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*G D mHIUT PS IJEFTP FOGPSDFT DBVTJOHD UP MPTF B QPSUJPO PG SFBMJ[FE ESVH QSPmUT 5P HBJO USBDUBCJMJUZ * NPEFM UIJT MPTT BT B DPOUFTU TVDDFTT GVODUJPO TPNF PG UIF SFTVMUT HFOFSBMJ[F UP B MBSHFS DMBTT PG GVODUJPOBM GPSNT 8IFUIFS IJEJOH PS mHIUJOHDT MPTT EFQFOET JO QBSU PO UIF BNPVOU PG SFQSFTTJPO IF GBDFT -FU s SFQSFTFOU UIF UPUBM BNPVOU PG SFQSFTTJWF GPSDF BWBJMBCMF UP UIF TUBUF XIJDI XF DBO DPODFJWF PG BT B DPNCJOBUJPO PG TIFFS USPPQ TUSFOHUI FRVJQNFOU BOE KVSJTEJDUJPO UP FOGPSDF MBXT BOE FOHBHF DBSUFMT ɨFO{s_h, s_f} ≤ s JT UIF BNPVOU PG SFQSFTTJWF GPSDF EJSFDUFE BUD DPOEJUJPOBM PO IJEJOH PS mHIUJOH SFTQFDUJWFMZ 8F DBO OPX HJWF UIF DPODFQU PG DPOEJUJPOBMJUZ PG SFQSFTTJPOB GPSNBM EFmOJUJPO SFQSFTTJWF QPMJDZ JT DPOEJUJPOBM UP UIF FYUFOU UIBU s_f >s_h 'PS DPODSFUFOFTT * BTTVNF UIBU SFQSFTTJPO JT ANBYFE PVU XIFOD mHIUT TP s_f = s BOE UIBU s_h = s(1−c) XIFSF c∈[0,1]QBSBNFUFSJ[FT DPOEJUJPOBMJUZ PG SFQSFTTJPO

/PX JGD mHIUT IF LFFQT B TIBSF PG ESVH QSPmUT HJWFO CZ UIF GVODUJPO _a+s^a *G IF IJEFTa EPFT OPU DPNF JOUP QMBZ 3BUIFS IF SFUBJOT _η+s^η

h PGy XIFSFηJT B TDBMJOH QBSBNFUFS UIBU DBQUVSFT UIF FċDBDZ PG IJEJOH SFMBUJWF UP mHIUJOH 4VCTUBOUJWFMZηDBO CF TFFO BT BO JOWFSTF NFBTVSF PG UIF UFSSJUPSJBMJUZ PG UIF ESVH USBEF XIFOη JT WFSZ MPX UFSSJUPSJBMJUZ JT IJHI TP UIBU IJEJOH GPSGFJUT B MBSHF TIBSF PG QSPmUT

* OPSNBMJ[FPT QBZPĊ UP FOGPSDJOH XIFODIJEFT UP0 BOE TBZ UIBU JGDmHIUTP JODVST EJTVUJMJUZ PG−_a+s^a φ XIFSFφJT B QBSBNFUFS WBMVF DBQUVSJOHPT SFMBUJWF EJTUBTUF GPS WJPMFODF ɨF GBDU UIBU HFOFSJDBMMZφ�=ySFnFDUT UIF JEFB UIBU JO DBSUFMTUBUF DPOnJDU BT JO XBST PG DPOTUSBJOU NPSF HFOFSBMMZ UIF UXP TJEFT BSF OPU mHIUJOH GPS DPOUSPM PWFS UIF TBNF AQJF *O UIJT DBTF XIJMF UIF QPMJDF JO TJUVBUJPOT PG SFHVMBSJ[FE DPSSVQUJPO BSF XJMMJOH UP UBLF PO UIF SPMF PG SFOU DPMMFDUPS QMBZFE CZ UIF .BmB JO 4DIFMMJOHT .JBNJ FYBNQMF UIFZ BSF OPU XJMMJOH UP QIZTJDBMMZ BENJOJTUFS UIF ESVH USBEF BOE XPVME HFU OPUIJOH GSPN BQQSPQSJBUJOHDT UVSG 'JHVSFQSFTFOUT UIF HBNF USFF XJUI UIFTF HFOFSBMGPSN QBZPĊT

