第五章：效用函数

Session5效用函数SessionTopic•期望货币损益值准则的局限•效用函数的定义和公理•效用函数的构成•风险和效用的关系•损失函数、风险函数和贝叶斯风险期望货币损益值准则的局限期望货币损益值准则的局限•以期望货币损益值为标准的决策方法一般只适用于下列几种情况：（1）概率的出现具有明显的客观性值，而且比较稳定；（2）决策不是解决一次性问题，而是解决多次重复的问题；（3）决策的结果不会对决策者带来严重的后果。•如果不符合这些情况，期望货币损益值准则就不适用，需要采用其他标准。•用期望值作为决策准则的根本条件是，决策有不断反复的可能。所谓决策有不断重复的可能，包括下列三层涵义：第一，决策本身即为重复性决策。第二，重复的次数要比较多，尤其是当存在对于决策后果有重大影响的小概率事件时，只有重复次数相当多时才能用期望值来作为决策标准，因为只有这样其平均后果才接近于后果的期望值。例如，要决定是否按月投保火险，而且需要决定的不是投保一个月（投保一次），而是决定10年内（即120个月）是否投保，这就重复120次了。但因为失火损失较大，而失火概率又非常小，比如说仅万分之一，即120个月也不一定会失一次火，所以其实际平均后果就和期望值相差很大。假定投保者投保资产为12万元，而保险费规定为万分之二（保险费征收率一定比失火概率大，否则保险公司就无盈利可图了），那么每月应交保险费24元，这是在投保情况下每月的支出。如果不去投保，则损失的期望值为120000（1/10000）=12元，比投保的支出小得多。如按期望值标准，则谁也不会去投保了。可是实际上决策者还是会去投保的，这是因为实际平均损失与其期望值大不一样，如果这120个月中没有失火，则1元损失也没有，但万一失火一次，则等于每月平均损失120000/120=1000元，比计算的损失期望值大80多倍。所以计算出来的损失期望值对决定是否投保的决策者来说毫无意义，决策者往往会按“不怕一万，只怕万一”的心里去投保。因为拥有12万元资产的决策者来说，每月支出24元同其资产额相比几乎等于零，而万一失火却会遭受惨重损失。第三，每次决策后果都不会给决策者造成致命的威胁，否则，如果有此威胁，一旦真的产生此种致命后果，决策者就不可能再作下一次决策，从而也失去了重复的可能性。这就像投机者把全部资本孤注一掷一样，一旦失败，资本赔光，下一次也就无法再投机了。对于有此致命危险的重复性决策，期望值标准的采用也就受到了限制。最后，采用期望值标准时，还得假定在不断重复作出相同决策时其客观条件不变，这一方面包括了个自然状态的概率不变，另一方面亦包括决策后果函数不变。Example1St.PetersburgparadoxAprimemotivatorforBernoulli’sworkontheevaluationorriskyventureswasthefamousSt.Petersburggame.Incurrentterms,afaircoinistosseduntilaheadappears.Ifthefirstheadoccursatthenthtoss,thepayoffis2n$.Supposeyouowntitletooneplayofthegame;thatis,youcanengageinitwithoutcost.Whatistheleastamountyouwouldsellyourtitlefor?AccordingtotheBernoullis,thisleastamountisyourequivalentmonetaryvalueofthegame.Heobservedthattheexpectedpayoff(1/2)2+(1/4)22+(1/8)23+…=1+1+1+…isinfinite,butmostpeoplewouldselltitleforarelativelysmallsum,andheaskedforanexplanationofsuchaflagrantviolationofmaximumexpectedreturn.Danielshowedhowhistheoryresolvestheissuebyprovidingauniquesolutionstotheequationforanyfinitew0,wheresistheminimumsellingpriceorequivalentmonetaryvalue.Moreover,exceptfortheveryrich,apersonwouldgladlyselltitleforabout$25or$30.Theeffectofw0canbeseenindirectlybyestimatingyourminimumsellingpricewhenthepayoffatnis2ncentsinsteadof2ndollarsandcomparing100timesthisestimatetoyouranswerfromtheprecedingparagraph.UnlikeBernoulli,Cramerpayslittleattentiontoinitialwealth,andforx0setsv(x)=.Inhisterms,theminimumsellingpriceisthevalueofsthatsatisfieswhichisalittleunder$6.)