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R9
Recipients’ Average Reward / Punishment to Allocators
−1
−.5
0
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1
1.5
Equal Recipients
Recipients’ Average Reward / Punishment to Allocators
−1
−.5
0
.5
1
1.5
All Recipients
Give to Me
Give to Other
Randomize
Give to Me
Randomize
Recipients’ Average Reward / Punishment to Allocators
−1
−.5
0
.5
1
1.5
Rich Recipients
Recipients’ Average Reward / Punishment to Allocators
−1
−.5
0
.5
1
1.5
Poor Recipients
Give to Other
Give to Me
Give to Other
Randomize
Give to Me
Give to Other
Randomize
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*>6 yXyyX h?mb- 2p2M i?Qm;? i?2 `2HiBp2Hv `B+?2` `2+BTB2Mib r2`2 ;Bp2M ?B;?2`
2M/QrK2Mib i?M `2+BTB2Mib rBi? 2[mH 2M/QrK2Mib- i?2v ;p2 mT H2bb KQM2v
iQ `2r`/ i?2 HHQ+iQ` 7Q` +?QQbBM; iQ `M/QKBx2 Ĝ BX2X r?2M i?2`2 rb BM@
2[mHBiv #2ir22M TQi2MiBH `2+BTB2Mib- i?2 `2HiBp2Hv `B+?2` `2+BTB2Mib bmTTQ`i2/
`M/QKBxiBQM H2bb i?M /B/ `2+BTB2Mib BM +QM/BiBQMb rBi? 2[mHBiv #2ir22M HH
TQi2MiBH `2+BTB2MibX bBKBH` `;mK2Mi +M #2 K/2 7Q` i?2 `2HiBp2Hv TQQ`2`
`2+BTB2Mib, 2p2M i?Qm;? i?2v r2`2 ;Bp2M HQr2` 2M/QrK2Mib i?M i?2 `2+BTB2Mib
rBi? 2[mH 2M/QrK2Mib- i?2v ;p2 mT KQ`2 KQM2v iQ TmMBb? i?2 HHQ+iQ` 7Q`
+?QQbBM; iQ `M/QKBx2 i?M /B/ `2+BTB2Mib rBi? 2[mH 2M/QrK2MibX h?mb- r?2M
i?2`2 rb BM2[mHBiv #2ir22M i?2 TQi2MiBH `2+BTB2Mib- #Qi? i?2 `2HiBp2Hv TQQ`2`
M/ i?2 `2HiBp2Hv `B+?2` `2+BTB2Mib bmTTQ`i2/ `M/QKBxiBQM H2bb i?M r?2M HH
Re
`2+BTB2Mib r2`2 ;Bp2M 2[mH 2M/QrK2MibX
h#H2 R bb2bb2b r?2i?2` i?2b2 /Bz2`2M+2b `2 biiBbiB+HHv bB;MB}+MiX q2
2biBKi2 i?2 7QHHQrBM; 2[miBQM b2T`i2Hv 7Q` 2+? TQbbB#H2 HHQ+iBQM +?QB+2
U:Bp2 iQ J2- :Bp2 iQ Pi?2`- M/ _M/QKBx2V,
Rewardic = β0 + β1 P ooric + β2 Richic + αi + εic .
URV
Rewardic Bb i?2 HBM2` `2r`/fTmMBb?K2Mi p`B#H2 T`2b2Mi2/ #Qp2- P ooric
Bb M BM/B+iQ` i?i i?2 `2+BTB2Mi bbmK2/ i?2 `QH2 Q7 i?2 `2HiBp2Hv TQQ`2` `2@
+BTB2Mi- r?BH2 Richic Bb M BM/B+iQ` i?i i?2 `2+BTB2Mi bbmK2/ i?2 `QH2 Q7 i?2
`2HiBp2Hv `B+?2` `2+BTB2MiX h?2 QKBii2/ +i2;Q`v Bb `2+BTB2Mib rBi? 2[mH 2M@
/QrK2MibX h?2 BMi2`T`2iiBQM Q7 +Q2{+B2Mi β0 Bb i?mb i?2 p2`;2 Reward
7Q` `2+BTB2Mib rBi? 2[mH 2M/QrK2Mib- r?BH2 +Q2{+B2Mib β1 M/ β2 +Tim`2 i?2
/Bz2`2M+2 BM p2`;2 Reward #2ir22M i?2 `2+BTB2Mib rBi? 2[mH 2M/QrK2Mib
M/ i?2 `2HiBp2Hv TQQ`2` M/ i?2 `2HiBp2Hv `B+?2` `2+BTB2Mib- `2bT2+iBp2HvX αi
+Tim`2b BM/BpB/mH H2p2H }t2/ 2z2+ib- εic Bb M 2``Q` i2`K +Hmbi2`2/ i i?2 T`@
iB+BTMi
BM/2t- i ∈ [1, 70] M/
! H2p2H- i Bb T`iB+BTMi
" BM/2t M/ c Bb +QM/BiBQM
#$
>B;? 2M/QrK2Mi
c ∈ (*2`iBM- lM+2`iBM) ×
X GBM2` +QK#BMiBQMb Q7
GQr 2M/QrK2Mi
+Q2{+B2Mib `2 `2TQ`i2/ BM i?2 HQr2` TM2H Q7 i?2 i#H2- HQM; rBi? +Q``2bTQM/@
BM; biM/`/ 2``Q`bX h?2 BMi2`T`2iiBQM Q7 i?2 HBM2` +QK#BMiBQM Q7 β0 M/ β1
Bb i?2 p2`;2 Reward +?Qb2M #v i?2 `2HiBp2Hv TQQ`2` `2+BTB2Mib- M/ i?2 HBM2`
+QK#BMiBQM Q7 β0 M/ β2 Bb i?2 p2`;2 Reward +?Qb2M #v i?2 `2HiBp2Hv `B+?2`
`2+BTB2MibX 6m`i?2`KQ`2- i?2 F @biiBbiB+ M/ i?2 +Q``2bTQM/BM; p@pHm2 7`QK qH/ i2bi Q7 2[mHBiv #2ir22M β1 M/ β2 Bb `2TQ`i2/ BM i?2 HQr2` TM2H Q7 i?2
i#H2X
*QHmKM (1) +QM}`Kb i?i HH i?`22 `2+BTB2Mi ivT2b- QM p2`;2- bB;MB}+MiHv
`2r`/2/ i?2 HHQ+iQ` 7Q` /B`2+i HHQ+iBQM iQ i?2Kb2Hp2b- M/ i?i i?2 /Bz2`@
2M+2b #2ir22M i?2 p2`;2 `2r`/b Q7 *>6 yXNy UEqualV- *>6 yXN3 UP oorV M/
*>6 RXyR URichV `2 MQi biiBbiB+HHv /Bz2`2Mi 7`QK 2+? Qi?2`X *QHmKM (2)
+QM}`Kb i?i HH i?`22 `2+BTB2Mi ivT2b- QM p2`;2- bB;MB}+MiHv TmMBb?2/ i?2
HHQ+iQ` 7Q` /B`2+i HHQ+iBQM iQ i?2 Qi?2` `2+BTB2Mi- M/ i?i p2`;2 TmMBb?@
K2Mi 7`QK i?2 `2HiBp2Hv TQQ`2` `2+BTB2Mi U*>6 @yXNdV Bb biiBbiB+HHv /Bz2`2Mi
Rd
h#H2 R, 1z2+i Q7 i?2 _2+BTB2MiǶb _2HiBp2 1M/QrK2Mi QM >2` _2@
r`/fSmMBb?K2Mi iQ i?2 HHQ+iQ`
.2T2M/2Mi p`B#H2, _2r`/
URV
:Bp2 iQ J2
UkV
:Bp2 iQ Pi?2`
UjV
_M/QKBx2
SQQ`
yXy3
(yXyN)
−yXkN
(yXRR)
−yX9d
(yXRk)
_B+?
