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T0M B$01! -T0M B$01! - The Ma/ima/ CriterionThe Ma/ima/ Criterion
The Ma/ima/ Criterion Ma/imumDecision "ar#e rise mall rise !o chan#e mall fall "ar#e fall Payoff&old /$%% $%% 0%% 1%% % 1%%Bond 02% 0%% $2% /$%% /$2% 0%%
toc' 2%% 02% $%% /0%% /3%% 2%%C(D 3% 3% 3% 3% 3% 3%
T h e o p t i m a l d e c i s i o n
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This criterion might appeal to a decision maker who
is neither pessimistic nor optimistic.: 't assumes all the states of nature are e,ually likely tooccur.
: The procedure to find an optimal decision.
?or each decision add all the payoffs. &elect the decision with the largest sum Afor profitsB.
Decision Ma'in# .nder .ncertainty -Decision Ma'in# .nder .ncertainty -
The Principle of Insufficient $easonThe Principle of Insufficient $eason
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T0M B$01!T0M B$01! // Insufficient $easonInsufficient $eason
&um of !ayoffs: +old 3%% Dollars
: "ond 12% Dollars: &tock 2% Dollars: ; D 1%% Dollars
"ased on this criterion the optimal decisionalternative is toinvest in #old5
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Decision Ma'in# .nder $is'Decision Ma'in# .nder $is'
The probability estimate for the occurrence of
each state of nature Aif availableB can be
incorporated in the search for the optimal
decision.
?or each decision calculate its expected payoff.
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Decision Ma'in# .nder $is' 2Decision Ma'in# .nder $is' 2
The /pected 7alue CriterionThe /pected 7alue Criterion
/pected Payoff 8 9Probability 9Payoff/pected Payoff 8 9Probability 9Payoff
?or each decision calculate the expected payoffas follows-
AThe summation is calculated across all the states of natureB
&elect the decision with the best expected payoff
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T0M B$01! -T0M B$01! - The /pected 7alue CriterionThe /pected 7alue Criterion
The /pected 7alue Criterion ExpectedDecision "ar#e rise mall rise !o chan#e mall fall "ar#e fall Value
&old /$%% $%% 0%% 1%% % $%%Bond )*+ )++ ,*+ -,++ -,*+ ,;+
toc' 2%% 02% $%% /0%% /3%% $02C(D 3% 3% 3% 3% 3% 3%Prior Prob5 %.0 %.1 %.1 %.$ %.$
E C A%.0BA02%B 9 A%.1BA0%%B 9 A%.1BA$2%B 9 A%.$BA/$%%
T h e o p t i m a l d e c i s i o n
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The expected value criterion is useful generally
in two cases-: *ong run planning is appropriate) and decision
situations repeat themselves.: The decision maker is risk neutral.
1hen to use the e/pected value1hen to use the e/pected value
approachapproach
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/pected $e#ret Criterion/pected $e#ret Criterion
Expected
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Applications-65;Applications-65;!ational foods has developed a new sports bevera#e it would li'e to advertise on!ational foods has developed a new sports bevera#e it would li'e to advertise on
uper Bowl unday5 !ational
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!umber of ;+ Perceived &ame /citement!umber of ;+ Perceived &ame /citement
ec5 Commercials ec5 Commercials
Purchased Dull Avera#e Above e/citin#Purchased Dull Avera#e Above e/citin#
avera#eavera#e
0ne -) ; ,;0ne -) ; ,;Two -* 6 ,) ,Two -* 6 ,) ,
Three -4 * ,; ))Three -4 * ,; ))
a5a5 1hat is the optimal decision if the national foods ad mana#er is optimistic1hat is the optimal decision if the national foods ad mana#er is optimistic
b5b5 1hat is the optimal decision if the national %oods advertisin# mana#er is1hat is the optimal decision if the national %oods advertisin# mana#er ispessimisticpessimistic
c5c5 1hat is the optimal decision if the !ational %oods ad mana#er wishes the minimi e1hat is the optimal decision if the !ational %oods ad mana#er wishes the minimi ethe firm
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Consider the data #iven in problem ; for national %oods5 Based on passed super bowlConsider the data #iven in problem ; for national %oods5 Based on passed super bowl#ames= suppose the Decision ma'er believes that the followin# probabilities hold#ames= suppose the Decision ma'er believes that the followin# probabilities holdfor the states of nature5for the states of nature5
P9Dull &ame 8+5)+P9Dull &ame 8+5)+P9Avera#e &ame 853+P9Avera#e &ame 853+
P9Above Avera#e #ame 85;+P9Above Avera#e #ame 85;+
P9e/citin# 8+5,+P9e/citin# 8+5,+
a5a5 .sin# the e/pected value criterion= determine how many commercials !ational.sin# the e/pected value criterion= determine how many commercials !ational%oods should purchaseE%oods should purchaseE
b5b5 Based on the probabilities #iven here= determine the e/pected value of perfectBased on the probabilities #iven here= determine the e/pected value of perfectinformation5information5
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653 /pected 7alue of Perfect Information653 /pected 7alue of Perfect Information
