Research Collection

Working Paper

A theory of reciprocity

Author(s): Falk, Armin; Fischbacher, Urs

Publication Date: 2000

Permanent Link: https://doi.org/10.3929/ethz-a-004372683

Rights / License: In Copyright - Non-Commercial Use Permitted

This page was generated automatically upon download from the ETH Zurich Research Collection. For moreinformation please consult the Terms of use.

ETH Library

Institute for Empirical Research in EconomicsUniversity of Zurich

Working Paper SeriesISSN 1424-0459

Working Paper No. 6

A Theory of Reciprocity

Armin Falk and Urs Fischbacher

July 2000

D Wkhru| ri Uhflsurflw|e|

Duplq Idon dqg Xuv Ilvfkedfkhu-

Xqlyhuvlw| ri ]xulfk

Lqvwlwxwh iru Hpslulfdo Hfrqrplf Uhvhdufk Eoxhpolvdosvwudvvh 43/ FK0;339 ] ulfk

idonClhz1xql}k1fk/ �edClhz1xql}k1fk

Iluvw yhuvlrq= Pdufk 4<<;

Wklv yhuvlrq= Mxo| 5333

Nh|zrugv= Uhflsurflw|/ Idluqhvv/ Frrshudwlrq/ Frpshwlwlrq/ Jdph Wkhru|1MHO= F:/ F<4/ F<5/ G97/ K74

Devwudfw

Wklv sdshu suhvhqwv d irupdo wkhru| ri uhflsurflw|1 Uhflsurflw| phdqv wkdw shrsoh

uhzdug nlqg dfwlrqv dqg sxqlvk xqnlqg rqhv1 Wkh wkhru| wdnhv lqwr dffrxqw wkdw

shrsoh hydoxdwh wkh nlqgqhvv ri dq dfwlrq qrw rqo| e| lwv frqvhtxhqfhv exw dovr e|

wkh lqwhqwlrq xqghuo|lqj wklv dfwlrq1 Wkh wkhru| h{sodlqv wkh uhohydqw vw|ol}hg idfwv ri

d zlgh udqjh ri h{shulphqwdo jdphv1 Dprqj wkhp duh wkh xowlpdwxp jdph/ wkh jliw0

h{fkdqjh jdph/ d uhgxfhg ehvw0vkrw jdph/ wkh glfwdwru jdph/ wkh sulvrqhu*v glohppd/

sxeolf jrrgv jdphv/ dqg wkh lqyhvwphqw jdph1 Ixuwkhu/ wkh wkhru| h{sodlqv zk|

vxemhfwv ehkdyh gl�huhqwo| lq wuhdwphqwv zkhuh wkh| h{shulhqfh wkh dfwlrqv ri uhdo

shuvrqv frpsduhg wr wuhdwphqwv zkhuh wkh| idfh cdfwlrqv* fdxvhg e| d udqgrp ghylfh1

Ilqdoo|/ wkh wkhru| h{sodlqv zk| lq elodwhudo lqwhudfwlrqv rxwfrphv whqg wr eh cidlu*

zkhuhdv lq frpshwlwlyh pdunhwv hyhq h{wuhpho| xqidlu glvwulexwlrqv pd| dulvh1

-Ilqdqfldo vxssruw e| wkh Vzlvv Qdwlrqdo Vflhqfh Irxqgdwlrq +Surmhfw 450768<31<8, dqg e| wkh

PdfDuwkxu Irxqgdwlrq +Qhwzrun rq Hfrqrplf Hqylurqphqwv dqg wkh Hyroxwlrq ri Lqglylgxdo Suhi0

huhqfhv dqg Vrfldo Qrupv, lv judwhixoo| dfnqrzohgjhg1 Zh zrxog olnh wr wkdqn Vdp Erzohv/ Pduwlq

Eurzq/ Vlprq Jçfkwhu/ Khuehuw Jlqwlv/ Oruhq} Jùwwh/ Pdwwkhz Udelq/ Duplq Vfkpxw}ohu dqg wkh

sduwlflsdqwv ri PdfDuwkxu Irxqgdwlrq phhwlqj lq Fklfdjr 4<<;1

4

Nlqgqhvv lv wkh sduhqw ri nlqgqhvv1

+Dgdp Vplwk,

4 Lqwurgxfwlrq

D odujh erg| ri hylghqfh lqglfdwhv wkdw uhflsurflw| lv d srzhuixo ghwhuplqdqw ri kx0

pdq ehkdylru1 H{shulphqwv dqg txhvwlrqqdluh vwxglhv shuiruphg e| sv|fkrorjlvwv dqg

hfrqrplvwv dv zhoo dv dq lpsuhvvlyh olwhudwxuh lq vrflrorj|/ hwkqrorj| dqg dqwkursro0

rj| hpskdvl}h wkh rpqlsuhvhqfh ri uhflsurfdo ehkdylru1 Wkh vrflrorjlvw Jrxogqhu

+4<93,/ iru h{dpsoh/ qrwhv wkdw wkh qrup ri uhflsurflw| lv �qr ohvv xqlyhuvdo dqg

lpsruwdqw dq hohphqw ri fxowxuh wkdq wkh lqfhvw wderr111� +s1 4:4,1

Wkh lpsruwdqfh ri uhflsurflw| iru hfrqrplfv kdv ehhq srlqwhg rxw e| pdq| vfkro0

duv1 Ndkqhpdq/ Nqhwvfk/ dqg Wkdohu +4<;9,/ h1j1/ vkrz wkdw idluqhvv qrupv

suhydlo lq d ydulhw| ri exvlqhvv frqwh{wv1 Lq wkh �hog ri oderu hfrqrplfv/ txhvwlrqqdluh

vwxglhv zlwk rzqhuv dqg pdqdjhuv ri �upv vxjjhvw wkdw d srvvleoh vrxufh iru uljlg

zdjhv lv wkdw wkh hpsor|huv duh xqzloolqj wr fxw zdjhv +Eolqghu dqg Fkrl +4<<3,/

Ehzoh| +4<<8,/ Djhoo dqg Oxqgeruj +4<<8,/ Fdpsehoo dqg Ndpodql +4<<:,

dqg Djhoo +4<<<,1 Dffruglqj wr wkhvh vwxglhv d pdmru uhdvrq iru �upv* uhixvdo wr

fxw zdjhv lv wkh ihdu wkdw sd| fxwv zloo dgyhuvho| d�hfw zrun prudoh1 Wkxv/ frqfhuqv

iru uhflsurflw| pd| sod| d nh| uroh lq wkh h{sodqdwlrq ri grzqzdugo| uljlg zdjhv1 Hfr0

qrplf frqvhtxhqfhv ri uhflsurflw| kdyh dovr ehhq vkrzq lq pdq| rwkhu duhdv/ vxfk dv

wd{ frpsoldqfh +Vplwk +4<<5,,/ rujdql}dwlrq wkhru| +Vwhhuv dqg Sruwhu +4<<4,/

Przgd| +4<<4,,/ frqwulexwlrqv wr sxeolf jrrgv +Vxjghq +4<;7,,/ frqwudfw hqirufh0

phqw +Ihku dqg Idon +4<<<,,/ jliw0jlylqj +Uxiioh +4<<8,,/ dqg vwulnh euhdnlqj

+Iudqflv +4<;8,,1

Wkh hvvhqfh ri uhflsurflw| lv yhu| qlfho| fdswxuhg lq d txrwh iurp Wkh Hggd/ d ph0

glhydo froohfwlrq ri Lfhodqglf hslf srhpv= �D pdq rxjkw wr eh iulhqg wr klv iulhqg dqg

uhsd| jliw zlwk jliw1 Shrsoh vkrxog phhw vplohv zlwk vplohv dqg olhv zlwk wuhdfkhu|1�

Wkh txrwh lqfoxghv srvlwlyh uhflsurflw| zklfk lv wkh uhzdug ri d nlqg wuhdwphqw dqg

qhjdwlyh uhflsurflw| zklfk phdqv wkh sxqlvkphqw ri dq xqnlqg wuhdwphqw1 Lpsru0

wdqwo|/ uhflsurflw| phdqv d ehkdylru wkdw fdqqrw eh mxvwl�hg lq whupv ri vho�vk dqg

sxuho| rxwfrph rulhqwhg suhihuhqfhv1 Wr dyrlg whuplqrorjlfdo frqixvlrq ohw xv/ wkhuh0

iruh/ foduli| wkdw uhflsurflw| vkduso| glvwlqjxlvkhv iurp cuhflsurfdo dowuxlvp* +Wulyhuv

+4<:4,,1 D uhflsurfdo dowuxlvw lv rqo| zloolqj wr uhflsurfdwh li wkhuh duh ixwxuh uhzdugv

dulvlqj iurp uhflsurfdo dfwlrqv14

H{shulphqwdolvwv kdyh irxqg hylghqfh iru erwk qhjdwlyh dv zhoo dv srvlwlyh uhfl0

4 Lq wkh sduodqfh ri jdph wkhru| wklv nlqg ri uhflsurfdo dfwlrq pd| eh vxssruwhg dv dq htxloleulxp

vwudwhj| lq lq�qlwho| uhshdwhg jdphv +iron wkhruhpv, ru lq �qlwho| uhshdwhg jdphv zlwk lqfrpsohwh

lqirupdwlrq +vhh Nuhsv/ Plojurp/ Urehuwv/ dqg Zlovrq +4<;5,,1

5

surflw|= Qhjdwlyh uhflsurflw| lv d yhu| urexvw revhuydwlrq/ h1j1/ lq wkh vr0fdoohg xowl0

pdwxp jdph15 Vwduwlqj zlwk wkh zrun ri J wk/ Vfkplwwehujhu/ dqg Vfkzdu}h

+4<;5, qxphurxv vwxglhv kdyh vkrzq wkdw shrsoh 0 frqwudu| wr wkhlu pdwhuldo vhoi0

lqwhuhvw 0 uhmhfw orz r�huv lq rughu wr sxqlvk wkh xqnlqgqhvv ri sursrvhuv1 Srvlwlyh

uhflsurflw| kdv ehhq irxqg/ h1j1/ lq wkh lqyhvwphqw jdph vwxglhg e| Ehuj/ Glfnkdxw

dqg PfFdeh +4<<8, dqg lq wkh jliw0h{fkdqjh jdph e| Ihku/ Nlufkvwhljhu/ dqg

Ulhgo +4<<6,1 Lq wkh odwwhu/ wkh dxwkruv dqdo|}h wkh lpsdfw ri uhflsurflw| lq dq h{0

shulphqwdo oderu pdunhw1 Wkh| uhsruw d vljql�fdqwo| srvlwlyh uhodwlrqvkls ehwzhhq

zdjhv sdlg e| �upv dqg wkh h�ruw ohyho surylghg e| zrunhuv1 Ehvlghv wklv reylrxv

hylghqfh iru uhflsurflw|/ wkhuh lv dovr hylghqfh zklfk vhhpv wr eh lqfrpsdwleoh zlwk

uhflsurfdo suhihuhqfhv= Pdq| pdunhw h{shulphqwv/ h1j1/ lpsuhvvlyho| vxssruw wkh rxw0

frph suhglfwhg e| wkh vwdqgdug hfrqrplf wkhru| zklfk dvvxphv frpsohwho| vho�vk

suhihuhqfhv1 Lq wklv sdshu zh suhvhqw d wkhru| ri uhflsurflw| zklfk h{sodlqv wkh uhoh0

ydqw vw|ol}hg idfwv ri d zlgh ydulhw| ri h{shulphqwdo jdphv1 Wklv lqfoxghv erwk/ �idlu�

rxwfrphv lq edujdlqlqj jdphv dv zhoo dv �xqidlu� rxwfrphv lq pdunhwv1

Dffruglqj wr rxu wkhru|/ d uhflsurfdo dfwlrq lv prghohg dv wkh ehkdylrudo uhvsrqvh

wr dq dfwlrq wkdw lv shufhlyhg dv hlwkhu nlqg ru xqnlqg1 Wkh pruh dq dfwlrq lv frq0

vlghuhg dv nlqg ru xqnlqg/ wkh pruh lw zloo eh uhzdughg ru sxqlvkhg/ uhvshfwlyho|1

Wkh fuxfldo txhvwlrq wr dvn lv= Krz gr shrsoh hydoxdwh zkhwkhu dq h{shulhqfhg df0

wlrq lv nlqg ru xqnlqgB Wzr dvshfwv duh hvvhqwldo khuh/ qdpho| +l, wkh rxwfrph ru

wkh frqvhtxhqfhv ri dq dfwlrq dqg +ll, wkh xqghuo|lqj prwlydwlrq/ l1h1/ wkh lqwhqwlrqv

lqyroyhg1 Lw lv qrw rqo| wkh frqvhtxhqfh wkdw ghwhuplqhv wkh nlqgqhvv ri dq dfwlrq1

Udwkhu/ �shrsoh ghwhuplqh wkhlu glvsrvlwlrqv wrzdug rwkhuv dffruglqj wr prwlyhv dw0

wulexwhg wr wkhvh rwkhuv/ qrw vroho| dffruglqj wr dfwlrqv wdnhq� +Udelq +4<<;,/ s1

55,1 Dv dq h{dpsoh wdnh wkh fulplqdo odz zklfk glvwlqjxlvkhv txlwh fduhixoo| eh0

wzhhq fulplqdo dfwlylwlhv wkdw zhuh lqwhqghg dqg wkrvh wkdw zhuh qrw1 D sduwlfxodu

fulph lv frqvlghuhg ohvv fulplqdo dqg zloo eh sxqlvkhg ohvv li lw zdv frpplwwhg qhj0

oljhqwo| dqg qrw zlwk fulplqdo lqwhqw +frpsduh/ h1j1/ wkh frqfhswv ri pdqvodxjkwhu

dqg vhfrqg0ghjuhh pdqvodxjkwhu,1 Dv dqrwkhu h{dpsoh wdnh djdlq sd| fxwv ri d �up=

Dv Ehzoh| +4<<8, vkrzv lq klv txhvwlrqqdluh vwxg|/ zrunhuv lqwhusuhw sd| fxwv dv

dq lqvxow zklfk dgyhuvho| d�hfw zrun prudoh1 Wklv/ krzhyhu/ pd| krog wr d pxfk

ohvvhu ghjuhh li zrunhuv nqrz wkdw wkhlu �up lv �irufhg� wr orzhu zdjhv/ h1j1/ wr dyrlg

edqnuxswf|1

Vhyhudo rwkhu dwwhpswv wr prgho rwkhu0uhjduglqj suhihuhqfhv kdyh ehhq pdgh1

Prvw uhfhqwo|/ Erowrq dqg Rfnhqihov +5333, dqg Ihku dqg Vfkplgw +4<<<,

kdyh vxjjhvwhg wkhrulhv zklfk hpskdvl}h lqhtxlw| dyhuvlrq dv d prwlydwlrqdo ghwhupl0

qdqw1 Wkhvh prghov duh sxuho| rxwfrph0rulhqwhg lq wkh vhqvh wkdw wkh| gr qrw dffrxqw

5Doo jdphv phqwlrqhg lq wklv vhfwlrq duh h{solflwo| ghvfulehg lq wkh dssolfdwlrqv vhfwlrq1

6

iru wkh uroh ri lqwhqwlrqv1 Frqvhtxhqwo|/ wkh| ljqruh dq lpsruwdqw dvshfw ri wkh sv|0

fkrorj| ri uhflsurflw|1 Pruhryhu/ wkh prghov idlo wr suhglfw lpsruwdqw h{shulphqwdo

uhvxowv +vhh rxu glvfxvvlrq lq Vhfwlrq 5,1

D gl�huhqw idluqhvv prgho kdv ehhq ghyhorshg e| Udelq +4<<6,16 Klv prgho h{0

solflwo| dffrxqwv iru wkh uroh ri lqwhqwlrqv1 Krzhyhu/ Udelq*v vwudwhjlf irup wkhru|

lv qrw phdqw wr suhglfw wkh gdwd ri vhtxhqwldo jdphv17 Lq wkh vhtxhqwldo sulvrqhu*v

glohppd/ h1j1/ Udelq*v wkhru| suhglfwv xqfrqglwlrqdo frrshudwlrq dqg uxohv rxw frqgl0

wlrqdo frrshudwlrq1 Lq wkh xowlpdwxp jdph klv prgho lv frpsdwleoh zlwk r�huv deryh

83 shufhqw ri wkh wrwdo slh1 Erwk suhglfwlrqv duh uhixwhg e| wkh gdwd1

Lq rxu wkhru|/ zh lpsohphqw dq htxlw| edvhg uhihuhqfh vwdqgdug iru wkh hydoxdwlrq

ri dq rxwfrph*v nlqgqhvv1 Krzhyhu/ zh dovr wdnh lqwr frqvlghudwlrq wkdw lq mxgjlqj

wkh nlqgqhvv ri dq dfwlrq/ shrsoh fduh iru idlu lqwhqwlrqv1 Wkh wkhru| |lhogv whvwdeoh

suhglfwlrqv1 Zh vkrz wkdw lw h{sodlqv wkh vw|ol}hg idfwv ri d zlgh ydulhw| ri h{shulphq0

wdo jdphv1 Zh ghulyh doo suhglfwlrqv zlwk d vlqjoh xwlolw| ixqfwlrq1 Ixuwkhupruh/ zh

xvh wkh vdph sdudphwhu frqvwhoodwlrq lq doo jdphv1 Wklv lv uhpdundeoh vlqfh zh gr qrw

rqo| h{sodlq rxwfrphv ri jdphv zkhuh shrsoh ehkdyh lq d �idlu� dqg rwkhu0uhjduglqj

zd|1 Udwkhu/ zh dovr vkrz dqg h{sodlq zk| wkh vdph vxemhfwv dfw dv li wkh| zhuh

vroho| gulyhq e| vhoi0lqwhuhvw1 Lq sduwlfxodu/ rxu wkhru| suhglfwv wkh vw|ol}hg idfwv ri

wkh iroorzlqj jdphv/ wkh xowlpdwxp jdph/ wkh glfwdwru jdph/ wkh jliw0h{fkdqjh jdph/

wkh sulvrqhu*v glohppd/ d uhgxfhg ehvw0vkrw jdph/ wkh lqyhvwphqw jdph/ sxeolf jrrg

jdphv dqg frpshwlwlyh pdunhw jdphv1

Wkh uhpdlqghu ri wklv sdshu lv rujdql}hg dv iroorzv= Lq Vhfwlrq 5 zh frqwudvw

wkh wzr glvwlqfw frqfhswv ri uhflsurflw| dqg lqhtxlw| dyhuvlrq1 Vhfwlrq 6 lqwurgxfhv

wkh irupdo wkhru|1 Vhfwlrq 7 surylghv dssolfdwlrqv ri wkh wkhru| lq wkh oljkw ri wkh

h{shulphqwdo hylghqfh1 Vhfwlrq 8 vxppdul}hv1

5 Zk| lqhtxlw| dyhuvlrq uhdfkhv wrr vkruw

Uhfhqwo| wzr prghov ri rwkhu0uhjduglqj suhihuhqfhv kdyh uhdfkhg frqvlghudeoh dwwhq0

wlrq= wkh lqhtxlw| dyhuvlrq prghov e| Erowrq dqg Rfnhqihov +5333, dqg Ihku

dqg Vfkplgw +4<<<,1 Lq wkhlu prghov lw lv wkh glvolnh ri lqhtxlwdeoh glvwulexwlrqv

zklfk wuljjhuv ehkdylrudo uhvsrqvhv1 Lqhtxlw| dyhuvlrq vkduso| frqwudvwv iurp uhfl0

surflw| iru dw ohdvw wzr uhdvrqv1 Iluvw/ lqhtxlw| dyhuvlrq lv d sxuho| frqvhtxhqwldolvwlf

frqfhsw/ l1h1/ lqwhqwlrqv ru prwlyhv sod| qr uroh1 Uhflsurflw| rq wkh rwkhu kdqg hp0

skdvl}hv wkh lpsruwdqfh ri lqwhqwlrqv1 Vhfrqg/ dq lqhtxlw| dyhuvh shuvrq sxqlvkhv ru

6Rwkhu prghov wkdw wdnh dffrxqw ri idluqhvv suhihuhqfhv duh Ohylqh +4<<;, dqg Gxizhqehuj

dqg Nlufkvwhljhu +4<<;,17Frpphqwlqj rq klv rzq prgho Udelq qrwhv wkdw �^h`{whqglqj wkh prgho wr vhtxhqwldo pryh

jdphv lv dovr hvvhqwldo iru dssolhg uhvhdufk1� +Udelq +4<<6,/ s1 45<9,1

7

YY ZZ

82

00

00

100

82

00

00

55

Game (b)Game (a)

X X

Iljxuh 4= Wzr jdphv wkdw vkrz wkh lpsruwdqfh ri lqwhqwlrqv1

uhzdugv dqrwkhu shuvrq li dqg rqo| li wklv uhgxfhv wkh lqhtxlw| ehwzhhq wkh shuvrq dqg

klv rssrqhqw+v,1 Uhflsurflw| rq wkh rwkhu kdqg phdqv wkh uhzduglqj ru sxqlvkphqw

ri nlqgqhvv ru xqnlqgqhvv1 Wkhuhiruh uhflsurflw| suhglfwv sxqlvkphqwv ru uhzdugv

hyhq lq vlwxdwlrqv zkhuh lqhtxlw| fdqqrw eh uhgxfhg1 Xowlpdwho| lw lv dq hpslulfdo

txhvwlrq zkhwkhu lqhtxlw| dyhuvlrq ru uhflsurflw| lv wkh pruh srzhuixo frqfhsw1 Lq

wklv vhfwlrq zh wkhuhiruh vnhwfk vrph ri wkh h{shulphqwdo hylghqfh zklfk dlpv dw

dqvzhulqj wklv txhvwlrq1

514 Lqwhqwlrqv pdwwhu

111wkh lqwhqwlrq/ qrw wkh pdwwhu pdnhv wkh ehqh�w1

+Vhqhfd,

Wkhuh lv dq rqjrlqj ghedwh zkhwkhu lqwhqwlrqv duh ehkdylrudoo| uhohydqw1 Wkh

frqvhtxhqwldolvwlf shuvshfwlyh wdnhq e| wkh lqhtxlw| dyhuvlrq dssurdfk fodlpv wkdw

idluqhvv gulyhq sxqlvkphqw ru uhzdug fdq ixoo| eh dffrxqwhg iru zlwkrxw wdnlqj lq0

whqwlrqv lqwr frqvlghudwlrq1 Dffruglqj wr Erowrq dqg Rfnhqihov +5333, dqg

Ihku dqg Vfkplgw +4<<<, wkh glvwulexwlyh frqvhtxhqfhv ri dq dfwlrq duh vx!flhqw

wr wuljjhu ehkdylru dqg qr frqvlghudwlrq ri lqwhqwlrqv lv qhhghg1 Rxu uhflsurflw|

prgho rq wkh rwkhu kdqg lqfrusrudwhv shrsoh*v frqfhuqv iru lqwhqwlrqv1 Wr looxvwudwh

wkh uroh ri lqwhqwlrqv frqvlghu wkh iroorzlqj wzr jdphv=

Erwk jdphv lq Iljxuh 4 duh uhgxfhg xowlpdwxp jdphv1 Lq jdph +d, wkh �uvw

pryhu lv dvnhg wr glylgh 43 grooduv ehwzhhq klpvhoi dqg d uhvsrqghu1 Kh kdv rqo|

wzr fkrlfhv/ hlwkhu wr r�hu 5 grooduv wr wkh uhvsrqghu dqg wr nhhs ; grooduv +;250r�hu,

ru wr r�hu dq htxdo vsolw1 Wkh uhvsrqghu fdq hlwkhu dffhsw ru uhmhfw wkh r�hu1 D

uhmhfwlrq ohdgv wr }hur lqfrphv iru erwk sod|huv1 Frqvlghu qrz jdph +e,1 Djdlq/ wkh

8

sursrvhu*v wdvn lv wr glylgh 43 grooduv1 Wkh wzr srvvleoh r�huv kh fdq fkrrvh iurp

duh wkh ;250r�hu/ mxvw dv lq jdph +d, dqg d 43230r�hu zklfk ohdyhv qrwklqj wr wkh

uhvsrqghu1

Lq rxu glvfxvvlrq zh frqfhqwudwh rq wkh uhmhfwlrq ehkdylru ri wkh ;25 r�hu1 Wkh

frqvhtxhqwldolvwlf shuvshfwlyh dgyrfdwhg e| Erowrq dqg Rfnhqihov +5333, dqg

Ihku dqg Vfkplgw +4<<<, lpsolhv wkdw wkh uhmhfwlrq udwhv ri wkh ;250r�hu lv wkh

vdph lq erwk jdphv1 Wkh uhdvrq lv wkdw wkhvh prghov glvuhjdug wkdw lghqwlfdo r�huv

pd| eh shufhlyhg dv pruh ru ohvv idlu/ ghshqglqj rq wkh lqwhqwlrqv lqyroyhg1 Wklv

ylhz lv fohduo| uhmhfwhg e| wkh h{shulphqwdo hylghqfh1 Hyhq wkrxjk wkh frqvhtxhqfhv

duh h{dfwo| wkh vdph/ uhvsrqghuv uhmhfw wkh ;250r�hu vljql�fdqwo| pruh riwhq lq jdph

