lecture 04 decision making under certainty: the …lecture 04 decision making under certainty: the...
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The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Lecture 04Decision Making under Certainty: The Tradeoff Problem
Jitesh H. Panchal
ME 597: Decision Making for Engineering Systems Design
Design Engineering Lab @ Purdue (DELP)School of Mechanical Engineering
Purdue University, West Lafayette, INhttp://engineering.purdue.edu/delp
September 5, 2014c©Jitesh H. Panchal Lecture 04 1 / 29
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Lecture Outline
1 The Multiattribute Value ProblemDefining the Tradeoff ProblemChoice Procedures Without Formalizing Value Trade-offs
2 Structuring PreferencesLexicographical OrderingIndifference CurvesValue Functions
3 Preference Structures for Two AttributesMarginal Rate of SubstitutionAdditive Value Functions
4 Preference Structure for More than Two AttributesConditional Preferences
Chapter 3 from Keeney, R. L. and H. Raiffa (1993). Decisions with Multiple Objectives: Preferences and Value Tradeoffs. Cambridge, UK,Cambridge University Press.
c©Jitesh H. Panchal Lecture 04 2 / 29
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Defining the Tradeoff ProblemChoice Procedures Without Formalizing Value Trade-offs
The Multiattribute Value Problem
Driving question for this lecture
How much achievement on objective 1 is the decision maker willing to give upin order to improve achievement on objective 2 by some fixed amount?
This is a tradeoff issue.
In this lecture, we will only focus on deterministic scenarios.This is a two-part problem:
1 What can we achieve in the multi-dimensional space (Achievability)?2 What are the decision maker’s preferences for the attributes (Preference
structure)?
c©Jitesh H. Panchal Lecture 04 3 / 29
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Defining the Tradeoff ProblemChoice Procedures Without Formalizing Value Trade-offs
Problem Statement
Act space: The space, A, defined by the set of feasible alternatives,a ∈ AConsequence space: The space defined by n evaluators X1, . . . ,Xn
A point in the consequence space is denoted by x = (x1, . . . , xn)Each point in the act space maps to a point in the consequence space,i.e., X1(a), . . . ,Xn(a)
a
Act space (A)
X1, …, Xn
x=(x1, …, xn)
Consequence space
Figure: 3.1 on page 67 (Keeney and Raiffa)
c©Jitesh H. Panchal Lecture 04 4 / 29
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Defining the Tradeoff ProblemChoice Procedures Without Formalizing Value Trade-offs
Problem Statement (contd.)
Decision maker’s problem
Choose a in A so that he/she is happiest with the payoff X1(a), . . . ,Xn(a)
Need an index that combines X1(a), . . . ,Xn(a) into a scalar index v ofpreferability or value, i.e.,
v(x1, . . . , xn) ≥ v(x ′1, . . . , x
′n)⇔ (x1, . . . , xn) & (x ′
1, . . . , x′n)
a
Act space (A)
X1, …, Xn
x=(x1, …, xn)
Consequence space
Figure: 3.1 on page 67 (Keeney and Raiffa)c©Jitesh H. Panchal Lecture 04 5 / 29
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Defining the Tradeoff ProblemChoice Procedures Without Formalizing Value Trade-offs
Choice Procedures Without Formalizing Value Trade-offs:a) Dominance
Assume:
Act a′ has consequences x′ = (x ′1, . . . , x
′n)
Act a′′ has consequences x′′ = (x ′′1 , . . . , x
′′n )
Preferences increase in each Xi
Definition (Dominance)
x′ dominates x′′ whenever
x ′i ≥ x ′′
i , ∀ix ′
i > x ′′i , for some i
c©Jitesh H. Panchal Lecture 04 6 / 29
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Defining the Tradeoff ProblemChoice Procedures Without Formalizing Value Trade-offs
Dominance with Two Attributes
The idea of dominance only exploits the “ordinal” character of the numbers inthe consequence space, and not the“cardinal” character
x’’
x1
x2
x’
Direction of
increasing
preferences
Figure: 3.2 on page 70 (Keeney and Raiffa)
Note: Dominance does not require comparisons between x ′i and x ′′
j for i 6= j
c©Jitesh H. Panchal Lecture 04 7 / 29
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Defining the Tradeoff ProblemChoice Procedures Without Formalizing Value Trade-offs
Choice Procedures Without Formalizing Value Trade-offs:b) The Efficient Frontier
Definition (Efficient Frontier / Pareto Optimal Set)
The efficient frontier consists of the set of non dominated consequences.
x1
x2
x1
x2
x1
x2
x1
x2
x’
x’’
x*
x(1)
x(2)
x(3)
Figure: 3.3 on page 71 (Keeney and Raiffa)c©Jitesh H. Panchal Lecture 04 8 / 29
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Defining the Tradeoff ProblemChoice Procedures Without Formalizing Value Trade-offs
A Procedure for Exploring the Efficient Frontier
The decision maker must select an act a ∈ A so that he/she will be satisfiedwith the resulting n−dimensional payoff.
