3-basic decision analysis

8/11/2019 3-Basic Decision Analysis

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Basic DecisionAnalysis


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DECISION THEORY


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Expected value criterion

Suppose you face a situation !ere you "ust c!oose#eteen alternativesA and B as follos$ AlternativeA$ %&'(''' for sure) Alternative B$ *'+ c!ance of receivin, %&-(''' and

.'+ c!ance of loosin, %/(''')

0!at is your personal c!oice1

Co"pare no Alternative B it!$ Alternative C$ *'+ c!ance of innin, %2/(3'' and

.'+ c!ance of loosin, %&4(/''

Note t!at EMV5B6 7 EMV5C6( #ut are t!ey8e9uivalent:1

Alternative C see"s to #e 8"ore ris;y: t!anAlternative B even t!ou,!t t!ey !ave t!e sa"e EMV)

Conclusion$ EMV does not take Risk into account


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T!e ? criterion$

T!is "eans t!at you s!ould #e illin, to pay up to an in=nite

a"ount of "oney to play t!e ,a"e( #ut !y people are

unillin, to pay "ore t!an a fe dollars1

=+

+

+

=

=

=

...)8($8

1)4($

4

1)2($

2

1)2($

2

1

1k

k

k

EMV


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T!e


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T!e rules of actional

t!ou,!t Ho a person s!ould acts or decides rationally underuncertainty1

Anser$ #y folloin, t!e folloin, rules or axio"s$

T!e orderin, rule

T!e e9uivalence or continuity rule T!e su#stitution or independence rule

Deco"position rule

T!e c!oice rule

T!e a#ove =ve rules for" t!e axio"s for DecisionTheory


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T!e orderin, rule

T!e decision "a;er "ust #e a#le to state !ispreference a"on, t!e prospects( outco"es( or priesof any deal

urt!er"ore( t!e transitivity property "ust #e

satis=ed$ t!at is( if !e prefers to Y( and Y to ( t!en!e "ust prefer to

>at!e"atically(

T!e orderin, rule i"plies t!at t!e decision "a;er canprovide a co"plete preference orderin, of all t!e

outco"es fro" t!e #est to t!e orst Suppose a person does not follo t!e transitivity

property$ t!e "oney pu"p ar,u"ent


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T!e e9uivalence or

continuity rule Fiven a prospectA( B( and Csuc! t!at ( t!ent!ere existsp!ere 0 < p < 1 suc! t!at t!e decision "a;erill #e indiGerent #eteen receivin, t!e prospect Bfor sureand receivin, a deal it! a pro#a#ilitypfor prospectAand apro#a#ility of 1 p for prospect C

Fiven t!at

B$ certain equivalent of t!e uncertain deal on t!e ri,!t

p$preference probability of prospect B it! respect toprospects A and C

CBA

CBA


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T!e su#stitution rule

0e can alays su#stitute a deal it! its certaintye9uivalent it!out aGectin, preference

or exa"ple( suppose t!e decision "a;er is indiGerent#eteen B and t!e A C deal #elo

T!en !e "ust #e indiGerent #eteen t!e to deals#elo !ere Bis su#stituted for t!eA C deal


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T!e deco"position rule

0e can reduce co"pound deals to si"ple ones usin,t!e rules of pro#a#ilities

or exa"ple( a decision "a;er s!ould #e indiGerent#eteen t!e folloin, to deals$


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T!e c!oice or

"onotonicity rule Suppose t!at a decision "a;er can c!oose #eteen todeals &and 2as follos$

If t!e decision "a;er prefers A to B( t!en !e "ust prefer& to 2 if and only if p& J p2) T!at is( if

In ot!er ords( t!e decision "a;er "ust prefer t!e dealt!at oGers t!e ,reater c!ance of receivin, t!e #etteroutco"e

BA


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>axi"u" expected

utility principle et a decision "a;er faces t!e c!oice #eteen touncertain deals or lotteries & and 2 it! outco"es A&(A2( K( An as follos$

T!ere is no loss of ,enerality in assu"in, t!at L1and L2!ave t!e sa"e set of outco"esA1, A2, , An#ecause ecan alays assi,n ero pro#a#ility to t!ose outco"est!at do not exist in eit!er L1and L2)

Its not clear !et!er L1or L2is preferred

By orderin, rule( letnAAA ...21


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>axi"u" expected

utility principle A,ain( t!ere is no loss of ,enerality as e can alaysrenu"#er t!e su#scripts accordin, to t!e preferenceorderin,

