PROBABILISTIC EXTENTION OF THE CUMULATIVE PROSPECT THEORY

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1 Il Zuler PROBABILISTIC EXTENTION OF THE CUMULATIVE PROSPECT THEORY BASIC RESEARCH PROGRAM WORKING PAPERS SERIES: ECONOMICS WP BRP 33/EC/23 Th Worng Per n ouu of reerch roec leened he Nonl Reerch Unver Hgher School of Econoc HSE. An onon or cl conned n h Worng Per do no necerl reflec he vew of HSE.

2 Il A. Zuler PROBABILISTIC EXTENTION OF THE CUMULATIVE PROSPECT THEORY A nuber of eeren ndce robblc reference n ce where no one lernve boluel ol. The of redcng he choce of one of he lernve ong ulle lernve hen rccll orn nd no rvl. I cn occur n uon of choce under r when no one loer ochcll done oher. For r loere here re everl colced odel of robblc bnr reference. For he fr e we heren rooe he robblc eenon of he cuulve roec heor CPT. The reened vul grhc ufcon of h odel nuvel cler nd doe no ue ohced cuulve ung or Choque negrl. Here we rooe odel of elecng fro e of lernve b connuou Mrov rndo wl. I e redcng he reul of choce e becue full ue de receved b robblc eenon of СPT. The rooed ehod re que le nd do no requre lrge oun of d for rccl ue. JEL Clfcon: D8 D83 C44. Keword: cuulve roec heor robblc choce connue Mrov roce. Nonl Reerch Unver Hgher School of Econoc. Fcul of Econoc Deren of Hgher Mhec Docen PhD; E-l: zuler@rbler.ru

3 . Inroducon Prccl eeren ofen how h n lr uon ronl ndvdul cn e dfferen choce fro r of lernve. Th en h bnr reference cnno be deerned unquel nd h we cn onl ndce he robbl of choong ech lernve. When dcung he ozon of ochc odel for choce over e of lernve een n Dgv [28] he nure of robblc choce re orgnll gven. Bu n h rcle we re nereed n he cue of robbl reference. Probblc bnr reference cn occur f no ngle lernve boluel ol. The fr lernve ore referble b oe reer for ele rce nd wegh nd he econd ore referble b oher reer qul nd ergonoc [Sw nd Mrle 23]. Probblc bnr reference cn hen n uon of choce under r f no one loer ochcll done noher. Aoc odel of robblc bnr choce under r n h uon hve been uded b Blv [22]. Here for he fr e we nroduce robblc bnr reference n eenon of he cuulve roecue heor of Tver nd Khnen [992]. We de b ung vul grhcl rereenon of roec h re cull ued b Wer [2.6 97] o eln he conce of rn-deenden ul. We lo defne he robblc bnr reference b grhc for whch nuvel cler nd doe no requre ohced cuulve ung or Choque negrl n connuou ce. The de of robblc eend bc o Fhburn [978] nd Kru e l [98]. In corng he lernve nd n ndvdul ee wh or b how uch he fr lernve beer hn noher nd wh or b how uch he econd lernve beer hn he fr. Thee vlue re referred o he corve ul 2. If no one lernve boluel done noher nd. The ro of hee vlue / deerne he ro of robble for choong ech lernve. The CPT defne ecfc ul u for ech roec. The rdg of ronl choce ue h he ndvdul wll chooe he o ueful lernve nd he elecon reul wll be unvlen. However n h er we deerne he corve ul funcon for r of loere o we ue o conruc he robblc eenon of CPT. There re everl que erou crc of CPT. Brnbu e l [999] reened ele of he volon of he fr-order ochc donnce rncl. Wu nd Mrle 2 Fhburn [978] cll ncreenl eeced ul dvnge bu hn o Kru e l [98] we wll here cll horer corve ul. 3