The gain in expected return obtained from knowing with certainty the future state of nature is called-
/pected 7alue of Perfect Information/pected 7alue of Perfect Information
9 7PI9 7PI
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/pected 7alue of Perfect Information/pected 7alue of Perfect Information
decision
making prior tooccurwill
natureof statewhichto
asninformatioadditional
houtreturn witE pected
decisionmaking
prior tooccurwillnatureof statewhichto
asninformatio perfect
hreturn witE pected
n!nformatio
"erfectof value
Expected
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The 6/pected 7alue of Perfect InformationDecision "ar#e rise mall rise !o chan#e mall fall "ar#e fall
&old /$%% $%% 0%% 1%% %Bond 02% 0%% $2% /$%% /$2%toc' 2%% 02% $%% /0%% /3%%
C(D 3% 3% 3% 3% 3%Probab5 %.0 %.1 %.1 %.$ %.$
'f it were known with certainty that there will be a =*arge in the mark
"ar#e rise
... the optimal decision would be to invest in...
/$%%
02% *++ 3%
&tock
&imilarly)F
T0M B$01! -T0M B$01! - 7PI7PI
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The 6/pected 7alue of Perfect InformationDecision "ar#e rise mall rise !o chan#e mall fall "ar#e fall
&old /$%% $%% 0%% 1%% %Bond 02% 0%% $2% /$%% /$2%toc' 2%% 02% $%% /0%% /3%%
C(D 3% 3% 3% 3% 3%Probab5 %.0 %.1 %.1 %.$ %.$
/$%%
02% *++ 3%
Expected
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65* Bayesian Analysis - Decision Ma'in#65* Bayesian Analysis - Decision Ma'in# with Imperfect Information with Imperfect Information
"ayesian &tatistics play a role in assessingadditional information obtained from varioussources.
This additional information may assist in refiningoriginal probability estimates) and help improvedecision making.
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Tom Brown Investment Decision 9ContinuedTom Brown Investment Decision 9Continued
Tom has learned that= for only >*+= he can receive the results of notedTom has learned that= for only >*+= he can receive the results of notedeconomist Milton amuelman
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Predicted ?ne#ative@Predicted ?ne#ative@
Tom would li'e to 'now whether it is worthwhile to pay >*+ for theTom would li'e to 'now whether it is worthwhile to pay >*+ for theresult of the amuelman forecast5result of the amuelman forecast5
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'f the e/pected #ain resulting from the decisions madewith the forecastexceeds $50 ) Tom should purchase
the forecast. The e/pected #ain 8/pected payoff with forecast 2 $ 7
To find /pected payoff with forecast Tom shoulddetermine what to do when-: The forecast is =positive growth>): The forecast is =negative growth>.
T0M B$01! 2 olutionT0M B$01! 2 olution
.sin# ample Information.sin# ample Information
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Conditional ProbabilitiesConditional ProbabilitiesP9forecast predicts@positive@(lar#e rise in the mar'et 8+5 +P9forecast predicts@positive@(lar#e rise in the mar'et 8+5 +
P9forecast predicts ? ne#ative@(lar#e rise in the mar'et 8+5)+P9forecast predicts ? ne#ative@(lar#e rise in the mar'et 8+5)+
P9forecast predicts Hpositive@( small rise in the mar'et 8+5 +P9forecast predicts Hpositive@( small rise in the mar'et 8+5 +
P9forecast predicts ?ne#ative@(small rise in the mar'et 8+5;+P9forecast predicts ?ne#ative@(small rise in the mar'et 8+5;+
P9forecast predicts ?positive@( no chan#e in the mar'et 8+5*+P9forecast predicts ?positive@( no chan#e in the mar'et 8+5*+
P9forecast predicts ?ne#ative@( no chan#e in the mar'et 8+5*+P9forecast predicts ?ne#ative@( no chan#e in the mar'et 8+5*+
P 9forecast predicts ?positive@( small fall in the mar'et 8+53+P 9forecast predicts ?positive@( small fall in the mar'et 8+53+
P9forecast predicts ? ne#ative@ ( small fall in the mar'et 8+56+P9forecast predicts ? ne#ative@ ( small fall in the mar'et 8+56+
P9forecast predicts Hpositive@( lar#e fall in the mar'et 8+P9forecast predicts Hpositive@( lar#e fall in the mar'et 8+
? @? @
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Tom needs to know the following probabilities: !A*arge rise H The forecast predicted =!ositive>B
: !A&mall rise H The forecast predicted =!ositive>B: !AIo change H The forecast predicted =!ositive >B: !A&mall fall H The forecast predicted =!ositive>B: !A*arge ?all H The forecast predicted =!ositive>B
: !A*arge rise H The forecast predicted =Iegative >B: !A&mall rise H The forecast predicted =Iegative>B: !AIo change H The forecast predicted =Iegative>B: !A&mall fall H The forecast predicted =Iegative>B:
!A*arge ?allB H The forecast predicted =Iegative>B
T0M B$01! 2 olutionT0M B$01! 2 olution
.sin# ample Information.sin# ample Information
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"ayes Theorem provides a procedure to calculatethese probabilities
P9BAi P9Ai
P9BA, P9A, J P9B A) P9A) JKJ P9BAn P9AnP9Ai B 8
!osterior !robabilities!robabilities determinedafter the additional infobecomes available.
