Duopoly Market Analysis within One-Shot Decision Framework with Asymmetric Possibilistic Information

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1 Itertl Jurl f Cmuttl Itellgee Sytem Vl3 N 6 Deember Duly Mrket Aly wth Oe-Sht De Frmewrk wth Aymmetr Pblt Ifrmt Peu Gu Fulty f Bue Admtrt Ykhm Ntl Uverty 79-4 Tkwd Hdgy-ku Ykhm J E-ml: gu@yu Rulg Y Shl f Bue d Em Id Uverty Nrthwet Gry IN USA E-ml: rulgy@gmlm Jh Wg Dertmet f Mgemet d Ifrmt Sytem Mtlr Stte Uverty Mtlr NJ USA hwg@gmlm Abtrt I th er ewly emergg duly mrket wth hrt lfe yle lyzed The rtlly kw frmt f mrket hrterzed by the blty dtrbut f the rmeter the demd fut Se the lfe yle f the ew rdut hrt hw my rdut huld be rdued by tw rvl frm tyl e-ht de rblem Wth the e-ht de frmewrk the blt Curt eulbrum bted fr the tml rdut level f eh frm duly mrket wth ymmetrl blt frmt The ly reult hw tht the red rhe re reble fr e-ht de rblem whh re etevely eutered bue d em Keywrd: Duly mrket fu t e-ht de blt Curt eulbrum blty thery Itrdut De ly hg mg ltertve derg frmt but the tte f ture whh re the ftr beyd the trl f de mker A lmtg e de mker my kw whh tte f ture wll ur I uh e kw de mkg uder ertty t ey t mke he beue the utme f eh ltertve determt Otherwe de mker t uder -ertty N-ertty dvded t three tegre -rk uertty d gre Stut vlvg rk re whe the rbblte f ll ble utme be etly lulted O the ther hd uert tut re ted wth rtlly kw frmt whe rbblte t be bted etly ] Stut vlvg gre ur whe frmt vlble t dtguh whh tte f ture mre lkely t ur De rule fr tut vlvg gre lude mm mm mm regret d Hurwz rter Ledg there f de lude Publhed by Atlt Pre Cyrght: the uthr 786

2 PGu et l the eeted utlty EU thery f v Neum d Mrgeter d the ubetve eeted utlty SEU thery f Svge EU ued de-mkg uder rk where SEU del wth de-mkg uder uertty Subetve rbblte re ued t reflet de-mker belef trdtlly lyzed term f bettg behvr The SEU m hw the dt uder whh referee be rereeted by umerl eeted utlty tht ue ubetve rbblte f the tte f ture t weght euee utlte Reerh h hw tht SEU de t rvde deute dert f dvdul he uder uertty 5] Altertve mdel uh regret thery 3] ed-rder rbblte 5 6] d -ddtve rbblty mdel 6 5] hve bee red th emrl hllege Pret thery develed by Khem d Tverky 6] ble -ddtve rbblty mdel Uder ret thery vlue ged t g d le bed referee t rther th t fl et EU d SEU Al rbblte re reled by de weght wthut ddtvty Vlue fut re rmlly ve fr g mlyg rk ver d ve fr le rk eekg Pblty thery tted by Zdeh 30] d dved by Dub Prde d Klr 3 0] e f the urret uertty there devted t hdlg f mlete frmt the rel wrld Pblty thery bed tw b -ddtve meure blty meure d eety meure De ly uder uertty l be fud the lterture ] It kw tht de there uder uertty re there f he uder uertty where the bet f he re rbblty dtrbut fr EU SEU d ther vrete ret frmed term f g d le fr ret thery r blty dtrbut regrded blt lttere 4] I ft fr e-ht de rblem whh fr the tut tht de eereed e ly there e d ly e he fr e tte f ture mg u Gu 9-] tlly red the e-ht de rhe fr uh de rblem The redure fr e-ht de erted t tw te The frt te vlve fug me tte f ture lled fu t Whh tte f ture fued hrterze de mker dfferet tttude but tft d blty I the ed te ltertve re evluted bed ther fu t where the tft level rvded by ll ltertve re mred wth eh ther t bt the tml ltertve Wth the e-ht de frmewrk duly mrket f ew rdut wth hrt lfe yle lyzed The blt Curt eulbrum red t lyze the tml rdut level f eh frm duly mrket 9] Th er t ete where the redure fr lultg the blt Curt eulbrum red d the ete f blt Curt eulbrum wth ymmetrl blt frmt mde Th er rgzed fllw: Set trdue e-ht de frmewrk I Set 3 blt de mdel fr duly mrket re ddreed where duly mrket wth ymmetrl blt frmt vetgted Flly ludg remrk fr th reerh re mde Set 4 Oe-Sht De Frmewrk Pblty dtrbut d Stft Fut I e-ht de rblem the et f ltertve A The et f tte f ture S The degree t whh tte f ture t ur the future hrterzed by blty dtrbut defed belw Deft Gve fut : S 0] f m S the the fut lled blty dtrbut where S {} the mle e the blty degree f me tht t rml tht ur d 0 me tht t brml tht ur The mller the blty degree f the mre urred fr the heg f The fllwg emle hw hw t etmte the uert demd by the blty dtrbut Emle The demd etmt f fh lthe We k 60 yug femle utmer whether they re tfed wth rtulr tye f fh lthe The Publhed by Atlt Pre Cyrght: the uthr 787

