Lecture 04 Decision Making under Certainty: The Tradeoff Problem

Transcription

1 Lecture 04 Decision Making under Certainty: The Tradeoff Problem Jitesh H. Panchal ME 597: Decision Making for Engineering Systems Design Design Engineering Purdue (DELP) School of Mechanical Engineering Purdue University, West Lafayette, IN September 5, 2014 c Jitesh H. Panchal Lecture 04 1 / 29

2 Lecture Outline The Multiattribute Value Problem 1 The Multiattribute Value Problem Defining the Tradeoff Problem Choice Procedures Without Formalizing Value Trade-offs 2 Lexicographical Ordering Indifference Curves Value Functions 3 Marginal Rate of Substitution Additive Value Functions 4 Conditional 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

3 The Multiattribute Value Problem Defining the Tradeoff Problem Choice Procedures Without Formalizing Value Trade-offs Driving question for this lecture How much achievement on objective 1 is the decision maker willing to give up in 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

4 Problem Statement The Multiattribute Value Problem Defining the Tradeoff Problem Choice Procedures Without Formalizing Value Trade-offs Act space: The space, A, defined by the set of feasible alternatives, a A Consequence space: The space defined by n evaluators X 1,..., X n A point in the consequence space is denoted by x = (x 1,..., x n) Each point in the act space maps to a point in the consequence space, i.e., X 1 (a),..., X n(a) X 1,, X n a x=(x 1,, x n ) Act space (A) Consequence space Figure: 3.1 on page 67 (Keeney and Raiffa) c Jitesh H. Panchal Lecture 04 4 / 29

5 Problem Statement (contd.) Defining the Tradeoff Problem Choice Procedures Without Formalizing Value Trade-offs Decision maker s problem Choose a in A so that he/she is happiest with the payoff X 1 (a),..., X n(a) Need an index that combines X 1 (a),..., X n(a) into a scalar index v of preferability or value, i.e., v(x 1,..., x n) v(x 1,..., x n) (x 1,..., x n) (x 1,..., x n) X 1,, X n a x=(x 1,, x n ) Act space (A) Consequence space Figure: 3.1 on page 67 (Keeney and Raiffa) c Jitesh H. Panchal Lecture 04 5 / 29

6 Defining the Tradeoff Problem Choice 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 X i Definition (Dominance) x dominates x whenever x i x i, x i > x i, i for some i c Jitesh H. Panchal Lecture 04 6 / 29

7 Dominance with Two Attributes Defining the Tradeoff Problem Choice Procedures Without Formalizing Value Trade-offs The idea of dominance only exploits the ordinal character of the numbers in the consequence space, and not the cardinal character x 2 x x Direction of increasing preferences x 1 Figure: 3.2 on page 70 (Keeney and Raiffa) Note: Dominance does not require comparisons between x i and x j for i j c Jitesh H. Panchal Lecture 04 7 / 29

8 x (2) x (3) The Multiattribute Value Problem Defining the Tradeoff Problem Choice 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. x 2 x 2 x* x x x 1 x 1 x 2 x 2 x (1) x 1 x 1 Figure: 3.3 on page 71 (Keeney and Raiffa) c Jitesh H. Panchal Lecture 04 8 / 29

9 A Procedure for Exploring the Efficient Frontier Defining the Tradeoff Problem Choice Procedures Without Formalizing Value Trade-offs The decision maker must select an act a A so that he/she will be satisfied with the resulting n dimensional payoff. Alternate procedures: 1 Goal programming: Set aspiration levels x o 1, x o 2,..., x o n 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., x2 o, x3 o,..., xn o ). Seek an a A that satisfies the imposed constraints X i (a) xi o, for i = 2, 3,..., n and maximizes X 1 (a). Pick another attribute and repeat. The above procedures are ad hoc. The procedures involve continuous interactions between what is achievable and what is desirable. The decision maker needs to constantly evaluate in his/her mind what he/she would like to get and what he/she thinks is feasible. c Jitesh H. Panchal Lecture 04 9 / 29

10 Using Weighted Averages Defining the Tradeoff Problem Choice Procedures Without Formalizing Value Trade-offs Pose an auxiliary mathematical problem which results in one point on the efficient frontier. Let λ = (λ 1, λ 2,..., λ n) λ i > 0, i n λ i = 1 Auxiliary Problem: Choose a A to maximize n λ i X i (a) i=1 Alternatively, choose x R to maximize n λ i x i. The solution to this problem must lie on the efficient frontier. i=1 i=1 c Jitesh H. Panchal Lecture / 29

