Decision theory

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DECISIONTHEORY
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DEFINITION1.1:
DECISIONTHEORY(DT)isasetofconcepts,principles,toolsand
techniquesthataidthedecisionmakerindealingwithcompl~xdecision
problemsunderuncertainty.
COMPONENTS OFADTPROBLEM:
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1.THEDECISIONMAKER
2.ALTERNATIVE COURSESOFACTION
Thisisthecontrollableaspectoftheproblem.
3.STATESOFNATUREOREVENTS
Thesearethescenariosorstatesoftheenvironmen.tnotunderthe
controlofthedecisionmaker.Theeventsdefmedshouldbe
mutuallyexclusiveandcollectivelyexhaustive.
4.CONSEQUENCES
Theconsequencesthatmustbeassessedbythedecisionmakerare
measuresofthenetbertefit,payoff,costorrevenuereceivedbythe
decisionmaker.Thereisaconsequence(orvectorof
consequences)associatedwitheachaction-eventpair.The
consequencesaresutilmarizedinadecisionmatrix. .
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DECISIONTHEORY EDGARL.DECASTRO PAGE1
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CLASSIFICATIONSOFDTPROBLEMS:
1.SingleStageDecisionProblems
Adecisionism~deonlyonce.
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2.MultipleStage/SequentialDecisionProblems
Decisionsaremadeoneafteranother.
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3.DiscreteDTProblems
Thealternativecoursesofactionsandstatesofnaturearefinite.
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4.ContinuousDTProblems
Thealternativecoursesofactionsandstatesofnatureareinfinite.
DTProblemscanalsobeclassifiedasthosewithorwithout
experimentation.Experimentationisperfoanedtoobtainadditional
informationthatwillaidthedecisionmaker.
I.DISCRETEDECISIONTHEORYPROBLEMS
DECISIONTREES
AdiscreteDTproblemcanberepresentedpictoriallyusingatree.
diagramordecisiontree.Itchronologicallydepictsthesequenceof
actionsandeventsastheyunfold.
Asquarenode(D)precedesthesetofpossibleactionsthatcan
betakenbythedecisionmaker.Aroundnode(0)precedesthesetof
eventsorstatesofnaturethatcouldbeencounteredafteradecisionis
made.Thenodesareconnectedbybranches. (""-)
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DECISIONTHEORY EDGARL.DECASTRO PAGE2
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3. '.
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EXAMPLE:
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DECISIONSUNDERRISKANDUNCERTAINTY
ConsideraDTproblemwithmalternativecoursesofactionsanda
maximumofneventsorstatesofnatureforeachalternativecourseof
action.
Defme:
Ai=alternativecourseofactioni;i =1,2,...,m
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DECISIONTHEORY EDGARL.DECASTRO PAGE3
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q)j=stateofnaturej;j=1,2,...,n
Thedecisionmatrixofpayoffsisgivenby:
q)l q)2 ...
q)11
Al v(A1,q)1)veAl,q)2) ... v(Abn)
A2 v(A2'1) v(A2'2) ... v(A2'11)
. . . . .
. . . . .
. . . . .
Am v(Am'1) V(Am'2) ... v(An"l1)

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A.LAPLACECRITERION
Thiscriterionisbasedonwhatismownastheprincipleof
insufficientreason.Her~theprobabilitiesassociatedwiththe'occurrence
oftheevent rjJjisuriknown.Wedonothave,Su(.ticie~~,leasonto
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concludethattheprobabilitiesaredifferent.Hencewe'assumethatall
eventsareequallylikely,i.e.
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P(
t/J=rjJ.)=-
} n
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Then,theoptimaldecisionruleistoselectaction '51tcorrespondingto
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B.MINIMAX(MAXIMIN)CRITERION
Thisisthemostconservativecriterionsinceitisbasedonmaking
thebestoutoftheworstpossibleconditions.Foreachpossibledecision
alternative,weselecttheworstconditionandthenselectthealternative
correspondingtothebestoftheworstconditions.
