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^s +p I . q @+4 ,, . +4 s, p ` @^4+4s,` g . q @+4,/ zkhuh
g ghqrwhv wkh rqh0vkrw ghyldwlrq surw1 Qrwlfh wkdw wkh suhylrxv frqglwlrq fdq
eh uhzulwwhq dv iroorzv= ^p s+I q , q ` @^4 +4 s,` g .1
Khqfh/ frooxvlrq lv vhoi0hqiruflqj li wkh orqj uxq h{shfwhg orvvhv gxh wr wkh
sxqlvkphqw duh qr vpdoohu wkdq wkh rqh vkrw h{shfwhg qhw jdlqv iurp ghyldwlrq1
Wkh kljkhu s dqg I wkh ohvv olnho| iru frooxvlrq wr eh vxvwdlqhg dw htxloleulxp/
rwkhu wklqjv ehlqj htxdo1
+e, Wkh ghvfulehg jdph/ ehlqj d vxshujdph/ dgplwv d frqwlqxxp ri vroxwlrqv1
Li zh frqvlghu h{dfwo| wkh vdph prgho exw dvvxph wkdw upv/ dorqj wkh frooxvlyh
htxloleulxp/ vhw wkh sulfh s 5 +f> sp ,/ zklfk jlyhv upv d surw / wkhq lw lv
vwudljkwiruzdug wr vkrz wkdw upv* lqfhqwlyh frqvwudlqw zrxog eh vlplodu wr wkh
rqh ghulyhg lq +d,/ wkh rqo| glhuhqfh ehlqj wkdw p zrxog eh vxevwlwxwhg e|
1
+f, \hv/ wzr0skdvh sxqlvkphqw vwudwhjlhv zrxog lqfuhdvh wkh lqwhuydo ri glv0
frxqw idfwruv xqghu zklfk frooxvlrq pljkw eh vxvwdlqhg1
Vroxwlrq iru h{huflvh 55 +d, Lq rughu iru frooxvlrq wr eh vhoi hqiruflqj/
wkh iroorzlqj lqfhqwlyh frqvwudlqw vkrxog eh vdwlvhg= +p @qf ,@+4, g p @q1
Vlqfh wkh OKV ri wkh suhylrxv frqglwlrq lqfuhdvhv zlwk dqg / rqh frqfoxghv
wkdw li wkh glvfrxqw idfwru dqg wkh suredelolw| ri frqwlqxdwlrq duh kljk hqrxjk/
wkhq wkhvh vwudwhjlhv irup dq htxloleulxp ri wkh jdph1
+e, Lw lv vwudljkwiruzdug wr vhh wkdw wkh lqfhqwlyh frqvwudlqw ghulyhg lq +d, lv
uhod{hg zkhq wkh sxqlvkphqw ehfrphv vwurqjhu +l1h1/ f ghfuhdvhv,/ wkh ghyldwlrq
surwv g ghfuhdvh dqg wkh qxpehu ri upv q ghfuhdvh1
+f, Qr1 Dqrwkhu ihdvleoh htxloleulxp ri wklv jdph lv wkh rqh lq zklfk upv
sod| wkh Qdvk htxloleulxp ri wkh vwdjh jdph dw hdfk shulrg ri wkh uhshdwhg
jdph1
;:
Vroxwlrq
ri h{huflvh 561 +d, Wkh mrlqw surw ri wkh wzr upv duh P @
+d tl tm F . {l . o{m ,tl j{5l @5 dqg wkh sureohp ri wkh up lv wr
l@4
fkrrvh tl > tm zklfk pd{lplvh lw1 Vroylqj wkh IRFv gP @gtl @ 3 dqg irfxvlqj
rq wkh v|pphwulf vroxwlrq jlyhv t @ ++d F, . +4 . o,{, @71 Dw wkh uvw vwdjh
5
ri wkh jdph/ wkh mrlqw surw ehfrphv P @ ++d F, . +4 . o,{, @7 j{5 1
