Economics of Risk and Uncertainty Summer Term 2011 Ulrich Schmidt

1. Economics of Risk and UncertaintySummer Term 2011Ulrich Schmidt Phone: +49 431 8801400 Email: us@bwl.uni-kiel.de

2. Outline • Basic Concepts • Utility Theory under Risk and Uncertainty • Applications of Utility Theory

3. Basic Literature • U. Schmidt, „Alternatives to Expected Utility: Some Formal Theories“, in: S. Barbarà, P.J. Hammond & C. Seidl (Hrsg.), Handbook of Utility Theory, Vol. II, Kluwer, Boston, 2004. • H. Bleichrodt & U. Schmidt, Applications of Non-Expected Utility, forthcoming in: P. Anand & C. Puppe, Handbook of Rational and Social Choice.

4. Outline • Basic Concepts • Utility Theory under Risk and Uncertainty • Applications of Utility Theory

5. Decision Theory: Descriptive Decision Theory Question: How do subjects typically behave in a given choice situation? Method: Empirical Investigations Goal: Assessment, Modelling, and Forecasting of Actual Behavior Prescriptive Decision Theory Question: How should a person behave in a given choice situation? Method: Derivation of Rationality Requirements and their Implications Goal: Decision Support and Development of Normative Models 1 Basic Concepts

6. 1 Basic Concepts Ordinal Value Functions: Definition: An ordinal value function is a numerical representation of a binary prefence relation, i.e. Axiom 1 (Completeness): Axiom 2 (Transitivity): Definition: A complete and transitive binary preference relation is called binary preference ordering.

7. 1 Basic Concepts Axiom 3 (Separability): There exists a countable set ,which is order dense with respect to on the set A, i.e. with Theorem 1: A binary preference relation can be represented by an ordinal utility function if and only if it satisfies axioms 1-3. The resulting ordinal value function is unique up to montonous transformations, i.e. a function v represens the same ordering as v*, if there exists a strictly increasing function ,such that

8. 1 Basic Concepts Cardinal Value Functions: We identify the preferences of a decision maker now by a quaternary preference relation , which means: : the transition from b to a is as least as good as the transition from d to c. A binary preference relation can be derive from a quaternary relation as follows:

9. 1 Basic Concepts Definition: A cardinal value function is a numerical representation of a quaternary preference relation, i.e. Theorem 2: A quaternary preference relation can be represented by a cardianl utility function, if and only if axioms 1, 2 and some further technical conditions are satisfied. The cardinal value function is unique up to positive linear transformations, i.e v represents the same preferences as v*, if there are two real numbers, and such that

10. 1 Basic Concepts Basic framework of risk and uncertainty: There is a set of possible states of the world and precisely one of these states will occur in the future but you do not know which one. Definitions: • Risk (Risiko): objective probabilities of the single states of the world are known. • Uncertainty (Unsicherheit): subjective probabilities are known • Ambiguity (Ambiguität): ordinal or upper and lower probabilities are known. • Complete Ignorance (Ungewissheit): no probability information.

11. 1 Basic Concepts Objective Probabilities • The classical notion of probabilites: • All elementary events have the same probability • Principle of insufficient reason of Laplace (1825) • Problem: Apart from games nearly never applicable in practice.

12. 1 Basic Concepts 2. Probabilities as relative frequencies: • Assume a process which can be replicated identically • Probability of an event equals then relative frequency after infinite repititions. • Law of large numbers • Problems: Processes cannot be replicated identically. Only finite number of replications possible

13. 1 Basic Concepts Subjective Probabilities: • Is the probability which a decision maker assigns to a particular event. • Is influenced by personal experiences but also by several biases • Can be derived from individual preferences

14. 1 Basic Concepts Subjective Probabilities: • Potential Biases: • Representativeness heuristic: People typically evaluate the probabilities by the degree to which A is representative of B or C and sometimes neglect base rates . • Availability heuristic: This heuristic is used to evaluate the frequency or likelihood of an event on the basis of how quickly instances or associations come to mind. • Anchoring and adjustment: People who have to make judgements under uncertainty use this heuristic by starting with a certain reference point (anchor) and then adjust it insufficiently to reach a final conclusion. Literature: Tversky, A. & Kahneman, D. (1974). Judgment under uncertainty: Heuristics and biases. Science185, 1124-1130.

15. 1 Basic Concepts Subjective Probabilities: • Example Representativeness: Linda is 31 years old, single, outspoken and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in antinuclear demonstrations. What is the probability that (Group a) Linda is a bank teller? (Group b) Linda is a bank teller and is active in the feminist movement? Literature: Tversky, A. & Kahneman, D. (1974). Judgment under uncertainty: Heuristics and biases. Science185, 1124-1130.

