DADSS Multiattribute Utility Theory

Administrative Details Homework Assignment 6 is due Monday. (slightly shorter) Homework Assignment 7 posted tonight will be due Monday, March 24 th Project meetings

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

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

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

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

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

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

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

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

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

Compensatory Strategies Additive Value Functions Two questions: How are the weights (  i ) determined? How are the individual attribute values (v i ) determined?

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? =+

Additive Value Functions in 5 Steps Step 1: Check the validity of the additive value model Step 2: Assess the single attribute value functions (v i ) Step 3: Assess the scaling constants (  i ) 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)

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

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

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

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

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 v 1 = 25 the same as v 2 = 25? We need an “exchange rate” to allow us to compare different attributes on the same scale?  i should reflect the relative importance of the ranges of outcomes on the different attributes

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

Method 1: Swing Weights Suppose A 1 is most important, A 2 is 1/2 as important, and A 3 is 1/3 as important If A 1 = 100, then A 2 = 50 and A 3 = 33.3 Normalize to sum to = Weights by swing weight method: A 1 = 100/183.3 = A 2 = 50/183.3 = A 3 = 33.3/183.3 = 0.182

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

Method 2: Direct Trade-Off Consider 2 alternatives: A 1 = {X +, Y - } A 2 = { ___, Y + } Y + is preferable to Y - What value makes the decision maker indifferent? The “blank” is typically monetary Indifference implies that V(A 1 ) = V(A 2 ) Since we know the attribute values (from Step 2), we can easily solve for the weights

Method 2: Direct Trade-Off Suppose A 1 = {$16K, 9 s.} and A 2 = {___, 6 s.} What price produces indifference between A 1 and A 2 ? Or phrased differently, how much would you pay to improve the acceleration from 9 sec to 6 sec? Suppose it’s $21,000 We have 3 unknowns:  1,  2, and  3

Method 2: Direct Trade-Off Now compare another 2 attributes A 3 = {$16K, 160 ft.} A 4 = {___, 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 A 1 = A 2, A 3 = A 4, and  i = 1

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

Step 4: Compute Overall Value

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)…

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 A 2 is 1/2 as important as A 1 ? 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

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 unreliable Cognitive biases?