11 Using Weighted Averages (contd.) Defining the Tradeoff Problem Choice Procedures Without Formalizing Value Trade-offs By moving along the efficient frontier, other points can be identified, until a satisfactory point is obtained. Local marginal rates of substitution of X 1 for X 2 are 1 : 4 and 3 : 7. x 2 0.8x x 2 = constant The decision maker must decide when to be satisfied my looking at the points on the efficient frontier. x R 0.7x x 2 = constant Note Impact of non-convexity! x 1 Figure: 3.5 on page 76 (Keeney and Raiffa) c Jitesh H. Panchal Lecture / 29

12 Lexicographical Ordering Indifference Curves Value Functions Structuring the preferences independent of whether points in the consequence space are achievable or not. Different approaches for structuring preferences 1 Lexicographical Ordering 2 Indifference Curves 3 Value Functions c Jitesh H. Panchal Lecture / 29

13 Lexicographical Ordering Lexicographical Ordering Indifference Curves Value Functions 1 Widely used 2 Simple and easily administered Lexicographic ordering - Definition Assuming that evaluators X 1,..., X n are ordered according to importance, a a if and only if: (a) X 1 (a ) > X 1 (a ) or (b) X i (a ) = X i (a ), i = i... k, and X k+1 (a ) > X k+1 (a ) for some k = 1,..., n 1 Only if there is a tie in X i does X i+1 come into consideration. Note: If x and x are distinct points in an evaluation space, they cannot be indifferent with a lexicographic ordering. c Jitesh H. Panchal Lecture / 29

14 Lexicographical Ordering Indifference Curves Value Functions Lexicographical Ordering with Aspiration Levels Lexicographic ordering with Aspiration levels For each evaluator X i, set an aspiration level xi o a a whenever: (a) X 1 overrides all else as long as X 1 aspirations are not met i.e., X 1 (a ) > X 1 (a ) and X 1 (a ) < x o 1 and posit the following rules: (b) If X 1 aspirations are met, then X 2 overrides all else as long as X 2 aspirations are not met, i.e., X 1 (a ) x o 1 X 1 (a ) x o 1 X 2 (a ) > X 2 (a ) and X 2 (a ) < x o 2 for some k = 1,..., n 1 Note: In this case, two distinct points x and x may be indifferent, provided that x j > xj o and x j > xj o, for all j. c Jitesh H. Panchal Lecture / 29

15 Indifference Curves The Multiattribute Value Problem Lexicographical Ordering Indifference Curves Value Functions x x x Assume that any two points are comparable in the sense that one, and only one, of the following holds: (a) x (1) 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 than x (2) Note: All the relations,, are assumed to be transitive. x 2 Indifference curves x x x Direction of increasing preference x 1 Figure: 3.6 on page 79 (Keeney and Raiffa) c Jitesh H. Panchal Lecture / 29

16 Value Functions The Multiattribute Value Problem Lexicographical Ordering Indifference Curves Value Functions Definition (Preference Structure) A preference structure is defined on the consequence space if any two points are comparable and no intransitivities exist. Definition (Value Function) A function v, which associates a real number v(x) to each point x in an evaluation space, is said to be a value function representing the decision maker s preference structure provided that x x v(x ) = v(x ) and x x v(x ) > v(x ) c Jitesh H. Panchal Lecture / 29

17 Value Functions - Examples Lexicographical Ordering Indifference Curves Value Functions Examples: v(x) = c 1 x 1 + c 2 x 2, c 1 > 0, c 2 > 0 v(x) = x α 1 + x β 2, α > 0, β > 0 v(x) = c 1 x 1 + c 2 x 2 + c 3 (x 1 b 1 ) α (x 2 b 2 ) β Using the value functions, the decision making problem can be formulated as a standard optimization problem: Find a A to maximize v[x(a)]. c Jitesh H. Panchal Lecture / 29

18 Strategic Equivalence Lexicographical Ordering Indifference Curves Value Functions The knowledge of v uniquely specifies an entire preference structure. However, the converse is not true: a preference structure does not uniquely specify a value function. Definition (Strategic Equivalence) The value functions v 1 and v 2 are strategically equivalent written v 1 v 2, if v 1 and v 2 have the same indifference curves and induced preferential ordering. Example: If x i is positive for all i, the following value functions are strategically equivalent: v 1 (x) = i k i x i, k i > 0 i v 2 (x) = k i x i ( ) v 3 (x) = log k i x i i i c Jitesh H. Panchal Lecture / 29

19 Marginal Rate of Substitution Marginal Rate of Substitution Additive Value Functions Question If Y is increased by units, how much does X have to decrease in order to remain indifferent? Definition (Marginal Rate of Substitution) If at (x 1, y 1 ), 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 (x 1, y 1 ) is λ. Negative reciprocal of the slope of the indifference curve at (x 1, y 1 ) Figure: 3.9 on page 83 (Keeney and Raiffa) c Jitesh H. Panchal Lecture / 29

