1. Dynamic Strategic Planning Richard de Neufville, Joel Clark, and Frank R. Field Massachusetts Institute of Technology Utility AssessmentSlide 1 of 33 Utility Analysis Dynamic Strategic Planning

2. Dynamic Strategic Planning Richard de Neufville, Joel Clark, and Frank R. Field Massachusetts Institute of Technology Utility AssessmentSlide 2 of 33 Value Functions l In General: Preference Measure PM = f(X) where X = vector of attributes l Semantic Caution: Value –Value in Exchange –Value in Use –“Fair Market Value”

3. Dynamic Strategic Planning Richard de Neufville, Joel Clark, and Frank R. Field Massachusetts Institute of Technology Utility AssessmentSlide 3 of 33 Value Function - V(X) l Definition: –V(X) is a means of ranking the relative preference of an individual for a bundle on consequences, X

4. Dynamic Strategic Planning Richard de Neufville, Joel Clark, and Frank R. Field Massachusetts Institute of Technology Utility AssessmentSlide 4 of 33 Basic Axioms of Value Functions - V(X) l Completeness or Complete Preorder –People have preferences over all X i l Transitivity –If X 1 is preferred to X 2 ; and X 2 is preferred to X 3 ; Then X 1 is preferred to X 3 –Caution: Assumed True for Individuals; NOT Groups

5. Dynamic Strategic Planning Richard de Neufville, Joel Clark, and Frank R. Field Massachusetts Institute of Technology Utility AssessmentSlide 5 of 33 Basic Axioms of Value Functions - V(X) (cont’d) l Monotonicity or Archimedean Principle –For any X i (X*  X i  X * ) there is a w (0 < w < 1) such that V(X i ) = w V(X*) + (1 - w) V(X * ) –That is, More is Better (or Worse)

6. Dynamic Strategic Planning Richard de Neufville, Joel Clark, and Frank R. Field Massachusetts Institute of Technology Utility AssessmentSlide 6 of 33 Another Preference Function l Represents a Benefit Which Ultimately Becomes Undesirable

7. Dynamic Strategic Planning Richard de Neufville, Joel Clark, and Frank R. Field Massachusetts Institute of Technology Utility AssessmentSlide 7 of 33 Consequence of V(X) Axioms l Existence of V(X) l Ranking Only Strategic Equivalence of Many Forms of V(X) Any Monotonic Transform of a V(X) is Still an Equivalent V(X) e.g., V(X 1, X 2 )= X 1 2 X 2 = 2 log (X 1 ) + log (X 2 )

8. Dynamic Strategic Planning Richard de Neufville, Joel Clark, and Frank R. Field Massachusetts Institute of Technology Utility AssessmentSlide 8 of 33 Value Functions

9. Dynamic Strategic Planning Richard de Neufville, Joel Clark, and Frank R. Field Massachusetts Institute of Technology Utility AssessmentSlide 9 of 33 Utility Function - U(X) l Definition: –U(X) Is A Special V(X), Defined In An Uncertain Environment l It Has A Special Advantage –Units of U(X) Do Measure Relative Preference –Can Be Used In Meaningful Calculations

10. Dynamic Strategic Planning Richard de Neufville, Joel Clark, and Frank R. Field Massachusetts Institute of Technology Utility AssessmentSlide 10 of 33 Some Conventional Definitions Unfortunately, Terms Can Be Misleading) l Risk Aversion Risk Averse if X is Preferred to an Uncertain Event Whose Expected Value is X l Risk Neutral

11. Dynamic Strategic Planning Richard de Neufville, Joel Clark, and Frank R. Field Massachusetts Institute of Technology Utility AssessmentSlide 11 of 33 Some Conventional Definitions (cont’d) l Risk Positive (Preference, or Prone!) l Risk Premium

12. Dynamic Strategic Planning Richard de Neufville, Joel Clark, and Frank R. Field Massachusetts Institute of Technology Utility AssessmentSlide 12 of 33 Representation of “Risk Averse” Behavior

13. Dynamic Strategic Planning Richard de Neufville, Joel Clark, and Frank R. Field Massachusetts Institute of Technology Utility AssessmentSlide 13 of 33 Utility Assessment l Basic Axioms l Example l Interview Process l Procedures –Conventional –New l Discussion