" GFX DIBSBDUFSJTUJDT PG UIFTF DPOUFTU GVODUJPOT XIJDI NPSF HFOFSBM GPSNT TIPVME TIBSF _{d a}^d � _a

a+s

�>0 _{d a}^d �

η η+s_h

�

=0 BSNJOH SBJTFTDT BOE MPXFSTPT QBZPĊT UP mHIUJOH BQBSU GSPN UIF DPTU PG BSNT CVU IBT OP FĊFDU PO UIFJS QBZPĊ UP IJEJOH

" LFZ BTTVNQUJPO UISPVHIPVU JT UIBUPDBOOPU BĊFDU UIFTF WBSJBCMFT TIF DBO POMZ EFDJEF UP FOGPSDF PS OPU

'JHVSF #SJCF /FHPUJBUJPO (BNF 5SFF

_{d s}^d � _a

a+s

�<0 _{d s}^d �

η η+s_h

�

<0 UIF NPSF TUBUF SFQSFTTJWF GPSDF BQQMJFE JO SFTQPOTF UP BO BDUJPO UIF TNBMMFS TIBSF PGy UIBUDDBO SFUBJO

_{d b}^d � _a

a+s

�=_{d b}^d

� η η+s_h

�

=0 4JODF OP CSJCF JT QBJE UIF WBMVF PG UIF CSJCF EFNBOEbIBT OP FĊFDU PO FJUIFS QMBZFST QBZPĊT UPH PSF

∃a(·)˜ >0 :_a+s˜^a˜ =_η+s^η

h 'PS BOZ QBSBNFUFS WBMVFT UIFSF FYJTUT TPNF MFWFM PG BSNBNFOU TVDI UIBU CZ mHIUJOH D XJOT B MBSHFS TIBSF PG yUIBO CZ IJEJOH 'JHIUJOH XIFOa=0PCUBJOT B TNBMMFS TIBSF PG yUIBO IJEJOH

%SVH QSPmUT y ɨF TJ[F PG UIF AESVH QJF‰UIF SFBMJ[FE QSPmUT GSPN ESVH USBċDLJOH‰ JT B SBOEPN WBSJBCMF y BOE JU JT VODFSUBJOUZ PWFS UIJT WBSJBCMF UIBU HFOFSBUFT CBSHBJOJOH CSFBLEPXO JO FRVJMJCSJVN JF IJEJOH PS mHIUJOH XJUI QPTJUJWF QSPCBCJMJUZ BT PQQPTFE UP BMXBZT QBZJOH UIF FRVJMJCSJVN CSJCF EFNBOE "MUIPVHI UIF NPEFM JT GBJSMZ SPCVTU UP EJĊFSFOU TQFDJmDBUJPOT GPS UIF EJTUSJCVUJPO PG y XIFO UIF EFHSFF PG VODFSUBJOUZ JT WFSZ TNBMMP PGUFO QSFGFST UP NBLF B MPXCBMM EFNBOE UIBU XJMM BMXBZT CF BDDFQUFE 5P TFF UIJT NPSF DMFBSMZ * GPMMPX 1PXFMM BOE %BM #Ø BOE TQFDJGZ y =µ+� XIFSFµJT mYFE BOE�JT B SBOEPN WBSJBCMF XJUI NFBO [FSP EJTUSJCVUFE PWFS[�,�¯] 'PS USBDUBCJMJUZ * GVSUIFS BTTVNF UIBU �JT EJTUSJCVUFE VOJGPSNBMMZ XIJDI JNQMJFT UIBU

� =−¯� /PX UIF MPXCBMM PVUDPNF NFOUJPOFE BCPWF JT SVMFE PVU XIFOFWFS � >¯ ^µ₃ 4JODF UIF QJF JT BMXBZT QPTJUJWF JU JT OBUVSBM UP BTTVNF� < µ ɨVT UP QJO EPXO UIF NPEFM BOE SVMF PVU UIF MPXCBMM PVUDPNF * BTTVNF¯

