Chapter4

DecisionAnalysis(2010)Decisionsoftenmustbemadeinenvironmentsthataremuchmorefraughtwithuncertainty.Example:TheGOFERBROKECOMPANYownsatractoflandthatmaycontainoil.Aconsultinggeologisthasreportedtomanagementthatshebelievesthereis1chancein4ofoil.Becauseofthisprospect,anotheroilcompanyhasofferedtopurchasethelandfor$90,000.However,Goferbrokeisconsideringholdingthelandinordertodrillforoilitself.Thecostofdrillingis$100,000.Ifoilisfound,theresultingexpectedrevenuewillbe$800,000,sothecompany'sexpectedprofit(afterdeductingthecostofdrilling)willbe$700,000.Alossof$100,000(thedrillingcost)willbeincurredifthelandisdry(nooil).Howtoapproachthedecisionofwhethertodrillorsellbasedjustonthesedata.(WewillrefertothisasthefirstGoferbrokeCblem.)However,beforedecidingwhethertodrillorsell,anotheroptionistoconductadetailedseismicsurveyofthelandtoobtainabetterestimateoftheprobabilityoffindingoil.1.Amanufacturerintroducinganewproductintothemarketplace.Whatwillbethereactionofpotentialcustomers?Howmuchshouldbeproduced?Shouldtheproductbetestmarketedinasmallregionbeforedecidinguponfulldistribution?Howmuchadvertisingisneededtolaunchtheproductsuccessfully?2.Afinancialfirminvestinginsecurities.Whicharethemarketsectorsandindividualsecuritieswiththebestprospects?Whereistheeconomyheaded?Howaboutinterestrates?Howshouldthesefactorsaffecttheinvestmentdecisions?3.Agovernmentcontractorbiddingonanewcontract.Whatwillbetheactualcostsoftheproject?Whichothercompaniesmightbebidding?Whataretheirlikelybids?5.Anoilcompanydecidingwhethertodrillforoilinaparticularlocation.Howlikelyisoilthere?Howmuch?Howdeepwilltheyneedtodrill?Shouldgeologistsinvestigatethesitefurtherbeforedrilling?

Thesearethekindsofdecisionmakinginthefaceofgreatuncertaintythatdecisionanalysisisdesignedtoaddress.Decisionanalysisprovidesaframeworkandmethodologyforrationaldecisionmakingwhentheoutcomesareuncertain.

Frequently,onequestiontobeaddressedwithdecisionanalysisiswhethertomaketheneededdecisionimmediatelyortofirstdosometesting(atsomeexpense)toreducethelevelofuncertaintyabouttheoutcomeofthedecision.Thefirstsectionintroducesaprototypeexamplethatwillbecarriedthroughoutthechapterforillustrativepurposes.Sections4.2and4.3thenpresentthebasicprinciplesofdecisionmakingwithoutexperimentationanddecisionmakingwithexperimentation.Wenextdescribedecisiontrees,ausefultoolfordepictingandanalyzingthedecisionprocesswhenaseriesofdecisionsneedstobemade.4.1APROTOTYPEEXAMPLETheGOFERBROKECOMPANYownsatractoflandthatmaycontainoil.Aconsultinggeologisthasreportedtomanagementthatshebelievesthereis1chancein4ofoil.Becauseofthisprospect,anotheroilcompanyhasofferedtopurchasethelandfor$90,000.However,Goferbrokeisconsideringholdingthelandinordertodrillforoilitself.Thecostofdrillingis$100,000.Ifoilisfound,theresultingexpectedrevenuewillbe$800,000,sothecompany'sexpectedprofit(afterdeductingthecostofdrilling)willbe$700,000.Alossof$100,000(thedrillingcost)willbeincurredifthelandisdry(nooil).Table4.1summarizesthesedata.Section4.2discusseshowtoapproachthedecisionofwhethertodrillorsellbasedjustonthesedata.(WewillrefertothisasthefirstGoferbrokeCblem.)However,beforedecidingwhethertodrillorsell,anotheroptionistoconductadetailedseismicsurveyofthelandtoobtainabetterestimateoftheprobabilityoffindingoil.Section4.3discussesthiscaseofdecisionmakingwithexperimentation,atwhichpointthenecessaryadditionaldatawillbeprovided.Foreachcombinationofanactionandastateofnature,thedecisionmakerknowswhattheresultingpayoffwouldbe.Thepayoffisaquantitativemeasureofthevaluetothedecisionmakeroftheconsequencesoftheoutcome.Iftheconsequencesoftheoutcomedonotbecomecompletelycertainevenwhenthestateofnatureisgiven,thenthepayoffbecomesanexpectedvalue(inthestatisticalsense)ofthemeasureoftheconsequences.Apayofftablecommonlyisusedtoprovidethepayoffforeachcombinationofanactionandastateofnature.Thedecisionanalysisframeworkcanbesummarizedasfollows:

1.Thedecisionmakerneedstochooseoneofthealternativeactions.2.Naturethenwouldchooseoneofthepossiblestatesofnature.3.Eachcombinationofanactionandstateofnaturewouldresultinapayoff;whichisgivenasoneoftheentriesinapayofftable.4.Thispayofftableshouldbeusedtofindanoptimalactionforthedecisionmakeraccordingtoanappropriatecriterion.Oneadditionalelementneedstobeaddedinthedecisionanalysisframework.Thedecisionmakergenerallywillhavesomeinformationthatshouldbetakenintoaccountabouttherelativelikelihoodofthepossiblestatesofnature.Suchinformationcanusuallybetranslatedtoaprobabilitydistribution,actingasthoughthestateofnatureisarandomvariable,inwhichcasethisdistributionisreferredtoasapriordistribution.

Priordistributionsareoftensubjectiveinthattheymaydependupontheexperienceorintuitionofanindividual.Theprobabilitiesfortherespectivestatesofnatureprovidedbythepriordistributionarecalledpriorprobabilities.4.2.2CharacteristicsofDecisionAnalysis1.DeterministicDecisionAnalysis2.IndeterminableDecisionAnalysis3.ProbabilityDecisionAnalysis(1)Goal(2)Actions(3)CertainNature(4)PayoffGoal(2)Actions(3)Uncertain

Nature(4)Noknowledgeaboutprobabilityofnature(5)Payoff(1)Goal(2)Actions(3)Nature(4)Payoff(5)Probability4.2.3FormulationofthePrototypeExampleinThisFramework

AsindicatedinTable4.1,theGoferbrokeCo.hastwopossibleactionsunderconsideration:drillforoilorselltheland.

Thepossiblestatesofnaturearethatthelandcontainsoilandthatitdoesnot,asdesignatedinthecolumnheadingsofTable4.1byoilanddry.Sincetheconsultinggeologisthasestimatedthatthereis1chancein4ofoil(andso3chancesin4ofnooil),thepriorprobabilitiesofthetwostatesofnatureare0.25and0.75,respectively.Therefore,withthepayoffinunitsofthousandsofdollarsofprofit,thepayofftablecanbeobtaineddirectlyfromTable4.1,asshowninTable4.2.

StateofNatureAlternativeOilDry1.DrillforOil700-1002.SelltheLand9090ChanceofStatus0.250.75(1)TheMaxiMaxPayoffCriterionMaximaxpayoffcriterion:Foreachpossibleaction,findthemaximumpayoffoverallpossiblestatesofnature.Next,findthemaximumofthesemaximumpayoffs.Choosetheactionwhosemaximumpayoffgivesthismaximum.

