1. Combining Bayesian Beliefs and Willingness to Bet to Analyze Attitudes towards Uncertainty by Peter P. Wakker, Econ. Dept., Erasmus Univ. Rotterdam (joint with Mohammed Abdellaoui & Aurélien Baillon) RUD, Tel Aviv, June 24 '07 Topic: Uncertainty/Ambiguity.

2. 2 Making uncertainty/ambiguity more operational: measuring, predicting, quantifying completely, in tractable manner. No (new) maths; but "new" (mix of) concepts:  uniform sources;  source-dependent probability transformation.

3. 1. Introduction Good starting point for uncertainty: Risk. Many nonEU theories exist, virtually all amounting to: x  y  0; xpy  w(p)U(x) + ( 1–w(p) ) U(y); Relative to EU: one more graph … 3

4. 4 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.

5. Now to Uncertainty (unknown probabilities); In general, on the x-axis we have events. So, no nice graphs … 5

6. Many advanced theories; mostly ambiguity-averse 6 CEU (Gilboa 1987; Schmeidler 1989) PT (Tversky & Kahneman 1992) Multiple priors (Gilboa & Schmeidler 1989) Endogeneous definitions (Epstein, Zhang, Kopylov, Ghirardato, Marinacci) Smooth (KMM; Nau) Variational model (Maccheroni, Marinacci, Rustichini) Many tractable empirical studies; also inverse-S Curley & Yates 1985 Fox & Tversky 1995 Biseparable (Ghirardato & Marinacci 2001) Choice-based Kilka & We- ber 2001 Cabantous 2005 di Mauro & Maffioletti 2005 Nice graphs, but x-axis- problem: choice-less probability-inputs there We connect Einhorn & Hogarth 1985 next p.p. 9 (theory)

7. Einhorn & Hogarth 1985 (+ 1986 + 1990). Over 400 citations after '88. For ambiguous event A, take "anchor probability" p A (c.f. Hansen & Sargent). Weight S(p A ): S(p A ) = (1 –  )p A +  (1 – p A  );  : index of inverse-S (regression to mean);   ½.  : index of elevation (pessimism/ambiguity aversion); 7

8. Einhorn & Hogarth 1985 Graphs: go to pdf file of Hogarth & Einhorn (1990, Management Science 36, p. 785/786). Problem of the x-axis … 8 p. 6 (butter fly-theories)

9. 2. Theory Only binary acts with gains. All popular static nonEU theories (except variational): x  y  0; xEy  W(E)U(x) + ( 1–W(E) ) U(y). (Ghirardato & Marinacci 2001). For rich S, such as continuum, general W is too complex. 9

10. Machina & Schmeidler (1992), probabilistic sophistication: x  y; xEy  w ( P(E) ) U(x) + ( 1–w ( P(E) ) ) U(y). Then still can get nice x-axis for uncertainty! However, 10

11. Common preferences between gambles for $100: (R k : $100)  (R u : $100) (B k : $100)  (B u : $100) > 11 Ellsberg paradox. Two urns with 20 balls. Ball drawn randomly from each. Events: R k : Ball from known urn is red. B k, R u, B u are similar. Known urn k 10 R 10 B 20 R&B in unknown proportion Unknown urn u ?20–?  P(R k ) > P(R u )  P(B k ) > P(B k ) + 1 + 1 > < Under probabilistic sophistication with a (non)expected utility model:

12. Ellsberg: There cannot exist probabilities in any sense. No "x-axis" and no nice graphs … 12 (Or so it seems?)

13. > Common preferences between gambles for $100: (R k : $100)  (R u : $100) (B k : $100)  (B u : $100) 20 R&B in unknown proportion Ellsberg paradox. Two urns with 20 balls. Ball drawn randomly from each. Events: R k : Ball from known urn is red. B k, R u, B u are similar. 10 R 10 B Known urn k Unknown urn u ?20–?  P(R k ) > P(R u )  P(B k ) > P(B k ) ++ 1 1 > < Under probabilistic sophistication with a (non)expected utility model: 13 two models, depending on source reconsidered.

