Tutorial on Risk and Uncertainty Peter P. Wakker Part 1: Introduction into Prospect Theory. Part 2: Using Prospect Theory to Better Describe and Prescribe Decisions (Medical Application): Bleichrodt, Han, José Luis Pinto, & Peter P. Wakker (2001), “Making Descriptive Use of Prospect Theory to Improve the Prescriptive Use of Expected Utility,” Management Science 47, 1498  Part 3: Behavioral Econometrics in Practice: Abdellaoui, Mohammed, Carolina Barrios, & Peter P. Wakker (2007), “Reconciling Introspective Utility With Revealed Preference: Experimental Arguments Based on Prospect Theory,” Journal of Econometrics 138, 336  378. Part 4: Elementary Introduction into the Maths of Prospect Theory: Why It Is a Natural Dual to EU. Wharton, June 15, 2007 Part 4 was not presented and, hence is dropped. Of part 3 only quickly the TO curve and CE2/3 curve were presented and nothing about SP or CE1/3..

2 Expected value Simplest way to evaluate risky prospects: x1x1 xnxn p1p1 pnpn  p 1 x p n x n Violated by risk aversion: x1x1 xnxn p1p1 pnpn  p 1 x p n x n Part 1: Introduction into Prospect Theory

3 Expected utility (EU) Bernoulli: x1x1 xnxn p1p1 pnpn  p 1 U(x 1 ) p n U(x n ) Theorem. EU: Risk aversion  U concave U x U concave: Measure of risk aversion: –U´´/U´ (Pratt & Arrow). Other often-used index of risk aversion: –xU´´/U´.

Rotate left and flipped horizontally: 1 = w(.10)100 + w(.90)0 U(1) = 0.10U(100) U(0) = (normalization) Psychology since 1950: Psychology: 9 = w(.30)100 Assume following data regarding choice under risk 0 (e) (d) (c) p $0 $ $70 $30 $ (a) (b) 0.7 (c) $ p $0 $100 $70 $ (a) (b) (d) (e) $ ~ $1 0 (a) $81 $ ~ $9 $ ~ $25 $ ~ $ ~ $ (b)(d)(c) (e) EU: EU: U(9) = 0.30U(100) = EU: U(x) = pU(100) = p. Below is graph of U. next p.p. 8 underidentiied 4 Psychology: x = w(p)100. Below is graph of w(p) (= x/100).

Intuitive problem: U reflects value of money; not risk !? U depends on specific nature of money outcome. Different for # hours of listening to music; # years to live; # liters of wine; … nonquantitative outcomes (health states) 5

Lopes (1987, Advances in Experimental  ): Risk attitude is more than the psychophysics of money. Empirical problems: Plentiful (Allais, Ellsberg) One more (Rabin 2000): For small amounts EU  EV. However, empirically not so! 6

Psychologists: What economists do with money, is better done with probabilities! 7 w increasing, w(0) = 0, w(1) = 1.  pU(x) Economists At first, for simplicity, we consider U linear. Is proper for moderate amounts of money. p. 4 U/w graph p x 1–p 0  w(p)x Psychologists p x 1–p 0 Joint x 0  w(p)U(x) p 1–p

w(p)U(x) + ( 1 – w(p) ) U(y) Data with one nonzero outcome is underidentified for measuring w and U. Fortunately, two-outcome data is sufficiently rich to identify the functions. Then: 8 p x 1–p y  w(p)U(x) + w – (1–p)U(y) if x > y  0 if x > 0 > y

9 inverse-S, (likelihood insensitivity) p w expected utility motivational cognitive pessimism extreme inverse-S ("fifty-fifty") prevailing finding pessimistic "fifty-fifty" Abdellaoui (2000); Bleichrodt & Pinto (2000); Gonzalez & Wu 1999; Tversky & Fox, 1997.

10 Part 2: Prospect Theory to Better Describe and Prescribe Decisions (Medical Application)

surgery Patient with larynx-cancer (stage T3). Radio-therapy or surgery? radio- therapy artificial speech  0.6 recurrency, surgery cure normal voice 1p1p p nor- mal voice  or artifi- cial sp eech Hypothetical standard gamble question: artificial speech  recurrency cure artificial speech For which p equivalence? Patient answers: p = 0.9. Expected utility: U(  ) = 0; U(normal voice) = 1; U(artificial speech) = 0.9   0 = 0.9. U p UpUp EU Answer: r.th!