4 [27] reened ele of he volon ne-deend rncle. The lcbl of CPT for ed roec een b u led. Th wh he robblc eenon of CPT nroduced here onl for non-negve roec. We hen ene he followng queon n h er. An ndvdul h robblc bnr reference on e of loere nd u chooe one lernve. How cn one redc he choce reul? There re everl wdel nown roche uch Brdle-Terr-Luce nd qu new ee Celn M. [22] Blv [29]. Bu n our ce he robbl bnr reference re genered b corve ul. So we rooe o ue odel of elecon b connuou Mrov rndo wl on e of lernve nroduced n Zuler [2]. Th odel cn ue vlue of corve ul nd llow u o el receve he requred redcon. I ude he coure of he elecon roce n e. Th ude how n ndvdul wll go hrough nd core lernve nd where led. The eenl fnl queon foruled follow: Whch lernve wll n ndvdul chooe? rher hn Wh he o ueful lernve o n ndvdul? Therefore he receved redcon concde wh our nuve de of choce reul. The rooed odel h he followng rereque. Frl how cn ndvdul cull chooe? The le odel of he elecon roce equenl cn h occur n he nd of he ndvdul. An ndvdul e ll he lernve n row nd hen e he lef-o lernve nd begn o core n order o oon on he rgh. If n ndvdul fnd beer lernve e nd leve he old lernve. When n ndvdul coe o he rghhnd end of row he fnll chooe he lernve h ren n hnd. Gven he horge of e he orng roce cn be ore errc. The ndvdul e oe lernve n he hoe h he be. He rndol nec he renng lernve. Soe of lernve h cch he ndvdul ee wll be wore hn he lernve n hnd. The ueror of he oher lernve gh no be noce nd ed. Bu f he ndvdul noce n lernve h beer he or he wll grb nd hrow he old choce bc no he le. If he ndvdul reference re rnve hen he or he wll evenull fnd beer lernve nd he roce of brue force o. For non-rnve reference h forl ochc roce cnno ever o. We hen ue h he reul of elecon wll be robblc nd rooronl o he e whn whch n ndvdul conder rculr lernve he be. We ue h he elecon roce leened b connuou hoogeneou Mrov rndo wl on e of lernve. Th he nen of he rnon fro he en lernve o n oher lernve deend onl on whch lernve he or he hold now. Th roce w clled Connuou Mrov Chn Choce CMCC. The e of 4

5 dfferenl equon for he CMCC odel obned b ndrd ehod. Regrdng he ergodc of he roce here wll be e of lner equon n ed e. The connuou Mrov odel h n dvnge over dcree odel. If we hve n nuercl rereenon of bnr relon on he e of lernve for he connuou odel doe no requre h we norlze he rnon robble. Thee nuercl vlue of dvnge cn be ued elcl rnon nene. In conr for dcree Mrov roce hould be de o norlze he rnon/no rnon n ech e. In ce of robblc eenon of CPT we wll e he nen of rnon equl o corve ul. The er orgnzed follow: Secon 2 nroduce he robblc eenon of he CPT nd Secon 3 reen elecon odel b connuou Mrov rndo wl for h ce. 2. Probblc eenon of he cuulve roec heor 2. Cuulve roec heor nd Proec dgr Suoe h we hve roec loer 3 wh ove nd negve oucoe gn nd loe where he robbl of oucoe. To clcule he ul of roec Tver nd Khnen [992] 4 rooed cuulve odel whch follow. Le he oucoe be nubered n cendng order of her vlue enng h > f he ul >. For ove oucoe ove ndce re ued for negve one negve ndce re ued for he zero oucoe u quo zero ndce re ued + nd - denoe he ove nd negve r of roec. The wegh funcon W reflec he ubecve een of robbl b n ndvdul. The vlue funcon V reflec he derbl b n ndvdul o receve over nohng. The ul of he loer clculed follow: u V V V n W W n V W n W V V... W n... W... n The fundenl of he CPT re uull reened n uon of uncern he leenon of he oble e of nure. Here we de he reenon doed n uon of r loer. 4 Here we reen onl bref nroducon o he CPT. The o deled decron of CPT vlble n Wer [2]. For oc foundon ee Wer nd Tver [993]. 5