¯

�=^µ₂ XIJDI ZJFMET y∼U�_µ

2,^3µ₂ �

&YQFDUFE TBODUJPO GSPN OPOFOGPSDFNFOU d ɨF QBSBNFUFSd SFQSFTFOUT GSPN UIF QFSTQFDUJWF PG UIF FOGPSDFS UIF TVN PG BMM FYQFDUFE DPOTFRVFODFT PG OPOFOGPSDFNFOU OFU PG BOZ CSJCF SFDFJWFE *U JT B SFEVDFEGPSN XBZ PG DBQUVSJOH UIF DPNQMFY IFODF VONPEFMFE EZOBNJDT PG QPMJDF DPSSVQUJPO "MUIPVHI OPU DSVDJBM UP UIF NPEFMT mOEJOHT * JOUFSQSFUd BT HFOFSBMMZ MPX XIFO DPSSVQUJPO JT SBNQBOU BOE IJHI XIFO DPSSVQUJPO JT SBSF ɨF MPHJD JT TUSBJHIUGPSXBSE 'JSTU EFDPNQPTFd UIVT

d =z_d·π_d+ψ; π_d= f(e_d,ξ_d,·)

XIFSFψSFQSFTFOUT UIF QTZDIPMPHJDBM PS NPSBM DPTUT PG CSJCFUBLJOHz_dJT UIF QVOJTINFOUP XJMM TVĊFS DPOEJUJPOBM PO CFJOH DBVHIU BOEπ_d JT UIF QSPCBCJMJUZ PG EFUFDUJPO BOE TVDDFTTGVM QSPTFDVUJPO π_d JT B GVODUJPO PG BNPOH PUIFS UIJOHT BOUJDPSSVQUJPO FĊPSUe_d ɨF HPWFSONFOU DBO TFU z_d BOEe_d EJSFDUMZ CVU B LFZ EFUFSNJOBOU PG

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π_d JT IPX NBOZ PUIFS FOGPSDFST BSF UBLJOH CSJCFT SFQSFTFOUFE IFSF CZξ_d 'SPN TVDI B TUBSUJOH QPJOU NPEFMT PG DPSSVQUJPO BOE MBXFOGPSDFNFOU FH5JSPMF IBWF EFSJWFE NVMUJQMF FRVJMJCSJB B AHPPE FRVJMJCSJVN JO XIJDI UIFSF BSF GFX MBXCSFBLFST TP UIBUπ_d BOE IFODF d BSF IJHI HJWJOH FBDI PċDFS BO JODFOUJWF UP PCFZ BOE B ACBE FRVJMJCSJVN JO XIJDI FWFSZCPEZ CSFBLT UIF MBX TP UIBU BOZ POF FOGPSDFS GBDFT B MPXπ_d BOE IFODF B MPX d &NQJSJDBM XPSL GVSUIFS TVQQPSUT UIF OPUJPO UIBU DPSSVQUJPO JT B NVMUJQMFFRVJMJCSJVN HBNF 'JTNBO BOE .JHVFM 0MLFO BOE #BSSPO 1PMJDF EFQBSUNFOUT EP PDDBTJPOBMMZ HFU DMFBOFE VQ TVHHFTUJOH UIBU TVċDJFOU TUBUF QSFTTVSF WJB DIBOHFT JOz_d BOEe_d DBO TIJGU UIF FRVJMJCSJVN PVUDPNF )PXFWFS DPSSVQUJPO SFNBJOT BO FOEFNJD QSPCMFN JO NBOZ QMBDFT BOE JU JT OPU DMFBS UIBU TUBUFT GBDF B TJNQMF DIPJDF PWFS XIFUIFS UP IBWF DPSSVQU QPMJDF PS OPU *O PUIFS XPSET UIF FNQJSJDBM SFMBUJPOTIJQ CFUXFFOπ_d BOEe_d JT OPU BU BMM DMFBS JOEFFE JU DPOTUJUVUFT BO FOUJSF SFTFBSDI BHFOEB JO JUT PXO SJHIU ɨJT NPEFM EPFT OPU BEESFTT UIFTF RVFTUJPOT BOE USFBUTd BT FTTFOUJBMMZ FYPHFOPVT ɨF NPEFM EPFT QSFEJDU UIBU B TUBUF UIBU TVDDFTTGVMMZ FMJNJOBUFE B DVMUVSF PG DPSSVQUJPO BOE TIJGUFEd GSPN B MPX UP B IJHI WBMVF XPVME BMTP NBLF QSPHSFTT BHBJOTU WJPMFOU DPSSVQUJPO #VU UIF NPSF SFMFWBOU BOE TBMJFOU mOEJOH JT UIBU TUBUF MFBEFST DBO UISPVHI EJSFDU NBOJQVMBUJPO PG UIF QBSBNFUFST UIFZ EP DPOUSPMs BOEc FMJNJOBUF WJPMFOU DPSSVQUJPO XJUIPVU BOZ DIBOHF JOePSd