+d, wkdq lq jdph +e, +frpsduh Idon/ Ihku/ dqg Ilvfkedfkhu +4<<<,,1 Zkloh wkh

;250r�hu lq jdph +d, lv uhmhfwhg lq 7717 shufhqw ri wkh fdvhv/ wkh vdph r�hu lv uhmhfwhg

lq rqo| ;1< shufhqw ri wkh fdvhv lq jdph +e,1 Wkh uhdvrq iru wkh gl�huhqw uhmhfwlrq

sdwwhuq lv wkdw uhvsrqghuv fduh derxw zk| wkh sursrvhu fkrvh wkh ;250r�hu/ l1h1/ wkh|

fduh derxw wkh sursrvhu*v lqwhqwlrqv1 Lq jdph +d, lw lv reylrxv wkdw wkh sursrvhu glg

qrw zdqw wr uhdfk d idlu doorfdwlrq ehwzhhq klpvhoi dqg wkh uhvsrqghu1 Kh frxog kdyh

fkrvhq dq htxdo vsolw exw kh glg qrw1 R�hulqj wkh ;250r�hu lq jdph +e,/ krzhyhu/

grhv qrw vljqdo edg lqwhqwlrqv1 Diwhu doo lw lv wkh prvw cidlu* r�hu wkh sursrvhu frxog

r�hu dw doo1

Wkh gl�huhqfh lq uhmhfwlrq udwhv ehwzhhq jdphv +d, dqg +e, fohduo| ghprqvwudwhv

wkh lpsruwdqfh ri lqwhqwlrqv18 Krzhyhu/ wkh idfw wkdw wkh uhmhfwlrq udwh ri wkh ;250

r�hu lv juhdwhu wkdq }hur hyhq li wklv r�hu lv dfwxdoo| d cidlu*/ lqglfdwhv wkdw lqwhqwlrqv

dorqh fdqqrw eh wkh zkroh vwru| hlwkhu19 Wklv ylhz lv vxssruwhg e| wkh h{shulphqwv

e| Eorxqw +4<<8, dqg Fkduqhvv +4<<9, zkr uhsruw wkdw uhflsurflw| lv +l, pxfk

zhdnhu exw +ll, qrw }hur lq d frqglwlrq zkhuh lqwhqwlrqv duh devhqw frpsduhg wr d

frqglwlrq zkhuh lqwhqwlrqv duh suhvhqw1 Lq rxu prgho zh wkhuhiruh lqfrusrudwh erwk/

wkh frqfhuq iru wkh rxwfrph shu vh dqg iru wkh xqghuo|lqj prwlydwlrq/ l1h1/ lqwhqwlrqv1

515 Shrsoh sxqlvk hyhq li wkh| fdqqrw uhgxfh lqhtxlw|

Dffruglqj wr wkh lqhtxlw| dssurdfk/ d shuvrq zloo sxqlvk dqrwkhu shuvrq li dqg rqo|

li wklv uhgxfhv wkh lqhtxlw| ehwzhhq wkh shuvrq dqg klv rssrqhqw+v,1 Uhflsurflw| rq

wkh rwkhu kdqg glfwdwhv wr sxqlvk lq rughu wr uhflsurfdwh dq xqnlqg dfw1 Wkh dlp ri

8H{shulphqwdo hylghqfh iru wkh lpsruwdqfh ri lqwhqwlrqv lv irxqg lq Erooh dqg Nulwlnrv +4<<;,/

Idon/ Ihku/ dqg Ilvfkedfkhu +4<<<,/ Fkduqhvv +4<<9,/ Eorxqw +4<<8,/ Juhhqvehuj dqg

Iulvfk +4<:5,/ dqg Jrudqvrq dqg Ehunrzlw} +4<99,1 Lq wkh odwwhu vwxg|/ h1j1/ wkh dxwkruv

uhsruw dq h{shulphqw zkhuh shrsoh frxog uhflsurfdwh hduolhu khos1 Uhflsurfdwlrq zdv vljql�fdqwo|

kljkhu zkhq wkh sulru khos zdv surylghg yroxqwdulo| frpsduhg wr lqyroxqwdu| khos19Wklv txhvwlrqv wkh dssurdfk wdnhq e| Gxizhqehuj dqg Nlufkvwhljhu +4<<<, ru Udelq +4<<6,

zklfk fodlpv wkdw uhflsurflw| lv h{foxvlyho| lqwhqwlrqv gulyhq1

9

wkh uhflsurfdwlqj vxemhfw lv qrw wr uhgxfh lqhtxlw| exw wr orzhu wkh rssrqhqw*v sd|0

r�1 Uhflsurflw| gulyhq sxqlvkphqwv duh wkhuhiruh qrw uhvwulfwhg wr vlwxdwlrqv zkhuh

lqhtxlw| fdq eh uhgxfhg1 Lqvwhdg lw rffxuv zkhqhyhu d shuvrq lv wuhdwhg xqnlqgo| dqg

lv jlyhq d fkdqfh wr sd| edfn1

H{shulphqwdo hylghqfh vxjjhvwv wkdw pdq| vxemhfwv lq idfw sxqlvk rwkhuv hyhq

li sxqlvkphqw grhv qrw uhgxfh lqhtxlw|1 Idon/ Ihku dqg Ilvfkedfkhu +5333,

suhvhqw wkuhh h{shulphqwv zklfk dgguhvv wklv txhvwlrq lq juhdw ghwdlo1 Dv lw wxuqv

rxw/ d vxevwdqwldo dprxqw ri sxqlvkphqw rffxuv hyhq lq vlwxdwlrqv zkhuh lqhtxlw|

fdqqrw eh uhgxfhg1 Rqh ri wkhvh jdphv lv d wkuhh shuvrq sulvrqhu*v glohppd zlwk d

vxevhtxhqw sxqlvkphqw rssruwxqlw|1 Lq wkh sxqlvkphqw vwdjh/ vxemhfwv frxog � dw d

frvw � ghgxfw srlqwv iurp hdfk ri wkh rwkhu wzr sod|huv1 Lq d vlwxdwlrq zkhuh sxq0

lvklqj hyhq lqfuhdvhg lqhtxlw|/ 791; shufhqw ri wkh vxemhfwv zkr kdg frrshudwhg lq

wkh sulvrqhu*v glohppd qhyhuwkhohvv sxqlvkhg1 Wkh frqfoxvlrq iurp wkh wkuhh h{shul0

phqwv lv wkdw pxfk ri wkh revhuyhg sxqlvkphqwv duh dfwxdoo| wuljjhuhg dv d uhvsrqvh

wr xqnlqgqhvv/ qrw dv dq dwwhpsw wr uhgxfh lqhtxlw|1

Wkh uhsruwhg hylghqfh vxjjhvwv wkdw uhflsurflw| lv d pruh srzhuixo frqfhsw wkdq

lqhtxlw| dyhuvlrq1 Wklv lv vr ehfdxvh lw ehwwhu fdswxuhv wkh sv|fkrorjlfdo prwlydwlrqv1

Dv d frqvhtxhqfh lw dovr |lhogv ehwwhu suhglfwlrqv1

6 D wkhru| ri uhflsurflw|

Rxu wkhru| irupdol}hv wkh edvlf vwuxfwxuh ri uhflsurflw| zklfk frqvlvwv ri d nlqg +ru

xqnlqg, wuhdwphqw e| dqrwkhu shuvrq +uhsuhvhqwhg e| wkh nlqgqhvv whup *, dqg d

ehkdylrudo uhdfwlrq wr wkdw wuhdwphqw +uhsuhvhqwhg e| wkh uhflsurfdwlrq whup �,1 Rxu

surfhgxuh lv wr wudqvirup d vwdqgdug jdph lqwr d sv|fkrorjlfdo jdph/ wkh vr0fdoohg

�uhflsurflw| jdph�1 Lq wklv qhz jdph wkh sod|huv* xwlolw| ghshqgv qrw rqo| rq wkh

sd|r�v ri wkh ruljlqdo vwdqgdug jdph exw dovr rq wkh nlqgqhvv dqg wkh uhflsurfdwlrq

whup1 Lq wkh iroorzlqj/ zh ghulyh erwk whupv1

Frqvlghu d wzr0sod|hu h{whqvlyh irup jdph zlwk d �qlwh qxpehu ri vwdjhv dqg zlwk

frpsohwh dqg shuihfw lqirupdwlrq1 +Iru qrwdwlrqdo vlpsolflw| zh ghyhors wkh wkhru|

iru wkh wzr0sod|hu fdvh1 Wkh h{whqvlrq wr q0shuvrq jdphv lv jlyhq lq Dsshqgl{ 51,

Ohw l eh d sod|hu lq wkh jdph1 Q� ghqrwhv wkh vhw ri qrghv zkhuh sod|hu l kdv wkh

pryh zlwk q ehlqj d qrgh ri wklv sod|hu1 Ohw D? eh wkh vhw ri dfwlrqv lq qrgh q1 Ohw

I eh wkh vhw ri hqg qrghv ri wkh jdph1 Wkh sd|r� ixqfwlrq iru sod|hu l lv jlyhq e|

�� = I $ U1

Ohw S +D?, eh wkh vhw ri suredelolw| glvwulexwlrqv ryhu wkh vhw ri dfwlrqv lq qrgh

q1 Wkhq V� @T

?M��S +D?, lv sod|hu l*v ehkdylru vwudwhj| vsdfh1 Wkxv/ d sod|hu*v

ehkdylru vwudwhj| sxwv d suredelolw| glvwulexwlrq rq hdfk ri wkh sod|hu*v ghflvlrq

:

qrghv1 Ohw sod|hu m eh wkh rwkhu sod|hu zlwk ehkdylru vwudwhj| vsdfh V�1: Iru v� 5V� dqg v� 5 V� zh gh�qh ��+v�> v�, dqg ��+v�> v�, dv wkh sod|huv* h{shfwhg sd|r�v/

jlyhq vwudwhjlhv v� dqg v�1 Ixuwkhupruh/ zh gh�qh ��+q> v�> v�, dv wkh h{shfwhg sd|r�

frqglwlrqdo rq qrgh q= Lw lv wkh h{shfwhg sd|r� ri sod|hu l lq wkh vxejdph vwduwlqj

iurp qrgh q/ jlyhq wkdw wkh vwudwhjlhv v� dqg v� duh sod|hg dqg jlyhq lw lv nqrzq wkdw

sod|hu l lv dw qrgh q1;

Ohw v�� ghqrwh wkh �uvw rughu eholhi ri sod|hu l1 Lw fdswxuhv l*v eholhi derxw wkhehkdylru vwudwhj| v� 5 V� zklfk sod|hu m zloo fkrrvh1 Vlploduo|/ wkh vhfrqg rughueholhi v��� ri sod|hu l lv gh�qhg dv sod|hu l*v eholhi derxw sod|hu m*v eholhi derxw zklfk

ehkdylru vwudwhj| sod|hu l zloo fkrrvh1 Lq rwkhu zrugv/ v��� lv l*v eholhi derxw v�� 1 Olnh

Udelq +4<<6,/ zh dvvxph wkdw v�� lv dq hohphqw ri V� dqg v��� lv dq hohphqw ri V�1 D

vhw ri eholhiv lv vdlg wr eh frqvlvwhqw/ li v� @ v�� @ v��� krogv iru l 9@ m1

614 Wkh nlqgqhvv whup *

Wkh nlqgqhvv whup * lv wkh fhqwudo hohphqw ri rxu wkhru|1 Lw phdvxuhv krz nlqg d

shuvrq shufhlyhv wkh dfwlrq e| dqrwkhu sod|hu1 Dv rxwolqhg lq wkh suhylrxv vhfwlrq

wkh shufhlyhg nlqgqhvv ghshqgv rq wkh rxwfrph dqg wkh lqwhqwlrq1 Wkh rxwfrph

lv phdvxuhg zlwk wkh rxwfrph whup � zkhuh � A 3 h{suhvvhv dq dgydqwdjhrxv

rxwfrph dqg � ? 3 h{suhvvhv d glvdgydqwdjhrxv rxwfrph1 Lq rughu wr ghwhuplqh

wkh ryhudoo nlqgqhvv/ � lv pxowlsolhg zlwk wkh lqwhqwlrq idfwru &1 Wklv idfwru lv dqxpehu ehwzhhq }hur dqg rqh/ zkhuh & @ 4 fdswxuhv d vlwxdwlrq zkhuh � lv lqgxfhg

ixoo| lqwhqwlrqdoo| dqg & ? 4 h{suhvvhv d vlwxdwlrq zkhuh ohvv ru qr lqwhqwlrqv duh

lqyroyhg1 Wkh nlqgqhvv whup * lv vlpso| wkh surgxfw ri � dqg &1 Doo whupv duh

ghulyhg lq wkh iroorzlqj1

Iluvw/ zh gh�qh wkh rxwfrph whup=

�+q, =@ ��+q> v��� > v

��,� ��+q> v

��� > v

��, +4,

Wr lqwhusuhw wklv h{suhvvlrq/ ohw xv �{ wkh lqwhqwlrq idfwru &1 Iru d jlyhq &/ wkh

rxwfrph whup � fdswxuhv wkh nlqgqhvv ri sod|hu m= Wkh nlqgqhvv ri sod|hu m lq qrgh

q lv/ fhwhulv sdulexv/ kljkhu/ wkh pruh vkh r�huv wr sod|hu l1 Wklv lv h{suhvvhg lq

wkh whup ��+q> v��� > v��,1 Iurp sod|hu m*v shuvshfwlyh/ sod|hu m lv r�hulqj ��+q> v��> v�, wr

sod|hu l1 Wklv lv wkh sd|r� sod|hu l lv h{shfwhg wr jhw/ li sod|hu m fkrrvhv wr sod| v�

dqg h{shfwv sod|hu l wr fkrrvh v�� 1 Sxw gl�huhqwo|/ wklv lv 0 iurp wkh shuvshfwlyh ri

sod|hu m 0 zkdw sod|hu m r�huv wr sod|hu l/ jlyhq khu h{shfwdwlrq derxw v�1 Sod|hu l*v

eholhi derxw wklv r�hu lv ��+q> v��� > v��,1

:Wkurxjkrxw wkh sdshu zh zloo xvh wkh pdoh irup iru sod|hu � +dqg iru �uvw pryhuv,1 Iru sod|hu �

+dqg vhfrqg pryhuv, zh zloo xvh wkh ihpdoh irup1;Ehfdxvh zh duh zrunlqj zlwk ehkdylru vwudwhjlhv/ wkh vwudwhjlhv r� dqg r� lqgxfh d vwudwhj| lq

wkh vxejdph1

;

Vlqfh & lv srvlwlyh/ wkh vljq ri wkh nlqgqhvv whup/ l1h1/ zkhwkhu dq dfwlrq lv frq0

vlghuhg dv nlqg ru xqnlqg/ lv ghwhuplqhg e| wkh vljq ri �1 Lq rughu wr ghwhuplqh wkh

vljq ri �/ sod|hu l qhhgv wr frpsduh wkh r�hu ��+q> v��� > v��, zlwk d uhihuhqfh vwdqgdug1

Lq sulqflsoh/ wkhuh duh pdq| srvvleoh uhihuhqfh vwdqgdugv djdlqvw zklfk vxemhfwv fdq

frpsduh d sduwlfxodu rxwfrph1 Iurp pdq| h{shulphqwv zh nqrz/ krzhyhu/ wkdw dq

htxlwdeoh vkduh ri sd|r�v lv d vdolhqw dqg frpprqo| khog vwdqgdug1 < Wkh h{suhvvlrq

��+q> v��� > v

��, vhuyhv wkdw sxusrvh1 Lw lv sod|hu l*v eholhi derxw zkdw sd|r� sod|hu m

zdqwv wr nhhs iru khuvhoi1 Wdnhq wrjhwkhu/ li ��+q> v��� > v��, A ��+q> v��� > v

��, krogv/ sod|hu

l wklqnv wkdw sod|hu m zdqwv klp wr jhw pruh rxw ri wkh h{fkdqjh wkdq sod|hu m zdqwv

iru khuvhoi/ l1h1/ sod|hu l eholhyhv wkdw sod|hu m lv dfwlqj nlqgo|1 Li/ rq wkh rwkhu kdqg/

��+q> v��� > v

��, ? ��+q> v

��� > v

��, krogv/ sod|hu l eholhyhv wkdw sod|hu m fodlpv pruh iru khuvhoi

wkdq vkh lv zloolqj wr ohdyh iru sod|hu l1 Lq wklv fdvh/ sod|hu l shufhlyhv sod|hu m dv

ehlqj xqnlqg1

Qrwh wkdw zh wdon d elw orrvho| derxw wkh nlqgqhvv ri dq dfwlrq1 Wkh zd| zh

prgho nlqgqhvv frpsulvhv erwk wkh nlqgqhvv ri dfwxdoo| rffxuuhg dfwlrqv dv zhoo dv

dqwlflsdwhg ixwxuh dfwlrqv1 Wkh dfwlrqv zklfk douhdg| rffxuuhg duh uhsuhvhqwhg lq

qrgh q ehfdxvh ehlqj lq qrgh q uhvxowv iurp sod|hu m*v dfwxdoo| fkrvhq dfwlrqv1 Wkh

dqwlflsdwhg dfwlrqv ri sod|hu m duh vlpso| fdswxuhg e| v��1

Wkh rxwfrph whup �+q, fdswxuhv wkh nlqgqhvv ru xqnlqgqhvv ri sod|hu m wrzdugv

sod|hu l iru d jlyhq &1 Lq mxgjlqj sod|hu m*v nlqgqhvv/ krzhyhu/ sod|hu l lv zhoo dzduh ri

wkh idfw wkdw sod|hu m pljkw qrw kdyh fdxvhg d sduwlfxodu rxwfrph lqwhqwlrqdoo|1 Wkh

�0whup grhv qrw gl�huhqwldwh ehwzhhq vlwxdwlrqv zkhuh sod|hu m kdg d uhdvrqdeoh

dowhuqdwlyh dqg wkrvh zkhuh kh kdg qrqh1 Lq rughu wr dffrxqw iru wklv gl�huhqfh/

�+q, lv pxowlsolhg zlwk & zklfk ghshqgv rq sod|hu m*v vhw ri dowhuqdwlyhv1 Ohw xv

vwdwh suhflvho| zkdw zh phdq e| dowhuqdwlyh sd|r� frpelqdwlrqv1 Ohw VR� eh wkh vhw

ri sxuh vwudwhjlhv ri sod|hu m1 Iru jlyhq vwudwhjlhv dqg eholhiv zh gh�qh=

�� =@q+��+v

��� > v

R�,> ��+v

��� > v

R� ,, m vR� 5 VR

�

r+5,

�� lv d vhw ri sd|r� frpelqdwlrqv �� dqg ��1 Wkhvh duh sd|r�v sod|hu m fdq lqgxfh e|

fkrrvlqj d sxuh vwudwhj| vR� jlyhq khu h{shfwdwlrq derxw sod|hu l*v vwudwhj|1 Vlqfh ��

lv ghwhuplqhg iurp sod|hu l*v shuvshfwlyh/ sod|hu l wdnhv lqwr dffrxqw klv eholhi derxw

zklfk vwudwhj| sod|hu m eholhyhv kh zloo fkrrvh/ qdpho| v��� 1 Lq vkruw/ �� lv sod|hu l*v

eholhi derxw doo sd|r� frpelqdwlrqv sod|hu m frqvlghuv dv khu rssruwxqlw| vhw/ l1h1/

jlyhq v��� lw frqwdlqv doo rswlrqv dydlodeoh wr sod|hu m1

<Wkh lghd wkdw htxlw| frqvlghudwlrqv duh lpsruwdqw lq wkh frqwh{w ri mxgjlqj nlqgqhvv zdv �uvw

ghyhorshg lq wkh vr0fdoohg htxlw| wkhru|1 Ehjlqqlqj lq wkh odwh vl{wlhv vrfldo sv|fkrorjlvwv ghyhorshg

htxlw| wkhru| dv d vshfldo irup ri vrfldo h{fkdqjh wkhru|1 Frpsduh/ h1j1/ Dgdpv +4<98, dqgZdovwhu

dqg zdovwhu +4<:;,1 Vhh dovr Orhzhqvwhlq/ Wkrpsvrq dqg Ed}hupdq +4<;<, dqg Vhowhq

+4<:;,1

<

Zkhwkhu sod|hu l h{shulhqfhv d sduwlfxodu rxwfrph dv fkrvhq lqwhqwlrqdoo| ghshqgv

rq wkh rswlrqv dydlodeoh wr sod|hu m1 Wkh txhvwlrq wr dvn lv= Krz lqwhqwlrqdo lv

d sduwlfxodu sd|r� glvwulexwlrq +�f� > �f� , lqgxfhg e| sod|hu m 0 jlyhq wkdw sod|hu m

kdg dq dowhuqdwlyh sd|r� glvwulexwlrq +��> ��, kh frxog kdyh fkrvhqB Wkh dqvzhu

wr wklv txhvwlrq lv jlyhq lq ixqfwlrq 1 Lq wklv ixqfwlrq/ wkh ydoxh ri h{suhvvhv

krz lqwhqwlrqdo sod|hu m*v fkrlfh ri +�f� > �f� , zdv/ jlyhq klv dowhuqdwlyhv +��> ��,1 Li

wkh fkrlfh zdv ixoo| lqwhqwlrqdo htxdov 4/ li wkh fkrlfh lv frqvlghuhg dv qrw ixoo|

lqwhqwlrqdo/ krzhyhu/ lv vpdoohu wkdq rqh1 Wkh 0ixqfwlrq lv gh�qhg dv iroorzv=

+��> ��> �f� > �

f�, =@

;AAAAAAAA?AAAAAAAA=

4 li �f� � �f� dqg �� ? �f�%� li �f� � �f� dqg �� � �f�4 li �f� ? �f� / �� A �f� dqg �� � ��

pd{

�4� Z�3Z�

Zf�3Zf�

> %�

�li �f� ? �f� / �� A �f� dqg �� A ��

%� li �f� ? �f� dqg �� � �f�

+6,

zkhuh %� lv dq lqglylgxdo sdudphwhu zlwk 3 � %� � 4143

Wkh �uvw wzr urzv fdswxuh vlwxdwlrqv zkhuh sod|hu m kdv wuhdwhg sod|hu l lq d nlqg

zd| +�f� � �f� ,1 Lq wkhvh vlwxdwlrqv wkh ydoxh ri ghshqgv rq zkhwkhu sod|hu m frxog

uhgxfh sod|hu l*v sd|r� +�� frpsduhg wr �f� , ru qrw1 Dvvxph/ dv vkrzq lq wkh �uvw

urz/ sod|hu m kdv wkh dowhuqdwlyh wr orzhu sod|hu l*v sd|r� +l1h1/ �� ? �f� ,1 Wkhq sod|hu

l frqvlghuv wkh nlqg dfwlrq dv ixoo| lqwhqwlrqdo1 Diwhu doo/ sod|hu m frxog kdyh wuhdwhg

sod|hu l zruvh exw fkrvh qrw wr gr vr1 Li/ krzhyhu/ sod|hu m*v rqo| dowhuqdwlyh lv wr

lqfuhdvh sod|hu l*v sd|r� +l1h1/ �� � �f� , wkhq sod|hu m kdv qr dowhuqdwlyh wr eh ohvv

nlqg1 Frqvhtxhqwo|/ klv nlqgqhvv lv frqvlghuhg dv ohvv lqwhqwlrqdo1 Diwhu doo/ lw lv qrw

sod|hu m*v phulw wkdw kh zdv cnlqg*1 Lq wklv vlwxdwlrq htxdov %�1

Wkh rwkhu wkuhh urzv uhsuhvhqw lqvwdqfhv zkhuh sod|hu m sxwv sod|hu l lq d glvdg0

ydqwdjhrxv vlwxdwlrq/ l1h1/ zkhuh �f� ? �f� krogv1 Zkhwkhu sod|hu l lq idfw shufhlyhv

wklv dv dq lqwhqwlrqdoo| xqnlqg dfwlrq ghshqgv djdlq rq sod|hu m*v dydlodeoh dowhuqd0

wlyhv1 Li sod|hu m kdv wkh dowhuqdwlyh wr lpsuryh sod|hu l*v sd|r� zlwkrxw sxwwlqj

klpvhoi lq d glvdgydqwdjhrxv vlwxdwlrq +�� A �f� dqg �� � ��,/ klv xqnlqgqhvv lv ixoo|

lqwhqwlrqdo1 Wkhuhiruh/ lv htxdo wr 41 Qrz vxssrvh wkdw wkhuh lv dq dowhuqdwlyh wr

lpsuryh sod|hu l*v sd|r� zklfk dw wkh vdph wlph ohdgv wr d glvdgydqwdjhrxv vlwxdwlrq

iru sod|hu m1 Wkh pruh wklv dowhuqdwlyh lv glvdgydqwdjhrxv iru sod|hu m/ wkh ohvv uhd0

vrqdeoh lw lv frqvlghuhg1 Diwhu doo lw zrxog lpso| dq xquhdvrqdeoh vdful�fh e| sod|hu

m1 Dv d frqvhtxhqfh/ wkh fkrlfh ri �f� lv qrw frqvlghuhg dv ixoo| lqwhqwlrqdoo| xqnlqg

dqg lv htxdo wr pd{

�4� Z�3Z�

Zf�3Zf�

> %�

�� 41 Wkh h{suhvvlrq 4� Z�3Z�

Zf�3Zf�

phdvxuhv ckrz

idu* sod|hu m pxvw vzlwfk wr wkh glvdgydqwdjhrxv vlgh li vkh zdqwv wr lpsuryh sod|hu l*v

43Wklv sdudphwhu lv fdoohg wkh sxuh rxwfrph frqfhuq sdudphwhu1 Lw lv lqwhusuhwhg ehorz1

43

sd|r� 0 uhodwhg wr wkh uhihuhqfh vlwxdwlrq +�f� > �f�,1 Li/ h1j1/ sod|hu m pxvw rqo| vzlwfk

d olwwoh elw wr wkh glvdgydqwdjhrxv vlgh +wkh qxphudwru lv vpdoo, wkh dowhuqdwlyh dfwlrq

zloo fhwhulv sdulexv eh frqvlghuhg dv udwkhu uhdvrqdeoh1 Li/ krzhyhu/ wkh qxphudwru lv

odujh +lq sduwlfxodu/ li wkh qxphudwru lv odujhu wkdq wkh ghqrplqdwru, lv htxdo wr

%�1 Ilqdoo|/ li sod|hu m*v rqo| dowhuqdwlyh lv wr fkrrvh dq hyhq orzhu sd|r� iru sod|hu

l/ l1h1/ �� � �f� / sod|hu l fdqqrw lqihu wkdw sod|hu m zdqwhg wr wuhdw klp lq dq xqnlqg

idvklrq1 Frqvhtxhqwo|/ wkh dfwlrq zdv xqlqwhqwlrqdoo| cxqnlqg* |lhoglqj @ %�1

Dv wkh uhihuhqfh glvwulexwlrq +�f� > �f� , zh xvh wkh sd|r�v wkdw ghwhuplqh wkh rxw0

frph whup �+q,/ qdpho| ��+q> v��� > v

��, dqg ��+q> v

��� > v

��,1 Wkxv/ zh gh�qh wkh lqwhqwlrq

idfwru=

&+q, @ pd{� +��> �� > ��+q> v

��� > v

��,> ��+q> v

��� > v

��,, m +��> ��, 5 ��

�+7,

Wkh pd{lpxp0rshudwru jxdudqwhhv wkdw d sduwlfxodu dfwlrq lv frqvlghuhg dv lqwhq0

wlrqdo li wkhuh lv dq| cwuxh* dowhuqdwlyh1 Zh qrz frph wr wkh gh�qlwlrq ri wkh nlqgqhvv

whup1

Gh�qlwlrq 4 Ohw vwudwhjlhv dqg eholhiv eh jlyhq1 Zh gh�qh wkh nlqgqhvv whup*+q, lq d qrgh q 5 Q� dv=