Analysis is based on EU!?!? “Classical Elicitation Assumption” I agree that EU is normative. Tversky, Amos & Daniel Kahneman (1986), “Rational Choice and the Framing of Decisions,” Journal of Business 59, S251  S278. P. S251: "Because these rules are normatively essential but descriptively invalid, no theory of choice can be both normatively adequate and descriptively accurate." p 1p1p  Perf. Health ~ artificial speech U = p Standard gamble question to measure utility: EU = p  1 + (1–p)  0 = p ? 12

13 Tversky, Amos & Daniel Kahneman (1986), “Rational Choice and the Framing of Decisions,” Journal of Business 59, S251  S278: “Indeed, incentives sometimes improve the quality of decisions, experienced decision makers often do better than novices, and the forces of arbitrage and competition can nullify some effects of error and illusion. Whether these factors ensure rational choices in any particular situation is an empirical issue, to be settled by observation, not by supposition (p. S273).” Common justification of classical elicitation assumption: EU is normative (von Neumann-Morgenstern). I agree that EU is normative. But not that this would justify SG (= standard gamble = “qol-probability measurement”) -analysis. SG measurement (as commonly done) is descriptive. EU is not descriptive. There are inconsistencies, so, violations. They require correction (? Paternalism!?).

14 Replies to discrepancies normative/descriptive in the literature: (1) Consumer Sovereignty ("Humean view of preference"): Never deviate from people's pref s. So, no EU analysis here! However, Raiffa (1961), in reply to violations of EU: "We do not have to teach people what comes naturally.“ We will, therefore, try more. (2) Interact with client (constructive view of preference). If possible, this is best. Usually not feasible (budget, time, capable interviewers …) (3) Measure only riskless utility. However, we want to measure risk attitude! (4) We accept biases and try to make the best of it.

15 That corrections are desirable, has been said many times before. Tversky & Koehler (1994, Psych. Rev.): “The question of how to improve their quality through the design of effective elicitation methods and corrective procedures poses a major challenge to theorists and practitioners alike.” E. Weber (1994, Psych. Bull.) “ …, and finally help to provide more accurate and consistent estimates of subjective probabilities and utilities in situations where all parties agree on the appropriateness of the expected-utility framework as the normative model of choice.” Debiasing (Arkes 1991 Psych. Bull. etc)

16 Schkade (Leeds, SPUDM ’97), on constructive interpretation of preference: “Do more with fewer subjects.” Viscusi (1995, Geneva Insurance): “These results suggest that examination of theoretical characteristics of biases in decisions resulting from irrational choices of various kinds should not be restricted to the theoretical explorations alone. We need to obtain a better sense of the magnitudes of the biases that result from flaws in decision making and to identify which biases appear to have the greatest effect in distorting individual decisions. Assessing the incidence of the market failures resulting from irrational choices under uncertainty will also identify the locus of the market failure and assist in targeting government interventions intended to alleviate these inadequacies.”

17 Million-$ question: Correct how? Which parts of behavior are taken as “bias,” to be corrected for, and which not? Which theory does describe risky choices better? Current state of the art according to me: Prospect theory, Tversky & Kahneman (1992).

First deviation from expected utility: probability transformation 18 p w+w Figure. The common weighting function (Luce 2000). w  is similar; Second deviation from expected utility: loss aversion/sign dependence. People consider outcomes as gains and losses with respect to their status quo. They then overweight losses by a factor = 2.25.

19 EU: U(x) = p. PT: U(x) = p p + (1  p) w + ( ) ww We: is wrong !! Have to correct for above “mistakes.” Not at all self-evident are: 1.value/utility of PT = normative utility for EU!? 2.probability weighting is bias to be corrected for!? 3.loss aversion is bias to be corrected for!? Still, these are my beliefs. Quantitative corrections proposed by Bleichrodt, Han, José Luis Pinto, & Peter P. Wakker (2001), "Making Descriptive Use of Prospect Theory to Improve the Prescriptive Use of Expected Utility," Management Science 47, 1498–1514.

Standard Gamble Utilities, Corrected through Prospect Theory, for p =.00,..., E.g., if p =.15 then U = 0.123

U p Corrected Standard Gamble Utility Curve 21

U SG  U CE ( at 1 st = CE(.10), …, at 5 th = CE(.90) ) 5 th 3d3d 1 st 2 nd 4 th *** * ** * ***   * Corrected (Prospect theory) U SG  U TO ( at 1 st = x 1, …, at 5 th = x 5 ) U CE  U TO ( at 1 st = x 1, …, 5 th = x 5 ) Classical (EU) 22

1 st utility measurement: Tradeoff (TO) method (Wakker & Deneffe 1996) Completely choice-based. Part 3: Behavioral Econometrics in Practice 23 Abdellaoui, Mohammed, Carolina Barrios, & Peter P. Wakker (2007), “Reconciling Introspective Utility With Revealed Preference: Experimental Arguments Based on Prospect Theory,” Journal of Econometrics 138, 336  378.