yXRR
(yXRy)
yXjN
(yXR8)
−yXk8
(yXRR)
*QMbiMi
yXNy
(yXy8)
−yXe3
(yXye)
yXke
(yXyd)
yXNe
−yXe8
yXyR
J2M Q7 /2TX p`B#H2
SQQ` Y *QMbiMi
UbX2XV
yXN3∗∗∗
(yXye)
_B+? Y *QMbiMi
UbX2XV
RXyR∗∗∗
(yXyd)
6@bii, SQQ` 4 _B+?
T@pHm2
P#b2`piBQMb
yXye
yX3y
−yXNd∗∗∗
(yXRy)
−yXkk∗∗∗
(yXy8)
−yXkN∗∗
(yXRk)
yXyy
(yXy8)
RyXRe
yXyy∗∗∗
39y
39y
NX3d
yXyy∗∗∗
39y
LQi2b, 1z2+i Q7 i?2 `2+BTB2MiǶi `2HiBp2 2M/QrK2Mi QM ?2` `2r`/fTmMBb?K2Mi iQ
i?2 HHQ+iQ`X h?2 +QMbiMi i2`K +Tim`2b `2+BTB2MibǶ p2`;2 _2r`/ r?2M 2M@
/QrK2Mib `2 2[mHX _2;`2bbBQMb +QMi`QH 7Q` BM/BpB/mH H2p2H }t2/ 2z2+ibc biM/`/
2``Q`b `2 ?2i2`QbF2/biB+Biv@`Q#mbi M/ +Hmbi2`2/ i i?2 T`iB+BTMi H2p2HX HBM2`
+QK#BMiBQM Q7 β1 M/ β0 - +Tim`BM; i?2 p2`;2 _2r`/ 7`QK i?2 `2HiBp2Hv TQQ`2`
`2+BTB2Mib- M/ HBM2` +QK#BMiBQM Q7 β2 M/ β0 - +Tim`BM; i?2 p2`;2 _2r`/
7`QK i?2 `2HiBp2Hv `B+?2` `2+BTB2MibǶ- b r2HH b F @biiBbiB+b M/ +Q``2bTQM/BM; p@
pHm2b 7`QK qH/ i2bib 7Q` +Q2{+B2Mi 2[mHBiv #2ir22M β1 M/ β2 - `2 `2TQ`i2/ BM
i?2 HQr2` TM2H Q7 i?2 `2;`2bbBQM i#H2X aB;MB}+Mi i i?2 RyW +QM}/2M+2 H2p2HaB;MB}+Mi i i?2 8W +QM}/2M+2 H2p2HaB;MB}+Mi i i?2 RW +QM}/2M+2 H2p2HX
R3
7`QK i?2 p2`;2 TmMBb?K2Mi 7`QK `2HiBp2Hv `B+?2` `2+BTB2Mi U*>6 @yXkNVX 6m`@
i?2`KQ`2- i?2 p2`;2 TmMBb?K2Mi 7`QK `2+BTB2Mib rBi? 2[mH 2M/QrK2Mib U*>6
@yXe3V Bb Bb HKQbi i?2 2t+i KB/TQBMi Q7 i?2 p2`;2 TmMBb?K2Mib 7`QK i?2 `2H@
iBp2Hv TQQ`2` M/ i?2 `2HiBp2Hv `B+?2` `2+BTB2Mib- r?BH2 biBHH #2BM; biiBbiB+HHv
/Bz2`2Mi 7`QK #Qi? p2`;2bX *QHmKM (3) `2p2Hb i?i i?2 p2`;2 `2bTQMb2b iQ
`M/QKBxiBQM 7`QK #Qi? `2+BTB2Mib rBi? 2[mH 2M/QrK2Mib U*>6 yXkeV M/ i?2
`2HiBp2Hv TQQ`2` `2+BTB2Mib U*>6 @yXkkV r2`2 biiBbiB+HHv /Bz2`2Mi x2`Q- r?BH2
i?2v r2`2 HbQ biiBbiB+HHv /Bz2`2Mi 7`QK 2+? Qi?2`X h?2 p2`;2 `2bTQMb2 iQ
`M/QKBxiBQM 7`QK i?2 `2HiBp2Hv `B+?2` `2+BTB2Mib U*>6 yXyyV rb MQi biiBbiB@
+HHv /Bz2`2Mi 7`QK x2`Q- #mi Bi /Bz2`2/ bB;MB}+MiHv 7`QK i?2 p2`;2 `2bTQMb2b
Q7 #Qi? i?2 2[mH `2+BTB2Mib M/ i?2 `2HiBp2Hv TQQ`2` `2+BTB2MibX h?mb- HH `2H2@
pMi /Bz2`2M+2b /Bb+mbb2/ BM `2HiBQM iQ 6B;m`2 k `2 biiBbiB+HHv bB;MB}+MiX
AM bmK- r2 }M/ i?i- QM p2`;2- `2+BTB2Mib `2r`/2/ `M/QKBxiBQM i +Qbi
iQ i?2Kb2Hp2b r?2M i?2`2 rb 2[mHBiv BM 2M/QrK2Mib #2ir22M TQi2MiBH `2+BTB@
