Decision making and problem solving Lecture 12. Scenario analysis Recap of technical course content Related courses

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1 Decision making and problem solving Lecture 12 Scenario analysis Recap of technical course content Related courses

2 Recap of what we have covered What have we covered Core techniques of decision analysis for modelling and solving decision problems After completing this course, you are pretty competent in methods of decision analysis What have we not covered yet Institutional and organizational matters in successfully interacting with clients Concerns of power, authority, legitimacy and influence on decision and policy making These are often decisive determinants of the impact of decision analytic studies Methods of seeking to systematically chart for what future context(s) the analysis being developed for Scenario analysis 2

3 Rationale for scenario analysis The impacts of decision alternatives are contingent on how the future will unfold This future state is inherently uncertain Technological and political disruptions Major accidents, natural catastrophies Unexpected behaviour by other parties Instead of seeking to optimize in view of a single future, it may be useful to explicate alternative possible futures plan systematically in view of these alternative futures 3

4 Emergence of scenario analysis History Pierre Wack s influentiel papers in Harvard Business Review (1975) Scenario analysis helped Shell navigate the oil crises better than its competitors Many methodological extensions in the late 1970 s and 1980 s (e.g., cross-impact analysis) Presently much emphasis on making use of data scources Scenarios can be used for Revitalize planning processess in order to expect for the unexpected Catalyze processes of organizational learning: No single expert is an expert in everything Collective co-creation contributes to sense-making and awareness-raising Testing existing strategies with regard to alternative assumptions about the future 4

5 Scenario definitions A hypothetical sequence of events constructed for the purpose of focusing attention on causall processes and decision points (Kahn and Wiener, 1967) A qualitative or quantitative picture of a given organization of group, developed with the framework of a set of specified assumptions MacNulty (1977) A description of a possible future state of an organization s environment considering possible developments of relevant interdependent factors in the environment Brauers and Weber (1988) 5

6 Approaches scenario building (1/2) Emphasis on the present vs. the future Exploratory scenarios explore the future by examining the logical consequences starting from the present Anticipatory scenarios seek to answer how a given state of the future could be reached (cf. backcasting) Treatment of stakeholders motives and interests Descriptive scenarios are portrayed as impartial and disinterested descriptions of the future with no judgements concerning the desirability of scenarios Prescriptive scenarios take a position on the stakeholders motives and interests, thus permitting judgements about the desirability of scenarios 6

7 Approaches to scenario building (2/2) Presence vs. absence of surprising content Trend-oriented scenarios portray the future as continuations of existing trends Peripheral scenarios highlight extreme realizations of variables and are therefore potentially more surprising (but possibly less relevant) Use of analysis in scenario building Deductive scenarios are built by starting from general themes and by elaborating these to furnish the scenarios with enough details Inductive scenarios are derived by examining logical interdependencies between variables and defining scenarios as logical combinations of these variables 7

8 Types of scenarios Qualitative verbal descriptions These are common and widely employed in strategic planning Cf. decision trees: What do the branches after the chance node really mean? Scenarios can be built to support the assessment of consequences Trend-oriented projections Widely employed in macroeconomic forecasting Often one of the scenarios is portrayed as the basic trajectory Cross-impact analysis Probabilities assigned to scenarios by consulting experts or by using statistical analysis Assessment of how a change in the value of one variable will impact that of another May lead to complex models with problems in guaranteeing model consistency 8

9 Scenario analysis as a process (1/2) Clarify the objectives For whom are the scenarios developed for and why? Who are the decision makers? Define the scope What is the time horizon? What kinds of scenarios best contribute to the objectives? Identification of key external variables PESTE: Political Economic Social Technological Environmental variables (essentially the same as STEEP, dimentions just in another order) Characterization of variables (ranges, variability) Choice of representative values 9

10 Scenario analysis as a process (2/2) Elaboration of scenarios Realistic combinations of key variables (consistency, credibility) Catchy naming, easy-to-remember descriptions, rich visual content Often the number of scenarios varies from three to five Use scenarios; there are many possible ways Assume that the most probable scenario will materialize Plan for the preferred scenario only (even if it may fail to occur) Prepare for the full range of eventualities portrayed by the scenarios Contibute actively to the realization of the most preferred scenarios Example: Decision analytic development of strategies for the Scottish food and drink sector; see separate slides by Dr. Leena Ilmola (IIASA) and 10 Prof. Juuso Liesiö (Aalto BIZ).

