Approaches for Multiattribute Assessment
Utility Independence
Multiattribute Assessment Procedure
Lecture 07
Multiattribute Utility Theory
Jitesh H. Panchal
ME 597: Decision Making for Engineering Systems Design
Design Engineering Lab @ Purdue (DELP)
School of Mechanical Engineering
Purdue University, West Lafayette, IN
http://engineering.purdue.edu/delp
September 25, 2014
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Lecture 07
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Approaches for Multiattribute Assessment
Utility Independence
Multiattribute Assessment Procedure
Lecture Outline
1
Approaches for Multiattribute Assessment
Conditional Assessments
Assessing Utility Functions over “Value” Functions
Direct Utility Assessment
Qualitative Structuring of Preferences
2
Utility Independence
Additive Independence and Additive Utility Function
Mutual Utility Independence
One Utility-Independent Attribute
General Case - No Utility Independence
3
Multiattribute Assessment Procedure
Multiattribute Assessment - Steps
Interpreting Scaling Constants
Keeney, R. L. and H. Raiffa (1993). Decisions with Multiple Objectives: Preferences and Value Tradeoffs. Cambridge, UK, Cambridge
University Press. Chapter 5.
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Lecture 07
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Approaches for Multiattribute Assessment
Utility Independence
Multiattribute Assessment Procedure
Conditional Assessments
Assessing Utility Functions over “Value” Functions
Direct Utility Assessment
Qualitative Structuring of Preferences
Focus of this Lecture
Assume that the attributes for the problem are X1 , X2 , . . . , Xn . If xi denotes a
specific level of Xi , the the goal is to find a utility function
u(x) = u(x1 , x2 , . . . , xn )
over the n attributes.
Property of the multiattribute utility function: Given two probability
distributions A and B over multi-attribute consequences x̃, probability
distribution A is at least as desirable as B if and only if
EA [u(x̃)] ≥ EB [u(x̃)]
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Lecture 07
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Approaches for Multiattribute Assessment
Utility Independence
Multiattribute Assessment Procedure
Conditional Assessments
Assessing Utility Functions over “Value” Functions
Direct Utility Assessment
Qualitative Structuring of Preferences
Conditional Unidimensional Utility Theory
Consider a two parameter scenario with consequence space defined by
(xi , wi ). If the decision maker knew that wi were to prevail then we can carry
out utility assessments for single parameter x.
x ∼ hx ∗ , πi (x), x o i
πi (x) is the conditional utility function for x values given the state wi ,
normalized by
πi (x o ) = 0 and πi (x ∗ ) = 1
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Lecture 07
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Approaches for Multiattribute Assessment
Utility Independence
Multiattribute Assessment Procedure
Conditional Assessments
Assessing Utility Functions over “Value” Functions
Direct Utility Assessment
Qualitative Structuring of Preferences
Using Value Functions
A utility function is a value function, but a value function is not a utility
function!
1
First assign a value function for each point in the consequence space, x
2
The utility function is monotonically increasing in V .
3
Assess unidimensional utility functions for u[v (x)]
x2*
x2 ’
X2
x2o
x1o
x1’
x1’’
x1*
X1
Figure : 5.1 on page 221 (Keeney and Raiffa)
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Lecture 07
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Approaches for Multiattribute Assessment
Utility Independence
Multiattribute Assessment Procedure
Conditional Assessments
Assessing Utility Functions over “Value” Functions
Direct Utility Assessment
Qualitative Structuring of Preferences
Direct Utility Assessment
Assigning utilities to possible consequences x1 , x2 , . . . , xR directly. Arbitrarily
set
u(xo ) = 0 and u(x∗ ) = 1
where xo is the least preferred and x∗ is the most preferred.
If lottery hx∗ , πr , xo i ∼ xr then
u(xr ) = πr
Shortcomings of the procedure:
1
it fails to exploit the basic preference structure of the decision maker
2
the requisite information is difficult to assess
3
the result is difficult to work with in expected utility calculations and
sensitivity analysis.
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H. Panchal
Lecture 07
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Approaches for Multiattribute Assessment
Utility Independence
Multiattribute Assessment Procedure
Conditional Assessments
Assessing Utility Functions over “Value” Functions
Direct Utility Assessment
Qualitative Structuring of Preferences
Qualitative Structuring of Preferences
Basic approach used in this lecture:
1
Postulate various assumptions about the preference attitudes of the
decision maker
2
Derive functional forms of the multiattribute utility function consistent with
these assumptions
Important property - Independence
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Lecture 07
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Approaches for Multiattribute Assessment
Utility Independence
Multiattribute Assessment Procedure
Additive Independence and Additive Utility Function
Mutual Utility Independence
One Utility-Independent Attribute
General Case - No Utility Independence
Utility Independence
Definition (Utility Independence)
Y is utility independent of Z when conditional preferences for lotteries on Y
given z do not depend on the particular level of z
z
If Y is utility independent of Z , all
conditional utility functions along
horizontal cuts would be positive linear
transformations of each other.