2Mibc TmMBb?2/ `M/QKBxiBQM i +Qbi iQ i?2Kb2Hp2b r?2M i?2v r2`2 `2HiBp2Hv
TQQ` +QKT`2/ iQ Qi?2` TQi2MiBH `2+BTB2Mibc M/ M2Bi?2` `2r`/2/ MQ` TmM@
Bb?2/ `M/QKBxiBQM r?2M i?2v r2`2 `2HiBp2Hv `B+? +QKT`2/ iQ Qi?2` TQi2MiBH
`2+BTB2MibX
_2+BTB2Mi "2?pBQ` 7`QK i?2 S2`bT2+iBp2 Q7 i?2 HHQ+iQ`
q2 MQr bF ?Qr `2+BTB2Mi #2?pBQ` HQQFb 7`QK i?2 T2`bT2+iBp2 Q7 i?2 HHQ+iQ`X
hF2M iQ;2i?2`- i?2 #Qp2 `2bmHib BKTHv i?i M HHQ+iQ` r?Q +?Qb2 iQ `M/QK@
Bx2 #2ir22M 2[mH `2+BTB2Mib +QmH/ 2tT2+i iQ #2 `2r`/2/ 7Q` i?Bb +?QB+2X PM
i?2 Qi?2` ?M/- M HHQ+iQ` r?Q +?Qb2 iQ `M/QKBx2 #2ir22M mM2[mH `2+BTB2Mib
+QmH/ 2tT2+i iQ #2 TmMBb?2/ 7Q` i?i bK2 +?QB+2X 6B;m`2 j b?Qrb i?2 HHQ+iQ`bǶ
2tT2+i2/ M2i Reward Ui?2 bmK Q7 `2r`/bfTmMBb?K2Mib 7`QK i?2 irQ `2+BTB2Mib
MQi iFBM; BMiQ ++QmMi i?2 R,k `2r`/fTmMBb?K2Mi i2+?MQHQ;vV 7Q` 2+? HHQ@
+iBQM +?QB+2 +`Qbb i?2 2[mH 2M/QrK2Mi +QM/BiBQMb M/ +`Qbb i?2 mM2[mH
2M/QrK2Mi +QM/BiBQMb- `2bT2+iBp2HvX
AM i?2 2[mH 2M/QrK2Mi +QM/BiBQMb- HHQ+iQ`bǶ KtBKBx2/ i?2B` 2tT2+i2/ M2i
Reward #v +?QQbBM; iQ `M/QKBx2- #mi i?2v HbQ `2+2Bp2/ TQbBiBp2 2tT2+i2/
M2i Reward r?2M HHQ+iBM; /B`2+iHv iQ 2Bi?2` _2+BTB2Mi Q` _2+BTB2Mi "X AM
RN
Randomize
Allocators’ Expected Net Reward / Punishment From Recipients
−.5
0
.5
1
Allocators’ Expected Net Reward / Punishment From Recipients
−.5
0
.5
1
Equal Recipients
Give to A/B
Unequal Recipients
Randomize
Give to Poor
Give to Rich
6B;m`2 j, HHQ+iQ`bǶ 1tT2+i2/ L2i Reward 7Q` 1+? SQbbB#H2 HHQ+iBQM *?QB+2
LQi2b, .`F ;`v #`b b?Qr HHQ+iQ`bǶ 2tT2+i2/ M2i _2r`/- /2}M2/ b i?2 bmK Q7 `2r`/b
M/ TmMBb?K2Mib 7`QK #Qi? `2+BTB2Mib rBi?BM ;Bp2M i`BH- r?2M i?2v +?Qb2 iQ `M/QKBx2
#2ir22M i?2 irQ `2+BTB2Mib BM i?2 2[mH 2M/QrK2Mi +QM/BiBQMb UH27i TM2HV M/ BM i?2 mM2[mH
2M/QrK2Mib +QM/BiBQMb U`B;?i TM2HVX J2/BmK ;`v #`b b?Qr HHQ+iQ`bǶ 2tT2+i2/ M2i _2@
r`/ r?2M i?2v +?Qb2 iQ HHQ+i2 /B`2+iHv iQ 2Bi?2` _2+BTB2Mi Q` _2+BTB2Mi " BM i?2 2[mH
2M/QrK2Mib +QM/BiBQMb UH27i TM2HV Q` iQ i?2 `2HiBp2Hv TQQ`2` `2+BTB2Mi BM i?2 mM2[mH 2M/Qr@
K2Mi +QM/BiBQMb U`B;?i TM2HVX h?2 HB;?i ;`2v #` U`B;?i TM2HV b?Qrb HHQ+iQ`bǶ 2tT2+i2/ M2i
_2r`/ r?2M i?2v +?Qb2 iQ HHQ+i2 /B`2+iHv iQ i?2 `2HiBp2Hv `B+?2` `2+BTB2Mi BM i?2 mM2[mH
√ X
2M/QrK2Mi +QM/BiBQMbX 1``Q` #`b `2 Mśp2 biM/`/ 2``Q`b- +H+mHi2/ b S.D.