T0M B$01! 2 olutionT0M B$01! 2 olution
Bayes< TheoremBayes< Theorem
!rior probabilities!robability estimatesdetermined based oncurrent info) before the
new info becomes available.
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Posterior Probability ? Positive@Posterior Probability ? Positive@%orecast for Tom Brown%orecast for Tom Brown
&tates of nature !rior!robability!A&iB
;onditional!robability!Apositive &iB
4oint !robability!Apositive &iB
!A&i positiveB
*& %.0% %.6% %.$3 %.063
&< %.1% %.G% %.0$ %.1G2
I; %.1% %.2% %.$2 %.036
&? %.$% %.@% %.%@ %.%G$
*? %.$% % % %
Total %.23
iS
P9positive 8+5*6P9positive 8+5*6
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Posterior Probability: ?ne#ative@Posterior Probability: ?ne#ative@forecast for Tom Brownforecast for Tom Brown
&tates of nature &i !rior probability!AsiB ;ondl !robability!Anegative &iB 4oint !robability!Anegative&iB
!Asi negativeB
*& %.0% %.0% %.%@ %.%J$
&< %.1% %.1% %.%J %.0%2
I; %.1% %.2% %.$2 %.1@$
&? %.$% %.3% %.%3 %.$13
*? %.$% $.%% %.$% %.00G
Total %.@@
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/pected value of sample informtion/pected value of sample informtionIf the amuelman 3
79Bond(, +79Bond(, +
79 toc'(@Positive@ 8>)3479 toc'(@Positive@ 8>)34
79C(D(@Positive@ 8>6+79C(D(@Positive@ 8>6+
Decision: o if the samuelman
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amuelman,)+
79 Bond( ne#ative 8>679 Bond( ne#ative 8>6
7 9 toc'( ?ne#ative@ 8->;;7 9 toc'( ?ne#ative@ 8->;;79C(D( ? !e#ative@ 8>6+79C(D( ? !e#ative@ 8>6+
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/pected value of sample Information/pected value of sample Information
=
ninformatio
samplewithout
returnE pected
ninformatiosamplewith
returnE pected
n!nformatio#ampleof
$alueE pected
E%E$&E%#!E$#! =
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DecisionDecision$ I85*69)345,, J5339,;+53* 8>,4)5*+$ I85*69)345,, J5339,;+53* 8>,4)5*+
$ 78>,;+$ 78>,;+
7 I8>,4)5*+-,;+8>6)5*+7 I8>,4)5*+-,;+8>6)5*+
Decision: Tom should acGuire itDecision: Tom should acGuire it
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/tension of Application 65;/tension of Application 65;
Consider the data #iven in problems ; and 3 for !ational %oods5 The firm can hire theConsider the data #iven in problems ; and 3 for !ational %oods5 The firm can hire thenoted sports pundit Lim 1orden to #ive his opinion as to whether or not the uper Bowlnoted sports pundit Lim 1orden to #ive his opinion as to whether or not the uper Bowl#ame will be interestin#5 uppose the followin# probabilities holds for Lim
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uestionsuestionsa5a5 If Lim predicts the #ame will be interestin# what is the probabilityIf Lim predicts the #ame will be interestin# what is the probability
that the #ame will be dull5that the #ame will be dull5
b5b5 1hat is the !ational
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The revised expected values for each decision-!ositive forecast Iegative forecast
E A+oldH!ositiveB C 6@ E A+oldHIegativeB C $0%E A"ondH!ositiveB C $6% E A"ondHIegativeB C 32E A&tockH!ositiveB C 02% E A&tockHIegativeB C /1E A; DH!ositiveB C 3% E A; DHIegativeB C 3%
If the forecast is ?Positive@Invest in toc'5
If the forecast is ?!e#ative@Invest in &old5
T0M B$01! 2 Conditional /pected 7aluesT0M B$01! 2 Conditional /pected 7alues
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&ince the forecast is unknown before it ispurchased) Tom can only calculate the expected
return from purchasing it. Expected return when buying the forecast C E
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The expected gain from buying the forecast is-E &' C E
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656 Decision Trees656 Decision Trees
The !ayoff Table approach is useful for a non/se,uential or single stage.