(2)2(00swvwvnnnxs8)8/1(4)4/1(2)2/1(Example2AGameillustratingthe‘St.Petersburgparadox’Acasinomakesrepeatedindependenttossesofafaircoinuntilatailoccurs.Agambler,startingwithastakeof$1,isofferedthefollowingwager.Aftereachtossthegamblerwillbegiventwochoices.Hemayeithertakeawayhiswinningsfromtheprevioustossesofthecoin.Inthiscasethegamewillend.Alternativelyhemayuseallhiswinningsfromprevioustossesplushisoriginalstakemoneyasastakeforthenexttossofthecoin.Thisstakewillbetripledbythecasinoifaheadistossedonthenextthrow.Ontheotherhand,ifatailisthrownthegamblerwillloseallhiswinningsfromprevioustossesofthecointogetherwithhisoriginalstakemoney.SupposethatthegamblerisinstructedtofollowtheEMValgorithmwhenplayingthisgame.Assumerconsecutiveheadshavebeenthrownanddenotethegambler’soriginalstakeplustotalwinningsasSr.Hisexpectedpay-offforwithdrawingfromthegameisclearlySr.However,hisexpectedpay-offforcontinuingtoplayisatleast(1/2)3Sr+(1/2)0=(3/2)Sr(theexpectedpay-offforplayingoncemore).SoundertheEMValgorithmthegamblershouldcontinuetostakehiswinningsuntilatailisthrown.Butsinceatailwillbethrowneventuallywithprobabilityone,byfollowingtheEMValgorithmthegamblerensuresthathewilllosehisoriginalstakemoneywithcertainty!Clearly,inthesimplegamegivenabove,veryrationalpeoplewillnotwanttofollowthedictatesoftheEMValgorithm.ItisthereforenecessarytomodifytheEMValgorithmsothat‘optimaldecisions’canbedefinedsensiblyforsituationsliketheonegivenabove.Itwillbeshowninthenextsectionthatsuchamodificationispossibleprovidedthatyourclientispreparedtocommithimselftofollowingcertainrules(oraxioms).ItalsogeneralizestheEMVapproachtoproblemswhenclient’sobjectivesarenotonlythemaximizationofpay-off.Homework:AmedicallaboratoryhastotestNsamplesofbloodtoseewhichhavetracesofararedisease.Theprobabilityanyonepatienthasthediseaseisp,andgivenp,theprobabilityofanygroupofpatientshavingthediseaseisuninfluencedbytheexistenceorotherwiseofthediseaseinanyotherdisjointgroupofpatients.Becausepisbelievedtobesmallitissuggestedthatthelaboratorycombinethebloodofdpatientsintoequalsizedpoolsofn=N/dsampleswheredisadivisorofN.Eachpoolofsampleswouldthenbetestedtoseeifitexhibitedatraceoftheinfection.Ifnotracewerefoundthentheindividualsamplescomprisingthegroupwouldbeknowntobeuninfected.Ifontheotherhandatracewerefoundinthepooledsampleitisthenproposedthateachofthensamplescomprisingthatpoolbetestedindividually.Ifitcosts1totestanysampleofblood,whetherpooledorunpooled,findtheBayesdecisionfortheoptimalsizeofthegroupsofpatientsforagivenvalueofp.AnswerYouaregiventhespaceofdecisionsyouaretoconsideristhesetofdivisionsofN.Alltheuncertaintyintheexperimentexistsbecauseyoudonotknow(d)thenumberoftestsyourclientwillneedtodoifhechoosestopoolthesamplesintogroupsifds