*+q, @ &+q,�+q, +8,

Lqwhusuhwdwlrq= Dffruglqj wr Gh�qlwlrq 4 dqg htxdwlrqv +6, wr +8,/ wkh nlqgqhvvwhup lv htxdo wr �+q, zlwk wkh h{fhswlrqv phqwlrqhg lq rxu glvfxvvlrq ri 1 Lq wkhvh

h{fhswlrqv wkh nlqgqhvv ru xqnlqgqhvv/ uhvshfwlyho|/ lv qrw ixoo| lqwhqwlrqdo lq wkh

vhqvh wkdw sod|hu m kdg qr +uhdvrqdeoh, dowhuqdwlyh+v,1 Wkhuhiruh/ lq wkhvh fdvhv/ wkh

nlqgqhvv whup lv uhgxfhg e| d idfwru ehwzhhq %� dqg 41 Ohw xv looxvwudwh wkh orjlf

ri wkh nlqgqhvv whup zlwk dq h{dpsoh= Lpdjlqh sod|hu m kdv wkuhh dowhuqdwlyh sd|r�

vkduhv +��/��, kh fdq fdxvh qdpho| +;/5,/ +:/6, dqg +9/7,1 Lq doo wkhvh vlwxdwlrqv

sod|hu l lv idyruhg/ l1h1/ wkh rxwfrph whup � lv srvlwlyh1 Vxssrvh sod|hu m fkrrvhv

hlwkhu +;/5, ru +:/6,1 Iurp sod|hu l*v shuvshfwlyh erwk fkrlfhv duh ixoo| lqwhqwlrqdo

+zlwk @ & @ 4, iru sod|hu m frxog kdyh fkrvhq d zruvh dowhuqdwlyh/ qdpho| +9/7,1

Ri frxuvh/ wkh �0whup +dqg wkhuhiruh wkh nlqgqhvv, lv kljkhu li m fkrrvhv +;/5, lqvwhdg

ri +:/6,1 Qrz vxssrvh sod|hu m fkrrvhv wkh dowhuqdwlyh +9/7,1 Vwloo sod|hu l lv idyruhg1

Krzhyhu/ wkh nlqgqhvv whup lv vpdoohu wkdq lq wkh vlwxdwlrq zkhuh +;/5, ru +:/6, lv

fkrvhq1 Pruhryhu/ sod|hu l nqrzv wkdw sod|hu m kdg qr dowhuqdwlyh wr wuhdw klp zruvh1

Wkxv kh fdqqrw lqihu dq| srvlwlyh lqwhqwlrqv/ l1h1/ @ & @ %�/ uhvxowlqj lq dq hyhq

orzhu nlqgqhvv whup1 Qrwh wkdw lq wkh vshfldo fdvh zkhuh sod|hu m kdv qr dowhuqdwlyh

dw doo +l1h1/ m��m @ 4,/ &+q, htxdov %�1 Lq prvw jdphv/ krzhyhu/ wkh sod|huv kdyh d

udwkhu xqolplwhg vwudwhj| lpso|lqj d &+q, ri 41

Wkh lqglylgxdo sdudphwhu %� lv fdoohg wkh sxuh rxwfrph frqfhuq sdudphwhu1Lw phdvxuhv d sod|hu*v sxuh frqfhuq iru dq htxlwdeoh rxwfrph= Li/ h1j1/ %� lv htxdo wr

44

}hur/ sod|hu l frqvlghuv d sduwlfxodu rxwfrph rqo| dv nlqg ru xqnlqg li lw zdv fdxvhg

lqwhqwlrqdoo|/ l1h1/ li wkh rwkhu sod|hu kdg dq dowhuqdwlyh wr dfw gl�huhqwo|1 Li/ rq wkh

rwkhu kdqg/ %� @ 4/ sod|hu l fduhv rqo| derxw dq htxlwdeoh rxwfrph/ l1h1/ kh grhv qrw

sd| dq| dwwhqwlrq wr wkh rwkhu sod|hu*v dowhuqdwlyhv dqg lqwhqwlrqv/ uhvshfwlyho|1 Lq

wklv fdvh +dqg wkhuhiruh &, lv dozd|v htxdo wr 41 Wkxv/ vd|lqj wkdw d shuvrq*v %�lv htxdo wr 4 lv htxlydohqw wr vd|lqj wkdw wklv shuvrq lv sxuho| rxwfrph rulhqwhg dv

lw lv vxjjhvwhg e| wkh prghov ri Erowrq dqg Rfnhqihov +5333, dqg Ihku dqg

Vfkplgw +4<<<,1 Lqvridu/ wkhlu prghov fdq eh ylhzhg dv d vshfldo fdvh ri rxu wkhru|1

615 Wkh uhflsurfdwlrq whup �

Wkh vhfrqg lqjuhglhqw ri rxu wkhru| frqfhuqv wkh irupdol}dwlrq ri uhflsurfdwlrq1 Ohw

xv �{ dq hqg qrgh i wkdw iroorzv qrgh q1 Wkhq zh ghqrwh e| �+q> i, wkh xqltxh qrgh

wkdw gluhfwo| iroorzv qrgh q rq wkh sdwk wkdw ohdgv iurp q wr i=

Gh�qlwlrq 5 Ohw vwudwhjlhv dqg eholhiv eh jlyhq dv deryh1 Ohw l dqg m eh wkh wzr

sod|huv dqg q dqg i eh gh�qhg dv deryh1 Wkhq zh gh�qh

�+q> i, =@ ��+�+q> i,> v��� > v

��,� ��+q> v

��� > v

��, +9,

dv wkh uhflsurfdwlrq whup ri sod|hu l lq qrgh q1

Wkh uhflsurfdwlrq whup h{suhvvhv wkh uhvsrqvh wr wkh h{shulhqfhg nlqgqhvv/ l1h1/

lw phdvxuhv krz pxfk sod|hu l dowhuv wkh sd|r� ri sod|hu m zlwk klv pryh lq qrgh q1

Jlyhq sod|hu l*v eholhi derxw sod|hu m*v h{shfwdwlrqv derxw khu sd|r� lq qrgh q +l1h1/

jlyhq ��+q> v��� > v

��,,/ sod|hu l fdq 0 lq qrgh q 0 fkrrvh dq dfwlrq1 Wkh uhflsurfdo lpsdfw

ri wklv dfwlrq lv uhsuhvhqwhg dv wkh dowhudwlrq ri sod|hu m*v sd|r� iurp ��+q> v��� > v

��, wr

��+�+q> i,> v��� > v

��, +dozd|v iurp sod|hu l*v shuvshfwlyh,1 Iru d jlyhq ��+q> v��� > v

��,/ sod|hu

l fdq/ wkxv/ fkrrvh wr hlwkhu uhzdug ru wr sxqlvk sod|hu m1 D uhzduglqj dfwlrq lpsolhv

d srvlwlyh/ zkhuhdv d sxqlvkphqw lpsolhv d qhjdwlyh uhflsurfdwlrq whup1

Qrwdwlrq= Ohw q� dqg q2 eh qrghv1 Li qrgh q2 iroorzv qrgh q� +gluhfwo| ru

lqgluhfwo|,/ zh ghqrwh wklv e| q� $ q21

616 Wkh xwlolw| ixqfwlrq

Kdylqj gh�qhg wkh nlqgqhvv dqg uhflsurfdwlrq whup zh fdq qrz ghulyh wkh sod|huv*

xwlolw| ri wkh wudqviruphg �uhflsurflw| jdph�=

Gh�qlwlrq 6 Ohw sod|hu l dqg m eh wkh wzr sod|huv ri wkh jdph1 Ohw i eh dq hqg

qrgh ri wkh jdph1 Zh gh�qh wkh xwlolw| lq wkh wudqviruphg uhflsurflw| jdph dv=

X�+i, @ ��+i, . ��[?<s?M��

*+q,�+q> i, +:,

45

Dffruglqj wr Gh�qlwlrq 6 sod|hu l*v xwlolw| lq wkh uhflsurflw| jdph lv wkh vxp ri

wkh iroorzlqj wzr whupv= Wkh �uvw vxppdqg lv vlpso| sod|hu l*v pdwhuldo sd|r���+i,1 Wkh vhfrqg vxppdqg 0 zklfk zh fdoo uhflsurflw| xwlolw| 0 lv frpsrvhg ri=

� Wkh srvlwlyh frqvwdqw ��> wkh uhflsurflw| sdudphwhu1 Wklv frqvwdqw lv dq

lqglylgxdo sdudphwhu zklfk fdswxuhv wkh vwuhqjwk ri sod|hu l*v uhflsurfdo suhi0

huhqfhv1 Lw wdnhv dffrxqw ri wkh idfw wkdw lqglylgxdov gl�hu lq wkhlu uhflsurfdo

lqfolqdwlrq1 Wkh kljkhu ��/ wkh pruh lpsruwdqw lv wkh uhflsurflw| xwlolw| dv frp0

sduhg wr wkh xwlolw| dulvlqj iurp wkh pdwhuldo sd|r�1 Qrwh wkdw li �� htxdov }hur/

sod|hu l*v xwlolw| lv htxdo wr klv pdwhuldo sd|r�/ l1h1/ lghqwlfdo wr wkh vwdqgdug

jdph1 Li/ lq dgglwlrq/ �� dovr htxdov }hur/ wkh uhflsurflw| jdph froodsvhv lqwr

wkh vwdqgdug jdph1

� Wkh nlqgqhvv whup *+q, zklfk phdvxuhv wkh nlqgqhvv sod|hu l h{shulhqfhv

iurp m*v +h{shfwhg, dfwlrqv1 Dv qrwhg deryh lw lv srvlwlyh li sod|hu m lv frqvlghuhg

dv nlqg dqg qhjdwlyh li sod|hu m lv frqvlghuhg dv xqnlqg1 Wkh nlqgqhvv whup

frpsulvhv rxwfrphv dqg lqwhqwlrqv1

� Wkh uhflsurfdwlrq whup �+q> i,1 Lw phdvxuhv wkh h�hfw ri wkh uhflsurfdo

dfwlrq1

� Wkh surgxfw ri wkh nlqgqhvv dqg wkh uhflsurfdwlrq whup phdvxuhv wkh uhflsurflw|

xwlolw| lq d sduwlfxodu qrgh1 Li wkh nlqgqhvv whup lq d sduwlfxodu qrgh q lv

juhdwhu wkdq }hur/ sod|hu l fdq fhwhulv sdulexv lqfuhdvh klv xwlolw| li kh fkrrvhv

dq dfwlrq lq wkdw qrgh zklfk lqfuhdvhv sod|hu m*v sd|r�1 Wkh rssrvlwh krogv li

wkh nlqgqhvv whup lv qhjdwlyh1 Lq wklv fdvh/ sod|hu l kdv dq lqfhqwlyh wr uhgxfh

sod|hu m*v sd|r�1

� Vlqfh nlqgqhvv lv phdvxuhg lq hdfk qrgh zkhuh sod|hu l kdv wkh pryh/ wkh ryhudoo

uhflsurflw| xwlolw| lv wkh vxp ri wkh uhflsurflw| xwlolw| lq doo qrghv +ehiruh wkh

frqvlghuhg hqg qrgh,/ zhljkwhg zlwk wkh uhflsurflw| sdudphwhu1

617 Wkh Uhflsurflw| Htxloleulxp

Wkh lqwurgxfhg suhihuhqfhv irup d sv|fkrorjlfdo jdph +Jhdqdnrsorv/ Shdufh dqg

Vwdffkhwwl +4<;<,,1 Lq sv|fkrorjlfdo jdphv/ wkh xwlolw| ri d sod|hu l grhv qrw rqo|

ghshqg rq wkh vhohfwhg vwudwhjlhv ri wkh sod|huv exw dovr rq wkh eholhiv sod|hu l kdv

+frpsduh Gh�qlwlrq 6,1 Qrwh/ krzhyhu/ wkdw eholhiv duh qrw sduw ri wkh dfwlrq vsdfh1

Sxw gl�huhqwo|/ eholhiv fdqqrw eh iruphg vwudwhjlfdoo|/ l1h1/ wkh| duh wdnhq dv jlyhq1

Jlyhq wkh eholhiv/ sod|hu l fkrrvhv klv rswlpdo vwudwhj|1 Wkh dgglwlrqdo uhtxluhphqw

lq d sv|fkrorjlfdo Qdvk htxloleulxp dv frpsduhg wr d Qdvk htxloleulxp lv wkdw doo

46

eholhiv pdwfk dfwxdo ehkdylru1 Wklv phdqv/ wkdw dq rswlpdo vwudwhj| lv rqo| sduw ri

dq htxloleulxp li wkh eholhiv duh dovr frqvlvwhqw zlwk wkh dfwxdo ehkdylru144

Jhdqdnrsorv/ Shdufh dqg Vwdffkhwwl +4<;<, vkrz wkdw wkh uh�qhphqw frq0

fhsw ri vxejdph shuihfwqhvv fdq dovr eh dssolhg wr sv|fkrorjlfdo Qdvk htxloleuld1 Lq

rxu uhflsurflw| jdph zh zloo fdoo d vxejdph shuihfw sv|fkrorjlfdo Qdvk htxloleulxp d

uhflsurflw| htxloleulxp1 Li �� @ �� @ 3/ wkh gh�qlwlrq ri d uhflsurflw| htxloleulxp

lv htxlydohqw wr wkh gh�qlwlrq ri d vxejdph shuihfw Qdvk htxloleulxp145

Wkh frqfhsw ri sv|fkrorjlfdo jdph wkhru| doorzv xv wr irupdol}h wkh frqfhsw ri

lqwhqwlrqv lq d vwudljkwiruzdug zd|1 Dv zh kdyh vhhq lq Vhfwlrq 5 dqg dv zh zloo

vkrz lq wkh qh{w vhfwlrq/ wkh irupdol}dwlrq ri lqwhqwlrqv lv lpsruwdqw vlqfh sxuho|

frqvhtxhqwldolvwlf prghov idlo wr h{sodlq uhohydqw dvshfwv ri wkh gdwd1

7 Dssolfdwlrqv

Pdq| vfkroduv kdyh suhvhqwhg frpshoolqj hylghqfh rq wkh uhohydqfh ri uhflsurflw| lq d

ydulhw| ri hfrqrplfdoo| lpsruwdqw vlwxdwlrqv1 Wkh prvw fohdu0fxw hylghqfh rq srvlwlyh

dqg qhjdwlyh uhflsurflw| vwhpv iurp frqwuroohg oderudwru| h{shulphqwv1 Lq wklv vhfwlrq

zh/ wkhuhiruh/ glvfxvv wkh suhglfwlrqv ri rxu wkhru| lq gl�huhqw h{shulphqwdo jdphv1

Wkh jdphv zh vwxg| duh wkh xowlpdwxp jdph/ wkh jliw0h{fkdqjh jdph/ d uhgxfhg

ehvw0vkrw jdph/ pdunhw jdphv zlwk sursrvhu ru uhvsrqghu frpshwlwlrq/ wkh glfwdwru

jdph/ wkh sulvrqhu*v glohppd/ sxeolf jrrgv jdphv/ dqg wkh lqyhvwphqw jdph1 Qrwlfh

wkdw lq doo �jxuhv zklfk looxvwudwh rxu sursrvlwlrqv zh xvh wkh vdph vhw ri sdudphwhuv

+3 � �� � 7 ru �� @ 5/ %� @ 3=5,1

714 Qhjdwlyh uhflsurflw|= Wkh xowlpdwxp jdph

Ohw xv vwduw rxu dssolfdwlrqv vhfwlrq zlwk dq h{dpsoh ri qhjdwlyh uhflsurflw|1 Wkh

prvw nqrzq jdph lq zklfk qhjdwlyh uhflsurflw| dssolhv lv wkh xowlpdwxp jdph1 Lq

wklv wzr shuvrq vhtxhqwldo pryh jdph/ wkh �uvw pryhu +�sursrvhu�, lv doorfdwhg dq

44Dvvxplqj udwlrqdolw| ri sod|huv jxdudqwhhv wkdw wkh rswlpdo vwudwhj| lv fkrvhq1 Wkh frlqflghqfh

ri dfwxdo ehkdylru dqg eholhiv/ rq wkh rwkhu kdqg/ fdqqrw vlpso| eh mxvwl�hg rq wkh edvlv ri udwlrqdolw|1

Lw lv yhu| sodxvleoh/ krzhyhu/ wr dvvxph wkdw srvvleoh lqfrqvlvwhqflhv ehwzhhq dfwxdo ehkdylru dqg

�uvw rughu eholhiv zloo glh rxw zlwk vrph h{shulhqfh1 Ohw xv wkhuhiruh dvvxph wkdw wkh �uvw rughu eholhi

ri sod|hu � +hyhqwxdoo|, pdwfkhv dfwxdo ehkdylru ri sod|hu �1 Qrz/ li sod|hu � irupv klv vhfrqg rughu

eholhi lq wkh vdph zd| dv sod|hu � irupv klv �uvw rughu eholhi +h1j1/ e| revhuylqj dfwxdo ehkdylru ru

lqwurvshfwlrq, wkhq sod|hu � zloo eh deoh wr pdwfk klv vhfrqg rughu eholhiv zlwk wkh �uvw rughu eholhiv

ri sod|hu �145D uhpdun rq wkh h{lvwhqfh ri uhflsurflw| htxloleuld= Lq wkh suhvhqwhg irup/ d uhflsurflw| htxlole0

ulxp grhv qrw dozd|v h{lvw ehfdxvh wkh ixqfwlrq l lv glvfrqwlqxrxv dw Z� ' Zf� 1 D plqru whfkqlfdo

prgl�fdwlrq ri l/ krzhyhu/ jxdudqwhhv wkh h{lvwhqfh ri d uhflsurflw| htxloleulxp1 Iru wkh hdvh ri

h{srvlwlrq zh ghohjdwh wkh h{lvwhqfh surri iru wkh prgl�hg l wr Dsshqgl{ 41

47

dprxqw ri prqh| +zklfk zh qrupdol}h wr 4,1 Wkh sursrvhu kdv wr glylgh wklv dprxqw

ehwzhhq klpvhoi dqg d vhfrqg pryhu +�uhvsrqghu�,1 Kh pd| r�hu dq| ihdvleoh dprxqw

f wr wkh uhvsrqghu/ l1h1/ 3 � f � 41 Diwhu wkh r�hu lv uhyhdohg wr wkh uhvsrqghu/ wkh

odwwhu fdq hlwkhu dffhsw ru uhmhfw lw1 Li vkh dffhswv/ wkh uhvxowlqj sd|r�v duh 4� f iru

wkh sursrvhu dqg f iru wkh uhvsrqghu1 Li wkh uhvsrqghu uhmhfwv wkh r�hu/ sd|r�v duh

}hur iru erwk sduwlhv1 Jlyhq wkh vwdqgdug dvvxpswlrqv/ wkh rxwfrph dffruglqj wr wkh

vxejdph shuihfw Qdvk htxloleulxp lv +f @ 3> dffhsw,1

Wkh xowlpdwxp jdph kdv lqwhqvlyho| ehhq vwxglhg1 Ryhuylhzv ri h{shulphqwdo

uhvxowv duh suhvhqwhg/ h1j1/ lq J wk/ Vfkplwwehujhu dqg Vfkzdu}h +4<;5,/

Wkdohu +4<;;,/J wk +4<<8,/ Fdphuhu dqg Wkdohu +4<<8, dqgUrwk +4<<8,146

Wkh uhsruwhg ehkdylrudo uhjxodulwlhv duh txlwh urexvw dqg fdq eh vxppdul}hg dv iro0

orzv= Wkhuh duh +l, sudfwlfdoo| qr r�huv wkdw h{fhhg 3=8/ +ll, wkh prgdo r�huv olh lq d

udqjh ehwzhhq 3=7 dqg 3=8/ +lll, r�huv ehorz 3=5 duh h{wuhpho| uduh/ dqg +ly, zkhuhdv

r�huv forvh wr 3=8 duh sudfwlfdoo| qhyhu uhmhfwhg/ wkh uhmhfwlrq udwh iru r�huv ehorz 3=5

lv udwkhu kljk1 Wkxv/ wkh vwdqgdug vxejdph shuihfw suhglfwlrq lv vwurqjo| uhixwhg e|

wkh vw|ol}hg idfwv1

Zh qrz vwdwh rxu suhglfwlrqv1 Xsrq dffhswdqfh/ pdwhuldo sd|r�v duh 4�f iru wkh

sursrvhu dqg f iru wkh uhvsrqghu/ uhvshfwlyho|1 Ohw s ghqrwh wkh suredelolw| wkdw wkh

uhvsrqghu dffhswv wkh r�hu1

Qrwdwlrq= Iru wkh iroorzlqj dssolfdwlrqv lw lv xvhixo wr lqwurgxfh wkh qrwdwlrqv

^f`Kod@ =@ plq+e>pd{+d> f,,> ^f`d@ =@ pd{+d> f,> dqg ^f`Ko =@ plq+e> f,1 Wkh h{suhvvlrq

^f`Kod@ lv vlpso| f li f olhv zlwklq wkh lqwhuydo ^d> e` dqg lv wkh qhduhvw hqg srlqw ri ^d> e`

rwkhuzlvh1

Sursrvlwlrq 4 Li �� dqg �2 duh srvlwlyh wkhuh lv d xqltxh uhflsurflw| htxloleulxp

+fW> sW, lq wkh xowlpdwxp jdph dv iroorzv=

sW @

;?=k

S42uE�32S�E�3S�

l�oli f ? �

2

4 li f � �2

+;,

fW @ pd{

%4 . 6�2 �

s4 . 9�2 . �22

7�2>4

5� +4� 4

��,

&+<,

Li hlwkhu �� ru �2 lv }hur sW dqg fW duh wkh olplwv ri wkh deryh irupxodv zkhuh �� dqg

�2 dssurdfk }hur iurp deryh1

Li �� dqg �2 duh erwk }hur/ sW @ 4 dqg fW @ 31

46Dq lqwhuhvwlqj h{whqvlrq ri wkh xowlpdwxp jdph wr pruh wkdq wzr sod|huv lv suhvhqwhg lq J wk/

Kxfn dqg Rfnhqihov +4<<9,1

48

Wkh surri ri Sursrvlwlrq 4 lv jlyhq lq Dsshqgl{ 61

Glvfxvvlrq= Ohw xv �uvw orrn dw wkh uhvsrqghu*v ehkdylru1 Htxdwlrq +;, uhyhdov

wkh frqglwlrqv wkdw ghwhuplqh wkh dffhswdqfh suredelolw| sW ri dq r�hu f lq wkh uhfl0

surflw| htxloleulxp= Iluvw/ li wkh sursrvhu*v r�hu lv htxdo ru kljkhu wkdq kdoi ri wkh

slh/ l1h1/ f � �2 / wkh uhvsrqghu zloo dozd|v dffhsw wkh r�hu1 Wklv krogv lqghshq0

ghqw rq wkh uhvsrqghu*v frqfhuq iru uhflsurflw| +frpsduh wkh vhfrqg urz ri Htxdwlrq

+;,,1 Vhfrqg/ iru r�huv vpdoohu wkdq kdoi ri wkh slh wkh zloolqjqhvv wr dffhsw dq r�hu

lv lqfuhdvlqj lq wkh ohyho ri wkh r�hu dqg ghfuhdvlqj lq wkh uhvsrqghu*v frqfhuq iru

uhflsurflw|/ �2 +vhh wkh �uvw urz ri Htxdwlrq +;,,1 Iljxuh 5 ghslfwv wkh dffhswdqfh

suredelolw| dv d ixqfwlrq ri wkh ohyho ri wkh r�hu iru d jlyhq uhflsurfdo lqfolqdwlrq ri

wkh uhvsrqghu1 Wkh �jxuh qhdwo| fdswxuhv wkh hvvhqfh ri qhjdwlyh uhflsurflw|= Wkh

orzhu wkh r�hu/ wkh kljkhu lv wkh zloolqjqhvv ri d uhflsurfdoo| prwlydwhg uhvsrqghu wr

sxqlvk wkh sursrvhu e| uhmhfwlqj wkh r�hu1 Qrwh wkdw dv �2 jhwv kljkhu +orzhu, wkh

dffhswdqfh0fxuyh vkliwv wr wkh uljkw +ohiw,1

0

0.2

0.4

0.6

0.8

1

0 0.25 0.5 0.75 1offer c

acce

ptan

ce p

roba

bili

ty

c0

Iljxuh 5= Dffhswdqfh suredelolw| lq wkh xowlpdwxp jdph ghshqghqw rq wkh r�hu ri

wkh sursrvhu iru d jlyhq uhflsurflw| sdudphwhu ri wkh uhvsrqghu ri �2 @ 51

Zh qrz wxuq wr wkh sursrvhu1 Htxdwlrq +<, vkrzv wkdw klv htxloleulxp fkrlfh ri f

ghshqgv rq wzr h{suhvvlrqv1 Zkhuhdv wkh �uvw h{suhvvlrq ghshqgv rq wkh uhvsrqghu*v

uhflsurfdo lqfolqdwlrq/ l1h1/ rq �2/ wkh vhfrqg h{suhvvlrq ghshqgv rq wkh sursrvhu*v

frqfhuq iru uhflsurflw|/ l1h1/ rq ��1 Wkh vhfrqg h{suhvvlrq wkh sursrvhu*v lqwulqvlf

frqfhuq iru dq htxlwdeoh rxwfrph1 Li kh lv dq hjrlvwlf sod|hu wkh h{suhvvlrq lv }hur1 Li