Many real/world decision problems consists of ase,uence of dependent decisions.
Decision Trees are useful in analyzing multi/stage decision processes.
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5 Decision Tree is a chronological representation of thedecision process.
The tree is composed of nodes and branches.
Characteristics of a decision treeCharacteristics of a decision tree
5 branch emanating from astate ofnature 9chance node corresponds to aparticular state of nature) and includesthe probability of this state of nature.
Decisionnode
Chancenode
D e c i s i o
n ,
C o s t
,
D e c i s i o n ) C o s t )
P9 )
P 9 , :
P 9 ; :
P9 )
P 9 , :
P 9 ; :
5 branch emanating from adecision node corresponds to adecision alternative. 't includes acost or benefit value.
"ill + l D l t
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"ill +alen Development ;ompany
The "ill +alen Development ;ompany A"+DB needs a variance from the cityof Lingston) Iew ork) in order to do commercial development on a property whose asking price is a firm #1%%)%%% "+D estimates that it can construcshopping center for an additional #2%%)%%% and cell the completed centeapproximately #J2%)%%%.
5 variance application cost #1%%)%%% in fees and expenses) and there is a @%N chance that the variance will be approved.
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!Aconsultant predicts approval approval grantedBC%.G%
!Aconsultant predicts denial approval deniedBC%.6%
"+D wishes to determine the optimal strategy regarding this parcel ofproperty.
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BI"" &A"" ! - olutionBI"" &A"" ! - olution
;onstruction of the Decision Tree
: 'nitially the company faces a decision about hiring the
consultant.
: 5fter this decision is made more decisions follow regardin
5pplication for the variance. !urchasing the option. !urchasing the property.
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BI"" &A"" ! - The Decision TreeBI"" &A"" ! - The Decision Tree
" e t u s c o n s i d e r t h e d e c i s i o n
t o n o t h i r e a c o n s u l t a n t
D o n o t h i r
e c o n s
u l t a n t
K i r e c o n s u l t a n t
; o s t C / 2 % % %
; o s t
C %
D o n o t h i n g
0
"uy land/1%%)%%%! u r c h a s e o p t i
o n
/ 0 % )% % %
5pply for variance
5pply for variance
/1%)%%%
/1%)%%%
%1
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5 p p r o v e d
D e n i e d
%. @
% .3
$0
5 p p r o v e d
D e n i e d %
. @
% .3
/1%%)%%% /2%%)%%% J2%)%%%
"uy land "uild &ell
/2%)%%%
$%%)%%%
/G%)%%%
03%)%%%&ell
"uild &ellJ2%)%%%/2%%)%%%
$0%)%%%"uy land andapply for variance
/1%%%%% : 1%%%% 9 03%%%% C
/1%%%%% : 1%%%% : 2%%%%% 9 J2%%%%
!urchase option andapply for variance
BI"" &A"" ! - The Decision TreeBI"" &A"" ! - The Decision Tree
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"uy land/1%%)%%%
5pply for variance
5pply for variance
/1%)%%%
/1%)%%%
%
61
$0
/1%%)%%% /2%%)%%% J2%)%%%
"uy land "uild &ell
/2%)%%%
$%%)%%%
/G%)%%%
03%)%%%&ell
"uild &ellJ2%)%%%/2%%)%%%
$0%)%%%"uy land andapply for variance
/1%%%%% : 1%%%% 9 03%%%% C
/1%%%%% : 1%%%% : 2%%%%% 9 J2%%%% C
!urchase option andapply for variance
This is where we are at this stage
*et us consider the decision to hire a consultant
BI"" &A"" ! - The Decision TreeBI"" &A"" ! - The Decision Tree
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D o n o t h i r e
c o n s u l t a n t
0
K i r e c o n s u l t a n t / 2 % % % !
