A game-theoretical analysis of the efficient
disposal of the industrial waste
著者
Kazuki Hirase
journal or
publication title
The Economic Review of Toyo University
volume
39
number
2
page range
53-61
year
2014-03-25
URL
http://id.nii.ac.jp/1060/00006417/
Creative Commons : 表示 - 非営利 - 改変禁止東洋大学「経済論集」39巻2号2014年12月
AgaIne-theoreticalanalysisoftheefficient
disposaloftheindustrialwaste
KazukiHirase
Abstract Wede6nethewastedisposalgamewheretheplayersarelocalgovernments.Playersarelocateddispersedly. Eachplayerisendowedwithoneunitoftheindustrialwasteandhastodecidetheplacewherehiswasteis disposed.Weanalyzetheplayers'decisionsintheequilibriuminthreecases,thatis,thepresentJapanese waste-endtaxsystem,thediscriminationwaste-endtaxsystem，andtheauction.Wecanobtainthatthe6rst casedoesnotalwaysleadtotheefficientwastedisposalandthatthesecondcaseandthethirdcaseleadtothe efficientwastedisposal. Keywords:IndustrialWasteDisposal,Taxation,Auctioll，NashEquilibirium,Efficiency JELclassification:C72;Q5811ntroduCtion
Wedefinethewastedisposalgamewheretheplayersarelocalgovernments.Theplayersarelocated dispersedly・Inthegame,eachplayerisendowedwiththeindustrialwastetobedisposedandthe facilitieswhichhasthedistinctdisposalcost.Eachplayerhastodecidetheplacewherehedisposehis waste.Wewouldliketoclarifytherelationbetweentheplayers,decisionsintheequilibriumandthe efficiencyoftheresultingdisposal.Weanalyzetherelationinthreecases.ThefirstoneisthepresentJapanesewaste-endtaxsystem.EachlocalgovernmentsetstheundifL ferentiatedwaste-endtaxrate.Typically,localgovernmentisettitotheothergovernments'disposal toi.Thatis，eachplayerwhosewasteisdisposedbyihastopaythecosttotransferthewastetoj
andthetaxti・Indeed,manylocalgovernmentsadoptthesametaxrateinJapan.'Inthiscase,we
canobtainthepossibilityoftheinefficiencyoftheequilibrium. Second)wefOcusonthediscriminationwaste-endtaxsystem.Inthiscase，eachplayerisallowed tosetdistincttaxratestotheotherplayers.Typically,playericansetthetaxratet"toplayerj.In thiscase,theequilibriumrealizestheefficiencyofthedisposal. Third)weconsidertheplayers'decisionmakingofthedisposalplaceinanauction.Inthiscase、 eachplayericanexecuteasecondpriceauction.Theotherplayersbidthepriceatwhichtheyare willingtodisposeplayerj'sindustrialwaste.Wecanobtainthatthiscasealsoleadstotheefficiency ofthedisposalattheequilibrium. Intheliterature,ShapleyandShubik(1969)andNakayama(2005)dealwiththegarbagegame. Inthegarbagegame,eachplayerhasgarbagewhichhemustdumpsomeone'syard.Theyfbcuson thesolutionconceptofthecooperativegames.Intheirpapas)neitherdisposalcostnorwaste-endtax arenotconsidered.Kanekoetal.(2010)discussesthepresentJapanesewaste-endtaxsystem,gives atheoreticalmodelwhichincludethewaste-endtax,andexecutesanexperimentinthelaboratoryto demonstratetheresultofthemodel.Inthispaper,wefOllowanddevelopKanekoetal.(2010).Using themodelinKanekoetal.(2010),thispaperinvestigatesthenon-cooperativedecisionmakingofthe disposalplaceinthethreecasesabove. TherestofthepaperiscomposedasfOllows.ThemodelisdefinedinSection2.Theresultsare summarizedinSection3.Section4showstheconcludingremarksandthefUtureproblems.2 T h e M o d e l
Inthissection,weintroducethenotationsanddefinethebasicmodelofthewastedisposalgame. Then,weapplythegametothreecases.Thatis,thepresentJapanesewaste-endtaxsystem,the discriminationwaste-endtaxsystem,andtheauction. LetJV={1,2,…,"}beasetofplayers,whichisinterpretedasthesetoflocalgovernments.We denotetypicalplayersasi,j,k,andsoon,Eachplayerisgiventheamountoftheindustrialwaste todisposewhichisdenotedbyei.Playerihasthefacilitywhichcandisposelunitofthewasteat 'Infact,mostofthelocalgovernmentinJapanadoptl,000yenperton(Kanekoetal.,2010)Agame-theoreticalanalysisoftheefficientdisposaloftheindustrialwaste Ifplayeridisposethewasteofplayerj,thegiven reHectsthedistancesamongtheplayers.Hence,we averagecostci.Playersarelocateddispersedly. transfercostdijisnecessary.Thetransfercosts assumethefOllowingproperties.