Alternate procedures:1 Goal programming: Set aspiration levels xo
1 , xo2 , . . . , x
on and find points
that are closest to the aspiration levels. Update aspiration levels. Repeat.2 Standard optimization: Set aspiration levels for all attributes but one
(e.g., xo2 , x
o3 , . . . , x
on ). Seek an a ∈ A that satisfies the imposed
constraints Xi (a) ≥ xoi , for i = 2, 3, . . . , n and maximizes X1(a). Pick
another attribute and repeat.
The above procedures are ad hoc. The procedures involve continuousinteractions between what is achievable and what is desirable. The decisionmaker needs to constantly evaluate in his/her mind what he/she would like toget and what he/she thinks is feasible.
c©Jitesh H. Panchal Lecture 04 9 / 29
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Defining the Tradeoff ProblemChoice Procedures Without Formalizing Value Trade-offs
Using Weighted Averages
Pose an auxiliary mathematical problem which results in one point on theefficient frontier. Let
λ = (λ1, λ2, . . . , λn)
λi > 0, ∀in∑
i=1
λi = 1
Auxiliary Problem: Choose a ∈ A to maximizen∑
i=1λiXi (a)
Alternatively, choose x ∈ R to maximizen∑
i=1λixi .
The solution to this problem must lie on the efficient frontier.
c©Jitesh H. Panchal Lecture 04 10 / 29
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Defining the Tradeoff ProblemChoice Procedures Without Formalizing Value Trade-offs
Using Weighted Averages (contd.)
By moving along the efficient frontier, other points can be identified, until a“satisfactory” point is obtained.
Local marginal rates of substitution ofX1 for X2 are 1 : 4 and 3 : 7.
The decision maker must decide whento be satisfied my looking at the pointson the efficient frontier.
Note
Impact of non-convexity! x1
x2
x’
R0.7x1+0.3x2 = constant
0.8x1+0.2x2 = constant
Figure: 3.5 on page 76 (Keeney and Raiffa)
c©Jitesh H. Panchal Lecture 04 11 / 29
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Lexicographical OrderingIndifference CurvesValue Functions
Structuring Preferences
Structuring the preferences independent of whether points in theconsequence space are achievable or not.
Different approaches for structuring preferences
1 Lexicographical Ordering2 Indifference Curves3 Value Functions
c©Jitesh H. Panchal Lecture 04 12 / 29
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Lexicographical OrderingIndifference CurvesValue Functions
Lexicographical Ordering
1 Widely used2 Simple and easily administered
Lexicographic ordering - Definition
Assuming that evaluators X1, . . . ,Xn are ordered according to importance,a′ � a′′ if and only if:
(a) X1(a′) > X1(a′′)or
(b) Xi (a′) = Xi (a′′), i = i . . . k , and Xk+1(a′) > Xk+1(a′′)for some k = 1, . . . , n − 1
Only if there is a tie in Xi does Xi+1 come into consideration.Note: If x′ and x′′ are distinct points in an evaluation space, they cannot beindifferent with a lexicographic ordering.
c©Jitesh H. Panchal Lecture 04 13 / 29
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Lexicographical OrderingIndifference CurvesValue Functions
Lexicographical Ordering with Aspiration Levels
Lexicographic ordering with Aspiration levels
For each evaluator Xi , set an aspiration level xoi and posit the following rules:
a′ � a′′ whenever:
(a) X1 overrides all else as long as X1 aspirations are not meti.e., X1(a′) > X1(a′′) and X1(a′′) < xo
1
(b) If X1 aspirations are met, then X2 overrides all else as long as X2
aspirations are not met, i.e.,X1(a′) ≥ xo
1X1(a′′) ≥ xo
1X2(a′) > X2(a′′) and X2(a′′) < xo
2for some k = 1, . . . , n − 1
Note: In this case, two distinct points x′ and x′′ may be indifferent, providedthat x ′
j > xoj and x ′′
j > xoj , for all j .
