulqj, phwkrg lv d pdsslqj * zklfk dvvljqv wr hdfk EQ> {> F 5 S d yhfwru ri frvw vkduhv *EQ> {> F 5 UQ

+ vdwlvi|lqj wkh exgjhw

edodqfh frqglwlrqPlMQ*lEQ> {> F ' FE{= Qrwh wkdw frvw vkduhv duh uhtxluhg wr eh

qrqqhjdwlyh1

Ohw SEf ' iEQ> {> F 5 S m { 5 if> jQj= Hyhu| sureohp EQ> {> F 5 SEf lv

lghqwlhg zlwk d xqltxh frrshudwlyh jdph EQ> K ghqhg e| KEV ' FE{V> fQ.V iru

hdfk V  Q> zkhuh {V 5 QV ghqrwhv wkh uhvwulfwlrq ri { wr V=Zh ghqrwh wkh Vkdsoh|

ydoxh ri wkdw jdph e| *VkEQ> K ru/ zlwk d voljkw dexvh ri qrwdwlrq/ *VkEQ> {> F1 Wkh

pdsslqj *Vk lv ghqhg rq SEf>wkh Dxpdqq0Vkdsoh| frvw0vkdulqj phwkrg h{whqgv

lw wr S dv iroorzv1

Ghqlwlrq1 Ohw EQ> {> F 5 S/ zulwh Q ' i> ===> qj> ohw Q1> ===> Qq eh sdlu0zlvh

glvmrlqw vhwv vxfk wkdw mQlm ' {l iru hdfk l 5 Q dqg ohw Q{ ' ^lMQQl= Iru hdfk

V  Q{ ghqh K{EV ' FEmV _ Q1m > ===> mV _ Qqm= Wkh Dxpdqq0Vkdsoh| phwkrg *DVk

frpsxwhv wkh frvw vkduhv lq wkh sureohp EQ> {> F dffruglqj wr wkh irupxod *DVk l EQ> {> F ' P mMQl *Vk m EQ{> K{ iru doo l 5 Q=

Wkh dqrq|plw| ri wkh Vkdsoh| ydoxh jxdudqwhhv wkdw wklv ghqlwlrq lv phdqlqjixo= wkh jdph EQ{> K{ lv qrw xqltxho| ghqhg exw doo srvvleoh fkrlfhv duh htxlydohqw1

(7)

Aumann-Shapley pricing: A reconsideration of the discrete case 8

3. Three axioms

Wklv vhfwlrq suhvhqwv rxu wkuhh d{lrpv iru frvw0vkdulqj phwkrgv1 Wkh uvw wzr/ Dg0 glwlylw| dqg Gxpp|/ duh vwdqgdug1 Wkh wklug surshuw|/ Qr Phujlqj ru Vsolwwlqj/ irupdol}hv dq dujxphqw riwhq +lqirupdoo|, lqyrnhg lq ghihqvh ri wkh Dxpdqq0Vkdsoh| phwkrg1 Zkloh vlplodu uhtxluhphqwv kdyh ehhq ghqhg lq uhodwhg prghov/ rxu irupx0 odwlrq lq wkh frvw0vkdulqj frqwh{w lv qhz/ dqg lwv lpsolfdwlrqv kdyh qrw ehhq vwxglhg1 Dgglwlylw|1 Iru doo EQ> { 5 G dqg F> F 5 FEQ> *EQ> F n F> { ' *EQ> F> { n

*EQ> F> {=

Wklv vwdqgdug surshuw| zlwk qr dssduhqw hwklfdo frqwhqw lv yhu| frqyhqlhqw1 Lw jxdudqwhhv wkdw zkhq d surgxfwlrq surfhvv fdq eh ghfrpsrvhg lqwr d olvw ri lqghshq0 ghqw vxe0surfhvvhv/ dsso|lqj wkh phwkrg wr wkh djjuhjdwh frvw ixqfwlrq lv htxlydohqw wr dsso|lqj lw wr hdfk vxe0surfhvv dqg vxpplqj xs wkh uhvxowlqj frvw vkduhv1 Wklv idflolwdwhv frpsxwdwlrq lq sudfwlfh>lw dovr pdnhv lw srlqwohvv iru wkh djhqwv wr dujxh derxw wkh surshu ohyho ri dssolfdwlrq ri wkh phwkrg= dq dgglwlyh phwkrg lv lqvhqvlwlyh wr wkh ghwdlov ri lpsohphqwdwlrq1

