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AxiomaticSystemsoftheExpectedUtilityTheoryTheExpectedutilitytheorywasproposedbyBernoulli(1738),Ramsey(1931),vonNeumann-Morgenstern(1947)andfullydevelopedbySavage(1954).inwhichchoiceoversubjectivelyuncertainprospectsischaracterizedbyexpectedutilityriskpreferencesandstandardprobabilisticbeliefs.ThedifferenceoftheseearlierstudiesandRamseyisthemannerwhichrepresentsuncertainty.Intheearlierstudies,theobjectiveframework,uncertaintycomesprepackagedintermsofnumericalprobabilities.Ontheotherhand,RamseytreatsthechoiceofprobabilitydistributionsoveroutcomesorlotteriesAsimilarandrelativelysimpleformulation,thatprovidesinsightsintothemostimportantassumptionsofexpectedutilityforstatisticalpurposes,wasproposedbyAnscombeandAumann(1963).Thefollowingaxiomsofthepreferencerelationonlotteries（≽𝐒;𝑓,𝑔）,(1)Completeness（𝒇≽𝐒𝒈or𝒈≽𝐒𝒇）,(2)Transitivity(𝒇≽𝐒𝒈𝒂𝒏𝒅𝒈≽𝐒𝒉⟹𝒇≽𝐒𝒉),(3)Relevance(𝒇(∙)=𝒈(∙)⟹𝒇∼𝐒𝒈),(4)Monotonicity,(𝒇≽𝐒𝒈𝒂𝒏𝒅𝟏≥𝜶𝛽𝟎⟹𝜶𝒇+(𝟏−𝜶)𝒈≽𝐒𝜷𝒇+(𝟏−𝜷)𝒈)(5)Continuity,(6)Sure-thingprinciple(Substitutionaxiom)aresatisfiedifandonlyifthefollowingrepresentationofaconditionalprobabilityfunctions𝑝𝑡𝑆andautilityfunction𝑢𝑥,𝑡exists:(A)Normalizationmaxx𝑢𝑥,𝑡=1,andminx𝑢𝑥,𝑡=0(B)Bayesformula𝑝𝑅𝑇=𝑝𝑅𝑆𝑝𝑆𝑇,𝑓𝑜𝑟𝑅⊆𝑆⊆𝑇⊆𝛺(C)Expectedutilityformula𝑓≽S𝑔⟺𝑝(𝑡|𝑆)𝑡∈𝑆𝑢𝑥𝑡𝑓(𝑥|𝑡)≥𝑝(𝑡|𝑆)𝑡∈𝑆𝑢𝑥𝑡𝑔𝑥𝑡.𝑥∈𝑋𝑥∈𝑋TheExpectedUtilityMaximizationTheorem(Savage,1954)SURE-THINGPRINCIPLE(Savage):ForalleventsEandacts(assignmentofanoutcometoeachstateofnature)f(･),f*(･),g(･),andh(･),𝑓∗(𝑠)𝑖𝑓𝑠∈𝐸𝑔(𝑠)𝑖𝑓𝑠∉𝐸≽𝑓(𝑠)𝑖𝑓𝑠∈𝐸𝑔(𝑠)𝑖𝑓𝑠∉𝐸⟹𝑓∗(𝑠)𝑖𝑓𝑠∈𝐸ℎ(𝑠)𝑖𝑓𝑠∉𝐸≽𝑓(𝑠)𝑖𝑓𝑠∈𝐸ℎ(𝑠)𝑖𝑓𝑠∉𝐸,thereexistsaunique,finitelyadditivenon-atomicprobabilitymeasure,andastate-independentutilityfunctionsuchasV(𝑓･)≡𝑈𝑓𝑠𝑑𝜇𝑠≡𝑈𝑥𝑖𝜇𝑓−1𝑥𝑖𝑖,thatis,theexpectedutilitypresentation.Sure-ThingPrincipleisalsocalledthe‘strongindependenceassumption’.Itstatesthatifoutcomexandoutcomex′areindifferentinthemselves,thenforanyoutcomey,aprobabilitymixofxandymustbeindifferenttoaprobabilitymixofx′andy.DenyingtheprincipleisapossibleresponsetotheAllaisparadox.Sure-ThingPrincipleworLotteryLotterywzxoryTAILSHEADSPossibleDecisions:(1)Takew.([w])(2)Refusew,andtakexiftails.(.5[x]+.5[z])(3)Refusew,andtakeyiftails.