�� lv odujh/ krzhyhu/ kh zloo r�hu d srvlwlyh f1 Wkh �uvw h{suhvvlrq fdq eh lqwhusuhwhg

dv dq h{wulqvlf frqvwudlqw wr r�hu d srvlwlyh f= Wklv h{suhvvlrq h{dfwo| gh�qhv wkh

vpdoohvw r�hu wkdw mxvw jxdudqwhhv dq dffhswdqfh suredelolw| ri 41 +Zh fdoo wklv

49

r�hu ff/ l1h1/ ff =@�n�423

s�nS42n4

22

e42/ frpsduh Iljxuh 5,1 Reylrxvo|/ wkh uhvsrqghu*v

frqfhuq iru uhflsurflw| lv fuxfldo iru wkh ydoxh ri ff1 Iru h{dpsoh/ li wkh uhvsrqghu lv

d vho�vk sod|hu +zlwk �2 @ 3, wkh ydoxh ri ff lv htxdo wr }hur1 Wkh kljkhu sod|hu 5*v

frqfhuq iru uhflsurflw|/ wkh kljkhu wkh ydoxh ri ff1 Dv �2 jhwv yhu| odujh/ ff dssurdfkhv�2 1 Wkh ghshqghqfh ri ff rq �2 lv vkrzq lq Iljxuh 61 Lw lv uhpdundeoh wkdw hyhq iru

vpdoo ydoxhv ri �2 wkh sursrvhu lv cirufhg* wr pdnh d udwkhu jhqhurxv r�hu lq rughu wr

dyrlg d uhmhfwlrq1

Wkh htxloleulxp r�hu fW lv wkh pd{lpxp ri wkh �uvw dqg wkh vhfrqg h{suhvvlrq

ri Htxdwlrq +<,= Wklv phdqv/ h1j1/ wkdw lq fdvh d vho�vk sursrvhu sod|v djdlqvw d

uhflsurfdo uhvsrqghu kh zloo r�hu wkh pruh/ wkh kljkhu �21 Lq wklv fdvh wkh h{wulqvlf

frqvwudlqw lv elqglqj1 Li/ krzhyhu/ wkh uhvsrqghu kdv d yhu| orz �2/ l1h1/ kh dffhswv

sudfwlfdoo| dq| r�hu/ wkh htxloleulxp r�hu lv ghwhuplqhg e| wkh sursrvhu*v frqfhuq

iru dq htxlwdeoh rxwfrph= Li kh lv udwkhu vho�vk wrr/ klv r�hu zloo eh +forvh wr, }hur1

Li/ rq wkh rwkhu kdqg/ �� lv udwkhu kljk/ wkh r�hu zloo eh kljk dv zhoo147

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4reciprocity parameter of the responder

Iljxuh 6= R�huv lq wkh xowlpdwxp jdph wkdw jxdudqwhh dq dffhswdqfh zlwk suredelolw|

4/ ghshqghqw rq wkh wkh uhflsurflw| sdudphwhu ri wkh uhvsrqghu +�2,1

71414 Wkh uroh ri lqwhqwlrqv

Ehiruh zh wxuq wr wkh qh{w jdph zh zloo eulh | glvfxvv wkh suhglfwlrqv ri rxu wkhru|

iru wkh qrq0lqwhqwlrqdo wuhdwphqw uhsruwhg e| Eorxqw +4<<8,1 Lq khu wuhdwphqw wkh

sursrvhuv* r�huv zhuh qrw fkrvhq e| kxpdq vxemhfwv exw lqvwhdg udqgrpo| vhohfwhg1

Frqvhtxhqwo|/ d orz r�hu glg qrw vljqdo dq| +edg, lqwhqwlrqv1 Dv wkh gdwd ri Eorxqw*v

47Wkh udqjh ri 4�0 dqg 420 frpelqdwlrqv zkhuh wkh htxloleulxp r�hu SW htxdov Sf lv jlyhq e|

42 D 44E443��

44n�1 Wklv krogv lq sduwlfxodu li 42 D 4�1

4:

h{shulphqw uhyhdov/ wkh dffhswdqfh udwh iru d jlyhq r�hu lv pxfk kljkhu wkdq lq wkh

cuhjxodu* wuhdwphqw1 Krzhyhu/ hyhq lq wkh devhqfh ri lqwhqwlrqv vrph vxemhfwv uhmhfw

h{wuhpho| glvdgydqwdjhrxv r�huv1 Rxu wkhru| suhglfwv h{dfwo| wkhvh wzr vw|ol}hg idfwv1

Lq wkh qrq0lqwhqwlrqdo wuhdwphqw wkh htxloleulxp dffhswdqfh udwh iru sW lv jlyhq e|=

sW @

;?=k

S0242uE�32S�E�3S�

l�oli f ? �

2

4 li f � �2

Iljxuh 7 ghslfwv wkh suhglfwhg dffhswdqfh suredelolwlhv lq wkh cuhjxodu* xowlpdwxp

jdph dqg lq Eorxqw*v wuhdwphqw iru d jlyhq �21 Wkh xsshu judsk fruuhvsrqgv wr

wkh Eorxqw0h{shulphqw zkhuhdv wkh orzhu judsk vkrzv wkh dffhswdqfh ehkdylru lq wkh

cuhjxodu* wuhdwphqw1 Dv fdq eh vhhq lq wkh �jxuh/ d uhvsrqghu*v dffhswdqfh suredelolw|

iru orz r�huv lv kljkhu li lqwhqwlrqv duh devhqw1

Wkh orzhu wkh rxwfrph frqfhuq sdudphwhu %2/ wkh pruh wkh xsshu judsk vkliwv wr

wkh ohiw1 Sxw gl�huhqwo|/ wkh pruh d uhvsrqghu lv frqfhuqhg zlwk lqwhqwlrqv uhodwlyh wr

rxwfrphv/ wkh pruh olnho| vkh zloo dffhsw yhu| orz r�huv lq Eorxqw*v qrq0lqwhqwlrqdo

wuhdwphqw1 Rq wkh rwkhu kdqg/ li d uhvsrqghu lv sxuho| rxwfrph rulhqwhg/ l1h1/ li %2 @ 4

krogv/ vkh h{klelwv wkh vdph ehkdylrudo sdwwhuq dv lq wkh cuhjxodu* wuhdwphqw1 Dv wkh

gdwd uhyhdov/ krzhyhu/ shrsoh fduh derxw lqwhqwlrqv +l1h1/ %� ? 4,1

0

0.2

0.4

0.6

0.8

1

0 0.25 0.5 0.75 1offer c

random offer

intentional offer

Iljxuh 7= Dffhswdqfh suredelolwlhv lq wkh xowlpdwxp jdph zlwk lqwhqwlrqv +orzhu

judsk, dqg zlwkrxw lqwhqwlrqv +xsshu judsk, ghshqghqw rq wkh r�hu1 Wkh sdudphwhuv

�2 @ 5 dqg %2 @ 3=5 duh jlyhq1

4;

715 Srvlwlyh uhflsurflw|= Wkh jliw0h{fkdqjh jdph

Pdq| h{shulphqwv kdyh ghprqvwudwhg wkh lpsruwdqfh ri srvlwlyh uhflsurflw|1 Rqh zhoo

nqrzq h{shulphqw lv wkh jliw0h{fkdqjh jdph1 Lq wklv wzr0shuvrq vhtxhqwldo jdph/

wkh �uvw pryhu +fdoohg dq hpsor|hu, r�huv d zdjh z wr d vhfrqg pryhu +fdoohg d

zrunhu,1 Wkh zrunhu fdq hlwkhu wdnh lw ru ohdyh lw +vhh h1j1/ Ihku/ Jçfkwhu dqg

Nlufkvwhljhu +4<<:,,1 Li wkh zrunhu uhmhfwv/ erwk hduq qrwklqj1 Li wkh zrunhu

wdnhv wkh zdjh r�hu/ vkh kdv wr pdnh dq h�ruw ghflvlrq h1 Surylglqj h�ruw deryh wkh

plqlpxp h�ruw ohyho lv frvwo| zlwk f+h, ehlqj d frqyh{ h�ruw frvw ixqfwlrq1 Sd|r�

ixqfwlrqv duh jlyhq e| �� @ yh � z iru hpsor|huv dqg �2 @ z � f+h, iru zrunhuv/

uhvshfwlyho|1 D udwlrqdo dqg vho�vk zrunhu zloo 0 luuhvshfwlyh ri wkh zdjh sdlg 0

fkrrvh wkh plqlpxp h�ruw ohyho1 Zlwk edfnzdug lqgxfwlrq/ wkh hpsor|hu zloo r�hu

rqo| wkh orzhvw srvvleoh zdjh1 Frqwudu| wr wklv suhglfwlrq wkh pdlq h{shulphqwdo

�qglqjv duh +l, wkdw zdjhv fohduo| h{fhhg wkh orzhvw srvvleoh zdjh dqg +ll, wkdw wkhuh

lv d srvlwlyh zdjh0h�ruw uhodwlrq1 Erwk �qglqjv duh uhpdundeo| urexvw1 Wkh| krog lq

elodwhudo lqvwlwxwlrqv dv zhoo dv lq frpshwlwlyh pdunhw lqvwlwxwlrqv +vhh/ h1j1/ Ihku/

Nlufkvwhljhu dqg Ulhgo +4<<6,/ Ihku dqg Idon +4<<<, dqg Jçfkwhu dqg

Idon +4<<:,,1

Lq rxu dqdo|vlv zh uhvwulfw z 5 ^3> 4`/ h 5 ^3> 4`/ zh qrupdol}h y wr 4 dqg zulwh

wkh frqyh{ h�ruw frvw ixqfwlrq dv f+h, @ �h2 zlwk � � �e 1 Wr hdvh wkh sursrvlwlrq ri

wkh htxloleuld lq wkh jliw0h{fkdqjh jdph/ zh xvh wkh iroorzlqj gh�qlwlrqv1 Iluvw/ ohwhz+�> ��> �2, eh wkh zdjh wkdw zrxog eh fkrvhq e| wkh hpsor|hu li z dqg h zrxog qrw

eh uhvwulfwhg wr eh vpdoohu wkdq 41 Wkh h{dfw irupxod iru hz+�> ��> �2, lv jlyhq lq wkh

surri ri wkh sursrvlwlrq1 Zh ixuwkhu gh�qh z+�> �2, @�n@2 . @

42zklfk lv wkh plqlpdo

zdjh wkdw jxdudqwhhv dq h�ruw fkrlfh ri rqh1 Wkh iroorzlqj sursrvlwlrq qrz vkrzv

wkdw wkh vw|ol}hg idfwv ri wkh jliw0h{fkdqjh jdph duh uhsolfdwhg e| rxu wkhru|=

Sursrvlwlrq 5 Lq wkh jliw0h{fkdqjh jdph wkhuh duh wkh iroorzlqj uhflsurflw| htxloleuld

+zW> hW,=

+l, Li �2 @ 3/ wkhq wkhuh h{lvwv d xqltxh htxloleulxp=

zW @ hW @ 3 +43,

+ll, Li �2 A 3/ wkhq

hW @

%�5�� �2 .

s+5�. �2,

2 . ;��22z

5��2

&�o+44,

Wkhuh lv dozd|v dq htxloleulxp jlyhq e|=

zW @ ^plq +z+�> �2,> hz+�> ��> �2,,`�odf +45,

4<

Li %��� � 42E342n2kn2k42�2kE32k342n2k42�

dqg z+�> �2, � 4> wkhq wkhuh lv d vhfrqg htxloleulxp

zklfk lv jlyhq e|=

zW @ z+�> �2,

Wkh surri ri Sursrvlwlrq 5 lv jlyhq lq Dsshqgl{ 61

Glvfxvvlrq= Zh looxvwudwh Sursrvlwlrq 5 zlwk wkh khos ri Iljxuhv 8 wr :1 Lq

Iljxuh 8 zh ghslfw krz 0 lq htxloleulxp 0 d zrunhu*v h�ruw fkrlfh ghshqgv rq wkh

zdjh sdlg e| wkh hpsor|hu jlyhq khu uhflsurfdo prwlydwlrq/ �2 +frpsduh htxdwlrq

+44, ri Sursrvlwlrq 5,1 Wklv �jxuh qhdwo| fdswxuhv wkh hvvhqfh ri srvlwlyh uhflsurflw|

dv uhsruwhg lq wkh jliw0h{fkdqjh h{shulphqwv1 Uhflsurfdo zrunhuv surylgh kljkhu

h�ruw ohyhov wkh kljkhu wkh zdjh sdlg e| cwkhlu* hpsor|huv1 Pruhryhu/ zrunhuv zloo 0

iru d jlyhq zdjh 0 fkrrvh kljkhu h�ruw ohyhov/ wkh vwurqjhu wkhlu uhflsurfdo lqfolqdwlrq

+frpsduh htxdwlrq 44,1 Wklv uhodwlrqvkls lv ghslfwhg lq Iljxuh 91 Wkh xsshu judsk

vkrzv wkh htxloleulxp h�ruw fkrlfh iru d zdjh z @ 3=8/ zkhuhdv wkh orzhu ghslfwv wkh

rswlpdo h�ruw iru d orzhu zdjh +z @ 3=6,1 Qrwh wkdw wkh h�ruw ohyho lv }hur li dqg

rqo| li �2 @ 3 ru z @ 31

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1wage

Iljxuh 8= H�ruw fkrlfh ghshqghqw rq wkh zdjh sdlg iru d jlyhq �2 dqg d jlyhq �

+�2 @ 5 dqg � @ 3=5,1

Krz fdq zh fkdudfwhul}h dq hpsor|hu*v rswlpdo fkrlfh/ jlyhq d zrunhu*v h�ruw

ehkdylruB Lq Iljxuh : zh vkrz krz wkh htxloleulxp zdjh ghshqgv rq �2 +iru d jlyhq

��,1 Lq htxloleulxp/ dq hpsor|hu zloo 0 iru d vpdoo �2 0 r�hu d zdjh htxdo wr }hur zklfk

dovr lpsolhv dq h�ruw fkrlfh ri }hur1 Li/ krzhyhu/ wkh zrunhu lv vx!flhqwo| uhflsurfdo

wkh zdjh r�hu lqfuhdvhv lq �2148 Qrwlfh lq Iljxuh : wkdw wklv lqfuhdvh vwrsv dv �2

48Dv fdq eh vhhq lq Iljxuh : wkhuh lv dq lqwhuphgldwh udqjh ri 42 zkhuh wzr htxloleuld h{lvw1 Wkh

htxloleulxp fruuhvsrqglqj wr wkh xsshu eudqfk lv ghvfulehg ehorz1

53

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4reciprocity parameter of the worker ( ρ2)

effo

rt

w=.5

w=.3

Iljxuh 9= Htxloleulxp h�ruw fkrlfh ghshqghqw rq wkh zrunhu*v uhflsurflw| sdudphwhu

�2 d jlyhq � dqg wzr zdjh ohyhov +� @ =5/ z @ =6 dqg z @ =8,1

uhdfkhv d sduwlfxodu ohyho1 Dw wklv ohyho wkh hpsor|hu jhwv wkh pd{lpdo h�ruw ohyho1

Wkxv/ lw pdnhv qr vhqvh wr sd| d kljkhu zdjh/ qhlwkhu iru d uhflsurfdo qru iru d vho�vk

hpsor|hu1 Wr wkh frqwudu|/ dv �2 jhwv kljkhu wkh orzhvw zdjh wkdw jxdudqwhhv wkh

pd{lpdo h�ruw ohyho dfwxdoo| ghfuhdvhv1

Krz grhv wkh hpsor|hu*v frqfhuq iru uhflsurflw| �� lq xhqfh lwv ghflvlrqB Wr

dqvzhu wklv txhvwlrq qrwlfh �uvw wkdw wkh zrunhu fdq dozd|v hqvxuh dw ohdvw dq htxdo

vkduh iru khuvhoi/ l1h1/ wkh hpsor|hu lv dozd|v lq d glvdgydqwdjhrxv vlwxdwlrq1 Zkhuhdv

d vx!flhqwo| vho�vk hpsor|hu grhv qrw erwkhu derxw wklv/ d uhflsurfdo hpsor|hu grhv1

Frqvhtxhqwo|/ iru vpdoo �� dq hpsor|hu sd|v wkh sur�w pd{lpl}lqj zdjh1 Dv �� jhwv

kljkhu/ krzhyhu/ wkh hpsor|hu dfwxdoo| sd|v d orzhu zdjh 0 gxh wr wkh lpsdfw ri

qhjdwlyh uhflsurflw|1

Iljxuh : uhyhdov dqrwkhu lqwhuhvwlqj ihdwxuh ri Sursrvlwlrq 5= Iru vx!flhqwo| kljk

�� wkhuh lv dq lqwhuphgldwh udqjh ri �2 zkhuh wzr htxloleulxp zdjhv h{lvw1 Lq wkh

htxloleulxp fruuhvsrqglqj wr wkh xsshu eudqfk wkh hpsor|hu sd|v d zdjh wkdw jxdu0

dqwhhv dq h�ruw ri 41 Wklv htxloleulxp fdq eh ghvfulehg dv iroorzv= Wkh hpsor|hu

h{shfwv wr jhw dq h�ruw ri 4/ l1h1/ wkh pd{lpxp h�ruw ohyho1 Wklv phdqv wkdw wkh

zrunhu frxog qrw kdyh fkrvhq d kljkhu h�ruw ohyho1 Wkxv/ hyhq wkrxjk wkh hpsor|hu

lv lq d glvdgydqwdjhrxv vlwxdwlrq kh fdqqrw lqihu wkdw wkh zrunhu zdqwhg wr wuhdw wkh

hpsor|hu xqnlqgo|1 Dv d frqvhtxhqfh wkh hpsor|hu grhv qrw kdyh wkh ghvluh wr sxqlvk

wkh zrunhu dqg wkhuhiruh fkrrvhv wkh sd|r� pd{lpl}lqj zdjh/ l1h1/ d udwkhu kljk zdjh

wkdw jxdudqwhhv dq h�ruw ohyho ri 41 Lq wkh htxloleulxp fruuhvsrqglqj wr wkh orzhu

eudqfk lq Iljxuh :/ wkh hpsor|hu h{shfwv wkdw wkh zrunhu surylghv dq h�ruw vwulfwo|

ehorz 41 Wklv htxloleulxp pd| frh{lvw zlwk wkh iruphu iru wkh iroorzlqj uhdvrq= Wkh

54

idfw wkdw wkh zrunhu fkrrvhv dq h�ruw vpdoohu wkdq 4 lv frqvlghuhg dv yhu| xqnlqg

vlqfh vkh frxog kdyh fkrvhq d +voljkwo|, kljkhu h�ruw ohyho1 Wkhuhiruh/ wkh +uhflsurfdo,

hpsor|hu kdv dq lqfhqwlyh wr sxqlvk wkh zrunhu/ l1h1/ wr sd| d orz zdjh1 Wklv orz

zdjh qrz lqgxfhv wkh zrunhu wr surylgh wkh orz h�ruw ohyho wkh hpsor|hu h{shfwhg1

Lq d fhuwdlq vhqvh wkh hpsor|hu*v htxloleulxp ehkdylru fdq eh xqghuvwrrg dv d nlqg ri

�vhoi ixo�oolqj surskhf|�= Li wkh hpsor|hu eholhyhv wkdw wkh zrunhu lv qlfh/ wkh zrunhu

zloo lqghhg eh qlfh1 Li/ rq wkh rwkhu kdqg/ wkh hpsor|hu eholhyhv wkdw wkh zrunhu lv

phdq/ wkh zrunhu zloo lq idfw ehkdyh dffruglqjo|1

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4

reciprocity parameter of player 2 (worker)

Iljxuh := Htxloleulxp zdjh r�hu ghshqghqw rq �2 iru d jlyhq �� dqg � +�� @ 5 dqg

� @ 3=5,1 Wkh orzhu eudqfk fruuhvsrqgv wr htxloleuld zlwk hW ? 4 zkhuhdv lq wkh

xsshu eudqfk hW @ 4 krogv1

71514 Wkh uroh ri lqwhqwlrqv

Frqfoxglqj rxu glvfxvvlrq ri wkh jliw0h{fkdqjh jdph zh suhvhqw rxu suhglfwlrqv iru

wkh wuhdwphqw uhsruwhg e| Fkduqhvv +4<<9,1 Lq klv h{shulphqw zdjhv duh qrw gh0

whuplqhg e| wkh shuvrq wkh zrunhu lv lqwhudfwlqj zlwk1 Udwkhu/ d wklug sduw| ru d

udqgrp phfkdqlvp ghwhuplqhv wkh zdjh1 Frpsduhg wr wkh cuhjxodu* wuhdwphqw/ wklv

ohdgv wr d zhdnhu fruuhodwlrq ehwzhhq zdjhv dqg h�ruw ohyhov1 Wkh uhdvrq lv wkdw

lq Fkduqhvv* wuhdwphqw d kljk zdjh grhv qrw vljqdo dq| +nlqg, lqwhqwlrqv1 Zkhuhdv

sxuho| rxwfrph rulhqwhg prghov fdqqrw glvwlqjxlvk ehwzhhq wkhvh wzr wuhdwphqwv/

rxu wkhru| grhv1 Iru d jlyhq uhflsurfdo prwlydwlrq �2 zh suhglfw d zhdnhu fruuhodwlrq

ehwzhhq zdjhv dqg h�ruw ohyhov1 Wklv phdqv wkdw wkh vdph zrunhu zloo vxsso| d

orzhu h�ruw iru d sduwlfxodu uhfhlyhg zdjh lq wkh udqgrp0wuhdwphqw frpsduhg wr wkh

cuhjxodu* wuhdwphqw1 Iljxuh ; vkrzv wkdw uhvxow1

55

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1wage

intentional wage

random wage

Iljxuh ;= Htxloleulxp h�ruw ghshqghqw rq wkh zdjh1 Lq wkh xsshu judsk/ wkh zdjh

lv fkrvhq e| d kxpdq ehlqj/ lq wkh orzhu judsk d udqgrp zdjh lv jlyhq1 +%2 @ 3=5>

� @ 3=5,

716 Lghqwlfdo rxwfrphv |lhog gl�huhqw uhvsrqvhv= D frpsdulvrq ehwzhhqehvw0vkrw dqg xowlpdwxp jdphv

Wkh ehvw0vkrw jdph zdv lqwurgxfhg e| Kduulvrq dqg Kluvkohlihu +4<;<, dqg

Sudvqlndu dqg Urwk +4<<5,1 Wkh lqwhuhvwlqj ihdwxuh ri wklv h{shulphqw lv wkdw

vhfrqg pryhuv duh zloolqj wr dffhsw d kljkhu ghjuhh ri lqhtxlw| wkdq lq wkh xowlpdwxp

jdph uhvxowv1 Wklv gl�huhqfh fdqqrw eh h{sodlqhg e| wkh lqhtxlw| dyhuvlrq prghov e|

Erowrq dqg Rfnhqihov +5333, dqg Ihku dqg Vfkplgw +4<<<,1 Rxu h{sodqdwlrq

iru wkh gl�huhqfh ehwzhhq ehkdylru lq wkh ehvw0vkrw jdph dqg wkh xowlpdwxp jdph

uhvwv rq wkh lpsruwdqfh ri lqwhqwlrq1 Wklv zloo ehfrph fohdu zlwk wkh khos ri wkh

iroorzlqj wzr jdphv1

Lq Iljxuh < zh suhvhqw d uhgxfhg ehvw0vkrw jdph zklfk fdswxuhv wkh nh| dvshfwv

ri wkh ruljlqdo jdph1 Wkh fuxfldo ihdwxuh ri wklv jdph +dqg wkh ulfkhu ruljlqdo jdph,

lv wkdw wkh �uvw pryhu fdq rqo| r�hu d sd|r� vkduh wkdw lv hlwkhu yhu| dgydqwdjhrxv

ru yhu| glvdgydqwdjhrxv wr klpvhoi +;25 ru 52;,1 Wklv lv gl�huhqw lq wkh uhgxfhg

xowlpdwxp jdph zkhuh 0 dowhuqdwlyh wr wkh yhu| dgydqwdjhrxv r�hu +;25, 0 wkh �uvw

pryhu fdq dovr fkrrvh wkh r�hu +828,1

Dv wkh iroorzlqj sursrvlwlrq uhyhdov/ rxu wkhru| suhglfwv d kljkhu dffhswdqfh sure0

delolw| iru wkh xqnlqg r�hu +;25, lq wkh uhgxfhg ehvw0vkrw jdph/ frpsduhg wr wkh

uhgxfhg xowlpdwxp jdph1

56

YY ZZ

82

00

00

28

82

00

00

55

Reduced best-shot gameReduced ultimatum game

X X

Iljxuh <= Wkh jdph wuhhv ri d uhgxfhg xowlpdwxp jdph dqg d uhgxfhg ehvw0vkrw

jdph1

Sursrvlwlrq 6 Lq wkh qrupdol}hg49 uhgxfhg xowlpdwxp jdph wkhuh lv d xqltxh uhfl0

surflw| htxloleulxp1 Lw lv jlyhq e|=

uW @ 4 +46,

tW @

�8

45

4

�2

��o+47,

sW @

�3=;tW � 3=8

3=9+3=8� 3=5tW,tW��

��odf

+48,

Li �2%2 ? 42� wkhuh lv d xqltxh uhflsurflw| htxloleulxp lq wkh uhgxfhg ehvw0vkrw jdph

zklfk lv jlyhq e|=4:

uW @ 4 +49,

tW @

�8

45

4

%2�2

��o+4:,

sW @

%4 . f�H^W3f�2

f�SEf�H3f�2^W�4�

4 . tW

&�odf

+4;,

Li �� ru �2 duh }hur/ sW dqg tW duh wkh olplwv ri wkh deryh irupxodv zkhuh �� dqg

�2 dssurdfk }hur iurp deryh149Wkh sd|r�v vkrzq lq wkh jdph wuhh ri Iljxuh 4 duh qrupdol}hg zlwk d idfwru ri �

�f 14: Lq Dsshqgl{ 5 zh ghvfuleh dovr wkh htxloleuld iru 4202 D � 2

�1 Wklv lv rplwwhg khuh iru wkuhh

uhdvrqv1 Iluvw/ ehfdxvh wklv fdvh lv udwkhu frpsolfdwhg1 Vhfrqg/ lq doo �jxuhv zkhuh zh vkrz wkh

suhglfwlrqv ri rxu prgho/ zh xvh wkh vdph sdudphwhu vhw +0� ' f�2 dqg 4� M dfc eo, zklfk zh frqvlghu

dv wkh uhohydqw udqjh iru 0� dqg 4�1 Iru 4202 D � 2� zh duh idu rxwvlgh ri wklv udqjh1 Wklug/ zh duh

prvwo| lqwhuhvwhg lq wkh frpsdulvrq ri vhfrqg pryhuv* ehkdylru lq wkh ehvw0vkrw dqg wkh xowlpdwxp

jdph dqg Htxdwlrq 4: dovr krogv iru 4202 D � 2�1

57

Wkh surri ri Sursrvlwlrq 6 lv jlyhq lq Dsshqgl{ 61

Glvfxvvlrq= Lq rxu glvfxvvlrq zh irfxv rq wkh vhfrqg pryhu ehkdylru1 Iluvw/

qrwh wkdw wkh nlqg r�hu +52;, lv dozd|v dffhswhg lq wkh ehvw0vkrw dv zhoo dv lq wkh

xowlpdwxp jdph +uW @ 4,1 Wkh dffhswdqfh suredelolw| tW ri wkh xqnlqg r�hu +;25,

gl�huv/ wkrxjk1 Wklv gl�huhqfh lv vkrzq lq Iljxuh 431

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4reciprocity parameter of player 2