TheMINIMAXstrategyisgivenby:
min
f
max{V(Ai,rjJj}
]
A; r/J.
L.J
TheMAXIMINstrategyisgivenby:
max
[
min{v(Ai,rjJ j}
]A; r/Jj
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DECISIONTHEORY EDGARL.DECASTRO PAGE4
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5. C.SAVAGEMINIMAXREGRETCRITERION
TheMINIMAXruleisanextremelyconservativetypeofdecision
rule.ThesavageMINIMAxregretcriterionassUmesthatanew.loss
matrixisconstructedidwhichv(Ai,(Jj)isreplacedbyr(A~,..(Jj)'which
isdefinedby:
,.
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r
,.
1.
max{v(Ak,(J j)}-v(Ai,(Jj),
Ak
v(Ai,(Jj) -min{v(Ak,(Jj)},
Ak .,/
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ifvisprofit
,
ifvi~loss II
r
t.
. .
Oncethelossmatrixisconstructedusingtheabovefonnula,wecan
nowapplytheMINIMAXcriteriondefinedinb.
D.HURWICZCRITERION
Thiscriterionrepresentsarangeofattitudesfromthemost
optimistictothemostpessimistic.
Underthemostoptimisticconditions,onewouldchoosetheaction
yielding:
max
{
max{v(Ai,(J j}
}Ai t/>I
Underthemostpessimisticconditions,thechosenactioncOlTesponds
to: .
max
{
min{v(Ai,(J j}
}
A.A..
t '1')
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6. TheHurwiczcriterionstrikesabalancebetweenextremepessimismand
extremeoptimismbyweighingtheaboveconditionsbyrespective
weightsaand(1-a),where0<a<1.Thatistheactionselectedis
thatwhichyields:'\ .
ma..'X
{
am~v(Ai,r/J j)+(1-a)minv(Ai,r/J j)
}
.,
Ai t/Jj t/Jj
I
t
,
..
[Notetheaboveformulasrepresentthecasewherepayoffsareexpressed
asprofits]
Ifa=1,thedecisionruleisreferredtoastheMAXIMAX RULE,andif
a=0,thedecisionrulebecomestheMAXIMINRULE.Forthecase
wherethepayoffrepresentcosts,thedecisionruleisgivenby:
min
{
aminv(Ai, f/Jj)+(1-a)maxv(Ai,f/J j)
}~~ ~
E.BAYES'RULE
Here,weasswnethattheprobabilitiesassociatedwitheachstate
ofnatureareknown.Let ..
P{ljJ=ljJj}=Pj
Theactionwhichminimizes(maximizes)thecost(profit)isselected.
Thisisgivenby:
Thebackwardinductionapproachisused.Withtheaidofadecision
tree,expectedvaluesarecomputedeachtimearoundnodeis
encounteredandtheabovedecisionruleisutilizedeachtimeasquare
nodeisencountered,i.e.,adecisionismadeeachtimeasquarenodeis
encountered.
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F.EXPECTEDVALUE-VARIANCE CRITERION
Thisisanextensionoftheexpectedvaluecriterion.Herewe
simultaneouslymaximi~eprofitandminimizethevarianceoftheprofit.
IfZrepresentsprofitasarandomvariablewithvarianceq:,thenthe
criterionisgivenby: ..
maximizeE(z) -Kvar(z)
whereKisanyspecifiedconstant.IfZrepresentscost:
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minimizeE(z)+Kvar(z)
G.DECISIONMAKINGWITHEXPERIMENTATION
Insomesituations,itmaybeviabletosecureadditional
informationtprevisetheoriginalestimatesoftheprobabilityof
occurrenceofthestateofnature.
DEFINITION1.2:
PreposteriorAnalysisconsidersthequestionofdecidingwhetherornot
itwouldbeworthwhiletogetadditionalinformationortoperfonn
furtherexperimentation.