ɨJT NPEFM BCTUSBDUT BXBZ GSPN JTTVFT PG OPSNT BOE JOEJWJEVBM QPMJDF PċDFST NPSBM DPNQVODUJPOT IFODF OP JOEFYJOH PGψ " EJĊFSFOU BQQSPBDI XPVME CF UP MFU WBSJBUJPO JOψ_i EFmOF NVMUJQMF UZQFT PG FOGPSDFST JU TFFNT MJLFMZ UIBU QSJWBUF JOGPSNBUJPO BCPVUψ_i DPVME QSPEVDF WJPMFOU FRVJMJCSJB GVODUJPOJOH JO B QBSBMMFM NBOOFS UP UIF QSJWBUF JOGPSNBUJPO BCPVU y JO UIJT NPEFM *O TVDI B NPEFM D XPVME TPNFUJNFT mHIU AIJHI OPODPSSVQUJCMF UZQFT UP LFFQ AMPX NPSF DPSSVQUJCMF UZQFT GSPN CMVċOH

"OBMZTJT

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˜ a=ηs_f

s_h = η 1−c

/PUF UIBU a˜ JT EFDSFBTJOH JO UIF UFSSJUPSJBMJUZ PG UIF ESVH USBEF JF JODSFBTJOH JOη BOE JODSFBTJOH JO UIF DPOEJUJPOBMJUZ PG SFQSFTTJPO c

-FNNB *O FRVJMJCSJVNDOFWFS QMBZTa∈(0,a)˜

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"O JNQPSUBOU BTTVNQUJPO NBEF IFSF JT UIBU UIF UPUBM FYQFDUFE TBODUJPOd JT OPU B GVODUJPO PG UIF TJ[F b PG UIF CSJCF UBLFO 0OF QPUFOUJBM PCKFDUJPO UP UIJT BTTVNQUJPO JT UIBU WFSZ MBSHF CSJCFT NJHIU BUUSBDU UIF BUUFOUJPO PG TVQFSJPST BOE IFODF IBWF B IJHIFS DIBODF PG EFUFDUJPO "OPUIFS JT UIBU UIFSF NJHIU CF BO JNQPSUBOU EJTDPOUJOVJUZ XIFOb=0 JF UIF MBX UIF QVCMJD PS CPUI NJHIU EJTUJOHVJTI CFUXFFO BO FOGPSDFS XIP TJNQMZ GBJMT UP FOGPSDF UIF MBX BOE POF XIP IBT SFDFJWFE NPOFZ GSPN DSJNJOBMT 3FMBYJOH UIFTF BTTVNQUJPOT CZ NPSF FYQMJDJUMZ NPEFMJOHd XPVME DPNQMJDBUF UIF NPEFM CVU NBLF BO JOUFSFTUJOH BWFOVF GPS GVSUIFS SFTFBSDI

1SPPGT GPS BMM GPSNBM SFTVMUT BSF HJWFO JO "QQFOEJY "

CFFO DIPTFO 4VCTUBOUJWFMZD DBOOPU DSFEJCMZ UISFBUFO UP mHIU VOMFTT TIF IBT TVċDJFOU mSFQPXFS a≥a˜ BOE PODF TIF IBT PCUBJOFE TVċDJFOU mSFQPXFS DBOOPU DSFEJCMZ UISFBUFO UP IJEF TIPVME B CSJCF OPU CF BHSFFE UP

5P GVSUIFS BOBMZ[F UIF NPEFM JU JT IFMQGVM UP TFQBSBUFMZ DBMDVMBUF UIFO DPNQBSF DT CFTUSFTQPOTF QBZPĊT GPS IJEJOH a=0 BOE mHIUJOH a >a˜ TUSBUFHJFT 'PS DMBSJUZ * EFOPUF UIPTF CFTUSFTQPOTF TUSBUFHJFTa_h^∗BOEa^∗_f SFTQFDUJWFMZ 8F UIFO UIJOL PGDT FRVJMJCSJVN DIPJDF BT

a^∗=arg max

{a^∗_h, a^∗_f}E[U^D(a)]