20 Marginal Rate of Substitution - Example Marginal Rate of Substitution Additive Value Functions λ c < λ a < λ b λ d < λ a < λ e Note: The marginal rate of substitution can be different for different points. Along the vertical line, the marginal rate of substitution decreases with increasing Y The more of Y we have, the less of X we are willing to give up to gain a given additional amount of Y. Figure: 3.10 on page 84 (Keeney and Raiffa) c Jitesh H. Panchal Lecture / 29

21 Special Cases The Multiattribute Value Problem Marginal Rate of Substitution Additive Value Functions 1 Constant Substitution (Linear Indifference Curves) v(x, y) = x + λy 2 Constant Substitution Rate with Transformed Variable v(x, y) = x + v Y (y) Here, λ(y) is a function of one variable (y) only. For some reference y 0, v Y (y) = y y 0 λ(y)dy Theorem The marginal rate of substitution between X and Y depends on y and not on x if and only if there is a value function v of the form v(x, y) = x + v Y (y) where v Y is a value function over attribute Y. c Jitesh H. Panchal Lecture / 29

22 Corresponding Tradeoffs Condition Marginal Rate of Substitution Additive Value Functions Assume 1 At (x 1, y 1 ) an increase of b in Y is worth a payment of a in X 2 At (x 1, y 2 ) an increase of c in Y is worth a payment of a in X 3 At (x 2, y 1 ) an increase of b in Y is worth a payment of d in X If, at (x 2, y 2 ) an increase of c in Y is worth a payment of d in X, then we say that the corresponding tradeoffs condition is met. y y 2 y 1 c a B b a A x 1 c b (?) Figure: 3.16 on page 90 (Keeney and Raiffa) d D C x 2 x c Jitesh H. Panchal Lecture / 29

23 Marginal Rate of Substitution Additive 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 that preference structure that can be expressed by v(x, y) = v X (x) + v Y (y) Theorem A preference structure is additive and therefore has an associated value function of the form v(x, y) = v X (x) + v Y (y), where v X and v Y are value functions if and only if the corresponding tradeoffs condition is satisfied. c Jitesh H. Panchal Lecture / 29

24 Marginal Rate of Substitution Additive Value Functions Conjoint Scaling: The Lock-Step Procedure 1 Define origin of measurement: v(x 0, y 0 ) = v X (x 0 ) = v Y (y 0 ) = 0 2 Choose x 1 > x 0 and arbitrarily set v X (x 1 ) = 1 3 Ask decision maker to provide value of y 1 such that (x 2, y 0 ) (x 1, y 1 ) (x 0, y 2 ) Define v X (x 2 ) = v Y (y 2 ) = 2. If the corresponding tradeoff condition holds, then (x 1, y 2 ) (x 2, y 1 ) 4 Ask the decision maker to provide value of x 3, y 3 such that Define v X (x 3 ) = v Y (y 3 ) = 3 (x 3, y 0 ) (x 2, y 1 ) (x 1, y 2 ) (x 0, y 3 ) 5 Continue in the same manner as above. Using the obtained points, define v(x, y) = v X (x) + v Y (y). c Jitesh H. Panchal Lecture / 29

25 Marginal Rate of Substitution Additive Value Functions Conjoint Scaling: The Lock-Step Procedure (Illustration) v X (x) y 3 y 2 E 2 1 c c 0 x 1 x 2 x 3 x y 1 b y 0x0 a a B b A x 1 d? x 2 D C x v Y (y) y 1 y 2 y 3 y Figure: on pages (Keeney and Raiffa) c Jitesh H. Panchal Lecture / 29

26 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 only if (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 the conditional 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 that if (x 1, y 1, z ) (x 2, y 2, z ) then (x 1, y 1, z) (x 2, y 2, z) z c Jitesh H. Panchal Lecture / 29

27 Mutual Preferential Independence Conditional Preferences Theorem A value function v may be expressed in an additive form v(x, y, z) = v X (x) + v Y (y) + v Z (z), where v X, v Y, and v Z 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, the attributes are pairwise preferentially independent. c Jitesh H. Panchal Lecture / 29

28 Summary The Multiattribute Value Problem Conditional Preferences 1 The Multiattribute Value Problem Defining the Tradeoff Problem Choice Procedures Without Formalizing Value Trade-offs 2 Lexicographical Ordering Indifference Curves Value Functions 3 Marginal Rate of Substitution Additive Value Functions 4 Conditional Preferences c Jitesh H. Panchal Lecture / 29

29 References The Multiattribute Value Problem Conditional Preferences 1 Keeney, R. L. and H. Raiffa (1993). Decisions with Multiple Objectives: Preferences and Value Tradeoffs. Cambridge, UK, Cambridge University Press. Chapter 3 c Jitesh H. Panchal Lecture / 29

30 THANK YOU! c Jitesh H. Panchal Lecture 04 1 / 1