StateofNatureMaximaxpayoffMaximuxSavageAlternativeOilDry1.DrillforOil700-1007002.SelltheLand909090700Action:DrillforOilTherationaleforthiscriterionisthatitprovidesthebestguaranteeofthepayoffthatwillbeobtained.Regardlessofwhatthetruestateofnatureturnsouttobefortheexample,thepayofffromsellingthelandcannotbelessthan90,whichprovidesthebestavailableguarantee.Thus,thiscriteriontakesthepessimisticviewpointthat,regardlessofwhichactionisselected,theworststateofnatureforthatactionislikelytooccur,soweshouldchoosetheactionwhichprovidesthebestpayoffwithitsworststateofnature.Thisrationaleisquitevalidwhenoneiscompetingagainstarationalandmalevolentopponent.However,thiscriterionisnotoftenusedingamesagainstnaturebecauseitisanextremelyconservative（保守的）criterioninthiscontext.Ineffect,itassumesthatnatureisaconsciousopponentthatwantstoinflictasmuchdamageaspossibleonthedecisionmaker.Natureisnotamalevolentopponent,andthedecisionmakerdoesnotneedtofocussolelyontheworstpossiblepayofffromeachaction.Thisisespeciallytruewhentheworstpossiblepayofffromanactioncomesfromarelativelyunlikelystateofnature.

Thus,thiscriterionnormallyisofinterestonlytoaverycautiousdecisionmaker.(3)TheSavageruleCriterion（后悔值法）Criterion:Foreachpossiblenaturestate,findthemaximumpayoffoverallpossiblealternativeactives.Next,findthedifferenceofeachactivepayoffcomparingwiththemaximumpayoff.Choosethemaximumdifferenceofeveryactive,afterthatselecttheactionwhosedifferencegivestheminimum.

StateofNatureSavageRuleAlternativeOilDryMax1.DrillforOil700(0)-100(190)1902.SelltheLand90(610)90(0)610Minimum190Action:DrillforOil

StateofNatureAlternativeOilDry1.DrillforOil700-1002.SelltheLand9090Probability0.50.5(4)TheEquivalentProbabilityCriterionSolutionofProbabilityanalysis:

TheMaximumLikelihoodCriterionMaximumlikelihoodcriterion:Identifythemostlikelystateofnature(theonewiththelargestpriorprobability).Forthisstateofnature,findtheactionwiththemaximumpayoff.Choosethisaction.Theappealofthiscriterionisthatthemostimportantstateofnatureisthemostlikelyone,sotheactionchosenisthebestoneforthisparticularlyimportantstateofnature.Basingthedecisionontheassumptionthatthisstateofnaturewilloccurtendstogiveabetterchanceofafavorableoutcomethanassuminganyotherstateofnature.Furthermore,thecriteriondoesnotrelyonquestionablesubjectiveestimatesoftheprobabilitiesoftherespectivestatesofnatureotherthanidentifyingthemostlikelystate.Themajordrawbackofthecriterionisthatitcompletelyignoresmuchrelevantinformation.Nostateofnatureisconsideredotherthanthemostlikelyone.Inaproblemwithmanypossiblestatesofnature,theprobabilityofthemostlikelyonemaybequitesmall,sofocusingonjustthisonestateofnatureisquiteunwarranted.Ineffect,thecriteriondoesnotpermitgamblingonalow-probabilitybigpayoff,nomatterhowattractivethegamblemaybe.

StateofNatureMaximinpayoffMaximuxlikelihoodAlternativeOilDry1.DrillforOil700-100-1002.SelltheLand909090900.250.75900.75Action:SelltheLandSolutionofProbabilityanalysis:Bayes'DecisionRulerBayes'decisionrule:Usingthebestavailableestimatesoftheprobabilitiesoftherespectivestatesofnature(currentlythepriorprobabilities),calculatetheexpectedvalueofthepayoffforeachofthepossibleactions.Choosetheactionwiththemaximumexpectedpayoff.Fortheprototypeexample,theseexpectedpayoffsarecalculatedasfollows:

E[Payoff(drill)]=0.25(700)+0.75(-100)=100.E[Payoff(sell)]=0.25(90)+0.75(90)=90.Since100islargerthan90,thealterativeactionselectedistodrillforoil.ThebigadvantageofBayes'decisionruleisthatitincorporatesalltheavailableinformation,includingallthepayoffsandthebestavailableestimatesoftheprobabilitiesoftherespectivestatesofnature.Itissometimesarguedthattheseestimatesoftheprobabilitiesnecessarilyarelargelysubjectiveandsoaretooshakytobetrusted.Thereisnoaccuratewayofpredictingthefuture,includingafuturestateofnature,eveninprobabilityterms.Thisargumenthassomevalidity.Thereasonablenessoftheestimatesoftheprobabilitiesshouldbeassessedineachindividualsituation.Nevertheless,undermanycircumstances,pastexperienceandcurrentevidenceenableonetodevelopreasonableestimatesoftheprobabilities.Usingthisinformationshouldprovidebettergroundsforasounddecisionthanignoringit.Furthermore,experimentationfrequentlycanbeconductedtoimprovetheseestimates,asdescribedinthenextsection.Therefore,wewillbeusingonlyBayes'decisionrulethroughouttheremainderofthechapter.4.3DECISIONMAKINGWITHEXPERIMENTATIONFrequently,additionaltesting(experimentation)canbedonetoimprovethepreliminaryestimatesoftheprobabilitiesoftherespectivestatesofnatureprovidedbythepriorprobabilities.Theseimprovedestimatesarecalledposteriorprobabilities.WefirstupdatetheGoferbrokeCo.exampletoincorporateexperimentation,thendescribehowtoderivetheposteriorprobabilities,andfinallydiscusshowtodecidewhetherifisworthwhiletoconductexperimentation.4.3.1ContinuingthePrototypeExampleAsmentionedattheendofSec.4.1,anavailableoptionbeforemakingadecisionistoconductadetailedseismicsurveyofthelandtoobtainabetterestimateoftheprobabilityofoil.Thecostis$30,000.Aseismicsurveyobtainsseismicsoundingsthatindicatewhetherthegeologicalstructureisfavorabletothepresenceofoil.Wewilldividethepossiblefindingsofthesurveyintothefollowingtwocategories:USS:Unfavorableseismicsoundings;oilisfairlyunlikely.FSS:Favorableseismicsoundings;oilisfairlylikely.Basedonpastexperience,ifthereisoil,thentheprobabilityofunfavorableseismicsoundingsis

P(USS|State=Oil)=0.4,soP(FSS|State=Oil)=1-0.4=0.6.Similarly,ifthereisnooil,thentheprobabilityofunfavorableseismicsoundingsisestimatedtobe

P(USS|State=Dry)=0.8,soP(FSS|State=Dry)=1-0.8=0.2.Wesoonwillusethesedatatofindtheposteriorprobabilitiesoftherespectivestatesofnaturegiventheseismicsoundings.PriorProbabilitiesConditionalProbabilitiesJointProbabilitiesPosteriorProbabilities

Fig4.6probabilitytreediagramshowingalltheprobabilitiesleadingtothecalculationofeachposteriorprobabilityofthestateofnaturegiventhefindingoftheseismicsurvey4.4DECISIONTREESDecisiontreesprovideausefulwayofvisuallydisplayingtheproblemandthenorganizingthecomputationalworkalreadydescribedintheprecedingtwosections.Thesetreesareespeciallyhelpfulwhenasequenceofdecisionsmustbemade.Decisiontreesmodelsequentialdecisionproblemsunderuncertainty.Adecisiontreedescribesgraphicallythedecisionstobemade,theeventsthatmayoccur,andtheoutcomesassociatedwithcombinationsofdecisionsandevents.Probabilitiesareassignedtotheevents,andvaluesaredeterminedforeachoutcome.Amajorgoaloftheanalysisistodeterminethebestdecisions.Decisiontreeshavethreekindsofnodesandtwokindsofbranches.NodesandBranches