0e note t!at A& is t!e "ost preferred outco"e( !ile

An is t!e least preferred outco"e By e9uivalent rule( for eac! outco"e A i5i 7&( K( n6

t!ere is a nu"#er uisuc! t!at ' L uiL & and

Note t!at u&7 & and un7 ') 0!y1


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>axi"u" expected

utility principle By t!e su#stitution rule( e replace eac! Ai 5i7&(K(n6in & and 2 it! t!e a#ove constructed e9uivalentlotteries


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>axi"u" expected

utility principle By t!e deco"position rule( &and 2"ay #e reducedto e9uivalent deals it! only to outco"es 5A&and An6

eac! !avin, diGerent pro#a#ilities

inally( #y t!e c!oice rule( since ( t!e decision

"a;er s!ould prefer lottery &to lottery 2if and only if

nAA 1

==

>

n

i

ii

n

i

ii qupu11


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Mtilities and utility

functions 0e de=ne t!e 9uantity ui5i=1,,n6 as t!e utility ofoutco"eAiand t!e function t!at returns t!e values ui

,ivenAias a utility function( i)e) u(Ai)7 ui

T!e 9uantities

are ;non as t!e expected utilities for lotteries & and

2respectively

Hence t!e decision "a;er "ust prefer t!e lottery it!a !i,!er expected utility

==

n

i

ii

n

i

ii AuqAup11

)(and)(


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Case for "ore t!an 2alternatives

T!e previous "ay #e ,eneralied to t!e case !en adecision "a;er is faced it! "ore t!an to uncertainalternatives) He s!ould c!oose t!e one it! "axi"u"expected utility

Hence

!ere is t!e pro#a#ility for t!e outco"eAiin t!e

alternativej

==

n

ii

j

ij AupMax 1 )(argealternativbest

j

ip


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Co"parin, expected utility criterionit! expected "onetary value criterion

T!e expected utility criterion ta;es into account #ot!return and ris; !ereas expected "onetary valuecriterion does not consider ris;

T!e alternative it! t!e "axi"u" expected utility is

t!e #est ta;in, into account t!e trade oG #eteenreturn and ris;

T!e #est preference tradeoG depends on a personsris; attitude

DiGerent types of utility function represent diGerent

attitudes and de,ree of aversion to ris; ta;in,


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BASIC DECISION ANAYSIS


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T!e party pro#le"

i" in,in "en,ada;anpesta ulan, ta!un) Dia"e"perti"#an,;an .lo;asi te"pat$ outdoor(indoor( teras 5porc!6

Decision

node

C!ance node


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Nodes

Decision node$

Ber#entu; perse,i

>ela"#an,;an titi; dala" tree yan, "enyata;antiti; pen,a"#ilan ;eputusan( decision "a;er

"e"punyai ;e#e#asan penu! untu; "en,a"#il;eputusan

C!ance node$

Ber#entu; #ulat

>ela"#an,;an uncertain varia#le( decision "a;ertida; "e"punyai ;ontrol ter!adap outco"e varia#leini


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Solvin, party pro#le"

Den,an "en,,una;an @ rules$

Orderin, rule

E9uivalence rule

Su#stitution rule

Deco"position rule

C!oice rule


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Orderin, rule


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>e"#uat decision tree

>isal;an$


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Su#stitution rule deco"position rule


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C!oice rule

Qadi dipili! lo;asi pesta indoor den,an pro#a#ilityuntu; "endapat;an #est outco"e ter#esar


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Expected utility untu;setiap alternatif


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E9uivalent >onetary orDollars ?alues

Cara lain selain"en,,una;an utilityvalue adala!"en,,una;an equivalent

mneta!y value untu;setiap outco"e)


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Dollar value vs utility

value

Mtility functionu5x6


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Certain e9uivalent

T!e certainty e9uivalent 5CE6 is t!e a"ount in!ic! a person is Pust indiGerent #eteenreceivin, it for sure and an uncertain or ris;yprospect t!at "i,!t eit!er pays "ore or less t!an

t!is a"ount)

T!e Certainty E9uivalent of a deal is t!e PersonalIndierent Selling Price (PISP

To =nd t!e CE of an alternative( e =rst co"puteits expected utility and t!en ta;e its inverse toconvert it #ac; into e9uivalent dollar value)


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>en,una;an utility functionuntu; ;asus lain >isal;an i" "en,!adapi deal se#a,ai #eri;ut$

EM 7 '(@

3-basic decision analysis

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