ɨFO CZ -FNNBa^∗_h=0BOEa^∗_f ≥a˜

$BTFha<a˜ A)JEF BOE #SJCF (JWFOa=0 GPS BOZbD XJMM IJEF JG

y<y^∗=b

� 1+ η

s_h

�

BOE QBZbPUIFSXJTF /PUF UIBU UIFSF JT B NJOJNVN BOE NBYJNVN CSJCF CFMPX BCPWF XIJDIDOFWFS BMXBZT IJEFT ɨFTF DBO CF EFSJWFE GSPN UIF FRVBUJPO BCPWF ZJFMEJOH

b_h^m=µ 2

s_h

s_h+η, b_h^M =3µ 2

s_h s_h+η

P EPFT OPU LOPX UIF SFBMJ[FE WBMVF PGy XIFO TIF DIPPTFTb TP UIF QSPCBCJMJUZ PGDIJEJOH DPOEJUJPOBM PObJT

Pr_H(b) =











1 GPS b≥b_h^M

b µ

�1+_s^η

h

�−¹₂ GPS b∈(b_h^m, b_h^M) 0 GPS b≤b_h^m PT FYQFDUFE VUJMJUZ JT B GVODUJPO PGPr_H

E[U_h^P(b)] =Pr_H(b)·0+ (1−Pr_H(b)) (b−d)

BOE TP JT LJOLFE BUb_h^mBOEb_h^M 5P EFUFSNJOFPT CFTU SFTQPOTF XF DBO NBYJNJ[F BO AVODPOTUSBJOFE WFSTJPO PG IFS VUJMJUZ GVODUJPO

b_h^∗=arg max

b





 b�

1+_s^η

h

�−¹₂µ

3

2µ−¹₂µ (0) +





1−b� 1+_s^η

h

�−¹₂µ

3 2µ−¹₂µ





(b−d)







ɨJT ZJFMET

b_h^∗= s_h s_h+η

3 4µ+1

2d

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ɨFO XF DIFDL XIBU IBQQFOT JG b_h^∗ MJFT PVUTJEF UIF JOUFSWBM(b_h^m, b_h^M) 'JSTU OPUF UIBU UIF BTTVNQUJPO PO UIF EJTUSJCVUJPO PG y HVBSBOUFFT UIBU b_h^∗>b_h^m SVMJOH PVU B TJUVBUJPO XIFSF P QSFGFST UP NBLF B MPXCBMM EFNBOE UIBU XJMM BMXBZT CF QBJE

-FNNB *G b_h^∗ > b_h^M P JT XFBLMZ CFUUFS Pą QMBZJOH b_h^M UIBO BOZ PUIFS b BOE TUSJDUMZ CFUUFS Pą UIBO GPS BOZ b<b_h^M

ɨJT ZJFMET UIF GPMMPXJOH TUSBUFHZ GPSP

�b_h^∗=











b_h^M =_s^sh

h+η 3

2µ JG b_h^∗≥b_h^M BOE Pr_H = 1 b_h^∗=_s^sh

h+η 3

4µ+¹₂d JG b_h^∗∈(b_h^m, b_h^M) BOE Pr_H = ¹₄+¹₂^sh_s^+η

h

d µ

b_h^m= _s^sh

h+η 1

2µ JG b_h^∗≤b_h^m BOE Pr_H = 0 ɨVT UIFSF JT BO JOUFSJPS TPMVUJPO JF b_h^∗<b_h^M XIFOFWFS

d <

�

1− η

s(1−c) +η

�3

2µ C_d

ɨJT DPOEJUJPO IBT B OBUVSBM JOUFSQSFUBUJPO UIF SJHIU IBOE TJEF JT UIF MBSHFTU MPTT UIBUP DBO JNQPTF POD JG OP CSJCF JT QBJE *G UIJT FYDFFETPT SFTFSWBUJPO CSJCF QSJDF d UIFO UIFSF JT TPNF SPPN GPS NVUVBMMZ CFOFmDJBM CSJCFT