14. Dynamic Strategic Planning Richard de Neufville, Joel Clark, and Frank R. Field Massachusetts Institute of Technology Utility AssessmentSlide 14 of 33 Utility Function - U(X) l Definition: –U(X) is a Special V(X), –Defined in an Uncertain Environment l It has a Special Advantage –Units of U(X) DO measure relative preferance –CAN be used in meaningful calculations

15. Dynamic Strategic Planning Richard de Neufville, Joel Clark, and Frank R. Field Massachusetts Institute of Technology Utility AssessmentSlide 15 of 33 Basic Axioms of U(x) l Probability –Probabilities exist - can be quantified –More is better A = p’ 1-p’ X1 X2 B = p” 1-p” X1 X2 If X1 > X2; A > B if p’ > p”

16. Dynamic Strategic Planning Richard de Neufville, Joel Clark, and Frank R. Field Massachusetts Institute of Technology Utility AssessmentSlide 16 of 33 Basic Axioms of U(x) (cont’d) l Preferences –Linear in Probability (substitution/independence) - Equals can be substituted if a subject is indifferent between A and B p A C = p B C Not a good assumption for small p (high consequences) !

17. Dynamic Strategic Planning Richard de Neufville, Joel Clark, and Frank R. Field Massachusetts Institute of Technology Utility AssessmentSlide 17 of 33 Cardinal Scales l Units of interval are equal, therefore averages and arithmetic operations are meaningful l Two types exist –Ratio Zero value implies an absence of phenomenon e.g., Distance, Time note: F’(x) = a F(x) defines an equivalent measure (e.g., meters and feet)

18. Dynamic Strategic Planning Richard de Neufville, Joel Clark, and Frank R. Field Massachusetts Institute of Technology Utility AssessmentSlide 18 of 33 Cardinal Scales (cont’d) l Ordered Metric Zero is relative, arbitrary e.g., Temperature define two points: 0 degrees C - freezing point of pure water 100 degrees C - boiling point of pure water at standard temperature and pressure 0 degrees F - freezing point of salt water 100 degrees F - What? Note: f’(x) = a f(x) + b (e.g. F = (9/5) C + 32 equivalent measures under a positive linear transformation

19. Dynamic Strategic Planning Richard de Neufville, Joel Clark, and Frank R. Field Massachusetts Institute of Technology Utility AssessmentSlide 19 of 33 Consequences of Utility Axioms l Utility exists on an ordered metric scale l To measure, sufficient to –Scale 2 points arbitrarily –obtain relative position of others by probability weighting e.g., Equivalent = (X*, p; X * ) Similar to triangulation in surveying

20. Dynamic Strategic Planning Richard de Neufville, Joel Clark, and Frank R. Field Massachusetts Institute of Technology Utility AssessmentSlide 20 of 33 Interview Issues l Put person at ease –this individual is expert on his values –his opinions are valued –there are no wrong answers –THIS IS NOT A TEST!! l Scenario relevant to –person –issues to be evaluated

21. Dynamic Strategic Planning Richard de Neufville, Joel Clark, and Frank R. Field Massachusetts Institute of Technology Utility AssessmentSlide 21 of 33 Interview Issues (cont’d) l Technique for obtaining equivalents: BRACKETING l Basic element for measurement: LOTTERIES

22. Dynamic Strategic Planning Richard de Neufville, Joel Clark, and Frank R. Field Massachusetts Institute of Technology Utility AssessmentSlide 22 of 33 Nomenclature l Lottery A risky situation with outcomes 0 j at probability p j Written as (0 1, p 1 ; 0 2, p 2 ;...) l Binary Lottery A lottery with only two branches, entirely defined by X u, p u, X L p(X L ) = 1 - P u Written as (Xu, Pu; X L ) p1 pn 01 02...... On p 1-p XuXu XLXL

23. Dynamic Strategic Planning Richard de Neufville, Joel Clark, and Frank R. Field Massachusetts Institute of Technology Utility AssessmentSlide 23 of 33 Nomenclature (cont’d) l Elementary Lottery Lottery where one outcome equals zero, i.e. status quo written as (X,p)

24. Dynamic Strategic Planning Richard de Neufville, Joel Clark, and Frank R. Field Massachusetts Institute of Technology Utility AssessmentSlide 24 of 33 Utility Measurement l Conventional Method l Certainty Equivalent - Balance Xi and a lottery –Define X* - best possible alternative on the range Define X * - worst possible alternative on the range –Assign convenient values - U(X*) = 1.0; U(X * ) = 0.0 –Conduct data collection/interview to find Xi and p Note: U(Xi) = p –Generally p = 0.5; 50:50 lotteries –Repeat, substituting new Xi into lottery, as often as desired e.g. X2 = (X1, 0.5; X) p 1-p X* U(X*) = 1.0 X * U(X * ) = 0.0 X i ~