Therefore,
z*
Z
0
u(y , z) = g(z) + h(z)u(y, z )
In other words, the conditional utility
function over Y given z does not
strategically depend on z.
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H. Panchal
zo
yo
y*
y
Y
Figure : 5.2 on page 225 (Keeney and
Raiffa)
Lecture 07
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Approaches for Multiattribute Assessment
Utility Independence
Multiattribute Assessment Procedure
Additive Independence and Additive Utility Function
Mutual Utility Independence
One Utility-Independent Attribute
General Case - No Utility Independence
Some examples
Which utility functions satisfy utility independence condition?
y αzβ
y +z
1
u(y , z) =
2
u(y , z) = g(z) + h(z)uY (y)
3
u(y , z) = k (y ) + m(y )uZ (z)
4
u(y , z) = k1 uY (y ) + k2 uZ (z) + k3 uY (y )uZ (z)
5
u(y , z) = [α + βuY (y )][γ + δuZ (z)]
6
u(y , z) = kY uY (y ) + kZ uZ (z)
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Lecture 07
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Additive Independence and Additive Utility Function
Mutual Utility Independence
One Utility-Independent Attribute
General Case - No Utility Independence
Approaches for Multiattribute Assessment
Utility Independence
Multiattribute Assessment Procedure
Benefits of Utility Independence in Utility Assessment
z*
z*
z*
z2
Z
Z
Z
z1
zo
zo
yo
Y
zo
yo
y*
(a)
yo
y*
Y
(b)
z*
y2
Y
y1
y*
(c)
z*
z*
z2
Z
Z
Z
z1
zo
zo
yo
Y
(d)
y*
yo
y2
Y
y1
zo
y*
(e)
yo
Y
y*
(f)
Figure : 5.3 on page 228 (Keeney and Raiffa) (a) No independence condition holds, (b)
Y is utility independent of Z , (c) Z is utility independent of Y , (d, e) Y , Z are mutually
utility independent, (f) Additivity assumption holds.
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Lecture 07
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Approaches for Multiattribute Assessment
Utility Independence
Multiattribute Assessment Procedure
Additive Independence and Additive Utility Function
Mutual Utility Independence
One Utility-Independent Attribute
General Case - No Utility Independence
Additive Utility Function
Additive utility function:
u(y , z) = kY uY (y ) + kZ uZ (z)
where kY and kZ are positive scaling constants.
Additive utility function implies that Y and Z are mutually utility independent.
But the converse is not true.
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Lecture 07
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Approaches for Multiattribute Assessment
Utility Independence
Multiattribute Assessment Procedure
Additive Independence and Additive Utility Function
Mutual Utility Independence
One Utility-Independent Attribute
General Case - No Utility Independence
Additive Independence
Definition (Additive Independence)
Attributes Y and Z are additive independent if the paired preference
comparison of any two lotteries, defined by two joint probability distributions
on Y × Z , depends only on their marginal probability distributions.
Alternate description for two attributes: For additive independence, the
following two lotteries must be equally preferable:
h(y, z), 0.5, (y 0 , z 0 )i ∼ h(y , z 0 ), 0.5, (y 0 , z)i
for all (y , z) given arbitrarily chosen y 0 and z 0 .
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Lecture 07
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Approaches for Multiattribute Assessment
Utility Independence
Multiattribute Assessment Procedure
Additive Independence and Additive Utility Function
Mutual Utility Independence
One Utility-Independent Attribute
General Case - No Utility Independence
Fundamental Result of Additive Utility Function
Theorem
Attributes Y and Z are additive independent if and only if the two-attribute
utility function is additive. The additive form may be either written as
u(y, z) = u(y , z o ) + u(y o , z)
or as
u(y , z) = kY uY (y ) + kZ uZ (z)
where
1
2
3
4
u(y , z) is normalized by u(y o , z o ) = 0 and u(y 1 , z 1 ) = 1 for arbitrary y 1
and z 1 such that (y 1 , z o ) (y o , z o ) and (y o , z 1 ) (y o , z o )
uY (y ) is a conditional utility function on Y normalized by uY (y o ) = 0 and
uY (y 1 ) = 1
uZ (z) is a conditional utility function on Z normalized by uZ (z o ) = 0 and
uZ (z 1 ) = 1
kY = u(y 1 , z o ) and kZ = u(y o , z 1 )
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Lecture 07
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Approaches for Multiattribute Assessment
Utility Independence
Multiattribute Assessment Procedure
Additive Independence and Additive Utility Function
Mutual Utility Independence
One Utility-Independent Attribute
General Case - No Utility Independence
Mutual Utility Independence
For mutual utility independence of Y and Z ,
1
Y must be utility independent of Z , i.e.,
u(y , z) = c1 (z) + c2 (z)u(y , z0 )
∀y , z
for an arbitrarily chosen z0 , and
2
Z must be utility independent of Y , i.e.,
u(y , z) = d1 (y ) + d2 (y )u(y0 , z)
∀y , z
for an arbitrarily chosen y0 .