11 Review of technical course content Preliminaries: Review of basic probability theory, Monte Carlo simulation Decision making under uncertainty: Decision trees, value of information, elicitation of probabilities, EUT, elicitation of utility functions, biases and behavioral aspects, modeling risk preferences, stochastic dominance Problems with multiple attributes: MAVT, elicitation of value functions and attribute weights, robust methods, MAUT, AHP, outranking methods Group decision making: group techniques for generating alternatives / objectives / attributes, voting, MAVT for groups Supporting decision making with optimization: MOO, Pareto-optimality, approaches to solving the Pareto-optimal set 11

12 Review of basic probability theory Independence: two events A and B are independent if P(AB)=P(A)P(B) Question: Is the weather independent of ice cream sales? 1. Yes 2. No Conditional probability P(A B) of A given that B has occurred is P A B P A B. P B Question: What is P(A A)? 1. P(A) Joint probability Ice cream sales Weather Low High Sum Cold Hot Sum Source: Wikipedia 12

13 Review of basic probability theory Law of total probability: If E 1,.,E n are mutually exclusive and collectively exhaustive events, then P(A)=P(A E 1 )P(E 1 )+ +P(A E n )P(E n ) E 2 E 1 E 3 E 4 A E 5 Bayes rule: P A B = P(B A)P(A) P(B) Question: Given that weather is cold, what is the probability of high ice cream sales? P(High)=0.8, P(Cold)= Conditional probability Ice cream sales Weather Low High Sum Cold Hot Sum

14 Monte Carlo simulation To support decision making, probabilistic models are often used to compute performance indices (expected values, probabilities of events) Such indices can easily be computed using Monte Carlo simulation = generation of random samples x 1,, x n from the probability model: n i=1 xi n n i=1 g(xi ) n i {1,,n} x i (a,b) estimates E X n estimates E g X estimates P(a < X b) Software tools: Excel, Matlab, etc. 14

15 Decision trees Decision trees consist of logically ordered Decision nodes Chance nodes Consequence nodes A decision tree is solved by starting from the leaves (consequence nodes) and going backward toward the root: At each chance node: compute the expected value at the node At each decision node: select the arc with the highest expected value Question: What is the optimal decision? 1. Take an umbrella 2. Do not take an umbrella Take an umbrella Do not take an umbrella EV=4.4 EV=6 It will rain, p= It will not rain, p=0.6 It will rain, p=0.4 0 It will not rain, p=

16 Expected value of perfect information What if you could make your decision only after all uncertainties had been resolved? Take an umbrella 5 Expected value of perfect information: EVPI=EVwPI-EVwoPI EVwPI is computed by putting All chance nodes at the beginning of the decision tree (in their logical order) All decison nodes after the chance nodes (in their logical order) It will rain, p=0.4 It will not rain, p=0.6 Do not take an umbrella Take an umbrella 4 0 Question: Expected value without perfect information (EVwoPI) was 6. What is EVwPI? Do not take an umbrella EVwPI=0.4*5+0.6*10=8 EVPI=8-6=2 16

17 Expected value of sample information EVSI=EVwSI-EVwoSI EVwSI=Expected value with sample information EVwoSI=Expected value without sample information EVSI = = 242 The benefits of consulting the economist can be expected to offset the costs only if her fee is less than

18 Estimation of probabilities If no historical data about the probabilites can be obtained, subjective assessments are needed Such assessments can be elicited through Bet for A Bet against A A Not A A Not A X -Y -X Y P(A) = X X + Y Betting approach (adjust X and Y) Reference lottery (adjust p) Direct judgment (e.g., quantiles of some distribution) Lottery Ref. lottery A Not A p 1-p X Y X Y P A = p 18

19 Biases in probability assessment Probability assessments are prone to biases, e.g., Representativeness: if x fits the description of A well, then P(xA) is assumed to be large regardless of the base-rate of A Conservativism: When information about some uncertain event is obtained, people typically do not adjust their initial probability estimate about this event as much as they should based on Bayes theorem. Availability: People asses the probability of an event by the ease with which occurences of this event can be brought to mind. Anchoring: Respondents are often anchored to some reference assessment Overconfidence: People tend to assign overly narrow confidence intervals to their probability estimates Desirability/undesirability: People tend to underestimate the probability of negative outcomes and overestimate the probability of positive outcomes 19

20 Expected Utility Theory (EUT) Given that preference relation between lotteries is complete and transitive and satisfies Archimedean and Independence axioms, can be represented by a real-valued utility function u(t) such that f X f Y t T f X t u t t T f Y t u(t) Implication: a rational DM selects the alternative with the highest expected utility Utility functions are assessed by asking the DM to choose between a simple lottery and a certain outcome X: Certain payoff t Y: Payoff t + t with probability p (1-p) E u X = E u Y u t = pu t + + (1 p)u t X Y p 1-p t t + t - 20