3 Duly Mrket Aly wth Oe-Sht De Frmewrk tem f uetre lude tegre uh re tyle lr d mfrt The ttl tft degree f utmer determed by the mmum f h tft level f ll tegre The umber f the utmer whe tft level re d the ther re d 0 reetvely Bed uh reult eereed lem ked the fllwg uet but h level f urre Q: Hw urred re yu f ly the utmer wth full tft level 0 buy the lthe? A: Very urred fr dfferet frm the eeree f th lem Q: Hw urred re yu f ly the utmer wth tft level t le th 08 buy the lthe? A: A ltter bt urred dfferet frm the eeree f th lem Q3: Hw urred re yu f ly the utmer wth tft level t le th 07 buy the lthe? A3: Nt urred mthe the eeree f th lem Q4: Hw urred re yu f ll the utmer wth tft level t le th 06 buy the lthe? A4: A ltter bt urred Q5: Hw urred re yu f ll the utmer wth tft level t le th 05 buy the lthe? A5: Very urred Dete the degree f urre t It the mm ee tht the mre urred the mre mble Thu t reble t defe the blty degree t If we et t 05 0 fr the tut f very urred ltter bt urred t urred reetvely the the blty degree f demd f 50 90= = = = re reetvely Frm the bve emle t ler tht the blty dtrbut be ued t rereet the kwledge r udgmet f hum beg I the lterture ] the uer d lwer eetl blty dtrbut re bted by blt ler rgrmmg rblem t reflet the feble reg f de vrble I the lterture 3-4] the uer d lwer eetl blty dtrbut re detfed frm the gve blty degree f mle by eert v ler rgrmmg rblem d the detfed blty dtrbut f the eurte retur re ued fr rtfl elet rblem The euee reultg frm the mbt f ltertve d tte f ture refereed t yff deted v The tft level f de mker fr yff be ereed by tft fut defed belw Deft Dete the et f yff v V The fllwg fut u : V 0] 3 wth v v fr v v 4 lled tft fut Beue the yff the fut f d we rewrte the tft fut u v Fr the ke f mlft metme we wrte v th er The frmt fr e-ht de be ummrzed udrule A S u Oe-ht de t he e ltertve bed A S u whe ly e de he gve Se e d ly e tte f ture wll ur fr e-ht de the de mker huld frt der whh tte f ture huld be fued fr mkg de The fued tte f ture re lled fu t The redure fr hg fu t trdued the fllwg ubet Determg Fu Pt f Altertve Let u beg wth the fllwg deft whh ued fr determg fu t Deft 3 Gve the vetr ] m ] d m ] re defed fllw: m ] ] 5 m ] ] 6 m ] d m ] re the lwer d uer bud f ] reetvely Fr emle m 0308] 0303] d m 0308] 0808] Che A de mker ly fue the rml e tht the tte f ture wth the blty degree deted tke t ut whh rg m 7 S Publhed by Atlt Pre Cyrght: the uthr 788