c©Jitesh H. Panchal Lecture 04 14 / 29
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Lexicographical OrderingIndifference CurvesValue Functions
Indifference Curves
Assume that any two points are comparable inthe sense that one, and only one, of thefollowing holds:
(a) x(1) v x(2), i.e., x(1) is indifferent to x(2)
(b) x(1) � x(2), i.e., x(1) is preferred to x(2)
(c) x(1) ≺ x(2), i.e., x(1) is less preferred thanx(2)
Note: All the relations v,�,≺ are assumed tobe transitive.
x ′′′ � x ′′ v x ′
x1
x2
x’’’
Direction of
increasing
preference
Indifference curves
x’
x’’
Figure: 3.6 on page 79 (Keeneyand Raiffa)
c©Jitesh H. Panchal Lecture 04 15 / 29
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Lexicographical OrderingIndifference CurvesValue Functions
Value Functions
Definition (Preference Structure)
A preference structure is defined on the consequence space if any two pointsare comparable and no intransitivities exist.
Definition (Value Function)
A function v , which associates a real number v(x) to each point x in anevaluation space, is said to be a value function representing the decisionmaker’s preference structure provided that
x′ v x′′ ⇔ v(x′) = v(x′′)
and
x′ � x′′ ⇔ v(x′) > v(x′′)
c©Jitesh H. Panchal Lecture 04 16 / 29
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Lexicographical OrderingIndifference CurvesValue Functions
Value Functions - Examples
Examples:
v(x) = c1x1 + c2x2, c1 > 0, c2 > 0
v(x) = xα1 + xβ
2 , α > 0, β > 0
v(x) = c1x1 + c2x2 + c3(x1 − b1)α(x2 − b2)β
Using the value functions, the decision making problem can be formulated asa standard optimization problem: Find a ∈ A to maximize v [X (a)].
c©Jitesh H. Panchal Lecture 04 17 / 29
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Lexicographical OrderingIndifference CurvesValue Functions
Strategic Equivalence
The knowledge of v uniquely specifies an entire preference structure.However, the converse is not true: a preference structure does not uniquelyspecify a value function.
Definition (Strategic Equivalence)
The value functions v1 and v2 are strategically equivalent written v1 v v2, if v1
and v2 have the same indifference curves and induced preferential ordering.
Example: If xi is positive for all i , the following value functions arestrategically equivalent:
v1(x) =∑
i
kixi , ki > 0 ∀i
v2(x) =
√∑i
kixi
v3(x) = log
(∑i
kixi
)
c©Jitesh H. Panchal Lecture 04 18 / 29
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Marginal Rate of SubstitutionAdditive Value Functions
Marginal Rate of Substitution
Question
If Y is increased by ∆ units, how much doesX have to decrease in order to remainindifferent?
Definition (Marginal Rate of Substitution)
If at (x1, y1), you are willing to give up λ∆units of X for ∆ units of Y , then for small ∆,the marginal rate of substitution of X for Y at(x1, y1) is λ.
Negative reciprocal of the slope of theindifference curve at (x1, y1)
Figure: 3.9 on page 83 (Keeneyand Raiffa)
c©Jitesh H. Panchal Lecture 04 19 / 29
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Marginal Rate of SubstitutionAdditive Value Functions
Marginal Rate of Substitution - Example
Note: The marginal rate of substitutioncan be different for different points.
Along the vertical line, the marginalrate of substitution decreases withincreasing Y ⇒The more of Y we have, the less of Xwe are willing to give up to gain agiven additional amount of Y .
λc < λa < λb
λd < λa < λe
Figure: 3.10 on page 84 (Keeney andRaiffa)
c©Jitesh H. Panchal Lecture 04 20 / 29
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Marginal Rate of SubstitutionAdditive Value Functions
Special Cases
1 Constant Substitution (Linear Indifference Curves)
v(x , y) = x + λy
2 Constant Substitution Rate with Transformed Variable
v(x , y) = x + vY (y)
Here, λ(y) is a function of one variable (y) only. For some reference y0,
vY (y) =
∫ y
y0
λ(y)dy
Theorem
The marginal rate of substitution between X and Y depends on y and not onx if and only if there is a value function v of the form
v(x , y) = x + vY (y)
where vY is a value function over attribute Y .
c©Jitesh H. Panchal Lecture 04 21 / 29
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Marginal Rate of SubstitutionAdditive Value Functions
Corresponding Tradeoffs Condition
Assume1 At (x1, y1) an increase of b in Y is
worth a payment of a in X2 At (x1, y2) an increase of c in Y is
worth a payment of a in X3 At (x2, y1) an increase of b in Y is
worth a payment of d in X
If, at (x2, y2) an increase of c in Y isworth a payment of d in X , then wesay that the corresponding tradeoffscondition is met.