21 Modeling risk preferences Certainty equivalent CE[X] of a random variable X is an outcome such that u CE X = E u X Risk premium for r.v. X is RP[X]=E[X]-CE[X] u(t) I. DM is risk neutral, iff E[X]=CE[X] RP[X]=0 u(x) is linear for all X II. III. DM is risk averse, iff E[X]CE[X] RP[X] 0 u(x) is concave for all X DM is risk seeking, iff E[X] CE[X] RP[X] 0 u(x) is convex for all X Question: Is the DM of the figure 1. Risk neutral? 2. Risk averse? 3. Risk seeking? u(5) E[u X ] u(3) CE[X] 3 5 t RP[X] E[X] 21

22 Stochastic dominance Def: X dominates Y in the sense of First-degree Stochastic Dominance (denoted X FSD Y), if F X t F Y t t T A DM who prefers more to less should not choose an FSD dominated alternative Question: Which alternative should you choose, if you prefer more to less? 1. X 2. Y Def: X dominates Y in the sense of Second-degree Stochastic Dominance (denoted X SSD Y), if z F X t F Y t dt 0 z T. A risk-averse or risk neutral DM who prefers more to less should not choose an SSD dominated alternative Question: Which alternative should you choose, if you are risk averse and prefer more to less? 1. X 2. Y t euros F Y (t) F X (t) F Y (t) F X (t) X FSD Y X SSD Y X SSD Y Neither X nor Y dominates the other in the sense of FSD 22

23 Risk measures Value-at-Risk (VaR α [X]) gives the outcome such that the probability of a worse or equal outcome is α: VaR α [X] f X t dt = F X VaR α [X] = α f X (t) VaR 10% X = 1.85 VaR 10% Y = 0.97 CVaR 10% X = 3.26 CVaR 10% Y = 4.23 Conditional Value-at-Risk (CVaR α [X]) gives the expected outcome given that the outcome is at most VaR α : CVaR X = E[X X VaR α X ] f Y (t)

24 Multiattribute value theory (MAVT) Given certain axioms, a DM s preferences about a single attribute can be represented by a cardinal value function v i x i such that v i x i v i y i x i y i v i x i v i x i v i y i v i y i x i x i d y i y i. Attribute-specific value functions are obtained by Defining measurement scales [x i 0, x i ] Asking a series of elicitation questions through, e.g., 1. Bisection method 2. Equally preferred differences 3. Giving a functional form; e.g., v i x i is linear and increasing Cardinal value functions are unique up to positive affine transformations, whereby they can be normalized such that v i x 0 i = 0 and v i x i = 1. 24

25 Axioms Attribute X is preferentially independent of the other attributes Y, if preference between different performance levels on X does not depend on the levels of the other attributes. Attributes A are mutually preferentially independent, if any subset of attributes X is preferentially independent of the other attributes Y=A\X Attribute X is difference independent of the other attributes Y, if preference over a change in attribute X does not depend on the levels of the other attributes. 25

26 Additive multiattribute value function If the attributes are mutually preferentially independent and each attribute is difference independent of the others, then there exists an additive value function V(x)= n i=1 w i v N i (x i ) such that V x V y x y V x V x V y V y x x d y y. Decision recommendation: choose the alternative with the highest overall value V x 26

27 Attribute weights Attribute weight w i reflects the increase in overall value when the performance level on attribute a i is changed from the worst level to the best relative to similar changes in other attributes Attribute weights should be elicited by trade-off methods: E.g., All else being equal, a change km/h in top speed is equally preferred to a change 14 7 s in acceleration time w 1 = v 2 N 7 v N 2 14 w 2 v N v N Other methods: SWING, SMART(S), ordinal weighting methods (not recommended) 27

28 Incomplete preference statements Sometimes the DM can only give incomplete preference statements E.g., A change km/h in top speed is preferred to a change 14 7 s in acceleration time w 1 v 2 N 7 v N 2 14 w 2 v N v N Incomplete preference statements result in a set of feasible attribute weights S The alternatives overall values are intervals Value intervals w 1 V V(x 1 ) V(x 2 ) V(x 3 ) w w 2 = 1 w 1 V 28

29 Non-dominated alternatives Preference over interval-valued alternatives can be established through a dominance relation Definition: x k dominates x j in S, denoted x k S x j, iff V x k, w, v V x j, w, v for all w S V x k, w, v > V x j, w, v for some w S Question: Which alternatives are dominated? 1. None of them 2. D 3. C and D The set of non-dominated alternatives is X ND = x k X j such that x j S x k If additional information is elicited such that S S, then X ND (S) X ND (S ) A B C D E w 1 = 0 w 2 = 1 w 1 = 0.5 w 2 = 0.5 w 1 = 1 w 2 = 0 E C B D A 29