4 PGu et l lled rml fu t rg m S ued t dete the vlue f whh mke mmze Che Fr ltertve de mker fue tte f ture deted fllw: rg m m ] 8 S It fllw frm 8 tht mmze g m ] I dert f 5 we kw tht m ] rereet the lwer bud f the vetr ] ; tht fr tte f ture bth f the blty degree d the tft level rvded by ltertve re t let m Ireg m ] m m ] wll S ree the lwer bud f the blty degree d the tft level multeuly Therefre rg m m u ] fr eekg the tte f S ture tht h the hgher blty degree d the hgher tft level The bted tte f ture lled tve fu t f ltertve Che 3 Fr ltertve de mker fue tte f ture deted fllw: rg m m ] 9 S It fllw frm 9 tht mmze k m ] Se m ] rereet the uer bud f the vetr u ] dereg m ] m m ] S wll deree the uer bud f d multeuly I ther wrd m m ] wll ree the lwer S bud f the blty degree d deree the uer bud f the tft level u Therefre rg m m ] fr eekg the tte S f ture tht h the hgher blty degree d the lwer tft level The bted tte f ture lled ve fu t f ltertve 3 Obtg the Otml Altertve Bed the bted dfferet tye f fu t the tml ltertve re determed fllw: rg m m 0 A X rg m m A X rg m m A X where X the et f rml fu t X d X re the et f tve d ve fu t f ltertve reetvely d re lled rml tve d ve tml ltertve reetvely d re lled tml tve d tml ve fu t reetvely De rule mm d mm re ued 0 d fr the e tht multle fu t et fr ltertve mm d mm reflet the ervtve d ggreve tttude reetvely A de mker lled rml de mker r tve de mker r ve de mker f he tke t ut the rml fu t r tve fu t r ve fu t fr mkg e-ht de We mge three tye f de mker fllw: whe mkg e-ht de the rml de mker fue the mt ble utme; the tve de mker tke t ut the er whh yeld the hgher tft wth hgher blty; the ve de mker der the er whh led t the lwer tft wth hgher blty Therem 9] Aume tht I Pblty dtrbut umdl tuu fut whh tfe l 0 u 0 d d ree wth ] d l deree wth u ] where l d u re the eter lwer d uer bud f II d m u re tuu trtly A reg fut f reetvely We hve m m m u A S m m m 3 S m A A Publhed by Atlt Pre Cyrght: the uthr 789

5 Duly Mrket Aly wth Oe-Sht De Frmewrk m m m A m where A S 4 the hrztl rdte f the uue teret f d m A wth l ] d the hrztl rdte f the uue teret f d m wth u ] A 3 Duly Mrket Aly wth Pblt Ifrmt 3 The B f Curt Eulbrum We mge dutry wth tw frm d duly mrket eh rdug d ellg gle gd Cumer d t re frm whh frm they urhe the gd The fllwg well-ued ler vere demd fut em dered ]: d 5 where 0 re d re rdut level f Frm d Frm reetvely 0 the -teret f 5 rereetg the lmt re f gd whe the ttl uly rhe t zer d d 0 the le f 5 hwg the re deree whe the ttl uly reg by e ut Here we der tht eh frm h Curt eture but t rvl: Eh f tw frm ume tht the ther frm wll t wy t kee fed the rdut level Wth Curt eture the rft f eh frm = fllw: w d d 6 where the rdut t gred fr mlty Ceturg tht Frm h the rdut level the uffet d eery dt fr mmzg 6 fllw: dw / d d d 0 7 whh bted by dfferettg 6 wth S tht the tml rdut level f Frm wth the rmeter d deted bted d / d 8 Tw frm re t eulbrum th dutry whe ether frm wt t hge wht t dg gve tht the rvl kee the me rdut level S tht th eulbrum lled Curt eulbrum the lut f the fllwg eut d / d 9 d / d whh led t 0 3d where lled the eulbrum rdut level f Frm 3 Curt Eulbrum wth Pblt Ifrmt Let u der ewly emergg duly mrket wth hrt lfe yle It reble t thk tht there ly e he fr thee tw frm t dede hw my rdut they huld rdue due t hrt lfe yle Beue the rdut ew relevt htrl dt be ued fr tttl ly th e Hwever mrketg reerh bt me frmt but wht the mt ble tut d wht mble e the future Suh luble frmt be hrterzed by the blty dtrbut f fllw: The blty dtrbut f et the fllwg tuu umdl fut : l r ] 0] where l d r re the lwer d uer bud f ] tht 0 d r l u 0 ree wth ] deree wth ] r l l d Remrk : Cderg 6 d 8 the mml rft f Frm wth fllw: d m w 4d hw tht the mmum rft f Frm fr gve The rft f Frm rehe t 4d mmum 0 whe 0 Cderg the reg f r the reg f the rft f Frm 0 ] It ey t 4d kw tht the reg f the rdut level f tw frm 0 r ] d Publhed by Atlt Pre Cyrght: the uthr 790

