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Lecture 15

Lecture 15. Decision Analysis Multiattribute Utility Theory. Administrative Details. Homework Assignment 6 is due Monday. (shorter – 3 questions) Homework Assignment 7 posted tonight will be due Monday, March 24 th Data collection?. Midterm 1 Results. Average 33.32 Max 52.59 Min 6.21

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Lecture 15

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  1. Lecture 15 Decision Analysis Multiattribute Utility Theory

  2. Administrative Details • Homework Assignment 6 is due Monday. (shorter – 3 questions) • Homework Assignment 7 posted tonight will be due Monday, March 24th • Data collection?

  3. Midterm 1 Results Average 33.32 Max 52.59 Min 6.21 Median 33.79 Stdev 17.89

  4. Average 33.32 Max 17.86 Min 6.21 Median 33.79 Stdev 17.89 Midterm 1

  5. Complex Choices • Multi-Objective Decision Making (MODM) • Multiple, Competing Goals • Maximize Tax Revenue • Minimize Tax Rate • Maximize Compliance • Multi-Attribute Decision/Utility Theory (MAUT) • Diverse Characteristics Aggregated to Single Value Measure • Price • Safety • Performance Municipal Fiscal Policy Buying a Car

  6. MAUT • MODM is typically dealt with using techniques such as Goal Programming and the Analytic Hierarchy Process • We will not cover MODM • MAUT involves an extension of our existing techniques to incorporate trade-offs • Trade-offs are expressions of preference

  7. Attributes • Basic Party Problem • Everything is reduced to dollars • MAUT Party Problem • U(x) is the utility of x • U(Party) = U(Cost) + U(Fun) + U(Attendance) • Multiple factors (attributes) influence our preferences for various outcomes • U(Party) is essentially a utility measure with multiple factors • MAUT Key: Can the attributes be traded-off? • Could the party still be “good” if the Cost goes up, provided that Fun and Attendance also go up? • THINK: Additive vs Multiplicative Value

  8. Choice Strategies • Non-Compensatory Strategies • Methods for choosing alternatives that do not allow for trade-offs between attributes • Compensatory Strategies • Decision maker can give up/get some of one attribute in exchange for another attribute or attributes to increase total value

  9. Non-Compensatory Strategies • Similar to simple heuristics • Easy to apply • Prone to biases and can be misleading • Lexicographic • Elimination-by-Aspects • Conjunctive • Disjunctive • Combinations

  10. Lexicographic Rule • Rank the attributes in order of importance • Rank all options on the most important attribute • Break ties by using next most important attribute • Pick option with best value on most important attribute • Problem: Only considers a single attribute when other attributes may also be important

  11. Elimination-by-Aspects Rule • Rank the attributes in order of importance • Establish a minimum acceptable level on each attribute • Eliminate alternatives that are unacceptable with respect to the most important attribute • Continue elimination with next most important attributes until only one alternative remains • Problem: Difficult to determine attribute importance independently of acceptability thresholds; “acceptability” can be arbitrary

  12. Conjunctive Rule • Establish a minimum or maximum acceptable level on each attribute • Alternatives found to be unacceptable on any attribute are eliminated • If no alternatives remain, weaken the acceptability level; if two or more remain, strengthen the acceptability level • Problem: Same issues as optimism/pessimism

  13. Disjunctive Rule • Establish a minimum or maximum excellence level on each attribute • Alternatives found to be excellent on any attribute are accepted • If no alternatives remain, weaken the excellence level; if two or more remain, strengthen the excellence level • Problem: Same issues as optimism/pessimism

  14. Compensatory Strategies • Additive Value Functions • Two questions: • How are the weights (wi) determined? • How are the individual attribute values (vi) determined?

  15. Additive Value Functions • Now, trade-offs are allowed: • Weights: How important is the car’s price relative to its performance? • Values: How much more valuable is a 0-100km time of 5.5 seconds over 6.5 seconds? = +

  16. Additive Value Functions in 5 Steps • Step 1: Check the validity of the additive value model • Step 2: Assess the single attribute value functions (vi) • Step 3: Assess the scaling constants (wi) • Step 4: Compute the overall value of each alternative • Step 5: Perform sensitivity analysis • Example: Buying a Car • 3 Choices • 3 Attributes (price, performance, braking)

  17. Step 1: Validity Check • With a linear AVF (additive value function) – what types of preference are ruled out? • “I only like high performance cars if they’re black” • “I would never work in a large city – unless it was for an investment bank” • For choices using an AVF to be rational, they must not only satisfy completeness and transitivity, but we will also require independence • Independence is an additional requirement for being able to use an additive value function to represent preferences

