Lecture 04 Decision Making under Certainty: The …Lecture 04 Decision Making under Certainty: The...

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

http://engineering.purdue.edu/delp



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



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)?





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)





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




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





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


Note: Dominance does not require comparisons between x ′i and x ′′

j for i 6= j





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




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.





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.





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





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





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.





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 .





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)





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′′)





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)].





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

)




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)





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)





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 .





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)





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.





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).





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)




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





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.





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





References

1 Keeney, R. L. and H. Raiffa (1993). Decisions with Multiple Objectives:Preferences and Value Tradeoffs. Cambridge, UK, Cambridge UniversityPress. Chapter 3


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