6 PGu et l The tft fut f Frm deted w trtly reg fut f w r wth the dt 0 0 d Let u 4d lyze the duly mrket rblem wth the e-ht de frmewrk Tw frm hve blty t rdue ew rdut wth hrt lfe yle The tye f frm re rml tve d ve Suh tye regrded heret feture f frm A frm kw h w tye but de t kw the tye f t rvl The frmt A S u mm kwledge fr bth f frm where A 0 r / d ] S l r ] Bed A S u d h w tye the frm huld mke e-ht de the rdut level t mmze t rft wth Curt eture but t rvl Frm Che we kw tht the rml fu t Cderg Che the tve fu t f the rdut level f Frm wth eturg the rdut level f Frm beg fllw: deted rg m m w ] 3 Lkewe derg Che 3 the ve fu t f the rdut level f Frm wth eturg the rdut level f Frm beg fllw: deted rg m m w ] 4 Cderg Therem d 8 the fllwg lu be bted 9] I Bed the tve fu t f he t tml rdut level Frm t mke w mmze wth eturg the rdut level f Frm beg Tht rg m w ˆ d 5 d where ˆ lled tve fu t f Frm wth eturg the rdut level f Frm beg the hrztl rdte f the uue teret f d w wth ] where bted by 8 u II Bed the ve fu t f he t tml rdut level Frm t mke w mmze wth eturg the rdut level f Frm beg Tht rg m w ˆ d 6 d where ˆ lled ve fu t f Frm wth eturg the rdut level f Frm beg the hrztl rdte f the uue teret f d w wth ] where bted by 8 l I wht fllw let u lyze the eulbrum tut f tw frm Fr the ke f mlft we ume tht the frm hve the me tye eh ther Sue tht bth f frm re rml t ey t udertd tht the Curt eulbrum fr th e deted the lut f eut 9 where tke the le f Sue tht bth f frm re tve r ve the tve d ve Curt eulbrum deted d reetvely be bted the fllwg eut: ˆ d d ˆ d d 7 ˆ d d ˆ d d 8 where d re the lut f 7 d 8 reetvely The redure fr lvg the eut 7: Ste Arbtrrly he ] r Publhed by Atlt Pre Cyrght: the uthr 79

7 Duly Mrket Aly wth Oe-Sht De Frmewrk Ste Set ĉ = d lve 7 t bt Ste 3 Clulte e w If e the ˆ d t; f e the g t Ste 4; f e the g t Ste 5 where mll tve rel umber Ste 4 Set 0 d g bk t Ste where mll tve rel umber Ste 5 Set 0 d g bk t Ste Th redure eled by Fg It ler frm Fg tht e 0 mle ˆ d e 0 mle ˆ Fg Grhl elt f the redure fr btg ĉ e l ĉ w r w e 33 Curt Eulbrum wth Aymmetrl Pblt Ifrmt Let u der ymmetrl frmt e Tht me ew frmt kw by Frm but t by Frm I th e the rdut level f Frm fut f tht The rft f Frm d re fllw: l ĉ r w d d 9 w d d 30 Fg Grhl elt f the redure fr btg ĉ The redure fr lvg the eut 8: Ste Arbtrrly he l ] Ste Tke ĉ = d lve 8 t bt Ste 3 Clulte e w If e the ˆ d t; f e the g t Ste 4; f e the g t Ste 5 Ste 4 Set 0 d g bk t Ste Ste 5 Set 0 d g bk t Ste Th redure eled by Fg It ler frm Fg tht e 0 mle ˆ d e 0 mle ˆ Fr Frm the ert vlue f ukw but t blty dtrbut S kw The blty dtrbut f the rdut level f Frm deted be lulted by the ete rle Q Q m t S Deft 4 Ceturg the rdut level f Frm beg d derg the blty degree f the tve fu t f the rdut level wth frmt deted defed fllw: rgm m Q Q 3 w ] where regrded dut effet f the tft level f Frm Publhed by Atlt Pre Cyrght: the uthr 79