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Lecture 07
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Approaches for Multiattribute Assessment
Utility Independence
Multiattribute Assessment Procedure
Additive Independence and Additive Utility Function
Mutual Utility Independence
One Utility-Independent Attribute
General Case - No Utility Independence
The Multilinear Utility Function
When Y and Z are mutually utility independent, then u(y , z) can be
expressed as
u(y , z) = kY uY (y ) + kZ uZ (z) + kYZ uY (y )uZ (z)
where kY > 0, and kZ > 0.
z
H
z1
G
Z
B
A
C
z
zo
D
yo
E
F
y
y1
y
Y
Figure : 5.4 on page 233 (Keeney and Raiffa)
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Lecture 07
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Approaches for Multiattribute Assessment
Utility Independence
Multiattribute Assessment Procedure
Additive Independence and Additive Utility Function
Mutual Utility Independence
One Utility-Independent Attribute
General Case - No Utility Independence
Special Case of Multilinear Utility Function - Two Attributes
Theorem
If Y and Z are mutually utility independent, then the two-attribute utility
function is multilinear. In particular, u can be written in the form
u(y , z) = u(y, z0 ) + u(y0 , z) + ku(y , z0 )u(y0 , z),
or
u(y , z) = kY uY (y ) + kZ uZ (z) + kYZ uY (y )uZ (z)
where
1
u(y , z) is normalized by u(y0 , z0 ) = 0 and u(y1 , z1 ) = 1 for arbitrary y1
and z1 such that (y1 , z0 ) (y0 , z0 ) and (y0 , z1 ) (y0 , z0 )
2
uY (y ) is conditional utility on Y normalized by uY (y0 ) = 0 and uY (y1 ) = 1
3
uZ (z) is conditional utility on Z normalized by uZ (z0 ) = 0 and uZ (Z1 ) = 1
4
kY = u(y1 , z0 )
5
kZ = u(y0 , z1 )
6
kYZ = 1 − kY − kZ and k =
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Jitesh
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kY Z
kY kZ
Lecture 07
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Additive Independence and Additive Utility Function
Mutual Utility Independence
One Utility-Independent Attribute
General Case - No Utility Independence
Approaches for Multiattribute Assessment
Utility Independence
Multiattribute Assessment Procedure
Use of Isopreference Curves
An isopreference curve may be substituted for one of the conditional utility
functions in the theorem above.
z
L
C
z0
B
J
K
H
D
z
z1
A
G
F
N
P
M
F
yo
y
y1
y
Figure : 5.5 on page 236 (Keeney and Raiffa)
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Lecture 07
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Approaches for Multiattribute Assessment
Utility Independence
Multiattribute Assessment Procedure
Additive Independence and Additive Utility Function
Mutual Utility Independence
One Utility-Independent Attribute
General Case - No Utility Independence
Use of Isopreference Curves
Theorem
If Y and Z are mutually utility independent, then
u(y, z) =
u(y0 , z) − u(y0 , zn (y ))
1 + ku(y0 , zn (y ))
where
1
u(y0 , z0 ) = 0
2
zn (y ) is defined such that (y , zn (y)) ∼ (y0 , z0 )
3
1 )−u(y1 ,z1 )−u(y0 ,zn (y1 ))
where (y1 , z1 ) is arbitrarily chosen such
k = u(y0 ,zu(y
1 ,z1 )u(y0 ,zn (y1 ))
that (y0 , z0 ) and (y1 , z1 ) are not indifferent.
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Lecture 07
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Approaches for Multiattribute Assessment
Utility Independence
Multiattribute Assessment Procedure
Additive Independence and Additive Utility Function
Mutual Utility Independence
One Utility-Independent Attribute
General Case - No Utility Independence
The Multiplicative Representation
If two attributes are mutually utility independent, their utility function can be
represented by either a product form, when k 6= 0, or an additive form, when
k = 0.