*Gd JT BCPWF UIJT DVUQPJOU UIFO QPMJDF MBDL TVċDJFOU MFWFSBHF UP FYUSBDU B CSJCF MBSHF FOPVHI UP DPNQFOTBUF UIFN GPS UIF FYQFDUFE QVOJTINFOU GSPN CSJCF UBLJOH BOE XF HFU QVSF FOGPSDFNFOU

3FDBMMJOH UIBU s_h=s(1−c)XF DBO OPX TFF UIBU BMMPXJOHcUP BQQSPBDI1OPU POMZ NBLFT mHIUJOH VOWJBCMF JU BMTP HVBSBOUFFT UIBU OP CSJCFSZ XJMM PDDVS BT MPOH BTd JT OPO[FSP

1SPQPTJUJPO *GC_d IPMET BOE BTTVNJOHa =0 UIF FRVJMJCSJVN QSPCBCJMJUZ PG IJEJOH JT ¹₄+¹₂^s(1−c_s(1−c^)+η₎ ^d_µ DzF QSPCBCJMJUZ PG CSJCFUBLJOH 1−Pr_H JT UIVT JODSFBTJOH JOµBOEs BOE EFDSFBTJOH JOd BOEc.

*U JT TUSBJHIUGPSXBSE UIBU UIF MBSHFS ESVH QSPmUT µ BSF UIF FBTJFS JU JT GPS CSJCFSZ UP PDDVS BOE MJLFXJTF UIBU B IJHIFS FYQFDUFE QVOJTINFOU GSPN CSJCFUBLJOH d EFUFST CSJCFSZ ɨF PUIFS UXP SFTVMUT IPXFWFS BSF DPVOUFSJOUVJUJWF DSBDLEPXOT JODSFBTFT JOs JODSFBTF UIF FRVJMJCSJVN QSPCBCJMJUZ PG CSJCFSZ CFDBVTF UIFZ SFEVDF DT PVUTJEF PQUJPO BOE UIVT NBLF IJN NPSF XJMMJOH UP QBZ B CSJCF #Z UIF TBNF UPLFO JODSFBTFT JO DPOEJUJPOBMJUZ c SFEVDF UIF FĊFDUJWF MFWFM PG GPSDFP DBO BQQMZ VOEFS B IJEJOH TUSBUFHZ UIVT HJWJOHD MFTT JODFOUJWF UP SFBDI B EFBM ɨJT MFBET UP UIF GPMMPXJOH

$PSPMMBSZ 'PS BOZ TFU PG QBSBNFUFS WBMVFT∃c^{N B}∈(0,1):c>c^{N B}=⇒b_h^∗≥b_h^M TVDI UIBU *OFRVBMJUZC_d EPFT OPU IPME BOE JGa=0 OP CSJCFT BSF QBJE

*OUVJUJWFMZ BT c BQQSPBDIFT 1 XF HFU B GPSN PG DPOEJUJPOBM EFDSJNJOBMJ[BUJPO XIFSF D GBDFT OFYU UP OP SFQSFTTJPO BT MPOH BT IF FTDIFXT WJPMFODF "T IJT QPUFOUJBM MPTT GSPN MBX FOGPSDFNFOU UIF SJHIUIBOE TJEF PG

*OFRVBMJUZC_d BQQSPBDIFT [FSP IF JT OP MPOHFS XJMMJOH UP QBZ B CSJCF UP BWPJE FOGPSDFNFOU

*G XF NPEFM UIF SBOHF PGyBT[µ−�,µ+�] UIFO� >^µ₃ HVBSBOUFFT UIBUb_h^∗>b_h^m ɨF MPXCBMM DPOEJUJPO DBO POMZ PDDVS XIFO UIF VODFSUBJOUZ PWFS UIF TJ[F PG UIF QJF JT SFMBUJWFMZ TNBMM "T OPUFE * IBWF DIPTFO�=^µ₂ UISPVHIPVU