16. 1 Basic Concepts Subjective Probabilities: • Example Availability:

17. 1 Basic Concepts Subjective Probabilities: • Example Anchoring: • Group A: • Are more than 45% of the African nations members of the UN? • How many percent of the African nations are member of the UN • Group B: • Are more than 75% of the African nations members of the UN? • How many percent of the African nations are member of the UN

18. 1 Basic Concepts Probability Measures: • X: Set of all consequences • P: Set of all probability measures (lotteries) which are defined on X. • A probability measure assigns to each subset of X a real number such that the following three axioms of Kolmogoroff (1933) are satisfied • Axiom 1: • Axiom 2: • Axiom 3: • In the following we mostly restrict attention to probability measures with finite support, i.e. there exists a finite set with

19. 1 Basic Concepts Multi-stage lotteries: A multi-stage lottery and ist reduced one-stage form characterize the same probability measure. Therefore, if we define preferences on probability measures we implicitly assume the axiom of „reduction of multi-stage lotteries, which demands that a multi-stage lottery and ist reduced form are indifferent. Multi-stage Lottery: Reduced Form: x1 x1 0,5 0,25 0,5 x2 0,25 x2 0,5 x3 x3 0,5 0,25 0,5 x4 0,25 x4 0,5

20. 1 Basic Concepts Mixtures of Probability Measures: • assigns to each consequence x the probability • x is the probability measure where you win x with certainty • A coin flip between 0 and 100 is thus represented by 0.50 + 0.5100, whereas 0.5*0 + 0.5*100 = 50

21. Outline • Basic Concepts • Utility Theory under Risk and Uncertainty • 2.1 Normative Utility Theory • 2.2 Descriptive Utility Theory • Applications of Utility Theory

22. 2.1 Normative Utility Theory • Expected Value and Expected Utility • Stochastic Dominance • Risk Aversion • Two-Outcome Diagrams • The Triangle Diagram

23. 2.1 Normative Utility Theory • Expected Value and Expected Utility • Stochastic Dominance • Risk Aversion • Two-Outcome Diagrams • The Triangle Diagram

24. 2.1.1 Expected Value and Expected Utility Expected value (EV): • The expected value E(p) of a lottery p is the average consequnce to be expected if the lottery would be played out infinitly many times, i.e..

25. 2.1.1 Expected Value and Expected Utility St. Petersburg Paradox: • Lottery with infinite EV • Calculation of EV: 2 0,5 4 0,5 8 0,5 0,5 16 0,5 0,5 0,5 … 0,5

26. 2.1.1 Expected Value and Expected Utility Moral Expectation: • Proposed by Bernoulli (1738) to overcome the St. Petersburg Paradox: • Calculation of ME:

27. 2.1.1 Expected Value and Expected Utility Moral Expectation: • But also ME can lead to infinite values [Menger (1934)] • Calculation: → Problem can only be avoided if utility is bounded from above 4 0,5 16 0,5 64 0,5 0,5 0,5 …

28. 2.1.1 Expected Value and Expected Utility Expected Utility Theroy: Ordering: is complete and transitive on P. Continuity: If , then there exists Independence: implies p q r

29. 2.1.1 Expected Value and Expected Utility • Justifying independence • Two-Stage lotteries • Dynamic consistency

30. 2.1.1 Expected Value and Expected Utility Expected Utility Theory: Theorem: Consider a binary preference relation on the set P. The following statements are equivalent: • satisfies ordering, continuity, and independence. • There exist functions V: and u: , such that represents on P. The function u is unique up to positive linear transformations, i.e. a function u* represents the same preferences as u, if there exist and such that (U. Schmidt, Entwicklungstendenzen in der Entscheidungstheorie unter Risiko, BFuP 47 (1996), S. 663-678.)

31. 2.1 Normative Utility Theory • Expected Value and Expected Utility • Stochastic Dominance • Risk Aversion • Two-Outcome Diagrams • The Triangle Diagram

32. 2.1.2 Stochastic Dominance First-order Stochastic Dominance: Definition: A cumulative distribution function (cdf) assigns to each consequence the probability that the outcome of the lottery is not bigger than this consequence, i.e. with or • The cdf is non-decreasing. Let and be the minimal resp. maximal element of X. Then for every cdf it must hold that: and • Each probability measure is uniquely characterized by a cdf and vice versa

33. 2.1.2 Stochastic Dominance First-order Stochastic Dominance: Definition: A risk profile assigns to each consequence in X the probability that the outcome of the lottery will be bigger as this consequence, i.e. with or Definition: A lottery p dominates q stochastically ( ), if the corresponding risk profiles and satisfy: and for at least one Analogously, SD can be defined by cdf´s

