Optimization under Uncertainty

Optimization under Uncertainty

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時間:2019-07-16

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1、Chapter7:OptimizationunderUncertaintySHI,HaoAntaiCollege,ShanghaiJiaoTongUniversityOctober30,20101ExpectedUtilityOftentimes,agentsinourrealeconomymakedecisionsinuncertainenviron-ments.Thischapterdiscussestheoptimizationproblemsunderuncertainty.Insuchproblems,(1)thechoiceorcontrolva

2、riablesleadtorandomoutcomeswithprobabilitydistributions;(2)theobjectivefunctionareappropriateprobability-weightedaverages(expectedvalues);(3)thechoicesarealsosubjecttosomeconstraints.Thegeneraltheorydevelopedinthe…rstsevenchapterscontinuestoapply.Nonewmathematicaltoolsarerequiredfo

3、rthetheoryofoptimizationunderuncertainty.Letusmaketheoptimizationproblemsunderuncertaintyalittlemorepre-cise.Insuchproblems,afterthedecisionathandhasbeentaken,theworldcouldevolveinanyofanumberofdi¤erentways.Thesearecalleddi¤erentstatesoftheworld.First,supposethestatesarediscreteand

4、…niteinnum-ber,indexedbyi=1;2;:::;m.IfeachstatecouldhappenwithaprobabilityXmpi,thenpi=1.Ineconomics,thestatesoftheworldaretypicallyde…nedi=1asthelevelsofincome,wealth,orpro…taccruingtothedecisionmaker,andaredenotesasyi,whichcouldbeascalaroravector.Thechoiceorcontrolvariablesxwilla¤

5、ectsomeoralloftheyiandthepi,andmayalsobesubjecttosomeadditionalconstraints.Wecanwriteageneralobjectivefunctionunderuncertaintyasf(y1;y2;:::;ym;p1;p2;:::;pm).Butoftentimes,weputsomerestrictionontheobjectivefunctionunderuncer-taintybyassumingthatthedecision-makershavevonNeumann-Morge

6、nsternutilityfunctions(ofincome,wealth,orpro…tsasthecasemaybe).Thatis,theobjectivefunctionstypicallytaketheformofexpectedutility:XmU=piU(yi)i=11Thisformulationisveryusefulinitssimplicityanditsabilitytocapturesomeeconomicallyinterestingaspectsofbehaviorsuchasrisk-aversion.1Sup-pose

7、therearejusttwostateswithdistinctoutcomesy1andy2,andpositiveprobabilitiespand1