N
ky
i?2 mM2[mH r2Hi? +QM/BiBQM- HHQ+iQ`b KtBKBx2/ i?2B` 2tT2+i2/ M2i Reward
#v HHQ+iBM; /B`2+iHv iQ i?2 `2HiBp2Hv TQQ`2` `2+BTB2MiX "mi ?2`2 i?2v `2+2Bp2/
M2;iBp2 2tT2+i2/ M2i Reward r?2M i?2v +?Qb2 iQ `M/QKBx2- M/ bKHH
#mi TQbBiBp2 2tT2+i2/ M2i Reward r?2M i?2v +?Qb2 iQ HHQ+i2 /B`2+iHv iQ i?2
`2HiBp2Hv `B+?2` `2+BTB2MiX
h#H2 k bb2bb2b r?2i?2` i?2b2 /Bz2`2M+2b `2 biiBbiB+HHv bB;MB}+MiX AM
+QHmKM (1) r2 2biBKi2 i?2 7QHHQrBM; 2[miBQM 7Q` i?2 2[mH 2M/QrK2Mi +QM/B@
iBQMb,
Rewardgc = β0 + β1 Give to A/B gc + αg + εgc ,
UkV
r?2`2 Reward Bb i?2 bK2 p`B#H2 b mb2/ BM i?2 T`2pBQmb `2;`2bbBQMb M/
Give to A/B Bb M BM/B+iQ` i?i i?2 HHQ+iQ` +?Qb2 iQ HHQ+i2 /B`2+iHv iQ
2Bi?2` _2+BTB2Mi Q` _2+BTB2Mi "X c Bb +QM/BiBQM BM/2t- g Bb M BM/2t i i?2
;`QmT H2p2H i?i B/2MiB}2b r?B+? irQ `2+BTB2Mib r2`2 TB`2/ rBi? r?B+? HHQ+iQ`
BM 2+? i`BHX α +Tim`2b ;`QmT H2p2H }t2/ 2z2+ib- M/ ε Bb M 2``Q` i2`K +Hmbi2`2/
i i?2 ;`QmT H2p2H- bBM+2 `M/QK bbB;MK2Mi iQ +QM/BiBQMb ?TT2M2/ i i?Bb H2p2HX
AM +QHmKM (2) r2 2biBKi2 bBKBH` 2[miBQM 7Q` i?2 mM2[mH 2M/QrK2Mi
+QM/BiBQMb,
Rewardgc = β0 + β1 Give to P oorgc + β2 Give to Richgc + αg + εgc ,
UjV
r?2`2 Give to P oor Bb M BM/B+iQ` i?i i?2 HHQ+iQ` +?Qb2 iQ HHQ+i2 /B`2+iHv
iQ i?2 `2HiBp2Hv TQQ`2` `2+BTB2Mi- M/ Give to Rich Bb M BM/B+iQ` i?i i?2
HHQ+iQ` +?Qb2 iQ HHQ+i2 /B`2+iHv iQ i?2 `2HiBp2Hv `B+?2` `2+BTB2MiX 1p2`vi?BM;
2Hb2 Bb b BM 1[miBQM kX AM #Qi? +QHmKMb (1) M/ (2)- i?2 BMi2`T`2iiBQM Q7 i?2
+Q2{+B2Mi β0 Bb i?mb HHQ+iQ`bǶ 2tT2+i2/ M2i Reward 7Q` `M/QKBxiBQMX AM +QH@
mKM (1) i?2 BMi2`T`2iiBQM Q7 +Q2{+B2Mi β1 Bb i?2 BM+`2b2f/2+`2b2 BM 2tT2+i2/
M2i Reward r?2M HHQ+iQ`b +?Qb2 iQ HHQ+i2 /B`2+iHv iQ 2Bi?2` _2+BTB2Mi Q`
_2+BTB2Mi " BM i?2 2[mH 2M/QrK2Mi +QM/BiBQMbX AM +QHmKM (2) i?2 BMi2`T`2i@
iBQM Q7 +Q2{+B2Mib β1 M/ β2 Bb i?2 BM+`2b2f/2+`2b2 BM 2tT2+i2/ M2i Reward
r?2M HHQ+iQ`b +?Qb2 iQ HHQ+i2 /B`2+iHv iQ i?2 `2HiBp2Hv TQQ`2` `2+BTB2Mi Q` i?2
kR
`2HiBp2Hv `B+?2` `2+BTB2Mi- `2bT2+iBp2Hv- BM i?2 mM2[mH 2M/QrK2Mi +QM/BiBQMbX
GBM2` +QK#BMiBQMb Q7 +Q2{+B2Mib β0 M/ β1 `2 `2TQ`i2/ BM i?2 HQr2` TM2H
Q7 i?2 i#H2 BM #Qi? +QHmKMbX AM +QHmKM (1) i?Bb HBM2` +QK#BMiBQM +Tim`2b
HHQ+iQ`bǶ 2tT2+i2/ M2i Reward 7Q` /B`2+i HHQ+iBQM iQ 2Bi?2` _2+BTB2Mi Q`
_2+BTB2Mi " BM i?2 2[mH 2M/QrK2Mi +QM/BiBQMbX AM +QHmKM (2) i?Bb HBM2` +QK@
#BMiBQM +Tim`2b HHQ+iQ`bǶ 2tT2+i2/ M2i Reward 7Q` /B`2+i HHQ+iBQM iQ i?2
`2HiBp2Hv TQQ`2` `2+BTB2Mi BM i?2 mM2[mH 2M/QrK2Mi +QM/BiBQMbX AM +QHmKM (2)
HBM2` +QK#BMiBQM Q7 β1 M/ β2 - +Tim`BM; HHQ+iQ`bǶ 2tT2+i2/ Reward 7Q`
/B`2+i HHQ+iBQM iQ i?2 `2HiBp2Hv `B+?2` `2+BTB2Mi- b r2HH b i?2 F @biiBbiB+ M/
+Q``2bTQM/BM; p@pHm2 7`QK qH/ i2bi Q7 2[mHBiv #2ir22M β1 M/ β2 `2 HbQ
`2TQ`i2/ BM i?2 HQr2` TM2H Q7 i?2 i#H2X
h#H2 k b?Qrb i?i r?2M M HHQ+iQ` +?Qb2 iQ `M/QKBx2 #2ir22M TQi2MiBH
`2+BTB2Mib r?Q ?/ 2[mH 2M/QrK2Mib BM 2tT2+iiBQM- i?2v +QmH/ BM/22/ 2tT2+i
bB;MB}+MiHv TQbBiBp2 M2i Reward Q7 *>6 yX8R Ĝ Q` KQ`2 i?M RyW Q7 i?2
T`Bx2 #2BM; HHQ+i2/X >Qr2p2`- 2p2M i?Qm;? i?Bb 2tT2+i2/ M2i Reward Bb KQ`2
i?M irB+2 b H`;2 b i?2 2tT2+i2/ M2i Reward 7Q` /B`2+i HHQ+iBQM iQ 2Bi?2`
_2+BTB2Mi Q` _2+BTB2Mi "- *>6 yXkk- i?2b2 irQ pHm2b `2 MQi bB;MB}+MiHv
/Bz2`2Mi 7`QK 2+? Qi?2`X
h#H2 k HbQ b?Qrb i?i M HHQ+iQ` r?Q +?Qb2 iQ `M/QKBx2 #2ir22M TQ@
i2MiBH `2+BTB2Mib r?Q ?/ mM2[mH 2M/QrK2Mib BM 2tT2+iiBQM +QmH/ 2tT2+i bB;MB}+MiHv M2;iBp2 M2i Reward Q7 *>6 @yXkR Ĝ Q` 9XkW Q7 i?2 T`Bx2 #2BM;