6 Here en he dfference beween he vlue of he wegh funcon for he oucoe no wore hn nd he weghng funcon of he oucoe rcl beer hn. For connuou e of lernve h forul rnfored no Choque negrl. Le u reen dgr of he roec. For ove roec loer he dgr of he roec he grh of relbl obned vlue decreng funcon V on he robbl nervl [ ]. For ele for he roec = $ 2.3; $.2; $ 5.5 dgr equl o V$ 2 on he nervl fro o.3 V$ on he nervl fro.3 o.5 nd V$ 5 on he nervl fro.5 o ee Fg.. V V$2 V$ V V$ Fg. A roec dgr llow u o clcule cuulve ul eer. For ove roec: where u V d dw d enng h ωр he den of he wegh funcon W. In he generl ce for ed roec we cn conruc dgr follow. For he ove r of he roec he grh of relbl obned vlue on he nervl [ ]. For he negve r of he roec he grh of relble lo vlue on he nervl [- ]. For ele he roec for z = -5$.2; -$.; $.4; 5$.3 hown on Fg. 2. 6

7 V V$2 V z V$ V-$ V-$5 ω ω - ω + For he ed roec: Fg. 2 u V d V d where V nd V grh for ove nd negve r dw d dw d he den of he wegh funcon for gn nd loe. Ele Le u conder volon of he Indeendence Ao of he All-e rdo n n ele fro Khnen nd Tver [979]. Of he wo lernve х = $ 4.8; $.2 nd у = $ 3. bou 8% of he reonden chooe he lernve у. Th o h 8% of he reonden refer o receve $3 for ure rher hn o rce n loer where he gn $4 wh robbl of.8. If he robbl of gn n boh loere 7

8 reduced b 4 e hen fro he obned lernve х 2 = $4.2; $.8 nd у 2 = $3.25 bou 65% of he reonden chooe he lernve х 2. To eln he rdo we conder he roec dgr gven n Fg. 3 nd forul. V V4$ V3$ V V у V V 2 V у2 ω.8 ω.2.25 b Fg. 3 Snce he den of he weghng funcon gnfcnl ncree n he vcn of = he reonden overvlued he lc of r hen:.8 4$ ω d V V 3$ ω d bu V 4$ ω d > V 3$ ω d whch eln h rdo Sochc donnce nd Probblc choce Of he wo lernve roec = $ 2.3; $.2; $ 5.5 nd ' = $ 9.3; $ 9.2; $ 4.5 n ndvdul choce clerl n fvor of he fr one. Conderng he e chnce enure greer gn. Th en ochc donnce. For coron of he roec n le obvou uon uch nd = $ 2.; $ 5.; $ 8.3; $ 5.2; $ 3.3 we cn conruc ulr dgr of he roec ee Fg. 4. The dgr of roec rcl hgher hn choce of he ndvdul wll be n fvor. A V х р V р for ll roec ochcll done oo. 8

9 V V$2 V V z V V V$5 V$ V$8 V$5 V$3 V V b b Fg. 4 Fnll f n ndvdul wll core roec nd z = $5.8; $.2 hen no one ochcll done he oher. There oe reon o chooe roec nd oe reon. I en h he choce cn be robblc. If roec ochcll done over he econd one he ronl choce of n ndvdul hould be unbguou. However f neher of he wo roec done ech roec n oe w beer hn he oher. The ndvdul h ncenve o elec ech of he roec nd cn be ued h he choce wll be robblc n nure. We defne he funcon of he corve cuulve ul on he r of ove roec: b V V ω d 2 b Th funcon correond o n re beween he roec dgr where roec hgher hn he roec b whch rereened b he crohched re n Fg. 4b - blue b - red. If he den of he weghng funcon were dencll equl o un he vlue of he corve ul would be equl o he qure of h re. I obvou h u u. 2.3 Probblc eenon of he cuulve roec heor We denoe here.5;] of he wo lernve nd n ndvdul chooe wh robbl nd chooe wh robbl -. Then noon wll en h n ndvdul refer over for ure. We denoe ~ hen of he wo lernve nd n ndvdul chooe or wh equl robble = ½. 9