Best-shot Game

Ultimatum Game

Iljxuh 43= Wkh dffhswdqfh suredelolw| li wkh xqnlqg r�hu +;/5, lv kljkhu lq wkh uhgxfhg

ehvw0vkrw jdph +xsshu judsk, wkdq lq wkh uhgxfhg xowlpdwxp jdph +orzhu judsk,1

Wkh xsshu judsk vkrzv wkh dffhswdqfh suredelolw| ri wkh xqnlqg r�hu lq wkh

uhgxfhg ehvw0vkrw jdph/ wkh orzhu vkrzv wkdw ri wkh uhgxfhg xowlpdwxp jdph1 Lq

erwk jdphv/ zh vhh wkdw wkh dffhswdqfh suredelolw| ri wkh xqnlqg r�hu ghfuhdvhv lq

sod|hu 5*v frqfhuq iru uhflsurflw|/ l1h1/ lq �21 Zh dovr vhh/ krzhyhu/ wkdw iru d jlyhq �2wkh dffhswdqfh suredelolw| lv orzhu lq wkh xowlpdwxp jdph frpsduhg wr wkh ehvw0vkrw

jdph1 Wkxv/ lq wkh ehvw0vkrw jdph d uhflsurfdo vhfrqg pryhu lv zloolqj wr dffhsw d

kljkhu ghjuhh ri lqhtxlw|1

Wkh uhdvrq lv wkdw lq wkh ehvw0vkrw jdph wkh rqo| dowhuqdwlyh wr r�hulqj +;25, lv

wr r�hu d yhu| glvdgydqwdjhrxv r�hu/ qdpho| +52;,1 Vlqfh vxfk dq dowhuqdwlyh lpsolhv

wkdw wkh �uvw pryhu vzlwfkhv iurp dq dgydqwdjhrxv wr d glvdgydqwdjhrxv vlwxdwlrq/

sod|hu 5 zloo qrw �qg lw yhu| xqnlqg li kh jhwv wkh cxqnlqg* r�hu/ ehfdxvh vkh fdqqrw

lqihu yhu| edg lqwhqwlrqv iurp wklv r�hu1 Wklqjv duh txlwh gl�huhqw lq wkh uhgxfhg

xowlpdwxp jdph1 Khuh/ wkh �uvw pryhu kdv wkh dowhuqdwlyh wr fkrrvh dq htxlwdeoh

rxwfrph1 Frqvhtxhqwo|/ lw grhv vljqdo edg lqwhqwlrqv dqg lv frqvlghuhg dv txlwh xqnlqg

li wklv rssruwxqlw| lv qrw fkrvhq1 Wkxv/ ghshqghqw rq wkh �uvw pryhu*v dowhuqdwlyhv wkh

vdph r�hu zloo eh dffhswhg zlwk d gl�huhqw suredelolw|1 Wklv gl�huhqfh lv frq�uphg

lq wkh h{shulphqwdo vwxg| e| Idon/ Ihku/ dqg Ilvfkedfkhu +4<<<,= Wkh uhmhfwlrq

58

udwh ri wkh +;25,0r�hu lv 5: shufhqw lq wkh uhgxfhg ehvw0vkrw jdph dqg 77 shufhqw lq

wkh uhgxfhg xowlpdwxp jdph1

717 Frpshwlwlrq

Lq wkh suhfhglqj jdphv zh kdyh dqdo|}hg rqo| elodwhudo lqwhudfwlrqv1 Lq sduwlfxodu/ zh

uhvwulfwhg rxu dqdo|vlv wr jdphv zlwkrxw dq| frpshwlwlrq dw doo1 Lq wklv vhfwlrq zh/

wkhuhiruh/ dsso| rxu wkhru| wr q0shuvrq jdphv dqg vkrz krz frpshwlwlyh suhvvxuh

lqwhudfwv zlwk uhflsurfdo suhihuhqfhv1

Lw lv d zhoo hvwdeolvkhg idfw lq h{shulphqwdo hfrqrplfv wkdw lq frpshwlwlyh lqvwlwx0

wlrqv pdunhw rxwfrphv frqyhujh yhu| zhoo wrzdugv wkh rxwfrph suhglfwhg e| vwdqgdug

hfrqrplf wkhru| +Vplwk +4<;5,/ Gdylv dqg Krow +4<<6,,1 Wklv krogv hyhq lq pdu0

nhwv zkhuh wkh htxloleulxp rxwfrph lv yhu| �xqidlu� lq wkh vhqvh wkdw doprvw wkh

zkroh vxusoxv lv uhdshg e| rqh vlgh ri wkh pdunhw1 Sxw gl�huhqwo|/ lw vhhpv wkdw lq

d frpshwlwlyh hqylurqphqw vxemhfwv ehkdyh dv li wkh| zhuh frpsohwho| vho�vk1 Krz

fdq d wkhru| ri uhflsurflw| srvvleo| dffrxqw iru wklv idfwB Lq wklv vhfwlrq zh vkrz

wkdw rxu wkhru| fdq h{sodlq qrw rqo| zk| lq elodwhudo lqvwlwxwlrqv rxwfrphv whqg wr

eh �idlu� exw dovr zk| lq pdunhwv uhflsurfdo vxemhfwv* ehkdylru jlyhv ulvh wr htxloleuld

wkdw lpso| dq h{wuhpho| xqhyhq glvwulexwlrq ri wkh jdlqv iurp wudgh1

Dv dq looxvwudwlrq zh dqdo|}h d pdunhw jdph zlwk sursrvhu frpshwlwlrq zklfk kdv

ehhq vwxglhg e|Urwk/ Sudvqlndu/ Rnxqr0Ixmlzdud/ dqg ]dplu +4<<4,1 Lq wklv

jdph wkhuh duh q� 4 sursrvhuv zkr vlpxowdqhrxvo| sursrvh dq r�hu f� 5 ^3> 4` wr wkh

uhvsrqghu zlwk l 5 ^4> q� 4`1 Wkhvh r�huv duh uhyhdohg wr wkh uhvsrqghu zkr kdv wr

ghflgh zkhwkhu wr dffhsw ru uhmhfw wkh kljkhvw r�hu f6@%1 Li pruh wkdq rqh sursrvhu

r�huv f6@% d udqgrp phfkdqlvp ghwhuplqhv zkrvh r�hu zloo eh vhohfwhg1 Sd|r�v

duh h{dfwo| dv lq wkh xowlpdwxp jdph/ l1h1/ wkh sursrvhu zkrvh r�hu lv dffhswhg

uhfhlyhv 4�f6@% dqg wkh uhvsrqghu jhwv f6@%1 D sursrvhu zkrvh r�hu lv qrw dffhswhg

uhfhlyhv d sd|r� ri }hur1 Li wkh uhvsrqghu uhmhfwv f6@%/ doo uhfhlyh qrwklqj1 Dvvxplqj

udwlrqdo dqg sxuho| vho�vk sod|huv wkh vxejdph shuihfw rxwfrph lv vwudljkwiruzdug=

Dw ohdvw wzr sursrvhuv r�hu f6@% @ 4 zklfk lv dffhswhg e| wkh uhvsrqghu1 Lq idfw/

wklv suhglfwlrq lv vxssruwhg e| wkh h{shulphqwdo gdwd1 Urwk/ Sudvqlndu/ Rnxqr0

Ixmlzdud/ dqg ]dplu +4<<4, uhsruw wkdw lq wkhlu h{shulphqwdo vhvvlrqv +zlwk <

sursrvhuv, wklv htxloleulxp rxwfrph zdv uhdfkhg diwhu d ihz shulrgv1 Pruhryhu/

wkhvh uhvxowv zhuh yhu| urexvw dqg vkrzhg xs lq irxu gl�huhqw frxqwulhv1 Jlyhq wkh

h{shulphqwdo uhvxowv ri wkh suhfhglqj elodwhudo jdphv wklv lv uhpdundeoh vlqfh lq wklv

htxloleulxp wkh sursrvhuv hduq sudfwlfdoo| qrwklqj zkloh wkh uhvsrqghu jhwv wkh zkroh

vxusoxv1 Dv lw wxuqv rxw/ wkh suhglfwlrq ri rxu wkhru| frlqflghv zlwk wkh vwdqgdug

suhglfwlrq1

59

Sursrvlwlrq 7 Lq d uhflsurflw| htxloleulxp lq wkh pdunhw jdph zlwk sursrvhu frp0

shwlwlrq dw ohdvw 5 sursrvhuv r�hu f6@% @ 4 zklfk lv dffhswhg e| wkh uhvsrqghu1

Wkh surri ri Sursrvlwlrq 7 lv jlyhq lq Dsshqgl{ 61 Lw uhtxluhv dq h{whqvlrq ri rxu

wkhru| wr q0shuvrq jdphv1 Wklv h{whqvlrq lv vlpso| qrwdwlrqdo dqg zloo eh glvfxvvhg

lq Dsshqgl{ 51

Glvfxvvlrq= Wkh vwulnlqj ihdwxuh ri Sursrvlwlrq 7 lv wkdw 0 lqghshqghqw ri wkhlu

uhflsurfdo lqfolqdwlrq 0 sursrvhuv zloo dffhsw d yhu| xqhyhq glvwulexwlrq ri wkh slh1

Wkh lqwxlwlrq ri wkdw uhvxow lv wkdw lq d frpshwlwlyh pdunhw d sursrvhu kdv qr fkdqfh

wr dfklhyh d �idlu� rxwfrph= Iurp wkh xowlpdwxp jdph zh nqrz wkdw d uhvsrqghu zloo

dffhsw dq| r�hu odujhu ru htxdo wr 3=8 zlwk fhuwdlqw|1 Vlqfh r�hulqj pruh ghfuhdvhv

d sursrvhu*v pdwhuldo dqg uhflsurflw| xwlolw| d sursrvhu zloo 0 lq wkh xowlpdwxp jdph

0 qhyhu r�hu pruh wkdq wkdw1 Dvvxph wkdw lq d pdunhw jdph zlwk wzr sursrvhuv d

uhflsurfdo sursrvhu l uhixvhv wr r�hu pruh wkdq 3=81 Zkdw zloo wkh rwkhu sursrvhu

grB E| lq�qlwhvlpdoo| ryhuelgglqj sod|hu l*v r�hu kh fdq rq wkh rqh kdqg lqfuhdvh klv

pdwhuldo sd|r� e| d srvlwlyh dprxqw +ehfdxvh kh fdq lqfuhdvh wkh zlqqlqj suredelolw|

iurp �?3� wr 4,1 Rq wkh rwkhu kdqg/ wkh uhflsurflw| glvxwlolw| uhvxowlqj iurp wkh xqidlu

uhodwlrq wr wkh uhvsrqghu grhv rqo| fkdqjh lq�qlwhvlpdoo|1 Wklv phdqv wkdw sod|hu l*v

uhixvdo wr sursrvh pruh wkdq 3=8 lv qrw dq h�hfwlyh wrro wr dfklhyh d �idlu� rxwfrph1

Dv d frqvhtxhqfh/ kh wulhv wr ryhuelg wkh rwkhu sursrvhu wr jhw dw ohdvw d plqlpdo

vkduh ri wkh slh1 Wklv �ryhuelgglqj�/ krzhyhu/ ohdgv lqhylwdeo| wr wkh htxloleulxp

ghvfulehg lq Sursrvlwlrq 71

Dqrwkhu h{shulphqw zklfk |lhogv dq �xqidlu� rxwfrph lv wkh pdunhw jdph zlwk

uhvsrqghu frpshwlwlrq +J wk/ Pdufkdqg/ dqg Uxoolëuh +4<<:,,1 Lq wklv

jdph rqh sursrvhu idfhv q uhvsrqghuv/ zkr pd| �xqghuelg� hdfk rwkhu lq dffhswlqj

orz r�huv1 Dv lq wkh pdunhw jdph zlwk sursrvhu frpshwlwlrq/ rxu wkhru| lv frpsdwleoh

zlwk wkh yhu| �xqidlu� rxwfrph suhglfwhg e| vwdqgdug hfrqrplf wkhru|1 Dffruglqj wr

wkh odwwhu suhglfwlrq dqg wkh vw|ol}hg h{shulphqwdo idfwv uhvsrqghuv dffhsw pxfk orzhu

r�huv wkdq lq wkh xowlpdwxp jdph1 Wklv krogv hyhq li doo uhvsrqghuv duh uhflsurfdoo|

prwlydwhg1 Wkh lqwxlwlrq lv vlplodu wr wkh pdunhw jdph zlwk sursrvhu frpshwlwlrq1

Zh vdz wkdw lq wkh xowlpdwxp jdph d uhvsrqghu fdq �irufh� wkh sursrvhu wr pdnh

d �idlu� r�hu e| fuhgleo| wkuhdwhqlqj wr uhmhfw yhu| orz r�huv1 Wklv sxqlvkphqw

ghylfh lv lqh�hfwlyh lq wkh pdunhw jdph zlwk uhvsrqghu frpshwlwlrq li wkhuh duh rwkhu

uhvsrqghuv zkr dffhsw orz r�huv1

Wkxv rxu wkhru| grhv qrw rqo| suhglfw �idlu� rxwfrphv1 Udwkhu/ lw fruuhfwo| h{0

sodlqv �rxw0ri0vdpsoh0idfwv�/ l1h1/ yhu| �xqidlu� htxloleuld wkdw rffxu lq frpshwlwlyh

lqvwlwxwlrqv1

5:

718 Ixuwkhu jdphv

Lq wklv vhfwlrq zh eulh | glvfxvv wkh lqwxlwlrq ri rxu wkhru|*v suhglfwlrqv lq vrph

ixuwkhu jdphv1 Wkh sursrvlwlrqv dqg surriv kdyh uljrurxvo| ehhq ghyhorshg lq d

suhylrxv yhuvlrq14; Lq wklv vhfwlrq zh uhsruw rqo| wkh pdlq uhvxowv1

71814 Wkh glfwdwru jdph

Wkh glfwdwru jdph lv d yhu| vlpsoh wzr shuvrq jdph1 Wkh wdvn ri wkh �uvw pryhu

+wkh vr0fdoohg �glfwdwru�, lv wr glylgh dq dprxqw ri prqh| ehwzhhq klpvhoi dqg d

frxqwhusduw +wkh �uhfhlyhu�,1 Ohw 4 eh wkh dprxqw ri prqh| dqg f wkh vkduh iru wkh

uhfhlyhu1 Wkh glfwdwru lv iuhh wr fkrrvh dq| ihdvleoh glylvlrq kh zdqwv wr +3 � f � 4,1

Wkh uhfhlyhu kdv qr fkrlfh wr pdnh/ l1h1/ vkh kdv wr dffhsw dq| dprxqw vhqw wr khu1

Wkh sd|r� iru wkh uhfhlyhu lv vlpso| wkh dprxqw f vkh kdv ehhq vhqw e| wkh glfwdwru1

Wkh glfwdwru*v sd|r� lv jlyhq e| wkh uhvlgxdo dprxqw 4� f1

Wkh glfwdwru jdph kdv ehhq vwxglhg/ h1j1/ e| Iruv|wkh/ Krurzlw}/ Vdylq dqg

Vhiwrq +4<<7, dqg e| Kriipdq/ PfFdeh/ Vkdfkdw dqg Vplwk +4<<7, dqg

Hfnho dqg Jurvvpdq +4<<;,1 Wkh vw|ol}hg idfwv fdq eh vxppdul}hg dv iroorzv1 +l,

Glfwdwru r�huv odujhu wkdq kdoi ri wkh slh/ l1h1/ f A �2 duh sudfwlfdoo| qhyhu revhuyhg1 +ll,

Urxjko| ;3 shufhqw ri wkh r�huv duh ehwzhhq }hur dqg kdoi ri wkh slh/ l1h1/ 3 ? f � �2 1

Krzhyhu/ frpsduhg wr wkh xowlpdwxp jdph/ wkh glvwulexwlrq ri r�huv lv vkliwhg wrzdugv

}hur1 +lll, Derxw 53 shufhqw ri wkh glfwdwruv r�hu wkh dprxqw suhglfwhg e| vwdqgdug

jdph wkhru|/ l1h1/ wkh| r�hu h{dfwo| }hur14<

Frpsdwleoh zlwk wkhvh vw|ol}hg idfwv rxu wkhru| suhglfwv d xqltxh uhflsurflw| htxl0

oleulxp1 Lq wklv htxloleulxp wkh glfwdwru r�huv fW @k�2 ��4� �

0�4�

�ldf153 Wkxv/ d

glfwdwru*v sursrvdo ghshqgv +l, rq ��/ l1h1/ rq krz vwurqj klv rwkhu0uhjduglqj suhihu0

hqfhv hqwhu klv xwlolw| dqg +ll, rq klv sxuh rxwfrph frqfhuq sdudphwhu %�1 Li %��� A 4/

wkh glfwdwru r�huv d srvlwlyh dprxqw ri prqh|1 Hyhq iru d yhu| kljk ydoxhv ri %���/

krzhyhu/ wkh glfwdwru*v r�hu zloo qhyhu h{fhhg �2 / l1h1/ kh zloo qhyhu r�hu pruh wkdq kh

nhhsv iru klpvhoi1 Li %��� � 4/ wkh glfwdwru fkrrvhv dq r�hu htxdo wr }hur1 Frpsdulqj

wkh htxloleulxp r�huv lq wkh glfwdwru jdph zlwk wkrvh ri wkh xowlpdwxp jdph/ zh vhh

wkdw wkh htxdwlrq wkdw ghwhuplqhv wkh htxloleulxp r�hu lq wkh glfwdwru jdph htxdov

wkh vhfrqg h{suhvvlrq lq htxdwlrq +<, ri Sursrvlwlrq 4 0 li zh uhsodfh �� e| %���1 Vlqfh

%� � 4/ wkh vdph shuvrq zloo dozd|v r�hu dw ohdvw dv pxfk lq wkh xowlpdwxp jdph dv

4;Wklv yhuvlrq/ lqfoxglqj sursrvlwlrqv dqg surriv lv dydlodeoh rq uhtxhvw14< Lw vkrxog eh qrwhg wkdw hvshfldoo| wkh uhvxowv ri wkh glfwdwru jdph duh qrw yhu| urexvw zlwk uhvshfw

wr wuhdwphqw yduldwlrqv1 Lqfuhdvlqj/ h1j1/ wkh vrfldo glvwdqfh dprqj sduwlflsdqwv ri dq h{shulphqw

dqg wkh h{shulphqwhu +grxeoh eolqg wuhdwphqw, lqfuhdvhv wkh shufhqwdjh ri }hur sursrvdov +frpsduh

Kriipdq/ PfFdeh dqg Vplwk +4<<9,,153Wklv krogv/ li 0�4� : f1 Li 0�4� htxdov wr }hur/ SW lv fkrvhq htxdo wr }hur1

5;

lq wkh glfwdwru jdph1

Wkxv/ frqvlvwhqw zlwk vw|ol}hg idfwv +l, wr +lll,/ rxu wkhru| suhglfwv wkdw glfwdwruv

r�hu ehwzhhq }hur dqg kdoi ri wkh slh dqg wkdw wkh glvwulexwlrq ri r�huv lq wkh glf0

wdwru jdph lv vkliwhg grzqzdugv frpsduhg wr wkh fruuhvsrqglqj glvwulexwlrq lq wkh

xowlpdwxp jdph1

71815 Wkh Sulvrqhu*v Glohppd dqg Sxeolf Jrrgv Jdphv

Wkh vhtxhqwldo sulvrqhu*v glohppd frqvlvwv ri wzr vwdjhv1 Dw wkh �uvw vwdjh/ sod|hu 4

fdq hlwkhu frrshudwh ru ghihfw1 Diwhu revhuylqj sod|hu 4*v fkrlfh sod|hu 5 dovr fkrrvhv

hlwkhu wr frrshudwh ru wr ghihfw1 Dvvxplqj udwlrqdo dqg vho�vk dfwruv/ erwk sod|huv

kdyh d grplqdqw vwudwhj| qrw wr frrshudwh1

Frqwudu| wr wklv suhglfwlrq/ rxu wkhru| suhglfwv wkh iroorzlqj= Iluvw/ sod|hu 5

dozd|v ghihfwv li sod|hu 4 kdv ghihfwhg ehiruhkdqg1 Wklv phdqv wkdw dffruglqj wr rxu

wkhru| zh zloo �qg qr dowuxlvwlf/ l1h1/ qrq0frqglwlrqdo frrshudwlrq1 Vhfrqg/ li sod|hu 5

lv vx!flhqwo| uhflsurfdoo| prwlydwhg/ wkhuh lv d srvlwlyh suredelolw| wkdw sod|hu 5 uhdfwv

wr frrshudwlrq zlwk frrshudwlrq1 Wkxv/ rxu wkhru| suhglfwv frqglwlrqdo frrshudwlrq

li sod|hu 5*v frqfhuq iru uhflsurflw| lv vwurqj hqrxjk1

H{shulphqwdo vwxglhv ri vhtxhqwldo yhuvlrqv ri wkh sulvrqhu* v glohppd duh uhsruwhg

lq Erooh dqg Rfnhqihov +4<<3, dqg Fodun dqg Vhiwrq +4<<;,1 Wkh uhvxowv

ri wkhlu vwxglhv duh lq olqh zlwk rxu suhglfwlrqv1 Lq sduwlfxodu/ xqfrqglwlrqdo frrshu0

dwlrq lv sudfwlfdoo| lqh{lvwhqw1 Lq wkhlu frqfoxvlrq Fodun dqg Vhiwrq/ h1j1/ qrwh wkdw

�frrshudwlrq lv ehwwhu uhjdughg dv uhflsurfdwlrq udwkhu wkdq xqfrqglwlrqdo dowuxlvp=

vhfrqg0pryhuv frrshudwh txlwh iuhtxhqwo| lq uhvsrqvh wr frrshudwlrq e| �uvw0pryhuv/

exw uduho| frrshudwh diwhu wkh �uvw0pryhu ghihfwv� +s1 4:,1

Zh dovr dqdo|}h wkh vlpxowdqhrxv sulvrqhu*v glohppd1 Dffruglqj wr rxu wkhru|/ zh

jhw ohvv frrshudwlrq li sod|huv fkrrvh vlpxowdqhrxvo| frpsduhg wr wkh vhtxhqwldo pryh

vwuxfwxuh1 Wkhuh lv vwurqj hylghqfh lq idyru ri wklv suhglfwlrq1 H{shulphqwdo vwxglhv

e| Zdwdeh/ Whudl/ Kd|dvkl/ dqg \dpdjlvkl +4<<9, dqg Kd|dvkl/ Rvwurp/

Zdonhu/ dqg \dpdjlvkl +4<<;, vkrz wkdw frrshudwlrq udwhv dprqj �uvw pryhuv

lq wkh vhtxhqwldo sulvrqhu*v glohppd duh pxfk kljkhu/ frpsduhg wr vlpxowdqhrxv pryh

jdphv1

Wkh vwudwhjlf vwuxfwxuh ri wkh sulvrqhu*v glohppd lv yhu| vlplodu wr wkdw ri sxeolf

jrrgv jdphv1 Wkh pdmru gl�huhqfh lv wkdw lq d sxeolf jrrgv jdph sod|huv kdyh pruh

vwudwhjlhv wkdq mxvw wr frrshudwh ru wr ghihfw1 Lq d w|slfdo sxeolf jrrgv jdph hdfk

sod|hu lv surylghg zlwk dq hqgrzphqw ri 53 wrnhqv/ vd|/ wkdw fdq eh doorfdwhg wr d

sxeolf jrrg ru wkdw fdq eh nhsw iru sulydwh frqvxpswlrq1 Wkh pdujlqdo shu fdslwd

uhwxuq ri dq lqyhvwphqw lqwr wkh sxeolf jrrg lv xvxdoo| vpdoohu wkdq rqh/ l1h1/ surylglqj

}hur wrnhqv wr wkh sxeolf jrrg lv d grplqdqw vwudwhj| +zkhuhdv lw lv lq wkh frpprq

5<

lqwhuhvw wr ixoo| frrshudwh,1 Prvw sxeolf jrrgv h{shulphqwv kdyh ehhq frqgxfwhg dv

vlpxowdqhrxv pryh jdphv1 Krzhyhu/ wkhuh lv d uhfhqw h{shulphqw e| Ilvfkedfkhu/

Jçfkwhu dqg Ihku +4<<<, zkhuh vxemhfwv frxog frqglwlrqdoo| lqglfdwh krz pdq|

wrnhqv wkh| zdqwhg wr lqyhvw lqwr wkh jurxs dffrxqw1 Ghvslwh wkh idfw wkdw wkh ehvw

uhso| lv wr surylgh }hur wrnhqv lqghshqghqw ri wkh rwkhu jurxs phpehuv* frqwulexwlrqv/

vxemhfwv rq dyhudjh frqwulexwhg pruh wkh kljkhu wkh frqwulexwlrqv ri wkh rwkhu jurxs

phpehuv1 Wklv cfrqglwlrqdo frrshudwlrq vwudwhj|* zdv lq prvw fdvhv vshfl�hg vxfk

wkdw vxemhfwv surylghg ohvv wkdq wkh jurxs dyhudjh1 Wklv lv h{dfwo| zkdw rxu wkhru|

suhglfwv= Lq d sxeolf jrrgv jdph vxemhfwv zlwk d vx!flhqwo| kljk uhflsurfdo lqfolqdwlrq

zloo frqglwlrqdoo| frrshudwh exw dozd|v frrshudwh voljkwo| ohvv wkdq wkh rwkhu sod|hu+v,1