The multilinear form
u(y , z) = u(y, z0 ) + u(y0 , z) + ku(y , z0 )(y0 , z)
Let
u 0 = ku(y , z) + 1
= ku(y0 , z) + ku(y , z0 ) + k 2 u(y0 , z)u(y , z0 ) + 1
= [ku(y , z0 ) + 1][ku(y0 , z) + 1]
= u 0 (y , z0 )u 0 (y0 , z)
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Lecture 07
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Approaches for Multiattribute Assessment
Utility Independence
Multiattribute Assessment Procedure
Additive Independence and Additive Utility Function
Mutual Utility Independence
One Utility-Independent Attribute
General Case - No Utility Independence
The Additive Representation
For k = 0, the utility function reduces to an additive function.
Theorem
If Y and Z are mutually utility independent and if
h(y3 , z3 ), (y4 , z4 )i ∼ h(y3 , z4 ), (y4 , z3 )i
for some y3 , y4 , z3 , z4 , such that (y3 , z3 ) is not indifferent to either (y3 , z4 ) or
(y4 , z3 ), then
u(y, z) = u(y , z0 ) + u(y0 , z)
where u(y , z) is normalized by
1
u(y0 , z0 ) = 0, and
2
u(y1 , z1 ) = 1 for arbitrary y1 and z1 such that (y1 , z0 ) (y0 , z0 ) and
(y0 , z1 ) (y0 , z0 ).
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Lecture 07
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Approaches for Multiattribute Assessment
Utility Independence
Multiattribute Assessment Procedure
Additive Independence and Additive Utility Function
Mutual Utility Independence
One Utility-Independent Attribute
General Case - No Utility Independence
Interpretation of Parameter k






 
 > 
 Y and Z are complements
hA, 0.5, Ci
∼
=
no interaction of preference
hB, 0.5, Di ⇔ k
0↔





≺
<
Y and Z are substitutes

z
B
C
A
D
z2
Z
z1
y1
y2
y
Y
Figure : 5.6 on page 240 (Keeney and Raiffa)
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Lecture 07
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Approaches for Multiattribute Assessment
Utility Independence
Multiattribute Assessment Procedure
Additive Independence and Additive Utility Function
Mutual Utility Independence
One Utility-Independent Attribute
General Case - No Utility Independence
One Utility Independent Attribute
Assuming that Z is utility independent of Y , for any arbitrary y0 ,
u(y , z) = c1 (y ) + c2 (y)u(y0 , z),
c2 (y ) > 0,
∀y
The two-attribute utility function can be specified by:
1
three conditional utility functions, or
2
two conditional utility functions and an isopreference curve, or
3
one conditional utility function and two isopreference curve
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Lecture 07
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Approaches for Multiattribute Assessment
Utility Independence
Multiattribute Assessment Procedure
Additive Independence and Additive Utility Function
Mutual Utility Independence
One Utility-Independent Attribute
General Case - No Utility Independence
1. Assessment using Three Conditional Utility Functions
z
E
B
z1
Theorem
Z
If Z is utility independent of Y , then
D
A
F
C
z
u(y , z) = u(y , z0 )[1−u(y0 , z)]+u(y , z1 )u(y0 , z)
z0
where u(y , z) is normalized by
u(y0 , z0 ) = 0 and u(y0 , z1 ) = 1
y0
y
y
Y
Figure : 5.7 on page 244 (Keeney
and Raiffa)
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Lecture 07
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Additive Independence and Additive Utility Function
Mutual Utility Independence
One Utility-Independent Attribute
General Case - No Utility Independence
Approaches for Multiattribute Assessment
Utility Independence
Multiattribute Assessment Procedure
2. Assessment using Two Conditional Utility Functions and One
Isopreference Curve
Theorem
If Z is utility independent of Y , then
u(y0 , z1 ) − u(y , z0 )
u(y , z) = u(y , z0 ) +
u(y0 , z)
u(y0 , zn (y ))
where u(y0 , z0 ) = 0, and zn (y ) is defined such that (y , zn (y)) ∼ (y0 , z1 ) for an
arbitrary z1 .