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theamountoftheindustrialwastebroughtfromianddisposedbyj.Playerd'sstrategyisshownby Figure2.Wedenote(",,"2,…;"")as". Fbrsimplicity,weusethefbllowingassumption. ej=1｡ViEN Thisassumptiondoesnotaffectthedecisionmakingoftheplayersbelow.Thisisbecauseonlythe costandtaxratefOroneunitdoesmatters.Andweregard"iastheplayerwhodisposesj'swasteif thereisnoconfUsionX
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Figure2:thestrategyofplayeri2．1Thewastedisposalgalnewithtax
Thissubsectiondefinesthewastedisposalgamewithtax.ThepresentJapanesewaste-endtaxsys-temisintroducedinsubsubsection2.1.1.ThediscriminaLiontaxsystemisconsideredinsubsubsection 2.1.2. Inthisgame,eachplayercaninducethewaste-endtaxratebefOrethedecisionofthedisposalplace. Playerj'staxrateisdenotedbyti=(til,ti2,…,ti").tijisthetaxrateplayeriinducestotheindustrial wastebroughtfromplayerj.Weassumethatt"iszerofOralliEJV.Wedenote(tl,t2),…,t7,,ast. Wecanconsiderthegameasa2stagegame. Inthesesettings,eachplayer'spayoffcanbedefinedasfbllows. De6nitionl(PayoH)."
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j ≠ i j ≠ i (1) Assumptionl(Tiebreakingrule). Ifsomestrategiesaboutthedisposalplacebringthesamepayofftoplayeri,hechoosestheplace whichhasthesmallerdisposalcost(despiteofthetransfercost)amongthem. Sincethewastedisposalgamewithtaxisde6nedasa2-stagegame,weusethesubgameperfectAgame-theoreticalanalysisoftheefficientdisposaloftheindustrialwaste Nashequilibrium(t*,"*)asasolutionconceptofthegame.Toexaminetherelationshipbetweenthe resultattheequilibriumandtheefficiency,weneedtodefinetheefficiencyconcept. Definition2(Efficiency).Theefficientdisposal〃零*isdefinedbythefOllowingequation
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＊ ＊ " = a r g m a x 釦 iEⅣ ui(t,") (2) Onthedefinitionoftheefficiency,onlythedisposalplace(x)isessential.Sincethetaxpaidby oneplayeristherewardofanotherplayer.TherefOre、thetaxisnotessentialtotheefficiency. Wecandescribethetwocasesbysettingtheconstraintonthetaxation. 2.1.1ThewastedisposalganmewiththepresentJapanesewaste-endtaxsystexn Definition3(ThepresentJapanesewaste-endtaxsystem).Thepresent-waste-endJapanesetax canbecharacterizedbythefOllowingequation. tij=tik=t'Vj,j､kEN (3) Morerestrictedtaxationcanbecharacterizedbythefbllowingequation Definition4(Uniquetaxation). t〃=tkj=tV/j,j,k,JEN (4) Indeed,inJapan,almostalllocalgovernmentsinducethesametaxratel,000yenperton 2.1.2ThewastedisposalganlewiththediscriIninationtaxsysteIn Inthiscase,eachplayerisallowedtoinducedistincttaxratetodifferentplayers.(Norestrictionis requiredonthebasicmodel.)2.2ThewastedisposalgalnewithauCtion
Inthisgame､Thedisposalplaceandtherewardpricetothedisposerisdecidedbyasecondprice auction. Withoutlossofgenerality,wefbcusonthecasethattheobjectistheindustrialwasteofplayeri. Eachplayer(includingi)bidstherewardhewantifhedisposestheobject.Basedonthesecondpriceauctionrule,thelowestbiddershouldbeawinner.Hedisposestheobjectandhegetstherewardas muchasthesecondlowestbid.