Adecisionnodeisapointwhereachoicemustbemade;itisshownasasquare.Thebranchesextendingfromadecisionnodearedecisionbranches,eachbranchrepresentingoneofthepossiblealternativesorcoursesofactionavailableatthatpoint.Thesetofalternativesmustbemutuallyexclusive(ifoneischosen,theotherscannotbechosen)andcollectivelyexhaustive(allpossiblealternativesmustbeincludedintheset).Aneventnodeisapointwhereuncertaintyisresolved(apointwherethedecisionmakerlearnsabouttheoccurrenceofanevent).Aneventnode,sometimescalleda"chancenode,"isshownasacircle.Theeventsetconsistsoftheeventbranchesextendingfromaneventnode,eachbranchrepresentingoneofthepossibleeventsthatmayoccuratthatpoint.Thesetofeventsmustbemutuallyexclusive(ifoneoccurs,theotherscannotoccur)andcollectivelyexhaustive(allpossibleeventsmustbeincludedintheset).Eacheventisassignedasubjectiveprobability;thesumofprobabilitiesfortheeventsinasetmustequalone.Thethirdkindofnodeisa

terminalnode,representingthefinalresultofacombinationofdecisionsandevents.Terminalnodesaretheendpointsofadecisiontree,shownastheendofabranchonhand-drawndiagramsandasatriangleorverticallineoncomputer-generateddiagrams.Ingeneral,Decisionnodesandbranchesrepresentthecontrollablefactorsinadecisionproblem;Eventnodesandbranchesrepresentuncontrollablefactors.Decisionnodesandeventnodesarearrangedinorderofsubjectivechronology.Forexample,thepositionofaneventnodecorrespondstothetimewhenthedecisionmakerlearnstheoutcomeoftheevent(notnecessarilywhentheeventoccurs).OtherTermsDecisiontreemodelsincludesuchconceptsasnodes,branches,terminalvalues,strategy,payoff

distribution,probabilitydistribution,andtherollbackmethod.Thefollowingproblemillustratesthebasicconcepts.Nodes:(1)DecisionNode(2)EventNode(3)TerminalNodeBranch:(1)DecisionBranch(2)EventBranch(Probability)Profit(Payoff/Outcome)Typeof

NodeWritten

SymbolComputer

SymbolNode

SuccessorDecisionsquaresquaredecisionbranchesEventcirclecircleeventbranchesTerminalendpointtriangleorverticallineterminalvalueSometimes,thenodesofthedecisiontreearereferredtoasforks,andthearcsarecalledbranches.Adecisionfork,representedbyasquare,indicatesthatadecisionneedstobemadeatthatpointintheprocess.Achancefork,representedbyacircle,indicatesthatarandomeventoccursatthatpoint.Theprototypeexampleinvolvesasequenceoftwodecisions:1.Shouldaseismicsurveybeconductedbeforeanactionischosen?2.Whichaction(drillforoilorselltheland)shouldbechosen?Thecorrespondingdecisiontree(beforeaddingnumbersandperformingcomputation)isdisplayedinFig..1ConstructingtheDecisionTree

DoSeismicNoSeismicUnfavorableFavorableDrillSellDrillSellSellDrillOilDryOilOilDryDry4.4.2PerformingtheAnalysis

Havingconstructedthedecisiontree,includingitsnumbers,wenowarereadytoanalyzetheproblembyusingthefollowingprocedure.1,Startattherightsideofthedecisiontreeandmoveleftonecolumnatatime.Foreachcolumn,performeitherstep2orstep3dependinguponwhethertheforksinthatcolumnarechanceforksordecisionforks.2.Foreachchancefork,calculateitsexpectedpayoffbymultiplyingtheexpectedpayoffofeachbranch(showninboldfacetotherightofthebranch)bytheprobabilityofthatbranchandthensummingtheseproducts.Recordthisexpectedpayoffforeachdecisionforkinboldfacenexttothefork,anddesignatethisquantityasalsobeingtheexpectedpayoffforthebranchleadingtothisfork.3.Foreachdecisionfork,comparetheexpectedpayoffsofitsbranchesandchoosethealternativewhosebranchhasthelargestexpectedpayoff.Ineachcase,recordthechoiceonthedecisiontreebyinsertingadoubledashasabarrierthrougheachrejectedbranch.Tobegintheprocedure,considertherightmostcolumnofforks,namely,chanceforksf,g,andh.Applyingstep2,theirexpectedpayoffs(EP)arecalculatedasEP=1/7(670)+6/7(130)=-15.7,forfortf,EP=1/2(670)+1/2(130)=270,forforkg,EP=1/4(700)+3/4(-100)=100,forforkh.