Pd{lplvdwlrq uhtxluhv gP @g{ @ 3/ zklfk lv vroyhg e|= {P @ +d F,+4 .
o,@^7j +4 . o,5 `/ dqg diwhu uhsodfhphqw wklv jlyhv wkh htxloleulxp rxwsxw= t P @
+d F,j@^7j +4 . o,5 `1
+e, Wklv lv grqh lq wkh wh{w1
+f, Frpsdulvrq ri wkh htxloleulxp ohyhov uhyhdov wkdw {P A {F iru o A :@4:/
zkhuhdv t P A t F iru 6j . 4 7o 8o5 A 31 Djdlq/ o pxvwseh kljk hqrxjk iru
prqrsro| wr jlyh ulvh wr d vxshulru rxwfrph= li o ? +5 . 5<,@8 * =9::/ wkhq
t P ? t F zkdwhyhu j1 Exw iru kljkhu ohyhov ri wkh vslooryhu sdudphwhu/ wkhuh h{lvw
frpelqdwlrqv ri j dqg o iru zklfk ixoo fr0rshudwlrq lv ehwwhu1 Lqwxlwlyho|/ wkhuh lv
d wudgh0r xqghu ixoo fr0rshudwlrq/ ehwzhhq kljkhu U)G gxh wr lqwhuqdolvdwlrq
ri wkh vslooryhu zklfk uhvwruhv U)G lqfhqwlyhv/ dqg odfn ri frpshwlwlrq lq wkh
surgxfw pdunhw zklfk uhgxfhv htxloleulxp rxwsxwv1 Wkh iruphu hhfw lv vwurqjhu
zkhq vslooryhuv duh yhu| kljk1
^Wkh yduldqw ri wkh h{huflvh lqyroyhv qglqj htxloleulxp U)G ohyhov dqg
rxwsxwv xqghu U)G fr0rshudwlrq/ zklfk lv grqh lq wkh wh{w/ dqg wkhq frpsduh
wkhp wr wkrvh rewdlqhg lq wkh ixoo fr0rshudwlrq uhjlph1 Lw lv hdv| wr fkhfn wkdw
{P A {M Y iru doo o/ zkhuhdv t P lv dozd|v vwulfwo| orzhu wkdq t M Y +t P ? t M Y
dprxqwv wr j A +4 . o,5 @6 zklfk lv dozd|v wuxh vlqfh j A 7@6 dqg o 4,1`
S5
Vroxwlrq ri h{huflvh 57 Dw wkh odvw vwdjh ri wkh jdph/ wkh Frxuqrw htxl0
oleulxp lv irxqg e| vroylqj hdfk up* sureohp pd{tl l @ +4 tl tm fO ,tl .
fO tm 1 Qrwh wkdw hdfk up kdv wr sd| wkh rwkhu d xqlw ur|dow| rq wkh rzq rxw0
sxw/ zklfk lpsolhv wkdw wkh ur|dowlhv dsshdu erwk dv d frvw dqg dv d uhyhqxh lq
wkh surw ixqfwlrq1 Iurp wkh IRFv rqh rewdlqv wkh htxloleulxp rxwsxw dw wkh
v|pphwulf htxloleulxp dv= t +fO , @ +4 fO ,@61 Wkh dvvrfldwhg shu0up surwv
duh +fO , @ +4 . 5fO ,+4 fO ,@<1
Dw wkh uvw vwdjh/ upv mrlqwo| ghflgh rq wkh furvv0olfhqvlqj djuhhphqw dqg
rq wkh ohyho ri fO 1 Qrwh wkdw wkh ixqfwlrq uhdfkhv lwv pd{lpxp lq fO @ 4@71
Wkh upv zloo wkhuhiruh kdyh dq lqfhqwlyh wr furvv0olfhqvh dqg zloo rswlpdoo|
fkrrvh fO @ 4@71 Wkhlu rxwsxwv dqg surwv zloo eh htxdo wr t +4@7, @ 4@7
dqg +4@7, @ 4@;1 Wkhvh ydoxhv fruuhvsrqg wr wkh mrlqw surw pd{lplvdwlrq
vroxwlrqv1 Lq rwkhu zrugv/ e| furvv0olfhqvlqj wkh upv pdqdjh wr uhdfk wkh
