Optimization under Uncertainty

《Optimization under Uncertainty》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

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=pi
U(yi)i=11Thisformulationisveryusefulinitssimplicityanditsabilitytocapturesomeeconomicallyinterestingaspectsofbehaviorsuchasrisk-aversion.1Sup-pose

7、therearejusttwostateswithdistinctoutcomesy1andy2,andpositiveprobabilitiespand1prespectively.Comparethiswithanalternativewherethemathematicalexpectationofy,namelypy1+(1p)y2,isreceivedwithcertainty.Adecision-makerwhoisrisk-aversewillpreferthesuresum.Thatis,U(py1+(1p)y2)>pU(y1)+(1

8、p)U(y2),whichjustsaysthatU