x
y
y1
x1
y2 a
c
(?)
c
a
b b
d
x2
A C
DB
Figure: 3.16 on page 90 (Keeney andRaiffa)
c©Jitesh H. Panchal Lecture 04 22 / 29
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Marginal Rate of SubstitutionAdditive Value Functions
Corresponding Tradeoffs Condition: An Additive Value Function
Definition (Additive preference structure)
A preference structure is additive if there exists a value function reflecting thatpreference structure that can be expressed by
v(x , y) = vX (x) + vY (y)
Theorem
A preference structure is additive and therefore has an associated valuefunction of the form
v(x , y) = vX (x) + vY (y),
where vX and vY are value functions if and only if the corresponding tradeoffscondition is satisfied.
c©Jitesh H. Panchal Lecture 04 23 / 29
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Marginal Rate of SubstitutionAdditive Value Functions
Conjoint Scaling: The Lock-Step Procedure
1 Define origin of measurement:
v(x0, y0) = vX (x0) = vY (y0) = 0
2 Choose x1 > x0 and arbitrarily set vX (x1) = 13 Ask decision maker to provide value of y1 such that
(x2, y0) ∼ (x1, y1) ∼ (x0, y2)
Define vX (x2) = vY (y2) = 2. If the corresponding tradeoff conditionholds, then (x1, y2) ∼ (x2, y1)
4 Ask the decision maker to provide value of x3, y3 such that
(x3, y0) ∼ (x2, y1) ∼ (x1, y2) ∼ (x0, y3)
Define vX (x3) = vY (y3) = 35 Continue in the same manner as above.
Using the obtained points, define v(x , y) = vX (x) + vY (y).
c©Jitesh H. Panchal Lecture 04 24 / 29
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Marginal Rate of SubstitutionAdditive Value Functions
Conjoint Scaling: The Lock-Step Procedure (Illustration)
x
y
y1
x1
y2
x2
A
D
E
y0x0
a
b
a
B
b
C
cc
?
d
x
vX(x)
x1
y0
vY(y)
x2 x3
y1 y2 y3
1
2
3
0
1
2
3
Figure: 3.17-18 on pages 90-91 (Keeney and Raiffa)
c©Jitesh H. Panchal Lecture 04 25 / 29
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Conditional Preferences
Conditional Preferences
Consider three evaluators: X ,Y , and Z
Definition (Conditionally Preferred)
Consequence (x ′, y ′) is conditionally preferred to (x ′′, y ′′) given z′ if and onlyif (x ′, y ′, z′) is preferred to (x ′′, y ′′, z′).
Definition (Preferentially Independent)
The pair of attributes X and Y is preferentially independent of Z if theconditional preferences in the (x , y) space given z′ do not depend on z′.
If the pair {X ,Z} is preferentially independent of Z , then we can say thatif (x1, y1, z′) & (x2, y2, z′) then (x1, y1, z) & (x2, y2, z) ∀z
c©Jitesh H. Panchal Lecture 04 26 / 29
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Conditional Preferences
Mutual Preferential Independence
Theorem
A value function v may be expressed in an additive form
v(x , y , z) = vX (x) + vY (y) + vZ (z),
where vX , vY , and vZ are single-attribute value functions, if and only if
{X ,Y} are preferentially independent of Z ,
{X ,Z} are preferentially independent of Y , and
{Y ,Z} are preferentially independent of X .
Definition (Pairwise preferentially independent)
If each pair of attributes is preferentially independent of its complement, theattributes are pairwise preferentially independent.
c©Jitesh H. Panchal Lecture 04 27 / 29
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Conditional Preferences
Summary
1 The Multiattribute Value ProblemDefining the Tradeoff ProblemChoice Procedures Without Formalizing Value Trade-offs
2 Structuring PreferencesLexicographical OrderingIndifference CurvesValue Functions
3 Preference Structures for Two AttributesMarginal Rate of SubstitutionAdditive Value Functions
4 Preference Structure for More than Two AttributesConditional Preferences
c©Jitesh H. Panchal Lecture 04 28 / 29
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Conditional Preferences
References
1 Keeney, R. L. and H. Raiffa (1993). Decisions with Multiple Objectives:Preferences and Value Tradeoffs. Cambridge, UK, Cambridge UniversityPress. Chapter 3
c©Jitesh H. Panchal Lecture 04 29 / 29
THANK YOU!
c©Jitesh H. Panchal Lecture 04 1 / 1