8 PGu et l Bed the tve fu t Frm he t tml rdut level w mmze Tht t mke rg m w h 3 Deft 5 Ceturg the rdut level f Frm beg d derg the blty degree f the ve fu t f the rdut level wth frmt deted defed fllw: rg m m 33 w ] Q Bed the ve fu t Frm he t tml rdut level w mmze Tht t mke rg m w h 34 Frm refe by ug the ddtl frmt The reewed frmt kd f dtl blty dtrbut deted C whh me tht f frmt the the blty dtrbut f C Deft 6 Ceturg the rdut level f Frm beg bed the dtl blty dtrbut C the tve fu t f the rdut level f Frm deted defed fllw: rgm m C 35 w ] Bed the tve fu t Frm he t tml rdut level t mke w mmze Tht rg m w k 36 Deft 7 Ceturg the rdut level f Frm beg bed the dtl blty dtrbut C the ve fu t f the rdut level f Frm deted defed fllw: rg m m C 37 w ] Bed the ve fu t Frm he t tml rdut level w mmze Tht t mke rg m w k 38 Deft 8 The lut f the fllwg eut 39 d 40 deted d re lled tve d ve Curt eulbrum wth frmt reetvely h 39 k h 40 k Lemm If hld fr the S Q hld fr I th e 3 d 33 re eul t 3 d 4 reetvely A el e we defe fllw: C C 4 C where the frmt ftr The mller the mre frmed C Therem Aume tht ereed by 4 C we hve I bed the tve fu t f ereed by 3 Frm he t tml rdut level mke w mmze wth t Publhed by Atlt Pre Cyrght: the uthr 793

9 Duly Mrket Aly wth Oe-Sht De Frmewrk eturg the rdut level f Frm beg Tht rg m w ˆ 4 d d where ˆ lled tve fu t f Frm wth ymmetr frmt the hrztl rdte f the uue teret f d w wth ] Q where bted by 8 u II bed the tve fu t f ereed by 35 Frm he t tml rdut level mke w mmze wth eturg the rdut level f Frm beg Tht rg m w ˆ 43 d d where ˆ lled tve fu t f Frm wth ymmetr frmt the hrztl rdte f the uue teret f C ereed by 4 d u w wth ] where bted by 8 u III bed the ve fu t f ereed by 33 Frm he t tml rdut level t t mke w mmze wth eturg the rdut level f Frm beg Tht rg m w ˆ 44 d d where lled ve fu t f Frm ˆ wth ymmetr frmt the hrztl rdte f the uue teret f d w wth ] Q where bted by 8 l IV bed the ve fu t f ereed by 37 Frm he t tml rdut level t mke w mmze wth eturg the rdut level f Frm beg Tht rg m w ˆ 45 d d where lled ve fu t f Frm ˆ wth ymmetr frmt the hrztl rdte f the uue teret f d C w wth l ] where bted by 8 Prf It trvl t rve 4-45 wth derg Therem d Deft re ued fr btg Curt eulbrum wth ymmetrl blt frmt by 39 d 40 4 Clu I th er wth the e-ht frmewrk blt de mdel re red fr lyzg duly mrket f ew rdut wth hrt lfe yle The uertty f mrket hrterzed by the blty dtrbut f the rmeter the demd fut Three fu t lled rml tve d ve fu t re trdued t hw whh vlue huld be dered fr mkg de wth blt frmt Bed three kd f fu t blt Curt eulbrum re red t lyze the tml rdut level f eh frm duly mrket wth ymmetrl blt frmt It be ee tht e-ht de rhe re ueful fr the tut where de eereed ly e d the rbblty dtrbut uvlble due t lk f eugh frmt Publhed by Atlt Pre Cyrght: the uthr 794