  18. Independence • Preferential Independence • Your preferences for more or less of one attribute are not influenced by the levels of other attributes • Choosing among job offers: Salary levels in NYC vs Erie • Difference Independence • The degree of preference among one attribute cannot be affected by another attribute • If you prefer NYC twice as much as Erie at a salary of $50K, then you must maintain that same degree of preference (2x) at a salary of $80K • Trade-Off Independence • How you trade-off any two attributes cannot be affected by a third

  19. Step 2: Value Functions • Determine the ranking of alternatives for a specific attribute (e.g., price) • Let the worst alternative be 0 and the best one be 100 • Determine intermediate values based on their relative similarity • Fit (or interpolate) a value function • Challenges? Checking all possible combinations/differences for inconsistencies and intransitivities

  20. Assessing Single Attribute Value Functions • Considering new cars costing $20-$50K • Set end points • v($50,000) = 0 • v($20,000) = 100 • Where should $35K be? • Suppose v($42K) = 50 • Ask: is v($42K) = 0.5 × v($20K)? • Ask: is v($42K) – v($50K) = v($20K) – v($42K)? • Elicit other points to complete curve

  21. Step 3: Comparing Attributes • Which of the attributes matters most? Are they equally influential? • Each individual attribute has now been measured on a scale of 0 – 100, but is v1 = 25 the same as v2 = 25? • We need an “exchange rate” to allow us to compare different attributes on the same scale? • wi should reflect the relative importance of the ranges of outcomes on the different attributes

  22. Method 1: Swing Weights • Consider an alternative having the worst level of each attribute • Suppose you could increase one attribute to its best level • Which one? Which would be second? • Assign a value of 100 to the most important attribute and values to the remaining attributes to reflect their relative importance

  23. Method 1: Swing Weights • Suppose A1 is most important, A2 is 1/2 as important, and A3 is 1/3 as important • If A1 = 100, then A2 = 50 and A3 = 33.3 • Normalize to sum to 1 • 100 + 50 + 33.3 = 183.3 • Weights by swing weight method: • A1 = 100/183.3 = 0.545 • A2 = 50/183.3 = 0.273 • A3 = 33.3/183.3 = 0.182

  24. Method 2: Direct Trade-Off • The direct trade-off method requires continuity (money – yes; city – no) • Swing weight method doesn’t • Infer values from comparative judgments • Tends to work best when money is one of the attributes and can be used as the “medium of exchange” • Suppose X and Y are two attributes • Let + and – reflect the best and worst levels for each attribute

  25. Method 2: Direct Trade-Off • Consider 2 alternatives: • A1 = {X+, Y-} • A2 = { ___, Y+} • Y+ is preferable to Y- • What value makes the decision maker indifferent? • The “blank” is typically monetary • Indifference implies that V(A1) = V(A2) • Since we know the attribute values (from Step 2), we can easily solve for the weights

  26. Method 2: Direct Trade-Off • Suppose A1 = {$16K, 9 s.} and A2 = {___, 6 s.} • What price produces indifference between A1 and A2? • Suppose it’s $21,000 We have 3 unknowns: w1, w2, and w3

  27. Method 2: Direct Trade-Off • Now compare another 2 attributes • A3 = {$16K, 160 ft.} • A4 = {___, 150 ft.} • How much more would you pay to move from 160 ft to 150 ft? • Suppose you’d pay $18K • Now we have 3 equations for our 3 unknowns • A1 = A2, A3 = A4, and Swi = 1

  28. Method 2: Direct Trade-Off • Use values elicited in Step 2 • Equation 1: • Equation 2: • Equation 3:

  29. Step 4: Compute Overall Value

  30. Step 5: Sensitivity Analysis • Compare rankings of alternatives using swing weights with those produced by direct trade-off • “Procedural Invariance” • Consistency/Biases in Elicitation? • Compare results with equal weighting • Do the weights matter? Robustness? • Is there a clear winner? • Suppose you are unsure about your preferences (at least expressed numerically)…

  31. Step 5: Sensitivity Analysis • Vary weights given to less important attributes • Suppose the decision maker is only confident of the importance ranking, but not necessarily the values • Does it matter if A2 is 1/2 as important as A1? 3/4? 1/4? 3/8? • Examine more trade-offs using the direct trade-off method • Search thoroughly for any intransitivities or inconsistencies in preferences

  32. Challenges to MAUT • GIGO (garbage in, garbage out) • These steps are all just meaningless calculations unless the elicitation is done properly • Also, if Step 1 (validity of the AVF) isn’t satisfied, the methodology is totally unreliable • Cognitive biases?

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