r e d i c t
5 p p r o
v a l
! r e d i c t
D e n i a l
% . @
% . 3
/2%%%
5pply for variance
5pply for variance
5pply for variance
5pply for variance
/2%%%
/1%)%%%
/1%)%%%
/1%)%%%
/1%)%%%BI"" &A"" ! 2BI"" &A"" ! 2
The Decision TreeThe Decision Tree
"et us consider thedecision to hire aconsultant
Done
D o I o t h i n g
"uy land/1%%)%%%
! u r c h a s e o p t i o n / 0 % )% % %
D o I o t h i
n g
"uy land/1%%)%%%
! u r c h a s e o p t i o n / 0 % )% % %
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BI"" &A"" ! - The Decision TreeBI"" &A"" ! - The Decision Tree
5 p p r o v e d
D e n i e d
; o n s u l t a n t p r e d i c t s a n a p p r o v a l
O
O
"uild &ellJ2%)%%%/2%%)%%%
03%)%%%&ell
/G2)%%%
$$2)%%%
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BI"" &A"" ! - The Decision TreeBI"" &A"" ! - The Decision Tree
5 p p r o v e d
D e n i e d
O
O
"uild &ellJ2%)%%%/2%%)%%%
03%)%%%&ell
/G2)%%%
$$2)%%%
The consultant serves as a source for additional information about denial or approval of the variance.
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O
O
BI"" &A"" ! - The Decision TreeBI"" &A"" ! - The Decision Tree
5 p p r o v e d
D e n i e d
"uild &ellJ2%)%%%/2%%)%%%
03%)%%%&ell
/G2)%%%
$$2)%%%
Therefore) at this point we need to calculate theposterior probabilities for the approval and denial
of the variance application
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BI"" &A"" ! - The Decision TreeBI"" &A"" ! - The Decision Tree
00
5 p p r o v e d
D e n i e d
"uild &ellJ2%)%%%/2%%)%%%
03%)%%%&ell
/G2)%%%)
)*$$2)%%%
); )3
)6
The rest of the Decision Tree is built in a similar manner.
!osterior !robability of Aapproval Hconsultant predicts approvalB C %.G%!osterior !robability of Adenial Hconsultant predicts approvalB C %.1%
O
O
5
5;
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Pork backward from the end of each branch.
5t a state of nature node) calculate the expected valueof the node.
5t a decision node) the branch that has the highestending node value represents the optimal decision.
The Decision TreeThe Decision Tree
Determinin# the 0ptimal trate#yDeterminin# the 0ptimal trate#y
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00 5 p p
r o v e d
D e n i e d
)
)*); )3
)6/G2)%%%
$$2)%%%$$2)%%%
/G2)%%%
$$2)%%%
/G2)%%%
$$2)%%%
/G2)%%%
$$2)%%%
/G2)%%%00
$$2)%%%
/G2)%%%
A $ $ 2) % % %
B A %. G B C 6 %
2 % %
A / G 2 )% % % B A % .1 B C / 0 0 2 % %
/ 0 0 2 % %
6 % 2 % %
6 % 2 % %
/ 0 0 2 % %
6 % 2 % %
/ 0 0 2 % %
* =+++ '
'%.1%
%.G%
"uild &ellJ2%)%%%/2%%)%%%
03%)%%%&ell
/G2)%%
$$2)%%%
Pith 26)%%% as the chance node value) we continue backward to evaluate
the previous nodes.
BI"" &A"" ! - The Decision TreeBI"" &A"" ! - The Decision Tree
Determinin# the 0ptimal trate#yDeterminin# the 0ptimal trate#y
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P r e d i c t s a p p
r o v a lN i r e
Do nothin#
BI"" &A"" ! - The Decision TreeBI"" &A"" ! - The Decision Tree
Determinin# the 0ptimal trate#yDeterminin# the 0ptimal trate#y
5 3
56
>,+=+++
>* =+++
>-*=+++
>)+=+++
>)+=+++Buy landO Applyfor variance
P r e d i c t s d e n i a l
D e n
i e d
Build=ell
ellland
D o n
o t
h i r e
>- *=+++
>,,*=+++
5B
5 ;
A p p
r o v e d
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BI"" &A"" ! - The Decision TreeBI"" &A"" ! - The Decision Tree
/cel add-in: Tree Plan/cel add-in: Tree Plan
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BI"" &A"" ! - The Decision TreeBI"" &A"" ! - The Decision Tree
/cel add-in: Tree Plan/cel add-in: Tree Plan
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