25. Dynamic Strategic Planning Richard de Neufville, Joel Clark, and Frank R. Field Massachusetts Institute of Technology Utility AssessmentSlide 25 of 33 Utility Measurement - New Method l Avoid Certainty Equivalents to Avoid “Certainty Effect” l Consider a “Lottery Equivalent” –Rather than Comparing a Lottery with a Certainty –Reference to a Lottery is Not a Certainty l Thus l Vary “Pe” until Indifferent between Two Lotteries. This is the “Lottery Equivalent” pepe 1-p e X* X*X* 0.5 XiXi X*X* vs

26. Dynamic Strategic Planning Richard de Neufville, Joel Clark, and Frank R. Field Massachusetts Institute of Technology Utility AssessmentSlide 26 of 33 Utility Measurement - New Method (contd) l Analysis (X*, Pe; X * ) ~ (Xi, P; X *) ý PeU(X*) + (1-Pe)U(X * ) = P U(Xi) + (1-P) U(X * ) Pe (U(X*) - U(X * )) = P (U(Xi) - U(X * )) Pe = P U(Xi) ý U(Xi) = Pe/P; or U(Xi) = 2 Pe when P = 0.5 l Graph l Big Advantage - Avoids Large Errors (+/- 25% of “Certainty Equivalent” Method)

27. Dynamic Strategic Planning Richard de Neufville, Joel Clark, and Frank R. Field Massachusetts Institute of Technology Utility AssessmentSlide 27 of 33

28. Dynamic Strategic Planning Richard de Neufville, Joel Clark, and Frank R. Field Massachusetts Institute of Technology Utility AssessmentSlide 28 of 33 Lotteries - Central to Utility Measurement l Uncertainty –Basis for Assessment of Utility –Motivates Decision Analysis l Lottery - Formal Presentation of Uncertain Situation l Utility Assessment - Compares Preferability of Alternative of Known Value with Alternative of Known Value l How Does One Extract Utility Information from Interview Data? l How Does One Construct Lottery Basis for Interview?

29. Dynamic Strategic Planning Richard de Neufville, Joel Clark, and Frank R. Field Massachusetts Institute of Technology Utility AssessmentSlide 29 of 33 “Buying and Selling Lotteries” l Observable Feature of Daily Existence l Obvious One Include: –Buying Lottery tickets –Gambling; Other Games of Chance –Purchase of Insurance l Subtler Ones Are: –Crossing a Street against the Lights –Exceeding the Speed Limit –Illegal Street Parking –Smoking; Overeating; Drug-Taking l Question: How to Analyze This Behavior?

30. Dynamic Strategic Planning Richard de Neufville, Joel Clark, and Frank R. Field Massachusetts Institute of Technology Utility AssessmentSlide 30 of 33 Two Basic Lottery Transactions l Buying of Lotteries –In Absence of Transaction, Subject “Holds” an Object of Value –In Exchange for the Lottery, Subject Gives Up Valued Object –Buying “Price” Defines Net Value of Purchased Lottery l Selling of Lotteries –In Absence of Transaction, Subject “Holds” a Lottery –In Exchange for the Lottery, Subject Receives a Valued Object

31. Dynamic Strategic Planning Richard de Neufville, Joel Clark, and Frank R. Field Massachusetts Institute of Technology Utility AssessmentSlide 31 of 33 Two Basic Lottery Transactions (contd) l Selling of Lotteries (contd) –Selling “Price” Defines Value of Sold Lottery l Analytically Distinct Transactions; Must be Treated Differently

32. Dynamic Strategic Planning Richard de Neufville, Joel Clark, and Frank R. Field Massachusetts Institute of Technology Utility AssessmentSlide 32 of 33 Selling Lotteries l Generally Easier to Understand l Initially, Subject Holds a Lottery Example, You Own a 50:50 Chance to Win $100 p = 0.5 $100 $0

33. Dynamic Strategic Planning Richard de Neufville, Joel Clark, and Frank R. Field Massachusetts Institute of Technology Utility AssessmentSlide 33 of 33 Selling Lotteries (contd) l Subject Agrees to Exchange (Sell) this Lottery for No Less Than SP = Selling Price Example: $30 $30 ~ p = 0.5 $100 $0 This is Called an “Indifference Statement”