Wkh qh{w surshuw| xvhv wkh iroorzlqj qrwdwlrq1 Iru hdfk Q 5 Q / F 5 FEQ/ l 5 Q> dqg } 5 QQ/ ohw C

lFE} ' FE} n hl  FE}> zkhuh hl lv wkh xqlw yhfwru hll ' >

hl

m ' f iru m 5 Qql= Qrwh wkdw zh zulwh l lqvwhdg ri ilj zkhq frqyhqlhqw1 Ghqh

F0

lEQ ' iF 5 FEQ m ClFE} ' f iru doo } 5 QQj G wklv lv wkh vhw ri frvw ixqfwlrqv

iru Q zkhuh djhqw l lv d gxpp|1 Wkh d{lrp ehorz lv wkh qdwxudo h{whqvlrq ri Vkdsoh|*v +4<86, gxpp| d{lrp wr rxu prgho1

Gxpp|1 Iru doo EQ> { 5 G> l 5 Q> dqg F 5 F0

lEQ/ *lEQ> {> F ' f=

Dgglwlylw| dqg Gxpp| flufxpvfuleh d zhoo xqghuvwrrg fodvv ri phwkrgv= d xvhixo fkdudfwhul}dwlrq lv surylghg e| Prxolq dqg Yrkud +5336, dqg uhfdoohg lq Vwhs 4 ri wkh surri ri rxu wkhruhp1 Zh hpskdvl}h wkdw wklv fodvv lv yhu| ulfk1 Rq SEf> wkh vxevhw ri sureohpv fruuhvsrqglqj wr vwdqgdug frrshudwlyh jdph wkhru|/ wkh wzr d{lrpv duh yhu| srzhuixo= lq sduwlfxodu/ dgglqj Vkdsoh|*v +4<86, v|pphwu| d{lrp lv hqrxjk wr slq grzq wkh Vkdsoh| ydoxh1 E| frqwudvw/ rq wkh odujh grpdlq ri doo frvw0 vkdulqj sureohpv/ S/ Dgglwlylw| dqg Gxpp| duh vdwlvhg e| d ydulhw| ri lqwhuhvwlqj v|pphwulf frvw0vkdulqj phwkrgv glhuhqw iurp wkh Dxpdqq0Vkdsoh| phwkrg1 Wkh Vkdsoh|0Vkxeln phwkrg ghvfulehg lq Vhfwlrq 4 dqg wkh vhuldo phwkrg ghvfulehg lq Prxolq +4<<8, duh mxvw wzr h{dpsohv1

Zh wxuq qrz wr wkh frqglwlrq sod|lqj wkh fhqwudo uroh lq wklv sdshu1

Qr Phujlqj ru Vsolwwlqj1 Ohw EQ> {> F 5 S/ l 5 Q> dqg L 5 Q eh vxfk wkdw Q _ L ' ilj = Zulwh Q ' EQql ^ L> ghqh F 5 FEQ e| FE} ' FE}

Q.l> P l0ML}l0 iru doo } 5 QQ0 / dqg ohw { 5 QQ0 = Wkhq iPl0ML{l0 ' {l dqg {m ' {m iru doo m 5 Qqlj , iPl0ML*l0EQ> {> F ' *lEQ> {> Fj=

(8)

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

(9)

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

(10)

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=

(11)

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+(})

(12)

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 '

(13)

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

(14)

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 PEn1 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 En3u+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|/

(15)

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 GEnnqGEn= 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|

(16)

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

(17)

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

(18)

^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

(19)

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

Références

Documents relatifs

MPI’s remote memory access interface defines one-sided communication operations, data consistency, and synchronization models for accessing memory regions that are exposed through

7KHSURGXFWLRQRIDGHVFULSWLRQRIWKH$87LVDNH\HOHPHQWLQ RXUSURFHVVDVWKLVGHVFULSWLRQLVXVHGLQERWKWHVWJHQHUDWLRQ DQGGHYHORSPHQWDFWLYLWLHV)RUWKHModelling of the

scientifique et politique du phénomène, l’instinct le cède au conditionnement qui opère par les stimuli du contexte (ici l’angoisse collective produite par la

Pour l’étude de la cinétique de diffusion du docosanol, deux membranes de polycarbonates de diamètre 19 mm et dont le diamètre de pores égale à 400 et 800 nm ont été utilisées et

We remark nevertheless, from qualitative analyses of use of the regulator in real driving conditions, and from the lived experience of this use by subjects, that this drop in trust in

$IWHUVHYHUDOIHUPHQWDWLRQVWKHORVVHVRIPDVVEHWZHHQWKH EHJLQQLQJDQGWKHHQGRIWKHIHUPHQWDWLRQHYROYHLQWKHVDPH SURSRUWLRQV LQGHSHQGHQWO\ RI WKH TXDQWLW\ XVHG

Il convient alors de revenir aux peintures charentaises et d’observer que le verset 2 Maccabées 11 : 8, celui dont nous nous sommes servi pour interpréter le cavalier blanc,

ETHNIC BOOKLETS SERIES FUND / FONDS DES BROCHURES