 ( U(t 1 )  U(t 0 ) ) =  ( U(2000)  U(1000) )  U(1000) +  U(t 1 ) =  U(2000) +  U(t 0 ); _ ( U(2000)  U(1000) ) Tradeoff (TO) method t2t t 1 ~     t6t t 5   ~     (= t 0 ) EU = U(t 2 )  U(t 1 ) = = = U(t 6 )  U(t 5 ) = U(t 1 )  U(t 0 ) =   _ ( U(2000)  U(1000) )    _ ( U(2000)  U(1000) )      6,000   ~ 200,000 t 1 26, 1 curve

? ? ? Tradeoff (TO) method 25 _ ( U(2000)  U(1000) ) t2t t 1 ~     t6t t 5   ~     (= t 0 ) EU = U(t 2 )  U(t 1 ) = = = U(t 6 )  U(t 5 ) = U(t 1 )  U(t 0 ) =   _ ( U(2000)  U(1000) )   _ ( U(2000)  U(1000) )     12,000   ~ 200,000 t 1 Prospect theory: weighted prob s (even unknown prob s ) 11 22 11 22 11 22 ! ! ! 29, curves; then 31, CE 1/3

1 0 U $ Normalize: U(t 0 ) = 0; U(t 6 ) = 1. t0t0 t1t1 t6t6 1/6 t5t5 5/6 t4t4 4/6 t3t3 3/6 t2t2 2/6 Consequently: U(t j ) = j/6. 26

2 nd utility measurement: Strength of Preference (SP) Based on direct judgment, not choice-based. 27

For which s 2 is ?s2s2 Strength of Preference (SP) For which s 6 is s 6 s 5 ~* t 1 t 0 ? We assume: U(s 2 ) – U(t 1 ) = U(t 1 ) – U(t 0 ) U(s 3 ) – U(s 2 ) = U(t 1 ) – U(t 0 ) U(s 6 ) – U(s 5 ) = U(t 1 ) – U(t 0 ) t1t0t1t0 t1t1 ~* For which s 3 is ?s3s3 t1t0t1t0 s2s2 ~*

CE 2/3 (EU) CE 2/3 (PT) corrects CE 2/3 (EU) FF CE 1/3 CE 2/3 (PT) SP TO Utility functions (group averages) 0 1/6 2/6 3/6 4/6 5/6 1 7/6 U t 0 = FF5, t 6 = FF26,068 30, nonTO,nonEU 32, power? 34, which th? PT! (then TO)) 36,concl 33, CE 2/3 31, CE 1/3 TO(PT) = TO(EU) CE 1/3 (PT) = CE 1/3 (EU) (gr.av.)

Question: Could this identity have resulted because the TO method does not properly measure choice-based risky utility? 30 (And, after answering this, what about nonEU?)

Certainty equivalent CE 1/3 (with good-outcome probability 1/3) 3 d utility measurement: t0t0 t 6   c2c2 ~ t0t0 c 2   c2c2 t 6   EU U(c 2 ) = 1/3 U(c 1 ) = 1/9 U(c 3 ) = 5/9 31 For which c 2 : ? c1c1 ~ For which c 1 : ? c3c3 ~ For which c 3 : ? 29, curves & RDU & PT (for gr.av.) 29, curves (Chris Starmer, June 24, 2005) on inverse-S: "It is not universal. But if I had to bet, I would bet on this one.".

32 Questions Could this identity have resulted because our experiment is noisy (cannot distinguish anything)? How about violations of EU?

Certainty equivalent CE 4 th utility measurement: t0t0 t 6   d2d2 ~ t0t0 d 2   d2d2 t 6   CE 2/3 (EU): U(d 2 ) = 2/3 U(d 1 ) = 4/9 U(d 3 ) = 8/9 CE 2/3 (PT) (gr.av): U(d 2 ) =.51 U(d 1 ) =.26 U(d 3 ) = d3d3 ~ For which d 3 : ? d1d1 ~ For which d 1 : ? For which d 2 : ? 29, curves 2/3 (with good-outcome probability 2/3)

And, EU is violated. 34 So, our experiment does have the statistical power to distinguish. Which alternative theory to use? Prospect theory.

p w /3 Fig. The common probability weighting function. w(1/3) = 1/3; 35 24,TOmethod 1/3 w(2/3) =.51 2/3.51 We re-analyze the preceding measurements (the curves you saw before) in terms of prospect theory; first TO.