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ke
.8
Proportion of Trials
.4
.6
.2
0
0
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Proportion of Trials
.4
.6
.8
1
Unequal Endowments
1
Equal Endowments
Randomize
Give to A
Give to B
Randomize
Give to Poor
Give to Rich
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iQ i?2 `2HiBp2Hv `B+?2` `2+BTB2Mi
% BM i?2 mM2[mH 2M/QrK2Mi +QM/BiBQMbX Lśp2
biM/`/ 2``Q`b- +H+mHi2/ b p(1−p)
- `2 `2TQ`i2/ BM T`2Mi?2b2bX t@biiBbiB+b
N
7`QK i@i2bib +QKT`BM; i?2 /Bz2`2Mi T`QTQ`iBQMb iQ 2+? Qi?2` `2 `2TQ`i2/ BM i?2
HQr2` TM2H Q7 i?2 i#2H iQ;2i?2` rBi? +Q``2bTQM/BM; p@pHm2bX
k3
r?Q +?Qb2 iQ HHQ+i2 /B`2+iHv iQ 2Bi?2` _2+BTB2Mi Q` _2+BTB2Mi " b22K iQ ?p2
dz`M/QKBx2/Ǵ #2ir22M i?2 irQ TQi2MiBH `2+BTB2Mib- b i?2v HHQ+i2/ /B`2+iHv iQ
_2+BTB2Mi BM ReW Q7 i`BHb M/ /B`2+iHv iQ _2+BTB2Mi " BM R8W Q7 i`BHbX h?mbi?2`2 b22Kb iQ #2 MQ H#2H 2z2+i BM HHQ+iQ` #2?pBQ`X
aBM+2 i?2 TQi2MiBH `2+BTB2Mib QM p2`;2 `2r`/2/ `M/QKBxiBQM r?2M i?2v
?/ 2[mH 2M/QrK2Mib- i?2 HHQ+iQ`bǶ +?QB+2b r2`2 `2HiBp2Hv ;QQ/ Ki+? Q7
`2+BTB2MibǶ #2?pBQ` BM i?2 2[mH 2M/QrK2Mi +QM/BiBQMbX h?Bb H2/b iQ ?B;?
HHQ+iQ` dz2{+B2M+vǴ Q7 3kX9W- +H+mHi2/ b i?2 p2`;2 M2i Reward Q#iBM2/
#v i?2 HHQ+iQ`b b 7`+iBQM Q7 i?2 p2`;2 M2i Reward i?2 HHQ+iQ`b +QmH/
?p2 Q#iBM2/- B7 i?2v ?/ HH +?Qb2M i?2 M2i Reward KtBKBxBM; bi`i2;v
U 0.69·0.51+0.31·0.22
· 100W = 82.4WVX
0.51
q?2M `2+BTB2Mib ?/ mM2[mH 2M/QrK2Mib BM 2tT2+iiBQM- i?2 KQ/H HHQ@
+iQ` +?QB+2- +?Qb2M BM dRW Q7 i`BHb- rb iQ HHQ+i2 /B`2+iHv iQ i?2 `2HiBp2Hv
TQQ`2` `2+BTB2MiX AM 7+i- R8 Qmi Q7 j8 HHQ+iQ`b U9kXNWV HHQ+i2/ iQ i?2 `2H@
iBp2Hv TQQ`2` `2+BTB2Mi BM HH i`BHb BM r?B+? `2+BTB2Mib ?/ mM2[mH 2M/QrK2MibX
HHQ+iQ`b HbQ `M/QKBx2/ BM bm#biMiBH T`QTQ`iBQM Q7 i`BHb- RNWX aBM+2 i?2
TQi2MiBH `2+BTB2Mib QM p2`;2 TmMBb?2/ i?2 HHQ+iQ` 7Q` `M/QKBxBM; r?2M
i?2v ?/ mM2[mH 2M/QrK2Mib- i?Bb T`QTQ`iBQM Bb bm`T`BbBM;Hv H`;2X h?2 bK2
?QH/b 7Q` i?2 T`QTQ`iBQM Q7 i`BHb BM r?B+? i?2 HHQ+iQ`b +?Qb2 iQ HHQ+i2 /B`2+iHv
iQ i?2 `2HiBp2Hv `B+?2` `2+BTB2Mi- RyWX q?2M i?2`2 rb BM2[mHBiv #2ir22M i?2
TQi2MiBH `2+BTB2Mib- i?2 HHQ+iQ`bǶ +?QB+2b i?2`27Q`2 /B/ MQi Ki+? i?2 #2?pBQ`
Q7 i?2 `2+BTB2Mib p2`v r2HHc BM i?Bb +b2 i?2 HHQ+iQ` dz2{+B2M+vǴ QMHv KQmMib
iQ e8X3W U 0.19·(−0.21)+0.71·0.69+0.10·0.04
· 100WVX
0.69
h?mb- [m2biBQM 2K2`;2b, ;Bp2M i?i HHQ+iQ`bǶ 2tT2+i2/ M2i Reward 7Q`
`M/QKBxiBQM Bb bB;MB}+MiHv M2;iBp2 r?2M i?2 TQi2MiBH `2+BTB2Mib r2`2 mM@
2[mH BM 2tT2+iiBQM- r?v /B/ RNW Q7 i?2 HHQ+iQ`b +?QQb2 iQ `M/QKBx2 #2ir22M
i?2K\
q?v /Q HHQ+iQ`b _M/QKBx2 #2ir22M SQi2MiBH _2+BTB2Mib rBi? lM@
2[mH 1M/QrK2Mib\ A7 i?2 HHQ+iQ`b i`B2/ iQ Ki+? i?2 T`272`2M+2b Q7 i?2
`2r`/fTmMBb?K2Mi #2?pBQ` Q7 i?2 TQi2MiBH `2+BTB2Mib- r2 rQmH/ 2tT2+i i?i
HH i?2 HHQ+iQ`b +?Qb2 i?2 M2i Reward KtBKBxBM; HHQ+iBQM bi`i2;vX 1[mBp@
kN
H2MiHv- r2 rQmH/ 2tT2+i i?i MQ HHQ+iQ` 2tT2+i2/ iQ dzH2p2 KQM2v QM i?2
i#H2Ǵ rBi? i?2B` HHQ+iBQM +?QB+2X aBM+2 r2 2HB+Bi2/ HHQ+iQ`bǶ #2HB27b #Qmi
i?2 #2?pBQ` Q7 i?2 TQi2MiBH `2+BTB2Mib BM i?2 2M/ Q7 i?2 2tT2`BK2Mi- r2 +M
/B`2+iHv i2bi r?2i?2` i?Bb Bb i?2 +b2X
h#H2 9 bb2bb2b r?2i?2` i?2 HHQ+iQ`b 2tT2+i2/ iQ H2p2 KQM2v QM i?2 i#H2
rBi? i?2B` HHQ+iBQM +?QB+2bX q2 2biBKi2 i?2 7QHHQrBM; 2[miBQM 7Q` #Qi? i?2
2[mH 2M/QrK2Mi +QM/BiBQMb U+QHmKM RV M/ i?2 mM2[mH 2M/QrK2Mi +QM/BiBQMb
U+QHmKM kV,
D(HHQ+iQ` 1tT2+i2/ iQ G2p2 JQM2v QM i?2 h#H2)ic = β ′ HHQ+iBQMic + εic .