Pruhryhu/ rxu wkhru| uhsolfdwhv dqrwkhu vw|ol}hg idfw ri sxeolf jrrgv jdphv/ qdpho|

wkdw wkh surshqvlw| wr frrshudwh lqfuhdvhv lq wkh pdujlqdo shu fdslwd uhwxuq ri dq

lqyhvwphqw lqwr wkh sxeolf jrrg +vhh Ohg|dug +4<<8,,1

71816 Wkh Lqyhvwphqw Jdph

Lq wklv jdph/ �uvw pryhuv duh hqgrzhg zlwk dq dprxqw ri prqh| zklfk wkh| fdq nhhs

ru wudqvihu wr wkh vhfrqg pryhu1 Dq| dprxqw | wudqvihuuhg lv wulsohg1 Wkh vhfrqg

pryhu wkhq ghflghv rq d frxqwhuwudqvihu }/ zlwk 3 � } � 6|1 Li erwk sod|huv duh

udwlrqdo dqg vho�vk prqh| pd{lpl}huv wkh suhglfwhg rxwfrph lv | @ } @ 31

Wkh lqyhvwphqw jdph zdv vwxglhg e| Ehuj/ Glfnkdxw dqg PfFdeh +4<<8,

dqg Ploohu +4<<:,1 Wkh dxwkruv uhsruw wkdw frqwudu| wr wkh vwdqgdug suhglfwlrq/

�uvw pryhuv rq dyhudjh wudqvihu srvlwlyh dprxqwv ri prqh| dqg gr uhfhlyh srvlwlyh

frxqwhuwudqvihuv1 Rxu wkhru| suhglfwv h{dfwo| wklv= Li vhfrqg pryhuv duh vx!flhqwo|

uhflsurfdo/ wkh| pdnh frqglwlrqdo frxqwhuwudqvihuv/ l1h1/ } lqfuhdvhv lq |1 Wklv krogv

wkh pruh/ wkh juhdwhu wkh vhfrqg pryhu*v uhflsurfdo lqfolqdwlrq1 Iluvw pryhuv fkrrvh

srvlwlyh wudqvihuv li wkh h{shfwhg frxqwhuwudqvihu lv vx!flhqwo| kljk/ l1h1/ li vhfrqg

pryhuv duh vx!flhqwo| uhflsurfdo1

8 Vxppdu|

Lq wklv sdshu/ zh suhvhqw d irupdo wkhru| ri uhflsurflw| zkhuh uhflsurflw| phdqv wkh

ehkdylrudo uhvsrqvh wr shufhlyhg nlqgqhvv ru xqnlqgqhvv1 Zh dujxh wkdw wkh shufhlyhg

nlqgqhvv ri dq dfwlrq lv frpsrvhg ri wzr hohphqwv/ qdpho| wkh rxwfrph wkdw uhvxowv

iurp wkh dfwlrq dqg wkh dfwlrq*v xqghuo|lqj lqwhqwlrqv1 Lq rxu wkhru| zh lpsohphqw

dq htxlw| edvhg uhihuhqfh vwdqgdug iru wkh hydoxdwlrq ri wkh nlqgqhvv ru xqnlqgqhvv ri

wkh fkrvhq rxwfrph1 Wklv rxwfrph/ krzhyhu/ zloo eh shufhlyhg gl�huhqwo| ghshqglqj

rq wkh lqwhqwlrqv lqyroyhg1 Li lqwhqwlrqv duh devhqw/ wkh ehkdylrudo uhvsrqvh zloo eh

ohvv lqwhqvh1 Krzhyhu/ hyhq lq wklv fdvh/ vxemhfwv vwloo h{shulhqfh wkh rxwfrph shu vh

63

dv hlwkhu dgydqwdjhrxv ru glvdgydqwdjhrxv dqg vkrzv vrph uhflsurfdo uhvsrqvh1

Rxu wkhru| suhglfwv wkh vw|ol}hg idfwv ri d zlgh ydulhw| ri h{shulphqwdo jdphv=

Lq wkh xowlpdwxp jdph/ sursrvhuv r�hu ehwzhhq }hur dqg kdoi ri wkh wrwdo slh1 Uh0

mhfwlrqv duh ghfuhdvlqj lq wkh ohyho ri wkh r�hu dqg lqfuhdvlqj lq wkh vwuhqjwk ri wkh

uhvsrqghu*v frqfhuq iru uhflsurflw|1 Lq wkh glfwdwru jdph/ wkh wkhru| suhglfwv r�huv

zklfk duh orzhu wkdq lq wkh xowlpdwxp jdph1 Zh dqdo|}h d uhgxfhg ehvw0vkrw jdph

dqg vkrz wkdw 0 dv lq wkh h{shulphqwdo gdwd 0 vxemhfwv duh zloolqj wr ehdu d kljkhu

ghjuhh ri lqhtxlw| wkdq lq wkh xowlpdwxp jdph1 Lq wkh jliw0h{fkdqjh jdph rxu wkh0

ru| suhglfwv d srvlwlyh uhodwlrqvkls ehwzhhq zdjhv dqg h�ruw ohyhov1 Pruhryhu/ �upv

r�hu deryh0plqlpxp zdjhv 0 dv uhsruwhg lq wkh h{shulphqwdo olwhudwxuh1 Lq wkh vh0

txhqwldo sulvrqhu*v glohppd zh suhglfw frqglwlrqdo frrshudwlrq1 Vlploduo|/ lq sxeolf

jrrgv jdphv vxemhfwv frqwulexwh pruh/ wkh pruh wkh| h{shfw rwkhuv wr frqwulexwh1

Pruhryhu/ frqwulexwlrqv lqfuhdvh lq wkh pdujlqdo shu fdslwd uhwxuq ri dq lqyhvwphqw

wr wkh sxeolf jrrg1 Lq wkh lqyhvwphqw jdph zh �qg d srvlwlyh uhodwlrq ehwzhhq wkh

�uvw pryhu*v wudqvihu dqg wkh vhfrqg pryhu*v frxqwhuwudqvihu1 Lq dq q0shuvrq pdu0

nhw jdph zlwk sursrvhu frpshwlwlrq wkh wkhru| suhglfwv wkdw dw ohdvw wzr sursrvhuv

r�hu wkh wrwdo slh wr wkh uhvsrqghu zkr dffhswv wklv r�hu1 Wkxv/ wkh zkroh vxusoxv

lv uhdshg e| wkh uhvsrqghu1 Wklv h{wuhpho| �xqidlu� htxloleulxp lv uhsruwhg lq wkh

h{shulphqwdo olwhudwxuh1

64

9 Dsshqgl{ 4= H{lvwhqfh ri uhflsurflw| htxloleuld

Lq wkh suhvhqwhg irup/ wkh h{lvwhqfh ri d uhflsurflw| htxloleulxp lv qrw dozd|v jxdu0dqwhhg1 D jdph zkhuh d uhflsurflw| htxloleulxp pd| qrw h{lvw lv vkrzq lq Iljxuh441

1-p

Y

p

q

1-aa

X

L

l

R

r

1-cc

1-q

1-bb

Iljxuh 44= Jdph wuhh ri d wzr shuvrq jdph wkdw pd| qrw kdyh d uhflsurflw| htxlole0ulxp1

Dvvxph wkdw �2 ? d ? e ? f krogv1 Lq wklv jdph/ doo srvvleoh sd|r� vkduhv uhvxow

lq dq lqhtxlw| lq idyru ri sod|hu 51 Wkhuhiruh/ sod|hu 4 lv dozd|v nlqg1 Lq qrgh \sod|hu 4*v nlqgqhvv lv lqwhqwlrqdo lq wklv jdph li dqg rqo| li t��d. +4� t��,f A e1 Irulqvwdqfh/ li t�� @ 4/ wkh nlqgqhvv lv qrw lqwhqwlrqdo1 Dv d frqvhtxhqfh/ wkh uhflsurfdouhvsrqvh zloo eh zhdn/ l1h1/ wkh vhfrqg pryhu fkrrvhv d orz t1 Li/ rq wkh rwkhu kdqg/t�� @ 3/ wkh nlqgqhvv lv lqwhqwlrqdo dqg d kljk t zloo eh fkrvhq1 Khqfh/ dq htxloleulxppxvw olh vrphzkhuh lq ehwzhhq1 Krzhyhu/ gxh wr wkh glvfrqwlqxlw| ri ixqfwlrq / zhfdq �qg sdudphwhuv iru zklfk wkhuh lv qr uhflsurflw| htxloleulxp/ dv fodlphg lq wkhiroorzlqj sursrvlwlrq1

Sursrvlwlrq 8 Li �2K3� � �2 ?

�E2K3��02

/ wkhuh h{lvwv qr uhflsurflw| htxloleulxp lq wkhjdph suhvhqwhg lq Iljxuh 441

Wkh surri ri Sursrvlwlrq 8 lv jlyhq lq Dsshqgl{ 61Iljxuh 45 vkrzv wkh htxloleulxp fkrlfh ri t ghshqghqw rq �2= Dv rqh fdq vhh/ wkhuh

lv d udqjh +ehwzhhq 42� dqg ;��, zkhuh wkhuh lv qr htxloleulxp1Dv srlqwhg rxw deryh/ wkh uhdvrq zk| wkh h{lvwhqfh ri dq htxloleulxp lv qrw

jxdudqwhhg kdv wr gr zlwk wkh glvfrqwlqxlw| ri ixqfwlrq = Wr vkrz wklv/ zh gh�qhiru d +vpdoo, srvlwlyh qxpehu � d frqwlqxrxv dssur{lpdwlrq b iru 154 Zh vhw

54Zh kdyh lEZ�c Z� c Zf� c Z

f�� ' *�4

b<flbEZ�c Z� c Z

f� c Z

f� � iru dq| fkrlfh ri Z�c Z�c Z

f� c Z

f� �

65

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20reciprocity parameter of player 2

Iljxuh 45= Htxloleulxp fkrlfh ri t ghshqghqw rq �2 iru wkh jlyhq sdudphwhuv d @ 4/e @ 3=;/ f @ 3=9/ %2 @ 3=51

b+��> ��> �f� > �

f�, =@

;AAAAAAA?AAAAAAA=

plq+4> %� .�b+�f� � ��,, li �f� � �f� dqg �� ? �f�

%� li �f� � �f� dqg �� � �f�plq+4> %� .

�b+�� � �f� ,, li �f� ? �f� / �� A �f� dqg �� � ��

pd{+%�>plq+4� Z�3Z�Zf�3Z

f�

>

%� .�b+�� � �f� ,,

li �f� ? �f� / �� A �f� dqg �� A ��

%� li �f� ? �f� dqg �� � �f�Jlyhq wkh frqwlqxrxv yduldqw b ri > zh gh�qh d prgl�hg nlqgqhvv ixqfwlrq *b

dqg d xwlolw| ixqfwlrq Xb1 Zh fdoo d �0uhflsurflw| htxloleulxp d vxejdph shuihfw htxl0oleulxp ri wkh sv|fkrorjlfdo jdph zlwk xwlolw| Xb1 Wklv prgl�fdwlrq qrz jxdudqwhhvwkh h{lvwhqfh ri dq htxloleulxp dv wkh iroorzlqj wkhruhp vkrzv=

Wkhruhp 9 +H{lvwhqfh Wkhruhp, Ohw � eh d �qlwh wzr shuvrq h{whqvlyh irupjdph zlwk frpsohwh lqirupdwlrq1 Ohw � A 3= Wkhq � kdv d �0uhflsurflw| htxloleulxp1

Surri ri wkh H{lvwhqfh Wkhruhp= Ohw q 5 Q� eh d qrgh ri wkh jdph1 WkhqV? @ i+s@,@M�?

m s@ 5 ^3> 4`>S

@ s@ @ 4j lv wkh vhw ri pl{hg vwudwhjlhv lq wklv qrgh1Ohw V @

T?M� V? eh wkh vhw ri ehkdylru vwudwhj| frpelqdwlrqv1 Lw lqfoxghv wkh

vwudwhjlhv ri erwk sod|huv1 Ohw V3? @T

6 �'? V6 eh wkh vwudwhjlhv dw doo rwkhu qrghv1

Ohw v @ +v?> v3?, eh d ehkdylru vwudwhj| frpelqdwlrq zlwk v? 5 V? dqg v3? 5 V3?1Ohw v� dqg v�� wkh eholhiv ri �uvw dqg vhfrqg rughu1 Zh gh�qh Y +q> +v?> v3?,> v�> v��, dvwkh xwlolw| Xb

� frqglwlrqdo rq qrgh q1 Li q 5 Q� zh gh�qh Y +q> +v?> v3?,> v�> v��, dv wkhxwlolw| Xb

� frqglwlrqdo rq qrgh q1 Zh qrz gh�qh wkh ehvw uhso| fruuhvsrqghqfh

E = V � V = v :$ E+v, � V

Wkh frpsrqhqw E? = V � V? lv gh�qhg dv

E?+v, =@ duj pd{�?M7?

Y +q> +�?> v3?,> v> v,

66

Zh jhwE+v, =@ i+e?,?M� m e? 5 E?+v,j

Wklv gh�qlwlrq ri wkh ehvw uhso| fruuhvsrqghqfh lv uhodwhg wr wkh djhqw0vwudwhjlfirup +vhh Ixghqehuj dqg Wluroh +4<<4,,= Wkh sod|hu ehkdyhv dv li wkhuh zdv dqdjhqw lq hyhu| ghflvlrq qrgh1 D ehkdylru vwudwhj| wkdw rswlpl}hv lq wkh djhqw0vwudwhjlfirup fruuhvsrqgv wr d vxejdph shuihfw Qdvk htxloleulxp1

Dv V lv wkh surgxfw ri frqyh{ dqg frpsdfw vhwv/ lw lv frqyh{ dqg frpsdfw1 VlqfhV lv frpsdfw dqg Y lv frqwlqxrxv lq v?> E+v, lv qrw hpsw|1 Ehfdxvh Xb lv olqhdu lqwkh vwudwhjlhv/ E+v, lv frqyh{1

Zh qrz vkrz wkdw E lv xsshu0khpl frqwlqxrxv= Iluvw/ zh vkrz wkdw Y ghshqgvfrqwlqxrxvo| rq wkh vwudwhjlhv dqg eholhiv= Wkh lqhtxlw| �+q, lv frqwlqxrxv dqg�+q, @ 3 krogv iru ��+q> v��� > v

��, @ ��+q> v

��� > v

��,1 Ehfdxvh

b lv erxqghg +e| 4,/ zh jhwwkh ghvluhg frqwlqxlw| ri &+q,�+q, dw ��+q> v��� > v

��, @ ��+q> v

��� > v

��,1 E| frqvwuxfwlrq ri

b/ &b+q, lv frqwlqxrxv lq vwudwhjlhv dqg eholhiv li ��+q> v��� > v��, 9@ ��+q> v��� > v

��,1 Wkhuh0

iruh/ wklv dovr krogv iru &b+q,�+q,1 Khqfh/ *b+q, lv frqwlqxrxv1 Wkh uhflsurfdwlrqwhup dqg wkh pdwhuldo sur�w duh reylrxvo| frqwlqxrxv dqg wkhuhiruh Y lv frqwlqxrxv1Li dq| ixqfwlrq i = [ � \ $ U lv frqwlqxrxv/ wkhq wkh ehvw uhso| fruuhvsrqghqfhU = [ � \ = { :$ dujpd{+ i+{> |, lv xsshu khpl frqwlqxrxv1 Khqfh/ ehfdxvh Y lvfrqwlqxrxv/ wkh ehvw uhso| fruuhvsrqghqfh E lv xsshu khpl0frqwlqxrxv1

Wkhuhiruh/ zh fdq dsso| wkh �{hg srlqw wkhruhp ri Ndnxwdql dqg jhw dq vW 5 Vzlwk vW 5 E+vW,1 Wklv vwudwhj| vW zlwk �uvw rughu eholhi vW dqg vhfrqg rughu eholhi vW

qrz irupv d uhflsurflw| htxloleulxp= Wkh wulsoh +vW> vW> vW, wulyldoo| vdwlv�hv wkh frq0vlvwhqf| ri wkh eholhiv1 E| frqvwuxfwlrq ri E hdfk sod|hu rswlpl}hv klv xwlolw| lq hdfkqrgh 0 jlyhq wkh eholhiv dqg jlyhq wkh vwudwhjlhv ri wkh rwkhu sod|huv1 Wklv lv h{dfwo|wkh gh�qlwlrq ri wkh vxejdph shuihfw sv|fkrorjlfdo htxloleulxp1 Wklv frqfoxghv wkhsurri1 �

: Dsshqgl{ 5= H{whqvlrq wr q0shuvrq jdphv

Wkh lghd ehklqg wkh jhqhudol}dwlrq wr q shuvrqv frqvlvwv lq lqghshqghqwo| frqvlghulqjwkh uhflsurflw| uhodwlrq wrzdugv hdfk ri wkh rwkhu sod|huv1 Zh gr wklv lq wklv dsshqgl{yhu| eulh |1

Ohw q eh wkh qxpehu ri sod|huv1 Ohw v� 5 V� eh sod|hu l*v ehkdylru vwudwhj| dqg ohwvE��� 5 V� eh sod|hu l*v �uvw rughu eholhi derxw sod|hu m*v vwudwhj|1 Ixuwkhu ohw v

E�&�� 5 V&

eh sod|hu l*v +vhfrqg rughu, eholhi derxw zkdw kh wklqnv lv sod|hu m*v eholhi derxw n*vvwudwhj|1 Iru q @ 5/ zh jhw v�� @ v

E��� dqg v��� @ v

E���� 1 Ixuwkhupruh/ zh xvh wkh qrwdwlrq

+�lm, wr h{suhvv wkh vhw ri sod|huv rwkhu wkdq sod|huv l dqg m1 Vr/ vE�E3����� lv sod|hul*v eholhi derxw zkdw m eholhyhv zkdw wkh rwkhu sod|huv zloo gr1 Lq dqdorj| wr wkhgh�qlwlrqv +4, wr +:, zh gh�qh 0 iru d jlyhq vhw ri �uvw dqg vhfrqg rughu eholhiv 0 wkhiroorzlqj h{suhvvlrqv=

��<�+q, =@ ��+q> vE���� > v

E��� > v

E�E3����� ,� ��+q> v

E���� > v

E��� > v

E�E3����� , +4<,

��� =@

q���+q> v

E���� > vR� > v

E�E3����� ,> ��+q> v

E���� > vR� > v

E�E3����� ,

�mvR� 5 VR

�

r+53,

67

&�<�+q, =@ pd{q +��> ��> ��+q> v

E���� > v

E��� > v

E�E3����� ,> ��+q> v

E���� > v

E��� > v

E�E3����� ,, m +��> ��, 5 ��

r+54,

*�<�+q, @ &�<�+q,��<�+q, +55,

��<�+q> h, =@ ��+�+q> h,> vE���� > v

E��� > v

E�E3����� ,� ��+q> v

E���� > v

E��� > v

E�E3����� , +56,

X�+h, @ ��+h, . ��[� �'�

[?<e?M��

*�<�+q,��<�+q> h, +57,

D uhflsurflw| htxloleulxp lv djdlq d vhw ri vwudwhjlhv dqg �uvw dqg vhfrqg rughueholhyhv vxfk wkdw wkh vwudwhjlhv pd{lpl}h wkh xwlolw| lq Htxdwlrq +57, dqg vxfk wkdwvwudwhjlhv dqg eholhiv duh frqvlvwhqw1

; Dsshqgl{ 6= Surriv ri wkh sursrvlwlrqv

Surri ri Sursrvlwlrq 4= Ohw s ghqrwh wkh suredelolw| wkdw wkh uhvsrqghu dffhswvwkh r�hu> s� lv wkh sursrvhu*v eholhi derxw s dqg s�� wkh uhvsrqghu*v eholhi derxw s�1Ohw &+f, eh wkh lqwhqwlrq idfwru dw wkh qrgh diwhu sod|hu 4*v fkrlfh ri f1 Dffruglqj wrGh�qlwlrq 6/ wkh uhvsrqghu*v xwlolw| lq wkh uhflsurflw| jdph lv jlyhq e| X2� lq fdvhvkh dffhswv wkh r�hu dqg e| X2- li vkh uhmhfwv wkh r�hu=

X2� @ f. �2&+f,+s��+f� +4� f,,, � ++4� f,� s��+4� f,,

X2- @ 3 . �2&+f,+s��+f� +4� f,,, � +3� s��+4� f,,

Wkh gl�huhqfh lq wkh uhvsrqghu*v xwlolw| ehwzhhq dffhswlqj dqg uhmhfwlqj dprxqwvwr=

X2� �X2- @ f. �2&+f,s��+5f� 4,+4� f, +58,

Iru f � �2 / wkh h{suhvvlrq X2� � X2- lv srvlwlyh1 Wkxv/ r�huv ri dw ohdvw kdoi ri

wkh slh duh dozd|v dffhswhg/ l1h1/ s @ 41 Wklv suryhv wkh vhfrqg sduw ri htxdwlrq +;,ri Sursrvlwlrq 41

Ohw xv qrz wxuq wr wkh fdvh zkhuh f ? �2 1 Lq wklv fdvh/ wkh vhfrqg vxppdqg lq

htxdwlrq +58, lv qhjdwlyh1 Zh vhw htxdwlrq +58, htxdo wr }hur/ vroyh lw iru s��/ dqggh�qh wklv vroxwlrq dv s��So�|1 Zh jhw=

s��So�| @f

�2&+f,+4� 5f,+4� f,+59,

Li s�� ? s��So�| wkhq X2� A X2-1 Wklv lpsolhv wkdw sod|hu 5 dffhswv/ l1h1/ s @ 41 Wklvlv dq htxloleulxp li dqg rqo| li 4 @ s @ s�� ? s��So�|1 Li/ rq wkh rwkhu kdqg/ s�� A s��So�|wkhq X2� ? X2-1 Wklv lpsolhv wkdw sod|hu 5 uhmhfwv/ l1h1/ s @ 31 Wklv/ krzhyhu/ lvlpsrvvleoh ehfdxvh lw zrxog lpso| wkdw 3 @ s @ s�� A s��So�| A 31 +Wkh odvw lqhtxdolw|krogv ehfdxvh f ? �

2 1, Li/ �qdoo|/ s�� @ s��So�|/ wkhq X2� @ X2-1 Wkhuhiruh/ lq wklv fdvh/

68

sod|hu 5 lv lqgl�huhqw ehwzhhq dffhswlqj dqg uhmhfwlqj dqg zh jhw dq htxloleulxps @ s�� @ s��So�|1 Wklv pl{hg htxloleulxp lv srvvleoh iru 3 � s��So�| � 41 Wdnhq wrjhwkhu/

zh kdyh vkrzq wkdw sW @k

S42iES�E�32S�E�3S�

l�o1

Ohw xv qrz glvfxvv wkh uroh ri lqwhqwlrqv1 Li f ? �2 wkhq sod|hu 5 lv lq wkh glvdgydq0

wdjhrxv vlwxdwlrq1 Wkhuhiruh/ sod|hu 4*v pryh lv frqvlghuhg dv ixoo| lqwhqwlrqdo li dqgrqo| li wkhuh lv dq dfwlrq ri sod|hu 4 wkdw ohdgv wr d kljkhu sd|r� iru sod|hu 5 zlwkrxweulqjlqj sod|hu 4 lqwr wkh glvdgydqwdjhrxv vlwxdwlrq1 D frqwulexwlrq ri f @ �

2 vdwlv�hvwklv frqglwlrq1 Wkhuhiruh/ &+f, @ 4 zklfk frpsohwhv wkh surri ri htxdwlrq +;,1

Htxdwlrq +;, |lhogv wkh dffhswdqfh suredelolw| dv d ixqfwlrq ri f1 Ohw xv ghqrwh

wklv ixqfwlrq dv s+f,1 Ixuwkhupruh/ zh gh�qh ff =@�n�423

s�nS42n4

22

e421 Lw lv wkh vpdoohvw

frqwulexwlrq f zkhuh s htxdov 41 Zh qrz wxuq wr wkh sursrvhu*v ehkdylru1 Klv xwlolw|dffruglqj wr Gh�qlwlrq 6 dprxqwv wr=

X� @ s+f, � +4� f, . ��& � s+f��, � +4� 5f��, � +s+f,f� s+f��,f��, +5:,

Lq Htxdwlrq +5:, zh dssolhg douhdg| wkh frqvlvwhqf| frqglwlrq iru s +s @ s� @ s��,lq rughu wr dyrlg frqixvlrq ehwzhhq s� dv eholhi dqg wkh ghulydwlrq ri s dv d ixqfwlrqri f1 Zh �uvw qrwh wkdw Htxdwlrq +5:, lv ghfuhdvlqj lq f iru f�� � �

2 = Wkhuhiruh/ zh fdqfrqfoxgh wkdw f � �

2 = Lq wklv fdvh/ sod|hu 4 lv lq wkh dgydqwdjhrxv vlwxdwlrq dqg sod|hu5*v pryh lv ixoo| lqwhqwlrqdo li vkh kdv dq| dowhuqdwlyh dfwlrq wkdw ohdgv wr d vpdoohusd|r� iru sod|hu 41 Iru f A 3> wklv lv dozd|v wkh fdvh ehfdxvh sod|hu 5 fdq ghfuhdvhsod|hu 4*v sd|r� zlwk uhvshfw wr sod|hu 4*v h{shfwdwlrqv e| uhmhfwlqj wkh r�hu1 Dvzh zloo vhh ehorz/ sod|hu 4*v frqwulexwlrq lv srvlwlyh li erwk sod|huv kdyh d srvlwlyhuhflsurflw| sdudphwhu1 Wkhuhiruh/ zh fdq dvvxph ixoo lqwhqwlrqdolw|/ l1h1/ & @ 41

Lw fdq eh vkrzq iurp htxdwlrq +5:, wkdw wkh sursrvhu*v xwlolw| lv lqfuhdvlqj lq f dvorqj dv klv r�hu lv vwulfwo| vpdoohu wkdq ff +dqg wkhuhiruh wkh dffhswdqfh suredelolw|vpdoohu wkdq rqh,1 Wkxv/ wkh rswlpdo r�hu fW ri wkh uhvsrqghu lv dw ohdvw dv kljk dvff/ l1h1/ fW � ff1

Zh qrz nqrz wkdw wkh sursrvhu zloo fkrrvh fW dw ohdvw dv kljk dv wr jxdudqwhhdq dffhswdqfh suredelolw| ri rqh1 Wkhuhiruh/ zh frqvlghu wkh vlwxdwlrq zkhuh wkhsursrvhu fkrrvhv dq r�hu deryh ff +zklfk lv dozd|v dffhswhg,1 Li s @ 4/ zh jhw=

X�ER'�� @ +4� f, . �� � +4� 5f��, � +f� f��,

CX�ER'��

Cf@ �4 . �� � +4� 5f��,

Zh gh�qh

f��So�| @4

5

�4� 4

��

�Wkh xwlolw| X� lv ghfuhdvlqj lq f iru f�� A f��So�|/ X� lv lqfuhdvlqj lq f iru f�� ?