(.5[y]+.5[z])supposeanindividualwouldpreferxovery,buthewouldalsoprefer.5[y]+.5[z]over.5[x]+.5[z],inviolationofsubstitution.Supposethatwissomeotherprizethathewouldconsiderbetterthan.5[x]+.5[z]andworsethan.5[y]+.5[z].Thatis,xybut.5[y]+.5[z][w].5[x]+.5[z].Therefore,thethirdstrategywouldbebest.However,ifshe/hetakesthethirdstrategy,andthecoincomesupTails,thenshe/hewouldchoosex!Thatisactuallyendupwiththesecondstrategy,whichisworst.RogerB.Myerson,GameTheory:AnalysisofConflict,1997.WhyweneedSure-ThingPrinciple?ABriefProofoftheTheoremSupposespeciallotteries;a:alwaysgivesthebestprizeforeverystate.z:alwaysgivestheworstprizeforeverystate.bt:givesaifstate=t,andgiveszotherwise.Defineβas[x]≈{t}βa+(1−β)zDefineγasbt≈Sγa+(1−γ)zWecanshowthatβisu(x|t),andγisp(t|S),satisfyingtheaxiom.NOTE≈{t}means“indifferentifadecisionmakerknewthatstatetoccurs,”and≈Smeans“indifferentifadecisionmakerknewthattruestateisinS.”[x]:alotterythatwillgivestheprizexwithprobability1.βa+(1−β)z:atwostagelottery.Supposethreelotteries,bR，bB，bYa，andz,whoseprizesgivenasthefollowingtableforeachstatesR,B,Y,say,RrepresentsastateinwhichRballcomesuprandomly,andBforblackball,YforYellowball.RBYa1054z600bR1000bB650bY604AnHeuristicExampleTheexpectedutilityforeachprizesaredefinedas,．RBYaVaRVaBVaYzVzRVzBVzYbRVaRVzBVzYbBVzRVaBVzYbYVzRVzBVaYIftheexpectedutilitytheoryapplies,Va=VaR+VaB+VaY，Vz=VzR+VzB+VzYVbR=VaR+VzB+VzY，VbB=VzR+VaB+VzY，VbY=VzR+VzB+VaYExpectedUtilitiesThinkaboutatwostagelotterywhoseprizesarethelotteryandthelotteryz,withprobabilityγR,and(1−γR),.NowaskasubjectwhatnumberofγRmakesthistwostagelotteryindifferenttothalotterybR.Thatis,γRsatisfying，VbR=γRVa+(1−γR)VzIntheexpectedutilitytheory,[(VaR−VzR)+(VzB−VzB)+(VzY−VzY)]=γR[(VaR−VzR)+(VaB−VzB)+(VaY−VzY)].Therefore,γR=VaR−VzR(VaR−VzR)+(VaB−VzB)+(VaY−VzY)=VaR−VzRVa−VzInthesameway,wegetVbB=γBVa+(1−γB)Vz，andVbY=γYVa+(1−γY)Vz，thereforeγB=VaB−VzBVa−VzγY=VaY−VzYVa−VzNoteherethatγR+γB+γY=1．SubjectiveProbabilityContingent(state-dependent)UtilityInthesameway,undertheconditionthatBwilloccuressurely,theutilityleveloftheprizeof5is1,andoneoftheprizeof0is0.;undertheconditionthatYwilloccuresurely,theutilityleveloftheprize4is1,andoneoftheprizeof0is0.Thisis“contingentutility!”UndertheconditionthatRwilloccurssurely,ifyouaskaexpectedutilitymaximizerwhatisβRtosatisfy,VbR=βRVa+(1R－βR)Vz,theanswershouldbeβR=1．Thismeans,undertheconditionthatRwilloccurssurely,theutilityleveloftheprize10is1,andoneoftheprize0is0.Therefore,alltheutilitylevelsofthelotteryaare1forexpectedutilitymaximizers,andweget,Va=VaR+VaB+VaY=γR×1＋γB×1＋γY×1．AndVz=0．Thisassuresthatγtcanbeinterpretedasthesubjectiveprobabilityforeachstates.Questions:(1)Howcanweknowwhoisanexpectedutilitymaximizer?(2)Howisthebehaviorofnon-expectedutilitymaximizerdifferentfromoneoftheexpectedutil