D(HHQ+iQ` 1tT2+i2/ iQ G2p2 JQM2v QM i?2 h#H2) Bb M BM/B+iQ` i?i i?2
HHQ+iQ` #2HB2p2/ i?i b?2 rb H2pBM; KQM2v QM i?2 i#H2 BM ;Bp2M i`BHX
HHQ+iBQM Bb p2+iQ` Q7 BM/B+iQ`b 7Q` i?2 /Bz2`2Mi HHQ+iQ` +?QB+2b- BX2X
HHQ+iBQM = (_M/QKBx2- :Bp2 iQ f"V BM i?2 2[mH 2M/QrK2Mi +QM/BiBQMb
M/ HHQ+iBQM = (_M/QKBx2- :Bp2 iQ SQQ`- :Bp2 iQ _B+?) BM i?2 mM2[mH 2M@
/QrK2Mi +QM/BiBQMbX h?2 `2;`2bbBQMb `2 `mM rBi?Qmi +QMbiMi i2`K- r?B+?
K2Mb i?i HH +Q2{+B2Mib +M #2 BMi2`T`2i2/ /B`2+iHv b T`QTQ`iBQMbd X AM
+QHmKMb (1) M/ (2) β1 +Tim`2b i?2 T`QTQ`iBQM Q7 i`BHb BM r?B+? i?2 HHQ+@
iQ`b +?Qb2 iQ `M/QKBx2 /2bTBi2 #2HB2pBM; i?i i?2v r2`2 H2pBM; KQM2v QM i?2
i#H2X AM +QHmKM (1) β2 +Tim`2b i?2 T`QTQ`iBQM Q7 i`BHb BM r?B+? HHQ+iQ`b
+?Qb2 iQ HHQ+i2 /B`2+iHv iQ `2+BTB2Mib Q` " BM i?2 2[mH r2Hi? +QM/BiBQM /2@
bTBi2 #2HB2pBM; i?i i?2v r2`2 H2pBM; KQM2v QM i?2 i#H2X AM +QHmKM (2) β2 M/
β3 +Tim`2 i?2 T`QTQ`iBQMb Q7 i`BHb BM r?B+? HHQ+iQ`b +?Qb2 iQ HHQ+i2 /B`2+iHv
iQ i?2 `2HiBp2Hv TQQ`2` `2+BTB2Mi Q` i?2 `2HiBp2Hv `B+?2` `2+BTB2Mi- `2bT2+iBp2Hv/2bTBi2 #2HB2pBM; i?i i?2v r2`2 H2pBM; KQM2v QM i?2 i#H2X aiM/`/ 2``Q`b
d
h?2 2[miBQM Bb 2biBKi2/ rBi?Qmi +Q``2+iBM; 7Q` BM/BpB/mH H2p2H }t2/ 2z2+ib- b kk Qmi
Q7 j8 HHQ+iQ`b 2tT2+i2/ iQ H2p2 KQM2v QM i?2 i#H2 BM 2Bi?2` HH Q` MQM2 Q7 i?2 i`BHb BM
r?B+? `2+BTB2Mib ?/ 2[mH 2M/QrK2Mib- M/ N Qmi Q7 j8 HHQ+iQ`b 2tT2+i2/ iQ H2p2 KQM2v
QM i?2 i#H2 BM 2Bi?2` HH Q` MQM2 Q7 i?2 i`BHb BM r?B+? `2+BTB2Mib ?/ mM2[mH 2M/QrK2MibX
_2;`2bbBQMb +QMi`QHHBM; 7Q` BM/BpB/mH H2p2H }t2/ 2z2+ib `2 i?mb- BM T`+iB+2- #b2/ QM i?2
#2?pBQ` Q7 QMHv i?2 Rj HHQ+iQ`b BM i?2 2[mH 2M/QrK2Mi +QM/BiBQMb Q` i?2 ke HHQ+iQ`b
BM i?2 mM2[mH 2M/QrK2Mi +QM/BiBQMb r?Q b?B7i2/ #2ir22M 2tT2+iBM; iQ H2p2 KQM2v QM i?2
i#H2 M/ 2tT2+iBM; iQ KtBKBx2 i?2B` M2i Reward i H2bi QM+2 +`Qbb i`BHbX L2p2`i?2H2bbr2 `2TQ`i `2bmHib 7`QK `2;`2bbBQMb +QMi`QHHBM; 7Q` BM/BpB/mH H2p2H }t2/ 2z2+ib BM TT2M/Bt
h#H2 X8c i?Bb /Q2b MQi +?M;2 i?2 Tii2`M Q7 bB;MB}+M+2X
jy
`2 ?2i2`QbF2/biB+Biv@`Q#mbi M/ +Hmbi2`2/ i i?2 T`iB+BTMi H2p2HX
h?2 }`bi +QHmKM BM h#H2 9 `2p2Hb i?i i?2 HHQ+iQ`b #2HB2p2/ i?i i?2v r2`2
H2pBM; KQM2v QM i?2 i#H2 BM 9yW Q7 i`BHb BM r?B+? i?2v +?Qb2 iQ `M/QKBx2
M/ BM k9W Q7 i`BHb BM r?B+? i?2v +?Qb2 iQ HHQ+i2 /B`2+iHv iQ 2Bi?2` _2+BTB2Mi
Q` _2+BTB2Mi " r?2M i?2 `2+BTB2Mib ?/ 2[mH 2M/QrK2Mib BM 2tT2+iiBQMX
JQ`2 BMi2`2biBM;Hv- i?2 b2+QM/ +QHmKM `2p2Hb i?i i?2 HHQ+iQ`b 2tT2+i2/ iQ
H2p2 KQM2v QM i?2 i#H2 BM eyW Q7 i?2 i`BHb BM r?B+? i?2v +?Qb2 iQ `M/QKBx2
r?2M `2+BTB2Mib ?/ mM2[mH 2M/QrK2Mib BM 2tT2+iiBQMX PM i?2 Qi?2` ?M/HHQ+iQ`b 2tT2+i2/ iQ H2p2 KQM2v QM i?2 i#H2 BM QMHv jyW Q7 i`BHb BM r?B+?