f��So�|/ dqg X� lv frqvwdqw lq f iru f�� @ f��So�|1 Frqvlghu wkh fdvh f��So�| ? ff= Vlqfh lqhtxloleulxp f @ f��/ zh jhw f��So�| ? ff � f @ f��1 Wkhuhiruh X� lv ghfuhdvlqj lq fdqg fW @ ff +@ pd{+ff> f��So�|,,1 Frqvlghu qrz wkh fdvh f��So�| � ff= Li f�� A f��So�|/ X� lvghfuhdvlqj lq f dqg wkhuhiruh f lv fkrvhq htxdo wr ff zklfk lv lqfrpsdwleoh zlwk f @ f��

ehfdxvh f�� A f��So�| � ff @ f1 Li f�� ? f��So�|/ X� lv lqfuhdvlqj lq f dqg wkhuhiruh f lv fkrvhq

69

htxdo wr 4 zklfk lv dovr lqfrpsdwleoh zlwk f @ f�� ehfdxvh f�� ? f��So�| ?�2 ? 4 @ f1

Wkhuhiruh fW @ f�� @ f��So�| +@ pd{+ff> f��So�|,,1

Wklv frpsohwhv wkh surri ri Sursrvlwlrq 41 �

Surri ri Sursrvlwlrq 5=Li �2 @ 3 wkh zrunhu fkrrvhv h @ 3 vlqfh khu pdwhuldo sd|r� lv ghfuhdvlqj lq h=

Jlyhq wklv/ wkh nlqgqhvv ri sod|hu 5 lv vpdoohu ru htxdo wr }hur1 Wkhuhiruh/ dq lqfuhdvhlq z uhgxfhv sod|hu 4*v pdwhuldo sd|r� dqg klv uhflsurflw| xwlolw|1 Khqfh/ zW @ hW @ 3lv wkh xqltxh htxloleulxp iru �2 @ 31

Ohw xv qrz dvvxph wkdw �2 A 3=Iru z @ 3> wkh zrunhu fkrrvhv dq h�ruw ri }hur1 Li z A 3> wkh zrunhu dozd|v fkrrvhv

dq h�ruw vxfk wkdw khu sd|r� lv vwulfwo| juhdwhu wkdq wkh sd|r� ri wkh hpsor|hu1 Li wkhzrunhu fkrrvhv hW @ 3> wkhq wklv lv reylrxvo| wkh fdvh1 Li wklv zrxog qrw eh wkh fdvhiru hW A 3/ wkh zrunhu frxog lqfuhdvh khu xwlolw| e| uhgxflqj khu h�ruw1 Wklv zrxoglqfuhdvh khu rzq sd|r� dqg ghfuhdvhv wkh sd|r� ri wkh hpsor|hu/ erwk ohdglqj wr dqlqfuhdvh lq khu xwlolw|1 Frqvhtxhqwo|/ wkh hpsor|hu lv dozd|v nlqg wr wkh zrunhu dqgwkh zrunhu lv dozd|v xqnlqg wr wkh hpsor|hu1

Wkh zrunhu*v xwlolw| lv=

X2 @ z � �h2 . �2&++z � �h��2,� +h�� �z,,+h�z � +h�� �z,,

Zh gl�huhqwldwh X2 zlwk uhvshfw wr h> wkhq vhw h @ h�� dqg zh jhw

hW @

%�+5�. �2&, .

s+5�. �2&,

2 . ;�+�2&,2z

5��2&

&�oZh vhh wkdw iru d srvlwlyh zdjh/ wkh zrunhu surylghv d srvlwlyh h�ruw1 Qrz/ zh

vkrz wkdw li wkh hpsor|hu sd|v d srvlwlyh zdjh/ wklv lv ixoo| lqwhqwlrqdo zklfk lpsolhv& @ 41 Wklv frqfoxghv wkh surri ri irupxod +44,= Lqghhg/ li zh wdnh wkh h�ruw ixqfwlrqhW+z, dv jlyhq/ lw fdq eh vkrzq wkdw li � ? �

H +zklfk lv wuxh vlqfh zh dvvxphg � � �e,/

wkh zrunhu*v sd|r� lv vwulfwo| lqfuhdvlqj lq wkh zdjh z sdlg1 +Wr vhh wklv/ fdofxodwhYZ2Y�

=, Wklv idfw lv lqghshqghqw ri wkh uhflsurflw| sdudphwhu �2 dqg wkh lqwhqwlrq idfwru&1 Wkhuhiruh/ li wkh hpsor|hu lv nlqg dqg sd|v d zdjh deryh }hur/ wkh zrunhu jhwvpruh wkdq li wkh hpsor|hu kdg sdlg d zdjh htxdo wr }hur1 Wkhuhiruh/ wkh hpsor|hufdq dozd|v eh ohvv nlqg e| sd|lqj d zdjh ri }hur zklfk lpsolhv wkdw wkh hpsor|hu*vnlqgqhvv lv ixoo| lqwhqwlrqdo1

Zh qrz wxuq wr wkh hpsor|hu*v ehkdylru1 Dv zh kdyh vhhq deryh/ wkh zrunhu lvxqnlqg li wkh hpsor|hu sd|v d srvlwlyh zdjh1 Wklv xqnlqgqhvv lv ixoo| lqwhqwlrqdo lidqg rqo| li wkh h�ruw surylghg lv vwulfwo| vpdoohu wkdq 4= +Li h ? 4/ wkh zrunhu frxog0 zlwkrxw vzlwfklqj lqwr d glvdgydqwdjhrxv vlwxdwlrq 0 lpsuryh wkh hpsor|hu*v sur�we| fkrrvlqj d voljkwo| kljkhu h�ruw1 Wklv lv lpsrvvleoh li h @ 41, Wkhuhiruh/ wkhuh duhwzr srvvleoh uhflsurflw| htxloleuld= +l, Wkh hpsor|hu fkrrvhv wkh plqlpdo zdjh wkdwjxdudqwhhv dq h�ruw ri 41 Wkh zrunhu lqghhg fkrrvhv dq h�ruw ri 41 Lq wklv fdvh/ wkhxqnlqgqhvv ri wkh zrunhu lv xqlqwhqwlrqdo1 +ll, Wkh hpsor|hu sd|v d zdjh vxfk wkdwwkh zrunhu surylghv dq h�ruw vpdoohu wkdq 41 Lq wklv fdvh wkh zrunhu dfwxdoo| fkrrvhvkhu h�ruw ixoo| lqwhqwlrqdoo|1

6:

Ohw xv �uvw frqvlghu wkh fdvh lq zklfk wkh hpsor|hu sd|v wkh plqlpdo zdjh wkdwjxdudqwhhv dq h�ruw ri 41 Zh kdyh gh�qhg wklv zdjh dv z1 Wr fkhfn xqghu zklfkfrqglwlrqv wkh hpsor|hu fkrrvhv wklv vwudwhj|/ zh fkhfn zkhwkhu d uhgxfwlrq ri wkhzdjh zrxog uhgxfh wkh hpsor|hu*v xwlolw|1 Zh fdofxodwh

z+�> �2, @4

5�2+5�. �2 . ��2,

Ch

Cz@

5�2t�7�2 . 7��2 . �22 . ;��22z

�Ch

Cz+z @ z, @

5�25�. �2 . 5��2

dqg jhw

X� @ h+z,�z . ��%�+h+z��, . �h+z��,� 5z,+z � �h+z,2 � +z�� � �h+z��,2,,

dqgCX�

Cz@

Ch+z,

Cz� 4 . ��%�+h+z

��, . �h+z��,� 5z,+4� 5�h+z,Ch+z,

Cz,

Qrz zh vhw YL�Y�

+z, @ 3 dqg jhw d fulwlfdo

+��%�,So�| @�2 +��2 . 5�. 5��2,

5� +�5�� �2 . 5��2,

Li wkh dfwxdo ydoxh ri ��%� lv ehorz wklv fulwlfdo ydoxh/ z lv d orfdo rswlpxp ri wkhhpsor|hu*v xwlolw|1 Wklv orfdo pd{lpxp lv joredo vlqfh h lv d frqfdyh ixqfwlrq lq z=

Wklv hvwdeolvkhv wkh h{lvwhqfh ri wkh �uvw w|sh ri htxloleuld1Lw qrz uhpdlqv wr frqvlghu wkh fdvh zkhuh wkh zrunhuv ehkdyh ixoo| lqwhqwlrqdo/

l1h1/ h+zW, ? 3 dqg & @ 41 Wkh fdofxodwlrq ri wklv htxloleulxp ohdgv wr hz+�> ��> �2,/zlwk

hz+�> ��> �2, @;AAA?AAA=

3ek23ek42n�422

Hk422li �� @ 3

��Sk . �

e4�. �k

e422. �

e42. 42

Hk4�.

422�Sk42�

��ESk4�n�4�42n4

22�s

ek242�nek42�42n�2k4�4

22n4

2�4

22324�4

�2n4

e2

�Sk42�422

li �� A 3

Li hz+�> ��> �2, � z+�> �2,/ zh jhw zW @ hz+�> ��> �2,1 Li hz+�> ��> �2, A z+�> �2, wkhhpsor|hu fkrrvhv zW @ z+�> �2, ehfdxvh iru zdjhv deryh z+�> �2, wkh hpsor|hu*vpdwhuldo sd|r� dv zhoo dv wkh uhflsurflw| xwlolw| duh ghfuhdvlqj lq z1 Wklv frqfoxghvwkh surri1 �

Sursrvlwlrq : +Ehvw0vkrw jdph, Lq wkh qrupdol}hg uhgxfhg ehvw0vkrw jdph wkh uhfl0surflw| htxloleuld duh ghvfulehg dv iroorzv=

Li �2%2 ?D� wkhq wkhuh lv d xqltxh uhflsurflw| htxloleulxp lq wkh uhgxfhg ehvw0vkrw

jdph zklfk lv jlyhq e|uW @ 4

6;

tW @

�8

45

4

%2�2

��o+5;,

Li �2%2 ?D�

sW @

%4 . f�H^W3f�2

f�SEf�H3f�2^W�4�

4 . tW

&�odf

+5<,

Li �2%2 @D� dqg �� @ 3 wkhq dq| sW 5 ^3> 4` lv sduw ri dq htxloleulxp1

Li �2%2 @D� dqg �� A 3 wkhq sW @ e

D 1Li �2%2 A

D� wkhq

sW @

;AAAAAAAAAAA?AAAAAAAAAAA=

%�n f�H^W3f�2

f�SEf�H3f�2^W�4��n^W

&�odf

li �� � �f3ef^W

�23�D^Wn�^W2%2n f�H^W3f�2

f�SEf�H3f�2^W�4��n^W

&�odf

li �f3ef^W

�23�D^Wn�^W2� �� � DE�3��^W3e^W2n0�E3�n�^Wne^W2��

�0�Ee3D^Wn^W2�%�n f�H^W3f�2

f�SEf�H3f�2^W�0�4��n^W

&�odf

li DE�3��^W3e^W2n0�E3�n�^Wne^W2���0�Ee3D^Wn^W2�

� ��

+63,

Surri ri Sursrvlwlrq 6 dqg : = Ohw xv �uvw frqvlghu wkh vlwxdwlrq ri sod|hu 5lq zklfk vkh kdv wr ghflgh zkhwkhu wr dffhsw ru uhmhfw vrph glvwulexwlrq zlwk d� irusod|hu 4 dqg d2 iru sod|hu 51 Ohw X� ghqrwh wkh xwlolw| ri sod|hu 5 li vkh dffhswv dqgX- wkh xwlolw| ri sod|hu 5 li vkh uhmhfwv wkh r�hu1 Ohw w eh wkh dffhswdqfh suredelolw|dqg q wkh qrgh zkhuh sod|hu 5 kdv wr ghflgh1 Zh jhw

X� @ d2 . �2&+q,+w��+d2 � d�,,+d� � ����+q,,

X- @ 3. �2&+q,+w��+d2 � d�,,+3� ����+q,,

dqgX� � X- @ d2 . �2&+q,+w

��+d2 � d�,,d�

Zh vhh lpphgldwho| wkdw li d2 � d� wkh r�hu lv dozd|v dffhswhg1 Wkhuhiruh/ uW @ 4 lqerwk uhgxfhg jdphv1

Iluvw/ tW/ lv ghulyhg dqdorjrxvo| wr wkh surri ri Sursrvlwlrq 4=

tW @

�3=5

3=;�2&+\ ,+3=;� 3=5,

��o@

�8

45�2&+\ ,

��oZh qrz kdyh wr fdofxodwh wkh ydoxh ri &+\ ,> wkh lqwhqwlrq idfwru1 Lq wkh uhgxfhg

xowlpdwxp jdph lw lv htxdo wr 4 ehfdxvh wkh sursrvhu kdv wkh srvvlelolw| wr r�hu+3=8> 3=8, zklfk lv ohvv xqnlqg wkdq +3=;> 3=5, dqg zklfk lv uhdvrqdeoh ehfdxvh wkhsursrvhu vwloo uhfhlyhv dw ohdvw dv pxfk dv wkh uhvsrqghu1 Wklv suryhv +47,1

Lq wkh uhgxfhg ehvw0vkrw jdph wkh dowhuqdwlyh vwudwhj| ri wkh sursrvhu +3=5> 3=;,sxwv klpvhoi lqwr d glvdgydqwdjhrxv vlwxdwlrq1 Wklv glvdgydqwdjhrxv lqhtxlw| lv dwohdvw dv odujh dv wkh dgydqwdjhrxv lqhtxlw| ri wkh uhihuhqfh fkrlfh ri +3=;> 3=5, +lw lv

6<

3=9 frpsduhg wr 3=9t�,1 Wkh dowhuqdwlyh +3=5> 3=;, kdv wkhuhiruh dq lqwhqwlrq idfwru ri%21 Khqfh/ lq wklv fdvh/ &+\ , @ %2 zklfk suryhv +4:,1

Lq wkh uhgxfhg xowlpdwxp jdph wkh vhfrqg pryhu lv dozd|v nlqg dqg frxog eh ohvvnlqg e| uhmhfwlqj wkh r�hu1 Wkhuhiruh/ sod|hu 4*v rswlpdo vwudwhj| fdq eh fdofxodwhgzlwk dq lqwhqwlrq idfwru ri 41 Zh jhw

XH2 @ 3=;t . ��+3=;s��t . 3=8+4� s��,� +3=5s��t . 3=8+4� s��,,,+3=5t � ���2,

XDD @ 3=8 . ��+3=;s��t . 3=8+4� s��,� +3=5s��t . 3=8+4� s��,,,+3=8� ���2,

XH2 � XDD @ +3=;t � 3=8, . ��+3=;� 3=5,s��t+3=5t � 3=8,

Wr jhw +48,/ zh vhw XH2 � XDD @ 3 dqg dsso| wkh vdph dujxphqwv dv lq wkh surriri Sursrvlwlrq 41 Wklv suryhv htxdwlrq +48,1

Lq wkh uhgxfhg ehvw0vkrw jdph/ wkh vlwxdwlrq lv vrphzkdw pruh frpsolfdwhg1 Zhjhw

XH2 @ 3=;t . ��&+[,+3=;s��t . 3=5+4� s��,� +3=5s��t . 3=;+4� s��,,,+3=5t � ���2,

X2H @ 3=5 . ��&+[,+3=;s��t . 3=5+4� s��,� +3=5s��t . 3=;+4� s��,,,+3=;� ���2,

XH2 � X2H @ +3=;t � 3=5, . ��&+[,++3=;� 3=5,+s��t � +4� s��,,,+3=5t � 3=;,

dqg wkhuhiruh wkh htxloleulxp dffhswdqfh suredelolw| sW lv=

sW @

%4 . f�H^W3f�2

f�SEf�H3f�2^W�iEf�4�

4 . tW

&�odf

Iru wkh ydoxh ri &+[,> zh kdyh wr frqvlghu wkh lqwhqwlrq ri sod|hu 51 Iluvw/ zh vkrzwkdw lq htxloleulxp �+[, lv srvlwlyh li dqg rqo| li 3=;tW � 3=5 lv srvlwlyh= Zh jhw�+[, @ 3=9+sWtW,� 3=9+4� sW,1 Li sW lv vwulfwo| zlwklq wkh lqwhuydo ^3> 4`/ wkhq �+[,lv srvlwlyh li dqg rqo| li 3 ? sWtW�+4�sW, @ sW+tW.4,�4 @ f�H^W3f�2

f�SEf�H3f�2^W�4�= Wkhuhiruh/

lq wklv fdvh/ �+[, lv srvlwlyh li dqg rqo| li 3=;tW � 3=5 lv srvlwlyh1 Li sW @ 4 wkhq�+[, @ 3=9�+sWtW�+4�sW,, @ 3=9tW> zklfk lpsolhv �+[, A 3= Qrwh wkdw sW @ 4 fdqrqo| rffxu li 3=;tW � 3=5 A 3= Li sW @ 3 wkhq �+[, @ 3=9 � +sWtW � +4� sW,, @ �3=9>wkhuhiruh �+[, ? 3= Wklv fdq rqo| eh wkh fdvh li 3=;tW � 3=5 ? 3=

Li 3=;tW � 3=5 A 3/ l1h1/ tW A �e > sod|hu 5 lv nlqg dqg fdq eh ohvv nlqg e| uhmhfwlqj

dq| r�hu1 Wkhuhiruh/ lq wklv fdvh/ &+[, @ 41 Wklv suryhv +5<,1Li tW @ �

e > wkhq sod|hu 4 lv pdwhuldoo| lqgl�huhqw ehwzhhq klv wzr srvvleoh dfwlrqv1Li �� @ 3> kh lv dovr lqgl�huhqw zlwk uhvshfw wr klv uhflsurflw| xwlolw|1 Li �� A 3> wkhqrqo| sW @ e

D fdq eh sduw ri dq htxloleulxp= Wklv fkrlfh ohdgv wr dq htxlwdeoh h{shfwhgsd|r� ri +3=5> 3=5,1 Dq| rwkhu fkrlfh ri sW zrxog ohdg wr vrph lqhtxlw|1 Ehfdxvhsod|hu 4 lv pdwhuldoo| lqgl�huhqw ehwzhhq wkh wzr fkrlfhv/ kh zrxog uhflsurfdwh vxfklqhtxlw| dqg wu| wr surgxfh dq lqhtxlw| lq wkh rssrvlwh gluhfwlrq1 Wkhuhiruh/ sW @ e

Dlv wkh rqo| htxloleulxp fkrlfh lq wklv fdvh1

Li 3=;tW � 3=5 ? 3/ l1h1/ tW ? �e > sod|hu 5 lv xqnlqg exw frxog eh ohvv xqnlqg li vkh

zrxog dffhsw erwk r�huv/ l1h1/ li vkh zrxog fkrrvh t @ 41 Wklv lv uhdvrqdeoh li s � �2

73

ehfdxvh lq wklv fdvh/ sod|hu 5 lv vwloo lq wkh dgydqwdjhrxv srvlwlrq1 Wkh lqhtxdolw|s � �

2 lv vdwlv�hg li �� � �f3ef^W

�23�D^Wn�^W2= Li s A �

2 > wkh lqwhqwlrq idfwru lv

&+[, @ pd{

�%�c4� �3=9s�� . 3=9+4� s��,

3=9s��t� � 3=9+4� s��,

�+64,

Li wkh �uvw whup dssolhv/ zh jhw

sW @

%4 . f�H^W3f�2

f�SEf�H3f�2^W�0�4�

4 . tW

&�odf

Wkh �uvw whup dssolhv li 4�3f�SRWnf�SE�3RW�f�SRW^W3f�SE�3RW� � %� zklfk lv wkh fdvh li �� � DE�3��^W3e^W2n0�E3�n�^Wne^W2��

�0�Ee3D^Wn^W2�1

Ilqdoo|/ li �� ? DE�3��^W3e^W2n0�E3�n�^Wne^W2���0�Ee3D^Wn^W2�

wkhq lq +64,/ wkh vhfrqg whup dssolhvdqg

sW @

575� Ef�23�H^W�f�SEf�H3f�2^W�4�

6 . tW

68�o

df

zklfk frqfoxghv wkh surri1 �Surri ri Sursrvlwlrq 7=Iluvw qrwh wkdw wkh uhvsrqghu zloo dffhsw dq| r�hu deryh 3=81 Wklv krogv ehfdxvh

dffhswlqj |lhogv d kljkhu pdwhuldo sd|r� wkdq uhmhfwlqj1 Pruhryhu/ dffhswlqj surylghvd kljkhu uhflsurflw| xwlolw| zlwk uhvshfw wr wkh sursrvhu zkr r�huv f4@ ehfdxvh wklvsursrvhu lv nlqg wrzdugv wkh uhvsrqghu1 Ixuwkhupruh/ wkh uhvsrqghu*v uhflsurflw|xwlolw| zlwk uhvshfw wr wkh rwkhu sursrvhuv lv }hur/ lqghshqghqw ri wkh uhvsrqghu*vghflvlrq1

Li dw ohdvw wzr sod|huv r�hu f @ 4 wkhq iru doo sursrvhuv dq| r�hu lv d ehvw uhvsrqvhvlqfh doo sd|r�v duh xqd�hfwhg e| wkhlu ghflvlrqv1 Wkhuhiruh/ wkh vwudwhjlhv ghvfulehglq Sursrvlwlrq 7 duh lqghhg dq htxloleulxp1 Rqo| rqh sursrvhu r�hulqj f @ 4 fdq0qrw eh dq htxloleulxp vlqfh wklv sod|hu kdv dq lqfhqwlyh wr orzhu klv r�hu voljkwo|=Wklv lqfuhdvhv klv pdwhuldo sd|r� dqg wkh uhflsurflw| xwlolw| wrzdugv wkh uhvsrqghu1Fkdqjlqj wkh r�hu iurp f @ 4 kdv qr h�hfw rq wkh uhflsurflw| xwlolw| wrzdugv wkhrwkhu uhvsrqghuv ehfdxvh � @ 31 Li f4@ ? 4 wkhq doo sursrvhuv kdyh dq lqfhqwlyhwr lqfuhdvh wkhlu r�hu wr f4@ 1 Li dw ohdvw wzr sursrvhuv r�hu f4@ ? 4 wkhq wkhvhsursrvhu kdyh dq lqfhqwlyh wr lqfuhdvh wkhlu r�hu lq�qlwhvlpdoo|1 Wklv lqfuhdvhv wkhpdwhuldo sd|r� iurp S4@

&wr f4@ / zkhuh n lv wkh qxpehu ri sursrvhuv r�hulqj f4@ 1

Rq wkh rwkhu kdqg/ lw fkdqjhv wkh uhflsurflw| xwlolwlhv rqo| lq�qlwhvlpdoo|1 Khqfhf4@ ? 4 fdqqrw eh dq htxloleulxp1 �

Surri ri Sursrvlwlrq 8=Wr surri Sursrvlwlrq 8 zh vkrz wkdw wkhuh lv qr t wkdw fdq eh sduw ri dq htxlole0

ulxp=Wkh xwlolw| ri sod|hu 5 lq wkh hqg qrghv Oo dqg Ou lv=

X2u, @ d. �2&+t��+d� +4� d, . +4� t��,+f� +4� f,,+4� d,

X2uo @ d. �2&+t��+d� +4� d, . +4� t��,+f� +4� f,,+4� f,

74

Zh vhw=

X2u, � X2uo @ +f� d, . �2&+t��+5d� 5f, . +5f� 4,+d� f,,

@ +f� d,�4� �2&+5f� 4� t��+5d� 5f,,

�Zh gh�qh=

t��So�| @f� �

2

f� d� 4

�2&5+f� d,

Wklv lv wkh vroxwlrq ri wkh htxdwlrq X2u, �X2uo @ 31 Sod|hu 5 zloo fkrrvh t @ 3 lit�� A t��So�| dqg t @ 4 li t�� ? t��So�|1 Zh vkrz wkdw wklv lv lpsrvvleoh iru wkh sdudphwhuvjlyhq lq wkh sursrvlwlrq1

Fdvh 4= Li t��d . +4 � t��,f A e krogv/ wkhuh lv ixoo lqwhqwlrq/ l1h1/ & @ 41 Wkhlqhtxdolw| t��d . +4 � t��,f A e lv htxlydohqw wr t�� ? S3K

S3@1 Ixuwkhupruh/ �

2K3� ? �2

krogv e| dvvxpswlrqv1 Khqfh/ zh jhw tSo�| � S3 �2

S3@� �

�2K3�

2ES3@�@

S3 �2

S3@� K3�

2S3@

@ S3KS3@

A t��1

Dv zh kdyh vhhq deryh wklv fdq rqo| eh wkh fdvh li t @ t�� @ 4 dqg wkhuhiruh/tSo�| A 41 Exw e| gh�qlwlrq ri d> e dqg f wkh lqhtxdolw| S3K

S3@? 4 krogv1

Fdvh 5= Li t��d. +4� t��,f � e krogv/ qr lqwhqwlrqv duh lqyroyhg dqg & @ %2 krogv1Wkh lqhtxdolw| t��d.+4�t��,f � e lv htxlydohqw wr t�� � S3K

S3@1 Ixuwkhupruh/ �

E2K3��02A �2

krogv e| dvvxpswlrqv dqg zh jhw tSo�| ?S3 �

2S3@

� ��

2K3�202ES3@�

@ S3KS3@

� S3KS3@

� t��1 Dv zh

kdyh vhhq deryh/ wklv fdq rqo| eh wkh fdvh li t @ t�� @ 3 dqg wkhuhiruh/ tSo�| � 31 Exwe| gh�qlwlrq ri d> e dqg f wkh lqhtxdolw| S3K

S3@A 3 krogv1 �

< Uhihuhqfhv

Dgdpv/ M1 V1 +4<98,= �Lqhtxlw| lq Vrfldo H{fkdqjh�/ lq= Ohrqkdug Ehunrzlw} +hg1,/Dgydqfhv lq H{shulphqwdo Sv|fkrorj| 5/ Qhz \run= Dfdghplf Suhvv/ 59:05<<1

Djhoo/ M1 +4<<<,= �Rq wkh Ehqh�wv iurp Uljlg Oderxu Pdunhwv= Qrupv/ PdunhwIdloxuhv/ dqg Vrfldo Lqvxudqfh�/ Hfrqrplf Mrxuqdo 43</ 47604971

Djhoo/ M1 dqg Oxqgeruj/ S1 +4<<8,= �Wkhrulhv ri Sd| dqg Xqhpsor|phqw= Vxuyh|Hylghqfh iurp Vzhglvk Pdqxidfwxulqj Ilupv�/ Vfdqglqdyldq Mrxuqdo ri Hfrqrplfv<:/ 5<8 0 63:1