34. Dynamic Strategic Planning Richard de Neufville, Joel Clark, and Frank R. Field Massachusetts Institute of Technology Utility AssessmentSlide 34 of 33 Selling Lotteries - Alternative View l Another way to look at lottery transactions is to express them as decision analysis situations. Selling a lottery can be represented as follows: l When the two alternative strategies are equally valued, then we can construct an indifference statement using the two sets of outcomes. C $30 $100 $0 D Keep Sell p = 0.5

35. Dynamic Strategic Planning Richard de Neufville, Joel Clark, and Frank R. Field Massachusetts Institute of Technology Utility AssessmentSlide 35 of 33 Selling Lotteries (contd) $30 ~ p = 0.5 $100 $0 l Based on this Indifference Statement, Utility Values can be determined –Set U($0) = 0.0 and U($100) = 1.0. –Translate the Indifference Statement into a Utility Statement: U($30) = 0.50 U($0)  0.50 U($100) –Solve for U($30) U($30) = 0.50 (0) + 0.50 U($100) = 0.50

36. Dynamic Strategic Planning Richard de Neufville, Joel Clark, and Frank R. Field Massachusetts Institute of Technology Utility AssessmentSlide 36 of 33 Selling Lotteries (contd)

37. Dynamic Strategic Planning Richard de Neufville, Joel Clark, and Frank R. Field Massachusetts Institute of Technology Utility AssessmentSlide 37 of 33 Buying Lotteries l The “Other” Side of the Transaction l Subtle, but Critical Analytical Difference l Source of Difference: Buying Price Changes Net Effect of Lottery l Example: Look at the Buyer in the Last Example This Lottery was Purchased for $30 $100 $0 p = 0.5 What is the Appropriate Indifference Statement?

38. Dynamic Strategic Planning Richard de Neufville, Joel Clark, and Frank R. Field Massachusetts Institute of Technology Utility AssessmentSlide 38 of 33 Buying Lotteries (contd) l Indifference Statement $0 ~ $70 -$30 p = 0.5 Must Explicitly Consider “Do Nothing” vs Net Outcomes l Note: Net Outcomes, Not Original Outcomes, Determine Indifference Statement – Set U(-$30) = 0; U($70) = 1 – U($0) = 0.5 U(-$30) + 0.5 U($70) – U($0) = 0.5

39. Dynamic Strategic Planning Richard de Neufville, Joel Clark, and Frank R. Field Massachusetts Institute of Technology Utility AssessmentSlide 39 of 33 Buying Lotteries (contd) l Again, recast the buying situation as a decision tree l If the buyer is just indifferent between the two decision outcomes, then the following indifference statement must hold C $0 $100 - $30 $0 - $30 D Buy Do Nothing p = 0.5 $0 ~ $70 -$30 p = 0.5

40. Dynamic Strategic Planning Richard de Neufville, Joel Clark, and Frank R. Field Massachusetts Institute of Technology Utility AssessmentSlide 40 of 33 Buying Lotteries (contd) l Resulting Utility Function is Different ý Seller ýBuyer l This Should Not be Surprising. If the Utility Functions were Not Different, the Transaction would Not Have Taken Place!

41. Dynamic Strategic Planning Richard de Neufville, Joel Clark, and Frank R. Field Massachusetts Institute of Technology Utility AssessmentSlide 41 of 33 Exercises: Buying and Selling Lotteries l Given a Transaction, Generate the Indifference Statement ý Buy this Lottery for $35 ý Sell this Lottery for $50 ý Pay Someone $30 to Take This Lottery 0.5 0.75 0.5 $0 -$50 $165 -$50 -$15 $200

42. Dynamic Strategic Planning Richard de Neufville, Joel Clark, and Frank R. Field Massachusetts Institute of Technology Utility AssessmentSlide 42 of 33 Indifference Statements Let U($165) = 1 U(-$50) = 0 Then U($0) = 0.50 U($50) = 0.75 U(-$30) = 0.25 $0 ~ 0.5 $50 ~ -$30 ~ 0.75 0.5 $165 -$50 $165 -$50 $0

43. Dynamic Strategic Planning Richard de Neufville, Joel Clark, and Frank R. Field Massachusetts Institute of Technology Utility AssessmentSlide 43 of 33 Utility Result