$BTF f a>a˜ A'JHIU BOE #SJCF

ɨF mSTU TUFQT PG UIF BOBMZTJT BSF TJNJMBS TJODF P UBLFT a BT HJWFO XIFO IF DIPPTFT b "T CFGPSF UIFSF BSF NJOJNVN BOE NBYJNVN CSJCFT b^m_f , b^M_f XIJDI BSF OPX GVODUJPOT PG a "T CFGPSF XF DBO DBMDVMBUF BO VODPOTUSBJOFEb^∗_f BOE TIPX UIBUP DBO EP OP CFUUFS UIBO UP QMBZ�b^∗_f(a) =max�

min�

b^∗_f, b^M_f � , b^m_f �

�b^∗_f(a) =











b^M_f (a) =_a+s^{s 3}₂µ JG b^∗_f ≥b_f^M BOEPr_F = 1 b^∗_f(a) = ^s(32µ+d_2(a+s^)+a(d−φ)

) JG b^∗_f ∈(b_f^m, b_f^M) BOEPr_F = ¹₄+^{a(d−φ)+s d}_2sµ b^m_f (a) =_a+s^{s 1}₂µ JG b^∗_f ≤b_f^m BOEPr_F = 0

#FDBVTFaJT B DIPJDF WBSJBCMF B USVF JOUFSJPS TPMVUJPO JT POF JO XIJDIb^∗_f(a^∗_f)∈�

b_f^m(a^∗_f), b^M_f (a^∗_f)�

XIFSFa^∗_f HFOFSBUFT UIF IJHIFTU QBZPĊ UPDT DPOEJUJPOBM POa>a 0ODF BHBJO JU JT TJNQMFTU UP DBMDVMBUF AVODPOTUSBJOFE˜ WBMVFT GPSa^∗_f UIFO DIFDL XIFUIFSb^∗_f(a^∗_f)MJFT XJUIJO�

b^m_f (a^∗_f), b^M_f (a^∗_f)�

'JSTU GPS BOZaD XJMM CF JOEJĊFSFOU CFUXFFO QBZJOH BOE mHIUJOH XIFO y−b^∗_f (a) =y_a+s^a "TTVNJOH BO JOUFSJPS TPMVUJPO BOE QMVHHJOH JO b^∗_f GSPN

&RVBUJPO XF DBO EFmOF UIF DVUQPJOU y^∗(a) =^{2ad+2d s}_4s^+3sµ−2aφ TVDI UIBUD mHIUT XIFO y< y^∗(a)BOE QBZT PUIFSXJTF DT AVODPOTUSBJOFE FYQFDUFE VUJMJUZ BT B GVODUJPO PGaJT UIVT HJWFO CZ

E[U_f^D(a)] =� y^∗(a)

µ 2

�y a

a+s −a� 1 2µd y+

� ^3µ₂

y^∗(a)



y−s(³₂µ+d) +a(d−φ) 2(a+s) −a



 1 2µd y 8F DBO OPX NBYJNJ[F UIJT AVODPOTUSBJOFE GVODUJPO XIJDI ZJFMET

a^∗_f =s





��8sµ−(d−φ)²��

23µ²+12µφ−4φ²� 2�

8sµ−(d −φ)²� −1





5P FBTF UIF OPUBUJPO XSJUFΩ≡8sµ−(d−φ)²BOEΥ≡23µ²+12µφ−4φ² ɨFO TVCTUJUVUJPO JOUP &RVBUJPO ZJFMET

b^∗_f(a^∗_f) =d−φ 2 Ω+

�3 2µ+φ

�� Ω·Υ

'PDVTJOH PO BSFBT PG UIF QBSBNFUFS TQBDF XIFSF BO JOUFSJPS TPMVUJPO FYJTUT XF DBO OPX TPMWF GPS Pr_F(a^∗) ɨJT ZJFMET

Pr_F =1 4+ φ

2µ+(d−φ) 4µ

�Ω·Υ

Ω 5BLJOH DPNQBSBUJWF TUBUJDT POPr_F(a^∗_f)

∂Pr_F

∂ s = (φ−d)

�Ω·Υ

Ω

/PUF UIBUa^∗_f =s^�Ω_Ω^·Υ −s TP JGa^∗_f >0 XIJDI JU NVTU CF UP IBWF mHIUJOH JO FRVJMJCSJVN UIFO UIF TJHO PG ^∂_{∂ s}^PrF EFQFOET EJSFDUMZ PO UIF SFMBUJWF TJ[F PGd BOEφ ɨJT JT B LFZ SFTVMU PG UIF QBQFS