34. 2.1.2 Stochastic Dominance First-order Stochastic Dominance: Theorem: If then it holds that for all functions , which are (strictly) increasing. • Consistency with stochastic dominance (i.e. ) is besides transitivity the mostly accepted criterion for rational choice under risk

35. 2.1.2 Stochastic Dominance Example for Stochastic Dominance: Lottery q: Lottery p: 100 100 0,25 0,5 0,25 0,25 50 50 0,25 0,5 0 0 F(0) = 0,25 R(0) = 0,75 F(50) = 0,5 R(50) = 0,5 F(100) = 1 R(100) = 0 F(0) = 0,5 R(0) = 0,5 F(50) = 0,75 R(50) = 0,25 F(100) = 1 R(100) = 0 → → p dominiates q

36. 2.1 Normative Utility Theory • Expected Value and Expected Utility • Stochastic Dominance • Risk Aversion • Two-Outcome Diagrams • The Triangle Diagram

37. 2.1.3 Risk Aversion Risk Aversion: Definition: The certainty equivalent x(p) of lottery p is that element of X, for which If X is a continuous set, then in expected utility theory each lottery has due to the continuity axiom a unique certainty equivalent. Definition: The risk premium of a decision maker for a lottery p is given by

38. 2.1.3 Risk Aversion Weak Risk Aversion: Definition: A decision maker is globally risk averse (resp. risk neutral, resp. risk loving) if Theorem: The following statements are equivalent:

39. 2.1.3 Risk Aversion Arrow-Pratt Measure: Definition: The Arrow-Pratt Measure (APM) of absolute risk aversion measures the degree of risk aversion and is invariant to linear transformations of the utility function. • Decision maker i is more risk averse than decision maker j, if their utility functions ui(x) and uj(x) imply:

40. 2.1.3 Risk Aversion Example: • Consider lottery p and ist certainty equivalent: 100 0,5 ~ x(p) 0 0,5 E(p) = 50 → RP(p) = E(p) – x(p) = 50 – x(p) RP(p) = 0 → (risk neutrality) RP(p) > 0 → (risk aversion) RP(p) < 0 → (risk seeking)

41. 2.1.3 Risk Aversion risk seeking (convex) u(x) risk averse (concave) 1 Risk neutral: RP(x) = 0 if u‘‘(x) = 0 Risk averse: RP(x) > 0 if u‘‘(x) < 0 Risk seeking: RP(x) < 0 if u‘‘(x) > 0 0,5 x x(p) 100 50

42. 2.1.3 Risk Aversion • Strong risk aversion (Rothschild and Stiglitz, 1970): • The right lottery can be obtained from the left one by a series of mean-preserving spreads is a mean-preserving spread of p if xi > xj • p dominates q by second-order stochastic dominance (SSD) if q can be obtained from p by a series of mean-preserving spreads

43. 2.1.3 Risk Aversion • Theorem: When p dominates q by SSD then EU(p) > EU(q) for all utility functions which are strictly concave • In EU weak and strong risk aversion are equivalent

44. 2.1 Normative Utility Theory • Expected Value and Expected Utility • Stochastic Dominance • Risk Aversion • Two-Outcome Diagrams • The Triangle Diagram

45. 2.1.4 Two-Outcome Diagrams Two-Outcome-Diagram: We consider lotteries with only two possible consequences, x1 and x2, where x1 occurs with fixed probability p and x2 with 1 – p. Indifference Curves:

46. 2.1.4 Two-Outcome Diagrams Indifference curves for risk neutrality:

47. 2.1.4 Two-Outcome Diagrams Example: • Consider the lottery p: 20 0,5 ~ x(p) 5 0,5 E(p) = 12,5 → RP(p) = E(p) – x(p) = 12,5 – x(p) If x(p) = 10 the risk premium is given by: RP(p) = E(p) – x(p) = 12,5 – 10 = 2,5

48. 2.1.4 Two-Outcome Diagrams • Lottery p in a two-outcome diagram: x1 30 Certainty line 25 20 15 Risk neutral (RP(p) = 0) 10 5 Risk averse (x(p) = 10) x2 30 10 15 20 25 5 RP(p) = E(p) – x(p) = 2,5

49. 2.1 Normative Utility Theory • Expected Value and Expected Utility • Stochastic Dominance • Risk Aversion • Two-Outcome Diagrams • The Triangle Diagram

50. 2.1.5 The Triangle Diagram • We consider lotteries with three possible consequences, x1 > x2 > x3. • Since p2 = 1 – p1 – p3, the set of all lotteries over these consequences can be represented in the (p1 , p3)-plane. • For a constant utility level we have: V* = p1u(x1) + (1 – p1 – p3)u(x2) + p3u(x3) which yields: • Since all utility values are constant this is the equation for a line which slope is independent of the utility level V*. • This means that indifference curves are parallel lines