prqrsro| vroxwlrq/ lpsurylqj xsrq wkh Frxuqrw vroxwlrq rewdlqhg zkhq fO @ 3=
t +3, @ 4@6 dqg +3, @ 4@<1
Vroxwlrq ri h{huflvh 58 +d, Lq wkh odvw vwdjh/ jlyhq wkdw pdqxidfwxuhuv
frpshwh lq sulfhv/ wkh Ehuwudqg htxloleulxp dssolhv= wkh pdunhw sulfh zloo eh
s @ f . zD . zE / dqg qdo ghpdqg t @ 4 +f . zD . zE ,1
Lq wkh uvw vwdjh/ hdfk sdwhqw0kroghu ghflghv wkh ur|dow| ihh vr dv wr pd{zl l @
zl +4 f zl zm ,1 Iurp g l @gzl @ 3 lw iroorzv wkdw wkh v|pphwulf htxlole0
;;
ulxp lv z @ +4 f,@6/ dqg wkh qdo sulfh +e| vxevwlwxwlrq, lv s @ +5 . f,@61
Sdwhqw0kroghuv* surwv duh @ +4 f,5 @<1
+e, Xqghu wkh sdwhqw srro/ wkhuh lv mrlqw surw pd{lplvdwlrq ri wkh sdwhqw0
kroghuv1 Wkh srro*v sureohp lv wkhuhiruh pd{zl >zm S @ zl +4 f zl zm , .
zm +4 f zl zm ,1 Vroylqj wkh IRFv jlyhv wkh v|pphwulf vroxwlrq zS @
+4 f,@71 E| vxevwlwxwlrq sulfhv dqg +shu0up, surwv duh rewdlqhg dv= sS @
+4 . f,@5 dqg S @ +4 f,5 @;1
+f, Lw lv vwudljkwiruzdug wr vhh wkdw wkh sdwhqw srro Sduhwr grplqdwhv wkh
vlwxdwlrq zkhuh wkh wzr sdwhqwv duh olfhqvhg lqghshqghqwo|1 Ilqdo sulfhv +dv zhoo
dv ur|dowlhv, duh orzhu +wkhuhiruh/ frqvxphuv duh ehwwhu r, dqg sdwhqw0kroghuv*
surwv duh kljkhu1 +Pdqxidfwxuhuv lq wklv h{dpsoh dozd|v jhw }hur surwv1,
Vroxwlrq ri h{huflvh 59 Frooxvlrq1 Ohw xv uvw ghulyh wkh pdunhw door0
fdwlrq vfkhph vroxwlrq zkhq hdfk up vhoov rqo| lq lwv rzq pdunhw1 Lq wklv
fdvh/ surwv duh jlyhq e| @ +4 5t@v,t1 Iurp g@gt @ 3 lw iroorzv wkdw
p
t4d @ t5e @ v@7/ dqg t4e @ t5d @ 3/ p
4 @ 5 @ v@;1
Sxqlvkphqw1 Li upv zhuh sod|lqj wkh rqh0vkrw Qdvk jdph lq txdqwlwlhv/
wkh| zrxog vhoo lq erwk pdunhwv1 Wkhlu surwv zrxog eh= 4 @ sd +t4d > t5d ,t4d .
+se +t4e > t5e , w,t4e dqg 5 @ se +t4e > t5e ,t5e . +sd +t4d > t5d , w,t5d / zkhuh sn @
4 5+t4n . t5n ,@v/ iru n @ d> e1 Ilup l*v surjudpph lv pd{tld >tle l 1 Wkhuhiruh/
qglqj wkh htxloleulxp vroxwlrqv lqyroyhv vroylqj wkh v|vwhp ri irxu htxdwlrqv
gl @gtld @ 3/ g l @gtle @ 3/ iru l @ 4> 51
Lw lv hdv| wr fkhfn wkdw wkh vroxwlrq lv jlyhq e| wkh irxu sulfhv= t4d @
t5d @ t4e @ v+4 5w,@9/ dqg wkdw upv* surwv duh jlyhq e|=
t5e @ v+4 . w,@9/
h4 @ h5 @ v +4 . w,5 . +4 5w,5 @4;1
