Subjective Measurement Issues: Utility/Value and Probabilities

1. Subjective Measurement Issues: Utility/Value and Probabilities Baird: Managerial Decisions Under Uncertainty: Chapter 10 (The Theory and Application of Utility) Chesley: Elicitation of Subjective Probabilities: A Review, Accounting Review 50 (2), April 1975

2. Contents of lecture About utility functions • Motivation for incorporating utility into decision analysis (EMV, EU maximization) • Attitudes towards risk • Eliciting a person’s utility function • How does a person’s utility function look like? • Additional utility theory About value functions • What is a value function?

3. Contents of lecture … Subjective probability elicitation • Quantifying uncertainty: objective vs. subjective probabilities • Reliability, validity of probability encoding • Probability axioms • Encoding methods for subjective probabilities • Direct asking techniques

4. Motivation for incorporating utility into decision analysis • What does use of the Expected Monetary Value (EMV) criterion imply? • Show with examples that people violate the EMV criterion! Why do they violate it? • Replace EMV criterion with Expected Utility (EU) maximization; it does often change the outcome of the analysis! • Show that risk neutrality = EMV maximization

5. Attitudes towards risk • Risk neutrality, risk aversion, risk proneness • Show the connection to the shape of the utility function for money (linear, concave, convex) Examples: • u(x)= • u(x)= log M, u(x) = 0.4M + 100 • Risk premium = EMV – CE (Certainty Equivalent). Figure 10.14 from Baird.

6. Attitudes towards risk … continued • Arrow-Pratt measure of local risk aversion: r(x) = - u’’(x)/u’(x), example: u(x) = • If r(x) is positive for all x, then u(x) is concave and the DM is risk-averse; if r(x) is negative for all x, u(x) is convex and the DM is risk prone • Constant, increasing, decreasing risk aversion for monotonically increasing utility functions • Examples

7. Eliciting a person’s utility function • Von Neumann & Morgenstern method of standard gambles: indifference probability = utility • Derive your utility function for M (range 0 to 100,000€), scaling: u(0)=0, u(100,000)=1 Consider points 25,000€, 50,000€ and 75,000€ What probability p makes you indifferent between 25,000€ for certain and a lottery (100,000€, p, 0€)? u(25,000€) = p! Repeat the questioning for 50,000€ and 75,000€

8. Eliciting a person’s utility function … continued • Fit in a curve u(x) (in reality need more than 5 points) • Qualitative characteristics of u(x): monotonicity (monotonically increasing), attitude towards risk (concave, convex, linear) • Consistency checks • Note: Utility function only defined up to a positive monotone transformation • Multiattribute utility functions

9. How does a person’s utility function for money look like? • Commonly S-shaped, economists work with concave utility functions • Not necessarily stable in time • Everyone has a different utility function • Utility in terms of time, number of people laid off • Utilities traditionally ‘sacred’ – decision analysts normally only worry about consistency

10. Value function v(x) • Definitions: (cardinal) utility functions under uncertainty, (ordinal) value functions under certainty • Value function represents DM’s preferences: • Note1 : value function is represented by an ordinal scale (does not measure strength of preference) • Note 2: could be modified to measure strength of preference • Economists’ terminology sometimes different

11. Encoding methods • Self-elicitation versus interviewer elicitation • Direct questioning versus inferring from choices • The process (interviewer recommended!) • Motivate the subject and investigate the subject’s biases • Structure the uncertain quantity by having it clearly defined • Make the subject think fundamentally about the problem and avoid biases (if possible) • Encode the probability judgments • Verify the responses by checking for consistency (multiple estimations)

12. Direct asking techniques • Magnitude or direct estimation • What is the p(x equals or is less than k) (k specified)? • What is k such that p(x equals or is less than k) is the provided cumulative probability (probability provided)? • Odds method Estimate the ratio of two probabilities: what are the odds that boxer A will win boxer B? • Quartile or fractile assessment Seeks to find equally likely points Quartile assessments can be made by three splits of the range of the distribution

13. Direct asking techniques • Graphical techniques Histograms or smooth curves; either ask the individual to provide a graphical distribution or solicit his or her reaction to a given graph • Direct specification of the parameters of the underlying distribution Make assumptions about the distribution (for example normal), assess its parameters (mean and variance). Note: difficulty of assessing variances!

14. Direct asking techniques, indirect techniques • Equivalent prior sample method Ask the subject to specify the number of successes (r) and the sample size (n) such that the subjective probability is equivalent to having observed r successes in n trials • Indirect asking techniques: Assume knowledge of the individual’s underlying utility function to be able to infer probabilities from betting behavior. The specification of the utility function equally difficult?