10 Le u conder he followng odel of robblc reference relon. For n r of lernve nd here re r of non-negve vlue и whch deonre he dvnge lernve over nd dvnge lernve over lernve. Boh of hee vlue cn be ove ulneoul f n oe reer where beer hn nd n oe reer lernve beer hn. The robbl h n ndvdul elec ech of he lernve deerned b he ro nd. We h he robblc reference relon rereened b ove funcon of corve ul f here e unque u o ove ullcon bnr funcon uch h: 3 Forll: ~ 2. Bed on he corve ul funcon defned b 2 we cn deerne robbl bnr relon on he e of roec b forul 3. For loere n Fg. 4 for lc we ue h he den of he wegh funcon equl o he un nd lner vlue funcon: w= nd V=: 27 nd z z nd z 65 4 or z 62.. Ele We reurn o he All rdo nd wh he reference of he reonden were dvded. Wh dd he choce of 2% of he reonden n he fr oll nd 35% n he econd oll no concde wh he choce of he or? Clerl he reonden hve dfferen vlue or wegh funcon. However ccordng o he robblc odel conruced bove even wh dencl vlue nd wegh funcon he reone could be dvded. In fc een n Fg. 3:

11 V V ω d V 3$ ω d nd.8.8 V V ω d V 4$ V 3$ ω d Th wh non-zero robbl nd у cn be eleced whch cn be clculed b forul 3. Slrl for х 2 = $ 4.2 $.8 nd у 2 = $3.25:.25 V 2 V2 ω d V 3$ ω d nd V 2 V2 ω d V 4$ V 3$ ω d 2 2 whch eln wh he reone were dvded n he econd oll.. 3. Model of elecng fro e of lernve b connuou Mrov rndo wl on e of lernve 3. Connuou Mrov Chn Choce Choong b en of connuou hoogeneou Mrov rndo wl erfored follow. A e of roce e concde wh e of lernve nd n ndvdul e n lernve he be. A n gven e he or he cn robblcll ove o noher e b ng noher lernve. A he e e he nen of he rnon deend onl on he curren e of he lernve h he or he hold. In he ce of ergodc of he wl roce he reul of he robblc choce of robblc correond o he onr drbuon. Le u recll h he Mrov roer e h he condonl robbl drbuon for he e he fuure deend onl on he curren e of he e nd no ddonll on he e of he e n he. Le S... } e of e nd he robbl of rnon fro o { 2.. durng e o where he non-negve conn or he nen of he rnon. Th roce clled connuou-e hoogeneou Mrov chn. The Mrov roce clled ergodc f for n nl e here re rgnl robble of he e roce whch do no deend on he nl e. In rculr h condon fed f he verce of he grh re conneced nd beween n wo verce here dreced h. Th he onr robbl drbuon of he roce.

12 2 In he ce of ergodc of he wl roce he reul of he robblc choce correond o he onr drbuon. 3.2 Governng equon of CMCC Le u deerne he equon of he CMCC odel. If e he roce n e wh robbl. Then e goe no wh robbl o. Thu we hve: o. Trnng o he l we obn e of dfferenl equon: d d ll for. In he ce h onr robbl drbuon e he dervve n he rgh-hnd de vnh when. Thu we obn e of lner equon cull deenden rn = +:. ll for 4 The reulng robbl drbuon... 2 n nerreed he robbl h n ndvdul wll elec eher of he lernve of e S. In he ce of robblc eenon of CPT we wll e he nen of rnon for CMCC odel equl o corve ul. Noe h n ce of wo lernve he robbl of elecng receved b 4 wll equl he robbl receved b 3. Ele Le u conder e of hree lernve roec = $5.5; $.5 b = $4.7; $.3 c = $2. nd d= $.5; $.5. The grh of he roec re hown n Fg. 5.