10 PGu et l Referee M All Le mrtemet de l'hmme rtel devt le rue: Crtue de tult et me de l'éle mére Emetr D Bell Regret de mkg uder uertty Oert Reerh D Dub d H Prde Pblty Thery Pleum Pre New Yrk D Dub H Prde d R Sbbd De-theret fudt f blty thery Eure Jurl f Oertl Reerh D Ellberg Rk mbguty d Svge m Qurterly Jurl f Em I Glb Eeted utlty wth urely ubetve ddtve rbblte Jurl f Mthemtl Em L Gd d A Z O the blt-bed de mdel: hrterzt f referee relt uder rtl tey Aled Itellgee T Gürbüz Multle rter hum erfrme evlut ug Chuet tegrl Itertl Jurl f Cmuttl Itellgee Sytem P Gu Oe-ht de rh d t lt t duly mrket Itertl Jurl f Ifrmt d De See P Gu Prvte rel ette vetmet ly wth e-ht de frmewrk Itertl Rel Ette Revew P Gu Oe-ht de thery IEEE Trt Sytem M d Cyberet: Prt A I Pre P Gu H Tk d H -J Zmmerm Uer d lwer blty dtrbut f fuzzy de vrble uer level de rblem Fuzzy Set d Sytem P Gu D Zeg d H Shhd Gru de wth tet kwledge IEEE Trt Sytem M d Cyberet Prt A: Sytem d Hum P Gu d H Tk De ly bed fued duble eetl blty dtrbut Eure Jurl f Oertl Reerh B E Kh d R K Sr Mdellg mbguty de uder uertty Jurl f Cumer Reerh D Khem d A Tverky Pret Thery: A ly f de uder rk Emetr T Ky Mult-ttrbute evlut f webte ulty E-bue ug tegrted fuzzy AHPTOPSIS methdlgy Itertl Jurl f Cmuttl Itellgee Sytem X L D Ru J Lu d Y Xu A lgut-vlued weghted ggregt ertr t multle ttrbute gru de mkg wth utttve d ulttve frmt Itertl Jurl f Cmuttl Itellgee Sytem V Lez M St d J Mter Fuzzy eft rel ette mrket de mkg Itertl Jurl f Cmuttl Itellgee Sytem G J Klr d T Flger Fuzzy Set Uertty d Ifrmt Prete-Hll New Jerey 988 F H Kght Rk Uertty d Prft Hught Mffl Bt 9 D A Kre Cure Mrem Thery Pret Uverty Pre New Jerey G Lme d R Sugde Regret thery: ltertve thery f rtl he uder uertty Eml Jurl A Phlv A ew fuzzy MADM rh d t lt t ret elet rblem Itertl Jurl f Cmuttl Itellgee Sytem D Shmedler Subetve rbblty d eeted utlty wthut ddtvty Emetr U Segl The Ellberg rd d rk ver: tted utlty rh Iterl Em Revew H Tk d P Gu Pblt Dt Aly fr Oert Reerh Hedelberg; New Yrk Phy- Verlg T Whle De mkg uder uertty wth vru umt but vlble frmt IEEE Trt Sytem M d Cyberet R R Yger Pblt de mkg IEEE Trt Sytem M d Cyberet L A Zdeh Fuzzy et b fr thery f blty Fuzzy Set d Sytem Aed: The rf f Therem Frt let u hw m m m A S A m m m S A Set h m u Cderg the fllwg tw eut m m h m m h A A S S A m m m S A A3 m m m S A we hve A The we hw Publhed by Atlt Pre Cyrght: the uthr 795

11 Duly Mrket Aly wth Oe-Sht De Frmewrk m m S A4 m m m u S A Beue the relt m hld wth derg three e u ; ; 3 we hve m m m u A A d S A A5 Wth derg the ft tht f f g the m f m g t ey t udertd A4 tuu dereg fut wth l ] d m u A tuu reg fut The fllwg eulte trtly whh the ed d thrd eulte f 3 Set rg m It be udertd tht A l the hrztl rdte f the teret f d u Smlrly Beue tuu trtly reg fut f tfe m m whh me f S we tke d the A4 beme eulty Thu the frt eulty f 3 be bted It l ey t udertd tht uue teret f d m et wth ] A u Dete the hrztl rdte f th teret t bvu tht 4 hld derg A It rve Therem m A6 A l l 0 m A7 A hw tht there e d ly e teret f d m wth ] Dete the A l hrztl rdte f th teret the m A8 A dereg fut wth l ] whh me fr ll ] l m m A9 m A u A reg fut wth ] whh me fr ll ] m u m m A A8 mke the fllwg hld A A0 m m m S A m A A Publhed by Atlt Pre Cyrght: the uthr 796