Ghyldwlrq1 Wkh rswlpdo ghyldwlrq ri/ vd|/ up 4 frqvlvwv ri vhwwlqj lwv
prqrsro| txdqwlw| lq wkh krph pdunhw d/ exw fkrrvlqj wkh +srvlwlyh, rxwsxw
t4ge wkdw pd{lplvhv lwv h{sruw surwv 4e jlyhq wkdw wkh ulydo vhwv t5e @ v@71 Lq
rwkhu zrugv/ t4ge lv wkh txdqwlw| t wkdw pd{lplvhv 4e @ +4 w 5 +t . v@7, @v, t1
Iurp g4e @gt @ 3 lw fdq eh irxqg wkdw t4ge @ v+4 5w,@;/ dqg wkdw g4 @
v@; . v+4 5w,5 @65 +wkh uvw frpsrqhqw ri wkh surw ehlqj wkh krph prqrsr0
olvwlf surw,1
Wkh lqfhqwlyh frqvwudlqw iru frooxvlrq lv jlyhq e| wkh vwdqgdug frqglwlrq
@+4
, gl .hl @+4 ,/ iurp zklfk wkh fulwlfdo glvfrxqw idfwru lv ghulyhg
p
l
h
@ gl hl 1 Diwhu vxevwlwxwlrq/ lw fdq eh fkhfnhg wkdw h @
dv @ gl p
l
< +4 5w, @ +46 . 55w,/ vr wkdw frooxvlrq fdq eh dfklhyhg rqo| li h 1
Qrwh wkdw g h @gw ? 3/ vr wkdw dv wudqvsruwdwlrq frvwv lqfuhdvh/ frooxvlrq
ehfrphv hdvlhu wr vxvwdlq1 Wklv lv ehfdxvh wkh kljkhu wkh wudqvsruwdwlrq frvw
wkh vpdoohu wkh pdunhw vkduh wkdw fdq eh rewdlqhg wkurxjk d ghyldwlrq1 Lq wkh
olplw fdvh zkhuh w @ 4@5/ wkh wudqvsruw frvw ehfrphv surklelwlyho| kljk/ vr wkdw
qr h{sruwv zrxog rffxu dw wkh rqh0vkrw htxloleulxp= d up zrxog qrw kdyh dq|
whpswdwlrq wr ghyldwh iurp wkh +prqrsrolvwlf, dxwdun| htxloleulxp1
;<
Table 4.1. The revelation game
firm 2
Reveal
Not Reveal
Reveal
?N
− R,
1− δ
?N
−R
1− δ
?N
− R,
1− δ
?N
−F
1− δ
Not Reveal
?N
− F,
1− δ
?N
−R
1− δ
firm 1
?N
?
− F ) + (1 − p ) M ,
1−δ
1− δ
?
?
p( N − F ) + (1 − p) M
1−δ
1 −δ
p(
δ P,δ c
δ P,δ c
δP
1
1
δP
0.5
0.5
δc
δc
0.2
q~ P 1 − c 1
n −1 n
(a)
q
P
1− c
n −1
1
q~ P
n
0.2
qP
(b)
Figure 4.1. Incentive Constraints along collusive and punishment paths
δ,δ cn
δ cn
1
δ
0.5
5
10
n
Figure 4.2. Conditions for collusion: Nash reversal (δ cn ) v. two-phase
(δ ) punishment strategies
AA
Investigation
f1
Reveal
 N

−R

 1− δ



 N −R
1 − δ

a
Reveal
Not
Reveal
f2
f2
Not Reveal
Reveal
 N

−R 

 1− δ



 N −F 
1 − δ

Figure 4.3. Game tree, at t = 2
 N

−F

 1− δ



 N −R 
1− δ

1-a
No Investigation


M
M


Not
Reveal
AA
Guilty
Not Guilty
p
1-p
 N

−F 

−
δ
1



 N −F 
1− δ

 M

 1− δ

 M
1− δ






1
a NC
NC
aCR
(1)
a
( 2)
CNR
CR
~
p
0
p
Figure 4.4. Equilibrium solutions for given policy parameters
1