13 V V$5 V$4 V$2 V$ V V d V b c d b b Fg. 5 For lc ue h W = nd V =. Corve ul re equl o: b 8 b 6 c 5 c 5 b c 4 c b 6 d 25 d b d 9 d b c d 5 d c. The grh of Mrov rndo wl hown n Fg.5b. 5 The defnng e of equon for he onr drbuon for 4 : 6 5 5c 8b 25d 8 6 b 6 4 c 9 d 5 4 c 5 6 b 5d d one of he equon cn be oed b c d nd h oluon of V c NOTE In h ele he order of robble of choce lernve concde wh he order of cuulve ule for lernve. Bu h no necerl he ce n ll ele. In he CMCC odel o robbl of choce cn be oeed b n lernve of he non-l ul. Th could occur n uon where oe of he be lernve re ver lr nd oe re fundenll dfferen. For ele ong he nonee for he fl of he er wrd here be wo hrller ronce ove nd coed. And n he ruggle for he ur enon he lr lernve hrller loe o he le ueful nereng bu ver dfferen lernve ronce ove. 5 In choce heor cceble o drw he rrow fro he edge of beer lernve o he wor nd h rdon reeced. The ndvdul n he roce of rndo wl goe fro he wor lernve o he be lernve. In oher word ove gn he drecon of he rrow. 3

14 Concluon Th er rooed: A robblc eenon of he cuulve roec heor wh grhc for. We reen le vul grhc odel of loere. In h grhc odel cler h he conce of ochc done one loer over noher. If n ndvdul chooe one lernve over wo loere nd no one rculr loer ochcll done he oher we cn e robblc reference. In he frewor of CPT we clcule corve ule; he dvnge of he fr loer over noher nd dvnge of he econd loer over he fr. Thee vlue deerne he robble of choce for ech loer. A odel of choce ung connuou Mrov rndo wl for robblc eenon of CPT. Here we conder he n ndvdul undere when elecng one lernve fro e of lernve beween whch he h bnr robblc reference. In ce of robblc eenon of CPT rooed o ue odel of connuou Mrov rndo wl. The rnon robble were e equl o corve ule whch were lred receved. The rooed ehod re e o undernd nd nuvel cler. Alhough he odel re bed on rher crude uon nd eggere he cul roce of choong he ndvdul he do no requre lrge oun of nl d for reercher. The obned reul be que cceble for rccl ue for ele n fnncl nl or reng. Lerure Brnbu M. Pon J. Lo M. 999 Evdence gn Rn-Deenden Ul Theore: Te of Cuulve Indeendence Inervl Indeendence Sochc Donnce nd Trnv // Orgnzonl Behvor nd Hun Decon Procee vol. 77 ue Blv P. 29 How o Eend Model of Probblc Choce fro Bnr Choce o Choce ong More Thn Two Alernve // Econoc Leer Blv P. 22 Probblc choce nd ochc donnce // Econ. Theor Celn M. 22 Model for Pred Coron D: A Revew wh Eh on Deenden D // Scl Scence Vol. 27 No Dgv J.K. 28 Aozon of ochc odel for choce under uncern // Mhecl Socl Scence Fhburn P. 978 A robblc eeced ul heor of r bnr choce // Inernonl Econoc Revew

15 Khnen D.Tver A. 979 Proec Theor: An Anl of Decon under R // Econoerc Vol. 47 No Kru A.Y. Rubnov A.M. Ynov E.B. 98 Oln vbor rredelen v lonh oclno-eonocheh zdchh n Run. L.:Nu. Lenngr.od-e. Sw J. Mrle A.A.J.23 Probblc choce odel reul of blncng ulle gol //Journl of Mhecl Pcholog Volue 57 Iue 2. 4 Tver A. Khnen D. 992 Advnce n roec heor: cuulve rereenon of uncern // J. R Uncern Wer P. Tver A. 993 An Aozon of Cuulve Proec Theor // Journl of R nd Uncern 7: Wer P. 2 Proec Theor for r nd bgu. Cbrdge unver re. Wu G. Mrle A. 28 An Ercl Te of Gn-Lo Serbl n Proec Theor // Mngeen Scence vol. 54 no Zuler I.A. 2 The equenl lernve erch connuou Mrov rndo wl // Auoon nd Reoe Conrol Il A. Zuler Nonl Reerch Unver Hgher School of Econoc Mocow Ru Fcul of Econoc Deren of Hgher Mhec Docen PhD; E-l: zuler@rbler.ru zuler@gl.co Tel An onon or cl conned n h Worng Per do no necerl reflec he vew of HSE. 5