DEFINITION1.3:
PosteriorAnalysisdealswiththeoptimalchoiceandevaluationofan
actionsubsequenttoallexperimentationandtestingusingthe
experimentalresults.
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8. ---.......-----
DEFINITION1.4:
Priorprobabilitiesaretheinitialprobabilitiesassumedwithoutthe
benefitofexperiment~tion.Posteriorprobabilitiesrefertotherevised
probabilityvaluesobtamedafterexperimentation.
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Let:Pj=priorprobabilityestimateofevent(Jj
P{Zkl(Jj}=conditionalprobabilityofexperimental(jutcomeZk
P{(J
jIZk}=posteriorprobabilityofevent(Jj
TheexperimentalresultsareassumedtobegivenbyZk,k=1,2,...1.
Theconditionalprobabilitycanbeconsideredtobeameasureofthe
reliabilityofthe~xperiment.Theideaistocalculatetheposterior
probabilitiesbycombiningthepriorprobabilitiesandtheconditional
probabilitiesofexperimentaloutcomeZk.Theposteriorprobabilitiesare
givenby:
m
LP{Zkl(Ji}P{(Ji}
i=1
Oncetheposteriorprobabilitiesarecalculated,theoriginalproblemcan
beviewedasamultiplestage/sequentialDTproblem.Thefirststage
involvesthedecisionofwhethertoperformadditionalexperimentation
ornot.Oncethisisdecided,theoutcomesoftheexperimentare
consideredtogetherwiththeoriginalsetofdecisionalternativesand
events.
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9. DEFINITION1.5:
Aperfectinfonnationsourcewouldprovide,with100%reliability,
whichofthestatesofnaturewouldoccur. .
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Define:EPPI=expectedprofitfromaperfectinformationsource
EVPI=expectedvalueoftheperfectinfonnationsource
EP
=Bayes'expectedprofitwithoutexperimentation
Then:
.,
EVPI=EPPI-EP
where:
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EVPI=LPj*max{v(Ai,fjJ j)}
.j=1 Ai
EVPIiseasilyseenasameasureofthemaximumamountadecision
makershouldbewillingtopayforadditionalinfonnation.
Define:EVSI=expectedvalueofsampleinformation
ENOS
=expectednetgainfromsampling
CAI=costofgettingadditionalinformation
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Then:
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ENGS=EVSI-CAI
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TheinformationsourcewouldbeviableifENGS>O.
II.CONTINUOUS DECISIONTHEORY.
Aspreviouslymentioned,continuousdecisiontheoryproblems
refertothosewherethenumberofalternativesand/orstatesofnature
canbeconsideredinfmite.Theoptimizationmodelinthiscaseisgiven
by: .
maxf(A)=J:v(A,t/J)htjJ(t/J)dt/J
where:
htjJ(t/J)=priordistributionfunctionofthestatesofnature
Intheabovemodel,itisassumedthatnoadditionalinfonnationis
availableandtheexpectationisevaluatedwithrespecttotheprior
distributionofthestatesofnature.Ifadditionalinfonnationisavailable,
weupdatethepriordistributionofthestatesofnaturebydetenniningits
posteriordistribution,whichisnothingbuttheconditionaldistribution
ofthestatesofnaturegiventheexperimentaloutcome.Hence,the
optimizationconvertsto:
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maxf(A) =f:v(A,f/J)ht/>IZ=z(f/J)df/J
where:
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hfj)IZ=z(rjJ) =conditionaldistributionofthestateofnature'giventhe
experimentaloutcome
hZIfj)(z)=conditionaldistributionoftheexperimentaloutcomegiventhe
stateofnature
hz(z)=marginaldistributionfunctionoftheexperimentaloutcomes
where:
LEIBNIZ'RULE
LEIBNIZ'Ruleisappliedtofindthederivative'ofafunction
whichcontainsintegrals.ConsiderafunctioninonevariableA:
d
f
b
f
big db da
-g(A,rjJ)drjJ=~rjJ+g(A,b)--g(A,a)-
ciAa a8A ciA ciA
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