i?2v +?Qb2 iQ HHQ+i2 /B`2+iHv iQ i?2 `2HiBp2Hv TQQ`2` `2+BTB2MiX
h?2 7+i i?i HHQ+iQ`b #2HB2p2/ i?i i?2v r2`2 KtBKBxBM; i?2B` M2i Reward
BM i?2 KDQ`Biv Q7 i`BHb BM r?B+? i?2v +?Qb2 iQ HHQ+i2 /B`2+iHv iQ i?2 `2HiBp2Hv
TQQ`2` `2+BTB2Mi /Q2b MQi BKTHv i?i i?2v r2`2 KQiBpi2/ #v i`vBM; iQ KtBKBx2
i?2B` M2i RewardX >Qr2p2`- i?2 7+i i?i HHQ+iQ`b #2HB2p2/ i?i i?2v r2`2
H2pBM; KQM2v QM i?2 i#H2 BM i?2 KDQ`Biv Q7 i`BHb BM r?B+? i?2v +?Qb2 iQ `M@
/QKBx2 BKTHB2b i?i i?2B` +?QB+2 Q7 `M/QKBxiBQM +MMQi ?p2 #22M KQiBpi2/
#v M ii2KTi iQ KtBKBx2 i?2B` 2tT2+i2/ M2i RewardX q2 i?mb +QM+Hm/2 i?i
HHQ+iQ`b Kmbi ?p2 ?/ T`272`2M+2b 7Q` `M/QKBxiBQM Qp2` M/ #Qp2 r?i +M
#2 2tTHBM2/ #v i?2B` 2tT2+iiBQMb 7Q` `2+BTB2MibǶ `2r`/fTmMBb?K2Mi #2?pBQ`X
AM bmK- r2 B/2MiB7v /Bb+QMM2+i #2ir22M HHQ+iQ`b +?QB+2b M/ r?i `2+BT@
B2Mib `2r`/2/ i?2 HHQ+iQ` 7Q` +?QQbBM;, HHQ+iQ`b `M/QKBx2/ iQ ;`2i2`
2ti2Mi i?M rb QTiBKH BM i2`Kb Q7 KtBKBxBM; i?2B` M2i RewardX h?Bb /Bb@
+QMM2+i +MMQi #2 2tTHBM2/ #v HHQ+iQ`b ?pBM; bvbi2KiB+HHv r`QM; #2HB27b
#Qmi `2+BTB2MibǶ `2r`/fTmMBb?K2Mi #2?pBQ`- bBM+2 i?2v +?QQb2 iQ `M/QK@
Bx2 /2bTBi2 2tT2+iBM; i?i i?Bb rb MQi i?2B` M2i Reward KtBKBxBM; bi`i2;vX
h?mb- r2 +QM+Hm/2 i?i HHQ+iQ`b ?/ T`272`2M+2b 7Q` `M/QKBxiBQM Qp2` M/
#Qp2 r?i +M #2 2tTHBM2/ #v KQiBpiBQM iQ KtBKBx2 i?2B` M2i Reward
7`QK i?2 `2+BTB2MibX
1z2+i Q7 lM+2`iBMiv QM HHQ+iQ` "2?pBQ`X :Bp2M i?2 `;mK2Mi K/2
#v Eii M/ a2i?m`KM UkyyeV M/ 1+F?Qz URN3NV- BX2X i?i HQii2`B2b b?QmH/
#2 i?2 K2i?Q/ Q7 +?QB+2 r?2M MQ /2iBH2/ BM7Q`KiBQM Bb pBH#H2 #Qmi TQi2M@
jR
h#H2 9, S`QTQ`iBQMb Q7 h`BHb BM q?B+? HHQ+iQ`b "2HB2p2/ h?i h?2v q2`2
G2pBM; JQM2v QM i?2 h#H2 qBi? h?2B` HHQ+iBQM *?QB+2b
.2T2M/2Mi p`B#H2,
.mKKv 7Q` q?2i?2` i?2 HHQ+iQ`
1tT2+i2/ iQ G2p2 KQM2v QM i#H2
URV
1[mH
1M/QrK2Mib
UkV
lM2[mH
1M/QrK2Mib
_M/QKBx2
yX9y
(yXy3)
yXey
(yXyN)
:Bp2 iQ f"
yXk9
(yXyN)
:Bp2 iQ SQQ`
yXjy
(yXye)
:Bp2 iQ _B+?
yXdy
(yXRj)
J2M Q7 /2TX p`B#H2
yXje
6@bii, _M/QKBx2 4 :Bp2 iQ f"
T@pHm2
RX9N
yXkj
yXjN
6@bii, _M/QKBx2 4 :Bp2 iQ SQQ`
T@pHm2
3XNR
yXyR∗∗∗
6@bii, _M/QKBx2 4 :Bp2 iQ _B+?
T@pHm2
yX9k
yX8k
6@bii, :Bp2 iQ SQQ` 4 :Bp2 iQ _B+?