15. Recommendations • Not fully clear! Hundreds of studies! • Use an interviewer • Take biases seriously (future lecture) • Use density functions rather than cumulative distribution functions

16. Behavioral Assumptions and Limitations of Decision Analysis Holloway: DecisionMakingUnderUncertainty, chapter 18, 1979 ornewer

17. Axioms Ordering of outcomes and transitivity (2 axioms) Reduction of compound uncertain events Continuity Substitutability Monotonicity

18. Ordering and transitivity A DM can order (establish preference or indifference between) any two alternatives. The ordering is transitive. Part 1 provides the assurance that the DM’s preferences exist over a complete set of alternatives. Paradox: do you then need decision aid? Note 1: preferences need not be explicitly available.Note 2: The DM must supply the preferences, no one else. Transitivity ensures a degree of consistency in the DM’s preferences. Illustrate transitivity!

19. Transitivity, Reduction of compound events Definition of transitivity: given that A is preferred to B, and that B is preferred to C, then transitivity implies that A is preferred to C Reduction of compound events: you are indifferent between a compound uncertain event and the simple uncertain event determined by reduction using standard probability calculus. See figure 18.2 in Holloway

20. Continuity For each possible consequence A, the DM can specify a number p (indifference probability) between 0 and 1, such that (s)he is indifferent between A and the lottery L(A1, p, A2), where A1 is preferred to A and A is preferred to A2. This axiom asserts that you can always find p, the probability of obtaining A1, such that you are indifferent between a1 and a2, where A is “between” A1 and A2 (in terms of preference) See figure 18.3 in Holloway

21. Substitutability (substitution axiom) If you are indifferent between some certain outcome A and some uncertain event when the two are considered separately, you are also indifferent between them when substituted into some uncertain event Note 1: this axiom is often called the independence axiom (choice or indifference between two alternatives is independent of the existence of a common, hence irrelevant prospect Note 2: this axiom can also be stated in terms of (weak) preference

22. Monotonicity Consider two lotteries L1 = (A1, p1, A2) and L2 = (A1, p2, A2). L1 is preferred to L2 if and only if p1 is greater than p2, where A1 is preferred to A2. Seems rather straightforward! Note: the outcomes in both lotteries are identical, but the probabilities different

23. Implications of the axioms Given any two uncertain events B1 and B2, if axioms 1 through 5 hold, there exist numbers (representing preferences or utilities) associated with payoffs for outcomes, such that the overall preference between the uncertain events is reflected by the expected values of the utilities for each event. In other words, if the behavioral assumptions are satisfied, the expected utility maximization is legitimate and will result in the most preferred choice. See Holloway for an example showing, how the axioms lead to the conclusion stated. Are you a utility maximizer?

24. Limitations imposed by the axioms Are the axioms violated in practice? Occasionally or in a systematic way? Which ones? What should we do about such violations? Transitivity axiom (for individuals) • There is empirical evidence that transitivity is often violated by decision makers (Tversky: Intransitivity of Preference, 1969 etc.) • What explanations can you offer? • What should we do about intransitivity? Correct it when detected?

25. Limitations continuity, monotonicity Continuity • It has been argued that with extreme outcomes (such as death), one alternative is strictly preferred to the other, unless p = 1. Counter example: envelope containing money left in a hazardous place! Monotonicity (difference in the time uncertainty is resolved) • The axiom implies that, for two uncertain events with identical payoffs, an individual’s preference will depend only on the probability of winning. • Suppose that C represents a flip of a coin today with a payoff in 1 year, and D represents a flip of the same coin 1 year from now with immediate payoff; would you still be indifferent? Assume the payoff is 1000,000 €!

26. Limitations substition (independence) axiom Violation of this axiom was first demonstrated by Allais (1953) – the Allais paradox (the common consequences effect) Compare A&B and A’&B’

27. Limitations substition (independence) axiom - continued A frequent choice pattern is A and B’! Let us show that such a pattern violates this axiom: • Let C be a lottery offering 5M with probability 10/11 and otherwise 0; let D offer 0 for certain • Consequently we have: • A = 0.11 A + 0.89A; B = 0.11C + 0.89A A’ = 0.11A + 0.89D; B’ = 0.11C + 0.89D Preference between A&B and A’&B’ should depend on preference between A and C (A and D represent common consequences)!