Ehuj/ M1/ Glfnkdxw M1 dqg PfFdeh N1 +4<<8,= �Wuxvw/ Uhflsurflw|/ dqg VrfldoKlvwru|�/ Jdphv dqg Hfrqrplf Ehkdylru 43/ 45504751

Ehzoh|/ W1 +4<<8,= �D Ghsuhvvhg Oderu Pdunhw dv H{sodlqhg e| Sduwlflsdqwv�/Dphulfdq Hfrqrplf Uhylhz ;8/ Sdshuv dqg Surfhhglqjv/ 583 0 5871

Eolqghu/ D1 V1 dqg Fkrl/ G1 K1 +4<<3,= �D Vkuhg ri Hylghqfh rq Wkhrulhv ri ZdjhVwlfnlqhvv�/ Txduwhuo| Mrxuqdo ri Hfrqrplfv 438/ 4336043491

Eorxqw/ V1 +4<<8,= �Zkhq Vrfldo Rxwfrphv duhq*w Idlu= Wkh H�hfw ri FdxvdoDwwulexwlrqv rq Suhihuhqfhv�/ Rujdql}dwlrqdo Ehkdylru ) Kxpdq Ghflvlrq Surfhvvhv96/ 46404771

Erooh/ I1 dqg Rfnhqihov/ S1 +4<<3,= �Sulvrqhu*v Glohppd dv d Jdph zlwk Lqfrp0sohwh Lqirupdwlrq�/ Mrxuqdo ri Hfrqrplf Sv|fkrorj|/ 44/ 9<0;71

Erooh/ I1 dqg Nulwlnrv/ D1 +4<<;,= �Vhoi0Fhqwhuhg Lqhtxdolw| Dyhuvlrq yhuvxv Uhfl0surflw| dqg Dowuxlvp� plphr/ 472<8/ Hxursh0Xqlyhuvlw| Yldgulqd/ Iudqnixuw2Rghu1

75

Erowrq/ J1 dqg Rfnhqihov/ D1 +5333,= �HUF 0 D Wkhru| ri Htxlw|/ Uhflsurflw|dqg Frpshwlwlrq�/ Dphulfdq Hfrqrplf Uhylhz <3/ 49904<61

Fdphuhu/ F1 dqg Wkdohu/ U1 +4<<8,= �Xowlpdwxpv/ Glfwdwruv/ dqg Pdqqhuv�/ Mrxu0qdo ri Hfrqrplf Shuvshfwlyhv </ 53<054<1

Fdpsehoo/ F1 P1 dqg Ndpodql/ N1 +4<<:,= �Wkh Uhdvrqv iru Zdjh Uljlglw|=Hylghqfh iurp d Vxuyh| ri Ilupv�/ Txduwhuo| Mrxuqdo ri Hfrqrplfv 445/ :8<0:;<1

Fkduqhvv/ J1 +4<<9,= �Dwwulexwlrq dqg Uhflsurflw| lq d Vlpxodwhg Oderu Pdunhw=Dq H{shulphqwdo Lqyhvwljdwlrq�/ plphr/ Xqlyhuvlw| ri Ehunhoh|1

Fodun/ N1 dqg Vhiwrq/ P1 +4<<;,= �Wkh Vhtxhqwldo Sulvrqhu*v Glohppd= Hylghqfhrq Uhflsurfdo Dowuxlvp�/ plphr/ Xqlyhuvlw| ri Pdqfkdvwhu1

Gdylv/ G1 dqg Krow/ F1 +4<<6,= H{shulphqwdo Hfrqrplfv/ Sulqfhwrq Xqlyhuvlw|Suhvv/ Sulqfhwrq1

Gxizhqehuj/ P1 dqg Nlufkvwhljhu/ J1 +4<<;,= �D Wkhru| ri Vhtxhqwldo Uhflsurf0lw|�/ plphr/ FhqwHU iru Hfrqrplf Uhvhdufk/ Wloexuj1

Hfnho/ F1 dqg Jurvvpdq/ S1 +4<<;,= �DuhZrphq Ohvv Vho�vk WkdqPhqB Hylghqfhiurp Glfwdwru H{shulphqwv�/ Hfrqrplf Mrxuqdo 43;/ :590681

Idon/ D/ Ihku H1/ dqg Ilvfkedfkhu X1 +4<<<,= �Rq wkh Qdwxuh ri Idlu Ehkdylru �Lqwhqwlrqv Pdwwhu�/ plphr/ Xqlyhuvlw| ri ]xulfk1

Idon/ D/ Ihku H1/ dqg Ilvfkedfkhu X1 +5333,= �Lqirupdo Vdfwlrqv � Krz Pxfk dqgZk|B�/ plphr/ Xqlyhuvlw| ri ]xulfk1

Ihku/ H1 dqg Idon/ D1 +4<<<,= �Zdjh Uljlglwlhv lq d Frpshwlwlyh/ LqfrpsohwhFrqwudfw Pdunhw�/ Mrxuqdo ri Srolwlfdo Hfrqrp| 43:/ 43904671

Ihku/ H1/ Jçfkwhu/ V1/ dqg Nlufkvwhljhu J1 +4<<:,= �Uhflsurflw| dv d FrqwudfwHqirufhphqw Ghylfh= H{shulphqwdo Hylghqfh�/ Hfrqrphwulfd 98/ ;660;931

Ihku/ H1/ Nlufkvwhljhu/ J1/ dqg Ulhgo/ D1 +4<<6,= �Grhv Idluqhvv Suhyhqw PdunhwFohdulqjB Dq H{shulphqwdo Lqyhvwljdwlrq�/ Txduwhuo| Mrxuqdo ri Hfrqrplfv 43;/ 76:07931

Ihku/ H1 dqg Vfkplgw/ N1 +4<<<,= �D Wkhru| ri Idluqhvv/ Frpshwlwlrq/ dqg Fr0rshudwlrq�/ Txduwhuo| Mrxuqdo ri Hfrqrplfv 447/ Dxjxvw 4<<</ ;4:0;9;1

Ilvfkedfkhu/ X1/ Jçfkwhu/ V1 dqg Ihku/ H1 +4<<<,= �Dqrpdorxv Ehkdylru lq SxeolfJrrgv H{shulphqwv= Krz Pxfk dqg Zk|= Frpphqw�/ plphr/ Xqlyhuvlw| ri ]xulfk1

Iruv|wkh/ U1/ Krurzlw}/ M1/ Vdylq/ Q1 dqg Vhiwrq/ P1 +4<<7,= �Idluqhvv lq VlpsohEdujdlqlqj H{shulphqwv�/ Jdphv dqg Hfrqrplf Ehkdylru 9/ 67:069<1

Iudqflv/ K1 +4<;8,= �Wkh Odz/ Rudo Wudglwlrq dqg wkh Plqlqj Frppxqlw|�/ Mrxu0qdo ri Odz dqg Vrflhw| 45/ 559:055:41

Ixghqehuj/ G1 dqg Wluroh/ M1 +4<<4,= Jdph Wkhru|1 Wkh PLW Suhvv/ Fdpeulgjh/Pdvvdfkxvvhwv1

Jçfkwhu/ V1 dqg Idon/ D1 +4<<:,= �Uhsxwdwlrq ru Uhflsurflw|B�/ plphr/ Xqlyhuvlw|ri ]xulfk1

Jhdqdnrsorv/ M1/ Shdufh/ G1 dqg Vwdffkhwwl/ H1 +4<;<,= �Sv|fkrorjlfdo Jdphv dqgVhtxhqwldo Udwlrqdolw|�/ Jdphv dqg Hfrqrplf Ehkdylru 4/ 93 0 :<1

Jrxogqhu/ D1 +4<93,= �Wkh Qrup ri Uhflsurflw|�/ Dphulfdq Vrflrorjlfdo Uhylhz58/ 494 0 4:;1

Jrudqvrq/ U1 H1 dqg Ehunrzlw}/ O1 +4<99,= �Uhflsurflw| dqg Uhvsrqvlelolw| Uhdf0wlrqv wr Sulru Khos�/ Mrxuqdo ri Shuvrqdolw| dqg Vrfldo Sv|fkrorj| 6/ 55:05651

76

Juhhqehuj/ P1 V1 dqg Iulvfk/ G1 P1 +4<:5,= �H�hfw ri Lqwhqwlrqdolw| rq Zloolqjqhvvwr Uhflsurfdwh d Idyrug�/ Mrxuqdo ri H{shulphqwdo Vrfldo Sv|fkrorj| ;/ <<04441

J wk/ Z1/ Vfkplwwehujhu/ U1 dqg Vfkzdu}h/ E1 +4<;5,= �Dq H{shulphqwdo Dqdo|0vlv ri Xowlpdwlxp Edujdlqlqj�/ Mrxuqdo ri Hfrqrplf Ehkdylru dqg Rujdql}dwlrq 6/69:0;;1

J wk/ Z1 +4<<8,= �Rq Xowlpdwxp Edujdlqlqj H{shulphqwv 0 D Shuvrqdo Uhylhz�/Mrxuqdo ri Hfrqrplf Ehkdylru dqg Rujdql}dwlrq 5:/ 65< 0 6771

J wk/ Z1/ Kxfn/ V1 dqg Rfnhqihov S1 +4<<9,= �Wzr0ohyho xowlpdwxp edujdlqlqjzlwk lqfrpsohwh lqirupdwlrq� Hfrqrplf Mrxuqdo 439/ 8<609371

J wk/ Z1/ Pdufkdqg/ Q1 dqg Uxoolëuh/ M1O1 +4<<:,= �Rq wkh Uholdelolw| ri Uhfls0urfdo Idluqhvv 0 Dq H{shulphqwdo Vwxg|�/ Glvfxvvlrq sdshu/ Kxperogw0Xqlyhuvlw|Ehuolq1

Kduulvrq/ J1 Z1 dqg Kluvkohlihu M1 +4<;<,= �Dq H{shulphqwdo Hydoxdwlrq ri Zhdn0hvw Olqn2Ehvw Vkrw Prghov ri Sxeolf Jrrgv�/ Mrxuqdo ri Srolwlfdo Hfrqrp| <:/ 53405581

Kd|dvkl/ Q1/ Rvwurp/ H1/ Zdonhu/ M1/ dqg \dpdjlvkl/ W1 +4<<;,= �Uhflsurflw|/Wuxvw/ dqg wkh Vhqvh ri Frqwuro= D Furvv0Vrflhwdo Vwxg|�/ Glvfxvvlrq sdshu/ LqgldqdXqlyhuvlw|/ Eorrplqjwrq1

Kr�pdq/ H1/ PfFdeh/ N1 dqg Vplwk/ Y1 O1 +4<<9,= �Vrfldo Glvwdqfh dqg Rwkhu0Uhjduglqj Ehkdylru lq Glfwdwru Jdphv�/ Dphulfdq Hfrqrplf Uhylhz ;9/ 98609931

Kr�pdq/ H1/ PfFdeh/ N1/ Vkdfkdw/ N1/ dqg Vplwk/ Y1 +4<<7,= �Suhihuhqfhv/ Surs0huw| Uljkwv/ dqg Dqrq|plw| lq Edujdlqlqj Jdphv�/ Jdphv dqg Hfrqrplf Ehkdylru:/ 67906;31

Ndkqhpdq/ G1/ Nqhwvfk/ M1 dqg Wkdohu/ U1 +4<;9,= �Idluqhvv dv d Frqvwudlqwrq Sur�w0Vhhnlqj= Hqwlwohphqwv lq wkh Pdunhw�/ Dphulfdq Hfrqrplf Uhylhz :9/ 7/:5;0:741

Nuhsv/ G1/ Plojurp/ S1/ Urehuwv/ M1 dqg Zlovrq/ U1 +4<;5,= �Udwlrqdo Frrshudwlrqlq wkh Ilqlwho| Uhshdwhg Sulvrqhu*v Glohppd�/ Mrxuqdo ri Hfrqrplf Wkhru| 5:/ 57805851

Ohg|dug/ M1 +4<<8,= �Sxeolf Jrrgv= D Vxuyh| ri H{shulphqwdo Uhvhdufk�/ lq=Kdqgerrn ri H{shulphqwdo Hfrqrplfv/ hg1 e| M1 Ndjho dqg D1 Urwk/ Sulqfhwrq=Sulqfhwrq Xqlyhuvlw| Suhvv1

Ohylqh/ G1 +4<<;,= �Prgholqj Dowuxlvp dqg Vslwhixoqhvv lq H{shulphqwv�/ Uhylhzri Hfrqrplf G|qdplfv 4/ 8<609551

Orhzhqvwhlq/ J1 I1/ Wkrpsvrq/ O1/ dqg Ed}hupdq/ P1 K1 +4<;<,= �Vrfldo Xwlolw|dqg Ghflvlrq Pdnlqj lq Lqwhushuvrqdo Frqwh{wv�/ Mrxuqdo ri Shuvrqdolw| dqg VrfldoSv|fkrorj| 8:/ 75907741

Ploohu/ V1 +4<<:,= �Vwudwhjlhxqwhuvxfkxqj }xp Lqyhvwlwlrqvvslho yrq Ehuj/ Glfn0kdxw xqg PfFdeh�/ Glsorpd wkhvlv/ Xqlyhuvlw| ri Erqq1

Przgd|/ U1 W1 +4<<4,= �Htxlw| Wkhru| Suhglfwlrqv ri Ehkdylru lq Rujdql}dwlrqv�/lq= U1 P1 Vwhhuv dqg O1 Z1 Sruwhu +hgv1,/ Prwlydwlrq dqg Zrun Ehkdylru/ Qhz \run=PfJudz0Kloo/ 44404631

Sudvqlndu/ Y1 dqg Urwk/ D1 H1 +4<<5,/ �Frqvlghudwlrqv ri Idluqhvv dqg Vwudwhj|=H{shulphqwdo Gdwd iurp Vhtxhqwldo Jdphv�/ Txduwhuo| Mrxuqdo ri Hfrqrplfv/ ;980;;;1

77

Udelq/ P1 +4<<6,= �Lqfrusrudwlqj Idluqhvv lqwr Jdph Wkhru| dqg Hfrqrplfv�/Dphulfdq Hfrqrplf Uhylhz ;6/ 45;4 0 46351

Udelq/ P1 +4<<;,= �Sv|fkrorj| dqg Hfrqrplfv�/ Mrxuqdo ri Hfrqrplf Olwhudwxuh69/ 44 0 791

Urwk/ D1 +4<<8,= �Edujdlqlqj H{shulphqwv�/ lq= Kdqgerrn ri H{shulphqwdo Hfr0qrplfv/ hg1 e| M1 Ndjho dqg D1 Urwk/ Sulqfhwrq= Sulqfhwrq Xqlyhuvlw| Suhvv1

Urwk/ D1/ Sudvqlndu/ Y1/ Rnxqr0Ixmlzdud/ P1 dqg ]dplu/ V1 +4<<4,= �Edujdlqlqjdqg Pdunhw Ehkdylru lq Mhuxvdohp/ Omxeomdqd/ Slwwvexujk/ dqg Wrn|r= Dq H{shul0phqwdo Vwxg|�/ Dphulfdq Hfrqrplf Uhylhz ;4/ 439;043<81

Ux"h/ E1 +4<<8,= �Jliw Jlylqj zlwk Hprwlrqv�/ plphr/ Ghsw1 ri Hfrqrplfv/Sulqfhwrq Xqlyhuvlw|1

Vhowhq/ U1 +4<:;,= �Wkh Htxlw| Sulqflsoh lq Hfrqrplf Ehkdylru�/ Lq= GhflvlrqWkhru| dqg Vrfldo Hwklfv1 Lvvxhv lq Vrfldo Fkrlfh/ K1 Jrwwlqjhu dqg Z1 Ohlqihooqhu+Hgv1,/ Uhlgho Gruguhfkw1

Vhqhfd/ O1D1 +4<4:,= �Rq Ehqh�wv� ^Gh Ehqh�fllv`/ lq= Vhqhfd*v Prudov/ Qhz \run/Kdushu dqg Urz1

Vplwk/ D1 +4<:9,= Wkh Wkhru| ri Prudo Vhqwlphqwv1 Hglwhg e| G1G1 Udskdho dqgD1F1 Pdf�h/ R{irug/ Foduhqgrq Suhvv1

Vplwk/ N1 Z1 +4<<5,= �Uhflsurflw| dqg Idluqhvv= Srvlwlyh Lqfhqwlyhv iru Wd{Frpsoldqfh�/ lq= Zk| Shrsoh Sd| Wd{hv 0 Wd{ Frpsoldqfh dqg Hqirufhphqw/ hg1 e|Mrho Vohpurg/ Wkh Xqlyhuvlw| ri Plfkljdq Suhvv/ Dqq Dueru/ 55605831

Vplwk/ Y1 O1 +4<;5,> �Plfurhfrqrplf V|vwhpv dv dq H{shulphqwdo Vflhqfh�/ Dphu0lfdq Hfrqrplf Uhylhz :5/ <560<881

Vwhhuv/ U1 P1 dqg Sruwhu/ O1 Z1 +4<<4,= Prwlydwlrq dqg Zrun Ehkdylru/ IliwkHglwlrq/ Qhz \run= PfJudz0Kloo1

Vxjghq U1 +4<;7,= �Uhflsurflw|= Wkh Vxsso| ri Sxeolf Jrrgv Wkurxjk Yroxqwdu|Frqwulexwlrqv�/ Wkh Hfrqrplf Mrxuqdo <7/ ::50:;:

Wkdohu/ U1 K1 +4<;;,= �Dqrpdolhv= Wkh Xowlpdwxp Jdph�/ Wkh Mrxuqdo ri Hfr0qrplf Shuvshfwlyhv 5/ 4<805391

Wulyhuv/ U1 +4<:4,= �Wkh Hyroxwlrq ri Uhflsurfdo Dowuxlvp�/ Txdwhuo| Uhylhz riElrorj| 79/ 6808:1

Zdovwhu/ H1 dqg Zdovwhu/ J1 Z1 +4<:;,= Htxlw| 0 Wkhru| dqg Uhvhdufk/ Ervwrq/Doo|q dqg Edfrq1

Zdwdeh/ P1/ Whudl/ V1/ Kd|dvkl/ Q1/ dqg \dpdjlvkl/ W1 +4<<9,= �Frrshudwlrq lqwkh Rqh0Vkrw Sulvrqhu*v Glohppd Edvhg rq H{shfwdwlrqv ri Uhflsurflw|�/ MdsdqhvhMrxuqdo ri H{shulphqwdo Vrfldo Sv|fkrorj|/ [[[YL/ 4;604<91

78

The Working Papers of the Institute for Empirical Research in Economics can be downloaded in PDF-format fromhttp://www.unizh.ch/iew/wp/

Institute for Empirical Research in Economics, Blümlisalpstr. 10, 8006 Zurich, Switzerland

Phone: 0041 1 634 37 05 Fax: 0041 1 634 49 07 E-mail: bibiewzh@iew.unizh.ch

Working Papers of the Institute for Empirical Research in Economics

No.1. Rudolf Winter-Ebmer and Josef Zweimüller: Firm Size Wage Differentials in Switzerland: Evidence from Job

Changers, February 19992. Bruno S. Frey and Marcel Kucher: History as Reflected in Capital Markets: The Case of World War II, February

19993. Josef Falkinger, Ernst Fehr, Simon Gächter and Rudolf Winter-Ebmer: A Simple Mechanism for the Efficient

Provision of Public Goods – Experimental Evidence, February 19994. Ernst Fehr and Klaus M. Schmidt: A Theory of Fairness, Competition and Cooperation, April 19995. Markus Knell: Social Comparisons, Inequality, and Growth, April 19996. Armin Falk and Urs Fischbacher: A Theory of Reciprocity, July 20007. Bruno S. Frey and Lorenz Goette: Does Pay Motivate Volunteers?, May 19998. Rudolf Winter-Ebmer and Josef Zweimüller: Intra-firm Wage Dispersion and Firm Performance, May 19999. Josef Zweimüller: Schumpeterian Entrepreneurs Meet Engel’s Law: The Impact of Inequality on Innovation-

Driven Growth, May 199910. Ernst Fehr and Simon Gächter: Cooperation and Punishment in Public Goods Experiments, June 199911. Rudolf Winter-Ebmer and Josef Zweimüller: Do Immigrants Displace Young Native Workers: The Austrian

Experience, June 199912. Ernst Fehr and Jean-Robert Tyran: Does Money Illusion Matter?, June 199913. Stefan Felder and Reto Schleiniger: Environmental Tax Reform: Efficiency and Political Feasibility, July 199914. Bruno S. Frey: Art Fakes – What Fakes?, An Economic View, July 199915. Bruno S. Frey and Alois Stutzer: Happiness, Economy and Institutions, July 199916. Urs Fischbacher, Simon Gächter and Ernst Fehr: Are People Conditionally Cooperative? Evidence from a Public

Goods Experiment, July 200017. Armin Falk, Ernst Fehr and Urs Fischbacher: On the Nature of Fair Behavior, August 199918. Vital Anderhub, Simon Gächter and Manfred Königstein: Efficient Contracting and Fair Play in a Simple

Principal-Agent Experiment, September 200019. Simon Gächter and Armin Falk: Reputation or Reciprocity?, September 199920. Ernst Fehr and Klaus M. Schmidt: Fairness, Incentives, and Contractual Choices, September 199921. Urs Fischbacher: z-Tree - Experimenter’s Manual, September 199922. Bruno S. Frey and Alois Stutzer: Maximising Happiness?, October 199923. Alois Stutzer: Demokratieindizes für die Kantone der Schweiz, October 199924. Bruno S. Frey: Was bewirkt die Volkswirtschaftslehre?, October 199925. Bruno S. Frey, Marcel Kucher and Alois Stutzer: Outcome, Process & Power in Direct Democracy, November

199926. Bruno S. Frey and Reto Jegen: Motivation Crowding Theory: A Survey of Empirical Evidence, November 199927. Margit Osterloh and Bruno S. Frey: Motivation, Knowledge Transfer, and Organizational Forms, November 199928. Bruno S. Frey and Marcel Kucher: Managerial Power and Compensation, December 199929. Reto Schleiniger: Ecological Tax Reform with Exemptions for the Export Sector in a two Sector two Factor Model,

December 199930. Jens-Ulrich Peter and Klaus Reiner Schenk-Hoppé: Business Cycle Phenomena in Overlapping Generations

Economies with Stochastic Production, December 199931. Josef Zweimüller: Inequality, Redistribution, and Economic Growth, January 200032. Marc Oliver Bettzüge and Thorsten Hens: Financial Innovation, Communication and the Theory of the Firm,

January 200033. Klaus Reiner Schenk-Hoppé: Is there a Golden Rule for the Stochastic Solow Growth Model? January 200034. Ernst Fehr and Simon Gächter: Do Incentive Contracts Crowd out Voluntary Cooperation? February 200035. Marc Oliver Bettzüge and Thorsten Hens: An Evolutionary Approach to Financial Innovation, July 200036. Bruno S. Frey: Does Economics Have an Effect? Towards an Economics of Economics, February 200037. Josef Zweimüller and Rudolf Winter-Ebmer: Firm-Specific Training: Consequences for Job-Mobility, March 2000

Working Papers of the Institute for Empirical Research in Economics

No.

The Working Papers of the Institute for Empirical Research in Economics can be downloaded in PDF-format fromhttp://www.unizh.ch/iew/wp/

Institute for Empirical Research in Economics, Blümlisalpstr. 10, 8006 Zürich, Switzerland

Phone: 0041 1 634 37 05 Fax: 0041 1 634 49 07 E-mail: bibiewzh@iew.unizh.ch

38. Martin Brown, Armin Falk and Ernst Fehr: Contract Inforcement and the Evolution of Longrun Relations, March2000

39. Thorsten Hens, Jörg Laitenberger and Andreas Löffler: On Uniqueness of Equilibria in the CAPM, July 200040. Ernst Fehr and Simon Gächter: Fairness and Retaliation: The Economics of Reciprocity, March 200041. Rafael Lalive, Jan C. van Ours and Josef Zweimüller: The Impact of Active Labor Market Programs and Benefit

Entitlement Rules on the Duration of Unemployment, March 200042. Reto Schleiniger: Consumption Taxes and International Competitiveness in a Keynesian World, April 200043. Ernst Fehr and Peter K. Zych: Intertemporal Choice under Habit Formation, May 200044. Ernst Fehr and Lorenz Goette: Robustness and Real Consequences of Nominal Wage Rigidity, May 200045. Ernst Fehr and Jean-Robert Tyran: Does Money Illusion Matter? REVISED VERSION, May 200046. Klaus Reiner Schenk-Hoppé: Sample-Path Stability of Non-Stationary Dynamic Economic Systems, Juni 200047. Bruno S. Frey: A Utopia? Government without Territorial Monopoly, June 200048. Bruno S. Frey: The Rise and Fall of Festivals, June 200049. Bruno S. Frey and Reto Jegen: Motivation Crowding Theory: A Survey of Empirical Evidence, REVISED

VERSION, June 200050. Albrecht Ritschl and Ulrich Woitek: Did Monetary Forces Cause the Great Depression? A Bayesian VAR Analysis

for the U.S. Economy, July 200051. Alois Stutzer and Rafael Lalive: The Role of Social Work Norms in Job Searching and Subjective Well-Being, July

200052. Iris Bohnet, Bruno S. Frey and Steffen Huck: More Order with Less Law: On Contract Enforcement, Trust, and

Crowding, July 200053. Armin Falk and Markus Knell: Choosing the Joneses: On the Endogeneity of Reference Groups, July 200054. Klaus Reiner Schenk-Hoppé: Economic Growth and Business Cycles: A Critical Comment on Detrending Time

Series, August 200055. Armin Falk, Ernst Fehr and Urs Fischbacher: Appropriating the Commons – A Theoretical Explanation, September

200056. Bruno S. Frey and Reiner Eichenberger: A Proposal for a Flexible Europe, August 200057. Reiner Eichenberger and Bruno S. Frey: Europe’s Eminent Economists: A Quantitative Analysis, September 200058. Bruno S. Frey: Why Economists Disregard Economic Methodology, September 200059. Armin Falk, Ernst Fehr, Urs Fischbacher: Informal Sanctions, September 200060. Rafael Lalive: Did we Overestimate the Value of Health?, October 200061. Matthias Benz, Marcel Kucher and Alois Stutzer: Stock Options: the Managers’ Blessing. Institutional Restrictions

and Executive Compensation, October 2000