z1
z0
Z
Isopreference
curve
y0
Y
Figure : 5.9 on page 247 (Keeney and Raiffa)
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Lecture 07
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Approaches for Multiattribute Assessment
Utility Independence
Multiattribute Assessment Procedure
Additive Independence and Additive Utility Function
Mutual Utility Independence
One Utility-Independent Attribute
General Case - No Utility Independence
3. Assessment using One Conditional Utility Functions and Two
Isopreference Curves
Theorem
If Z is utility independent of Y , then
u(y , z) =
u(y0 , z) − u(y0 , zm (y )))
u(y0 , zn (y )) − u(y0 , zm (y ))
Z
(y, zn(y))
z1
(y, zm(y))
where
1
u(y , z) is normalized by
u(y0 , z0 ) = 0 and u(y0 , z1 ) = 1
2
zm (y ) is defined such that
(y , zm (y)) ∼ (y0 , z0 )
3
zn (y) is defined such that
(y , zn (y )) ∼ (y0 , z1 )
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Isopreference
curves
z0
y0
Y
Figure : 5.11 on page 250 (Keeney and
Raiffa)
Lecture 07
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Approaches for Multiattribute Assessment
Utility Independence
Multiattribute Assessment Procedure
Additive Independence and Additive Utility Function
Mutual Utility Independence
One Utility-Independent Attribute
General Case - No Utility Independence
Degrees of Freedom for Assigning Utility Functions
z1
z1
Z
Z
zo
z0
y0
Y
(a) Additive
y0
y1
Y
(b) Multilinear
y1
z1
z1
Z
Z
zo
z0
y0
Y
y1
(c) Three conditional utility functions
y0
Y
y1
(d) General utility independence
Figure : 5.12 on page 253 (Keeney and Raiffa)
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Approaches for Multiattribute Assessment
Utility Independence
Multiattribute Assessment Procedure
Additive Independence and Additive Utility Function
Mutual Utility Independence
One Utility-Independent Attribute
General Case - No Utility Independence
General Case - No Utility Independence
Possibilities:
1
Transformation of Y and Z to new attributes that might allow exploitation
of utility independence properties
2
Direct assessment of u(y , z) for discrete points and then
interpolation/curve fitting.
3
Application of independence results in subsets of the Y × Z space.
4
Development of more complicated assumptions about the preference
structure that imply more general utility functions.
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Lecture 07
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Approaches for Multiattribute Assessment
Utility Independence
Multiattribute Assessment Procedure
Multiattribute Assessment - Steps
Interpreting Scaling Constants
Assessment Procedure for Multiattribute Utility Functions
1
Introducing the terminology and ideas
2
Identifying relevant independence assumptions
3
Assessing conditional utility functions or isopreference curves
4
Assessing the scaling constants
5
Checking for consistency and reiterating
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Lecture 07
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Approaches for Multiattribute Assessment
Utility Independence
Multiattribute Assessment Procedure
Multiattribute Assessment - Steps
Interpreting Scaling Constants
Interpreting Scaling Constants
Consider an additive utility function
u(y , z) = kY uY (y ) + kZ uZ (z)
where
u(y o , z o ) = 0,
uY (y o ) = 0,
uZ (z o ) = 0
u(y ∗ , z ∗ ) = 1,
uY (y ∗ ) = 1,
uZ (z ∗ ) = 1
and
Important!
The scaling constants do not indicate the relative importance of attributes.
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Lecture 07
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Approaches for Multiattribute Assessment
Utility Independence
Multiattribute Assessment Procedure
Multiattribute Assessment - Steps
Interpreting Scaling Constants
Interpreting Scaling Constants
z
z
u = 0.75
z*
u=1
z*
Z
Z
u = 0.25
z+
z0
u=0
u = 0.25
y0
y*
u’ = 0.5
z+
u = 0.5
y
z0
u’ = 1
u’ = 0
u’ = 0.5
y0
Y
y*
y
Y
Figure : 5.18 on page 272 (Keeney and Raiffa)
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Approaches for Multiattribute Assessment
Utility Independence
Multiattribute Assessment Procedure
Multiattribute Assessment - Steps
Interpreting Scaling Constants
Summary
1
Approaches for Multiattribute Assessment
Conditional Assessments
Assessing Utility Functions over “Value” Functions
Direct Utility Assessment
Qualitative Structuring of Preferences
2
Utility Independence
Additive Independence and Additive Utility Function
Mutual Utility Independence
One Utility-Independent Attribute
General Case - No Utility Independence
3
Multiattribute Assessment Procedure
Multiattribute Assessment - Steps
Interpreting Scaling Constants
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Approaches for Multiattribute Assessment
Utility Independence
Multiattribute Assessment Procedure
Multiattribute Assessment - Steps
Interpreting Scaling Constants
References
1
Keeney, R. L. and H. Raiffa (1993). Decisions with Multiple Objectives:
Preferences and Value Tradeoffs. Cambridge, UK, Cambridge University
Press. Chapter 5.
2
Clemen, R. T. (1996). Making Hard Decisions: An Introduction to
Decision Analysis. Belmont, CA, Wadsworth Publishing Company.
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THANK YOU!
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