Theseexpectedpayoffsthenareplacedabovetheseforks,asshowninFig.4.11.Next,wemoveonecolumntotheleft,whichconsistsofdecisionforksc,d,ande.Theexpectedpayoffforabranchthatleadstoachanceforknowisrecordedinboldfaceoverthatchancefork.Therefore,step3canbeappliedasfollows.Forkc:DrillalternativehasEP=-15.7.SellalternativehasEP=60.60>-15.7,sochoosetheSellalternative.Forkd:DrillalternativehasEP=270.SellalternativehasEP=60.270>60,sochoosetheDrillalternative.Forke:DrillalternativehasEP=100.SellalternativehasEP=90.100>90,sochoosetheDrillalternative.Theexpectedpayoffforeachchosenalternativenowwouldberecordedinboldfaceoveritsdecisionnode,asalreadyshowninFig.4.11.Thechosenalternativealsoisindicatedbyinsertingadoubledashasabarrierthrougheachrejectedbranch.Next,movingonemorecolumntotheleftbringsustoforkb.Sincethisisachancefork,step2oftheprocedureneedstobeapplied.Theexpectedpayoffforeachofitsbranchesisrecordedoverthefollowingdecisionfork.Therefore,theexpectedpayoffisEP=0.7(60)+0.3(270)123,forforkb,asrecordedoverthisforkinFig.15.11.Finally,wemovelefttoforka,adecisionfork.Applyingstep3yieldsForka:DoseismicsurveyhasEP=123NoseismicsurveyhasEP=100123>100,sochooseDoseismicsurvey.4.5CaseStudy1.DriveTekProblemDriveTekResearchInstitutediscoversthatacomputercompanywantsanewtapedriveforaproposednewcomputersystem.Sincethecomputercompanydoesnothaveresearchpeopleavailabletodevelopthenewdrive,itwillsubcontractthedevelopmenttoanindependentresearchfirm.Thecomputercompanyhasofferedafeeof$250,000forthebestproposalfordevelopingthenewtapedrive.Thecontractwillgotothefirmwiththebesttechnicalplanandthehighestreputationfortechnicalcompetence.DriveTekResearchInstitutewantstoenterthecompetition.Managementestimatesacostof$50,000toprepareaproposalwithafifty-fiftychanceofwinningthecontract.However,DriveTek'sengineersareuncertainabouthowtheywilldevelopthetapedriveiftheyareawardedthecontract.Threealternativeapproachescanbetried.Thefirstapproachisamechanicalmethodwithacostof$120,000,andtheengineersarecertaintheycandevelopasuccessfulmodelwiththisapproach.Asecondapproachinvolveselectroniccomponents.Theengineersestimatethattheelectronicapproachwillcostonly$50,000todevelopamodelofthetapedrive,butwithonlya50percentchanceofsatisfactoryresults.Athirdapproachusesmagneticcomponents;thiscosts$80,000,witha70percentchanceofsuccess.DriveTekResearchcanworkononlyoneapproachatatimeandhastimetotryonlytwoapproaches.Ifittrieseitherthemagneticorelectronicmethodandtheattemptfails,thesecondchoicemustbethemechanicalmethodtoguaranteeasuccessfulmodel.Whatwilltheydo?ThemanagementofDriveTekResearchneedshelpinincorporatingthisinformationintoadecisiontoproceedornot.CASE1:ACompanywantstoupgradeoneoftheirproducts.Therearetwoapproachesfortheactions.Oneofthealternativeactionsisbuyingapatentfromthethirdpartwithan80percentchanceofsatisfactoryresults;theotherapproachisdevelopingtheresearchthemselveswitha60percentchanceofsuccessful