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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


.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.


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


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60

"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


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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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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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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


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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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.



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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#



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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/cel add-in: Tree Plan/cel add-in: Tree Plan

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