30 Multiattribute utility theory (MAUT) If the alternatives attribute-specific outcomes are uncertain MAUT Axioms: Attribute X is preferentially independent of the other attributes Y, if the preference order of certain outcomes that differ only in X does not depend on the levels of attributes Y Attributes A are mutually perferentially independent (MPI), if any subset of attributes X A is preferentially independent of the other attributes Y=A\X. Subset of attributes X A is additive independent (AI), if 0.5 I the DM is indifferent between lotteries I and II for any 0.5 x, y, (x, y ) A 0.5 II 0.5 (x, y) (x, y ) (x, y ) (x, y) 30

31 Multiattribute utility theory (MAUT) The attributes are MPI and single attributes are AI iff preference relation is represented by an additive MAU function U x = n w i u N i (x i ) i=1 Attribute-specific utility functions are elicited as in EUT for single attributes Attribute weights are elicited as in MAVT (i.e., by comparing certain outcomes) 31

32 Other multicriteria methods Analytic Hierarchy Process (AHP) Verbal preference statements are used to derive pairwise comparison matrices between criteria / alternatives Normalized eigenvectors of these matrices are used as local priority vectors for the criteria Total priorities between alternatives are obtained recursively from the local priorities Outranking methods Attribute-specific preferences between alternatives are determined by indifference and preference thresholds These preferences are used to determine whether an alternative outranks some other alternative AHP and outranking mehtods do not build on the axiomatization of preferences Suffer from rank reversals and intransitivity Attribute weights have no clear interpretation 32

33 Group techniques A group of experts can be used to generate objectives, attributes and/or decision alternatives Methods: Brainstorming Nominal group technique Delphi method etc. 33

34 Aggregation of preferences: voting A common way to aggregate the (ordinal) preferences of K DMs w.r.t. N alternatives is voting (e.g., plurality, Condorcet, Borda count, approval voting) Arrow s impossibility theorem: Ordinal individual preferences cannot be aggregated into a complete and transitive group preference which would satisfy Universality: any individual preference orderings should yield a unique and complete group preference ordering Independence of irrelevant alternatives: the group s preference between x and y should not change if other alternatives are removed or added Pareto principle: if all group members prefer x to y, then the group should, too Non-dictatorship: the preferences of one group member should not dictate the group s preference 34

35 Aggregation of preferences: MAVT Individual cardinal preferences can be aggregated into a group perference through value function V G : V G K (x)= k=1 W k V N K k (x), W k 0, k=1 W k = 1. This can be done for multiattribute cardinal value functions as well: V G K (x)= k=1 W n k i=1 w ki v N ki (x i ) Problems in defining W k : How to make trade-offs between people? Who gets to set x 0 and x*? W 1 V G (x) V N 1 (x) w 11 w 12 w 21 W 2 V 2 N (x) v N 11 (x 1 ) v N 12 (x 2 ) v N 21 (x 1 ) v N 22 (x 1 ) DM 1 DM 2 w 22 35

36 MOO for decision support Sometimes alternatives are not explicit, but defined implicitly through constraints multiobjective optimization (MOO) Def: Solution x*x is Pareto-optimal for a multiobjective optimization problem with objectives f i x, i {1,, n} if there does not exist xx such that f i x f i x for all i {1,, n} f i x > f i x for some i {1,, n} f 2 A C B D E f 1 Question: Which of the solutions are Pareto-optimal? 1. A and D 2. A, B, D, and E 3. All of them 36

37 MOO for decision support MOO problems are solved by Computing the set of all Pareto-optimal solutions Introducing preference information about trade-offs between objectives to support the selection of one of the PO-solutions f 2 1 (0.2,0.8) T 2 (0.7,0.3) T f 1 Methods for solving the set of Paretooptimal solutions: Weighted sum Weighted max-norm f 2 1 (0.5,0.5) T f * λ 2 =(0.9,0.1) T Value function methods (PO-set = ND-set with no attribute weight information) f 1 37

38 What next? Related courses: MS-E Investment Science (IV period) MS-E Riskianalyysi (III-IV period; in Finnish) MS-E Multiple Ccriteria Optimization (independent studies) MS-E Special Topics in Decision Making MS-E Seminar on Case Studies in Operations Research Starts in January 2017, students will be working on real life problems in project groups consisting of 3-5 students 38

39 THANK YOU! Please help me develop the course by giving feedback. 39