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1、1ElementsofDecisionunderUncertaintyWeintroduceamodelfordecisionmakingunderuncertaintythatwillbeourworkhorsethroughoutthebook.Uncertaintyismodeledwithasetofstatesofnature,oneofwhichwilloccur.Thedecisionmakerorindividualhasaprobabilitydistributionoverthestatesofnaturethatrepresentshis(orher)
2、subjectivebeliefsaboutthelikelihoodofdifferentstatesofnature.Thisindividualchoosesactionsandactionshaveconsequences.Theconsequencefortheindividualdependsonthestateofnatureandhischoiceofaction.Thestatesofnaturearerepresentedinawaythattheprobabilitiesofstatesareunaffectedbytheindividualsacti
3、ons.Theindividualspreferencesoverconsequencesarecapturedbyautilityfunction.Theprobabilitydistributionoverstatesofnatureandtheutilityfunctionoverconsequences,bothofwhicharesubjective,1arecombinedbytheexpected-utilityruletoinduceanexpectedutilityoveractions.Anindividualmustchooseamongactsors
4、ynonymously,heorshemustmakedecisions,orselectamongactions,options,ormoves.And,wherethereisuncertainty,naturemaybesaidtochoosethestateoftheworld(orstate,forshort).Youdecidewhetherornottocarryanumbrella;naturedecidesonrainorsunshine.Table1.1picturesanespeciallysimple2×2situation.Youralternat
5、iveactsx=(1,2)areshownalongtheleftmargin,andnaturesalternativestatess=(1,2)acrossthetop.Thebodyofthetableshowstheconsequencescxsresultingfromyourchoiceofactxandnatureschoiceofstates.1Subjectiveinthesensethatanotherindividual,facedwiththesamedecisionproblem,mayhaveadifferentprobabilitydistr
6、ibutionandadifferentutilityfunction:beliefsandtastesmaydifferfrompersontoperson.78ElementsofDecisionunderUncertaintyTable1.1.ConsequencesofalternativeactsandstatesStatess=1s=2x=1ccActs1112x=2cc2122Moregenerally,theindividualunderuncertaintywill,accordingtothisanalysis,specifythefollowingel
7、ementsofhisdecisionproblem:(1)asetofacts(l,...,x,...,X)availabletohim;(2)asetofstates(l,...,s,...,S)availabletonature;(3)aconsequencefunctioncxsshowingoutcomesunderallcombinationsofactsandstates.And,inaddition:(4)aprobabilityfunctionπ(s)expressinghisbeliefs(as