T@pHm2
3Xjk
yXyR∗∗∗
P#b2`piBQMb
dy
R9y
LQi2b, S`QTQ`iBQMb Q7 i`BHb BM r?B+? HHQ+iQ`b #2HB2p2/ i?i i?2v r2`2 H2pBM; KQM2v
QM i?2 i#H2 rBi? i?2B` HHQ+iBQM +?QB+2bX _2;`2bbBQMb `2 `mM rBi?Qmi +QMbiMi i2`KX
aiM/`/ 2``Q`b `2 ?2i2`QbF2/biB+Biv@`Q#mbi M/ +Hmbi2`2/ i i?2 T`iB+BTMi H2p2HX F @
biiBbiB+b M/ +Q``2bTQM/BM; p@pHm2b 7`QK qH/ i2bib 7Q` +Q2{+B2Mi 2[mHBiv #2ir22M β1
M/ β2 BM +QHmKM URV M/ #2ir22M β1 M/ β2 - β1 M/ β3 b r2HH b β2 M/ β3 BM +QHmKM
UkV `2 `2TQ`i2/ BM i?2 HQr2` TM2H Q7 i?2 i#H2X aB;MB}+Mi i i?2 RyW +QM}/2M+2 H2p2HaB;MB}+Mi i i?2 8W +QM}/2M+2 H2p2HaB;MB}+Mi i i?2 RW +QM}/2M+2 H2p2HX
jk
iBH `2+BTB2Mib- r2 ?vTQi?2bBx2/ i?i HHQ+iQ`b rQmH/ #2 KQ`2 HBF2Hv iQ `M/QK@
Bx2 BM i?2 mM2[mH 2M/QrK2Mi +QM/BiBQMb r?2M i?2v ?/ mM+2`iBMiv #Qmi i?2
2t+i /Bbi`B#miBQM Q7 2M/QrK2Mib #2ir22M i?2 TQi2MiBH `2+BTB2MibX AM h#H2 8r2 bb2bb ?Qr HHQ+iQ`b `2+i2/ iQ i?2 BMi`Q/m+iBQM Q7 mM+2`iBMivX q2 2biBKi2
i?2 7QHHQrBM; 2[miBQM b2T`i2Hv 7Q` i?2 2[mH 2M/QrK2Mi +QM/BiBQMb U+QHmKMb
R M/ kV M/ mM2[mH 2M/QrK2Mi +QM/BiBQMb U+QHmKMb j@8V,
Allocationjc = β0 + β1 U ncertainjc + αj + εjc ,
U9V
r?2`2 Allocation Bb M BM/B+iQ` i?i i?2 HHQ+iQ` +?Qb2 i?2 HHQ+iBQM K2+?@
MBbK x ∈ &_M/QKBx2- :Bp2 iQ f"- :Bp2 iQ SQQ`- :Bp2 iQ _B+?}- M/ U ncertain
Bb M BM/B+iQ` i?i i?2`2 rb mM+2`iBMiv #Qmi i?2 2t+i /Bbi`B#miBQM Q7 i?2 `2@
+BTB2MibǶ 2M/QrK2MibX h?2 BMi2`T`2iiBQM Q7 +Q2{+B2Mi β0 Bb i?mb i?2 T`QTQ`iBQM
Q7 i`BHb BM r?B+? i?2 HHQ+iQ`b +?Qb2 i?2 HHQ+iBQM K2+?MBbK x r?2M i?2v
r2`2 +2`iBM #Qmi i?2 /Bbi`B#miBQM Q7 `2+BTB2MibǶ 2M/QrK2Mib- M/ +Q2{+B2Mi β1
+Tim`2b i?2 BM+`2b2f/2+`2b2 BM i?Bb T`QTQ`iBQM r?2M mM+2`iBMiv rb BMi`Q@
/m+2/X αj +Tim`2b BM/BpB/mH H2p2H }t2/ 2z2+ib- εjc Bb i?2 `M/QK 2``Q` i2`Kj Bb T`iB+BTMi
BM/2t 7Q` HHQ+iQ`b!
" M/ c Bb +QM/BiBQM
#$ BM/2tX j ∈ [1, 35]
>B;? 2M/QrK2Mib
M/ c ∈ (*2`iBM- lM+2`iBM) ×
X aiM/`/ 2``Q`b
GQr 2M/QrK2Mib
`2 +Hmbi2`2/ i i?2 T`iB+BTMi H2p2H #2+mb2 HHQ+iQ`b r2`2 `M/QKBx2/ BMiQ
i?2 Q`/2` Q7 +QM/BiBQMb rBi?BM #HQ+FbX
jj
j9
yXeN
J2M Q7 /2TX p`B#H2
R9y
yXjR
yXje
(yXyj)
−yXyN
(yXy8)
UkV
:Bp2 iQ f"
k3y
yXRN
yXRR
(yXyk)
yXR8
(yXy8)
UjV
_M/QKBx2
k3y
yXdR
yXd3
(yXyj)
−yXR9
(yXye)
U9V
:Bp2 iQ SQQ`
k3y
yXRy
yXRR
(yXyk)
−yXyR
(yXyj)
U8V
:Bp2 iQ _B+?
lM2[mH 1M/QrK2Mib
LQi2b, 1z2+i Q7 mM+2`iBMiv QM i?2 HHQ+iQ`bǶ T`Q##BHBiv Q7 `M/QKBxBM; Q` HHQ+iBM; /B`2+iHv iQ QM2 Q7 i?2 irQ
TQi2MiBH `2+BTB2MibX h?2 +QMbiMi i2`K +Tim`2b HHQ+iQ`bǶ T`Q##BHBiv Q7 `M/QKBxBM; Q` HHQ+iBM; /B`2+iHv iQ QM2
Q7 i?2 irQ TQi2MiBH `2+BTB2Mib r?2M i?2B` 2M/QrK2Mib r2`2 FMQrM rBi? +2`iBMivX _2;`2bbBQMb +QMi`QH 7Q` BM/BpB/mH
H2p2H }t2/ 2z2+ibc biM/`/ 2``Q`b `2 ?2i2`QbF2/biB+Biv@`Q#mbi M/ +Hmbi2`2/ i i?2 T`iB+BTMi H2p2HX aB;MB}+Mi
i i?2 RyW +QM}/2M+2 H2p2HaB;MB}+Mi i i?2 8W +QM}/2M+2 H2p2HaB;MB}+Mi i i?2 RW +QM}/2M+2 H2p2HX
R9y
yXe9
(yXyj)
*QMbiMi
P#b2`piBQMb
yXyN
(yXy8)
lM+2`iBM
URV
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