3Theresults
Inthissectionwecanobtaintheresults:ThepresentJapanesewaste-endtaxsystemmayleadthe inefficiencyattheequilibrium.Inthecasethatplayerscanusediscriminatetaxsystemortheauction、 theequilibriumalwaysrealizesthee缶ciency.3．1Thewastedisposalganlewithtax
3･1．1ThepresentJapanesewaste-endtaxsysteIn Propositionl(Kanekoetal.,2010)．加肋el"asjedispos(zI"mefiﾉ肋肋eprese"〃叩α"eseI1ﾉaste‐ en(Itm7sI/stem,,Tﾉzee9MMr"m(Ioesnota如叩s雁α"zetﾉleaβ”ent(jispos(zI. P7℃qfThefOllowingexampleshowsacounterexample Exanmplel. LetN={1,2:3},dl2=d23=(I13=3,cl=1,c2=5，c3=9.Then)intheequilibrium,the playersinducesthewaste-endtaxasfOllows. tl=5,t2=5,t3>0 (5) Hence,theplayerschoosethedisposalplaceas "1=1,"2=2,Q""3=1 (6)Inthisequilibrium,EIE,vui(t,")=3-5-8=−mHowever,if"1="2="3=1holds､then
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disposal.Intheequilibrium,playerlwhohasthelowestdisposalcostaimstogethighertaxfrom player3whohasthehighestdisposalcostevenifplayerlcannotgetanytaxfromplayer2.Thus,the wastefromplayer2isnotdisposedefficiently・Therefbre,theequilibriumfailstoachievetheefficiency. Figure.3summarizesthisexample. □Agame-theoreticalanalysisoftheefficientdisposaloftheindustrialwaste
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Figure3:example Remarkl.Inthewastedisposalgamewithuniquetax(suchastij=tik=tfbralli,j,kEJV) theefficientdisposalcannotbealwaysachieved. ProqfConsidertl=t2=t3=5inExamplel □ 3.1.2Thediscriminationtax Proposition2・加tﾉzeuﾉasteddspos(zI9qmeMt/z仇e"scr伽z伽α〃on切aste-endtaZ,tllee9M肋γ伽、， le(z(istotileaβMemdispos(zI.Pγ､oQfWithoutlossofgenerality,wefOcusonplayeri,swaste.De伽ej'=argminjENcj+dijand
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Thus,thewastefromjisdisposedatthelowestcost・Sincejisarbitrary,allthewastearedisposed atthelowestcostattheequilibrium.Therefbre，theequilibriumcanachievethee缶ciencyinthis c a s e . D Themeritofthiscaseistheefficiency.However,inthiscase,theplayers'strategychoiceiscomplex. Theyhavetodecidentypesofthetaxrate(Hence,basically:n×ntypesofthetaxrateareneeded.) anditdependontheknowledgeoftheotherplayers'disposalcostsandtransfercosts.3.2Thewastedisposalganlewithauction
Inthissubsectionweusethesecondpriceauctionandcanderivethesameresultasthediscrimination taxcase Proposition3･肋油eUﾉ"tedisposal9(zme伽"l仇e(zuctionα、Ⅳas/ze9伽肋r"mc(z7zJe(z(it/zeノ Gげ耐emdispos(zI. ProqfWithoutlossofgenerality,wefbcusonthecaseplayeri'swasteistheobject.Andwedenote playerj'sbidfOrplayeri'swasteasbjd.Bythewellknownpropertyofthesecondpriceauction,each playerhasaweeklydominantstrategytobidthetruecosttodisposetheobject.Thatis,6ji=cj+dj# fOralljEN. AsarguedbefOre,theamountoftherewarddoesnoteffectontheefficiencybecausethepayment fromiistherevenueofj・Theessentialelementontheefficiencyisthedisposeroftheobject.Inthe equilibrium,allplayersbidthetruecostandthelowestbidderdisposestheobject.Hence,theefficient d i s p o s a l i s a c h i e v e d ． □ Besidestheefficiency,themeritofthesecondpriceauctionisthesimplicityofthestrategy.Each playercanchooseweaklydominantstrategyevenifhedoesnotknowanyotherplayers'disposalcost. However,toachievetheefficient，basicallynauctionsandn×"bidsareneeded.Ontheexecution， thecomplexityofthesecondpriceauctionrulemaybematter.4ConcludingReInarks
Wecanconcludethat(unlikethepresentJapanesetax-endsystem,)thewastedisposalgamewiththe discriminationtaxandwiththeauctionleadtothee缶ciency.However,inthesecases,the(equilibrium) strategiesarecomplex.Wehadbettersuggestthemoresimplesystemwhichleadstotheefficientwaste disposal.Hence,asfUtureproblem、wewouldliketoconsidertwothings. OneistoconnectourmodeltoNIMBY(notinmybackyard)problemssuchasSakai(2012).Ifit ispossible,wecanmakeuseoftheresultsinthestudiesofNIMBYproblems.Theotheristoexecute theempiricaltestandunderstandthebehaviorsoftheplayerstokeepbalancebetweentheefficiency andthesimplicityofthesystemtosuggest. Byaccumulatingtheresearchesintheliterature,wewouldliketosuggesttheappropriateway todisposethewastesandtocoverthecostofthedisposal.ItwouldcontributetosolvetheexistingAgame-theoreticalanalysisofthee価cientdisposaloftheindustrialwaste problemssuchastheradioactivewastefromFukushimaDaiichiNuclearPowerPlantinJapan