14. Step 1 of our approach (to operationalize uncertainty/ambiguity): Distinguish between different sources of uncertainty. Step 2 of our approach: Define sources within which probabilistic sophistication holds. We call them Uniform sources. 14

15. Step 3 of our approach: Develop a method for (theory-free) * eliciting probabilities within uniform sources; empirical elaboration of Chew & Sagi's exchangeability. * Important because we will use different decision theories for different sources 15

16. Step 4 of our approach: Decision theory for uniform sources S, source- dependent. E denotes event w.r.t. S. x  y; xEy  w S ( P(E) ) U(x) + ( 1– w S ( P(E) ) ) U(y). w S : source-dependent probability transformation. (Einhorn & Hogarth 1985; Kilka & Weber 2001) Ellsberg: w u (0.5) < w k (0.5) u: k: unknown known (Choice-based) probabilities can be maintained. We get back our x-axis, and those nice graphs! 16

17. We have reconciled Ellsberg 2-color with Bayesian beliefs! (Also KMM/Nau did partly.) We cannot do so always; Ellsberg 3-color (2 sources!?). 17

18. 18 ` c = 0.08 w(p) Fig.a. Insensitivity index a: 0; pessimism index b: 0. Figure 5.2. Quantitative indexes of pessimism and likelihood insensitivity 0  0.11 = c 1 0.89 d = 0.11 Fig.b. Insensitivity index a: 0; pessimism index b: 0.22. c = 0.11 d = 0.11 Fig.c. Insensitivity index a: 0.22; pessimism index b: 0. 0 d = 0.14 Fig.d. Insensitivity index a: 0.22; pessimism index b: 0.06. d = 0 1 p c = 0 0 Theory continued: (Chateauneuf, Eichberger, & Grant 2005 ; Kilka & Weber 2001; Tversky & Fox 1995)

19. 3. Let the Rubber Meet the Road: An Experiment Data: 19 4 sources: 1.CAC40; 2.Paris temperature; 3.Foreign temperature; 4.Risk.

20. Method for measuring choice-based probabilities 20 EEEEEE Figure 6.1. Decomposition of the universal event a 3/4 E a 1/2 a 1/4 a 1/8 a 3/8 E b1b1 a 5/8 a 7/8 b0b0 a 3/4 a 1/2 a 1/4 EE b1b1 b0b0 E E a 1/2 E b1b1 b0b0 E E = S b1b1 b0b0 The italicized numbers and events in the bottom row were not elicited.

21. 21 30 25 Median choice-based probabilities (real incentives) Real data over 1900  2006 0.0 35 20 15 10 0.8 0.6 0.4 0.2 1.0 Figure 7.2. Probability distributions for Paris temperature Median choice-based probabilities (hypothetical choice) 0.0 Median choice-based probabilities (real incentives) Real data over the year 2006 01 23 11 22 33 0.8 0.6 0.4 0.2 1.0 Figure 7.1. Probability distributions for CAC40 Median choice-based probabilities (hypothetical choice) Results for choice-based probabilities Uniformity confirmed 5 out of 6 cases.

22. Certainty-equivalents of 50-50 prospects. Fit power utility with w(0.5) as extra unknown. 22 0 Hypothetical Real 1 23 0 1 0.5 Figure 7.3. Cumulative distribution of powers Method for measuring utility Results for utility

23. 23 Results for uncertainty ("ambiguity?")

24. 24 0.125 0 0 Figure 8.3. Probability transformations for participant 2 Fig. a. Raw data and linear interpolation. 0.25 0.875 0.75 1 0.50 0.1250.875 0.25 0.50 0.751 Paris temperature; a = 0.78, b = 0.12 foreign temperature; a = 0.75, b = 0.55 risk: a = 0.42, b = 0.13 Within-person comparisons Many economists, erroneously, take this symmetric weighting fuction as unambiguous or ambiguity- neutral.

25. 25 participant 2; a = 0.78, b = 0.69 0 * Fig. a. Raw data and linear interpolation. * Figure 8.4. Probability transformations for Paris temperature 0.25 0.125 0.875 0.75 1 0.50 0.125 0.8750.2500.500.751 participant 48; a = 0.21, b = 0.25 Between-person comparisons

26. Example of predictions [Homebias; Within- Person Comparison; subject lives in Paris]. Consider investments. Foreign-option: favorable foreign temperature: $40000 unfavorable foreign temperature: $0 Paris-option: favorable Paris temperature: $40000 unfavorable Paris temperature: $0 Assume in both cases: favorable and unfavo- rable equally likely for subject 2; U(x) = x 0.88. Under Bayesian EU we’d know all now. NonEU: need some more graphs; we have them! 26

27. 27 Paris temperature Foreign temperature decision weight expectation certainty equivalent uncertainty premium risk premium ambiguity premium 0.490.20 20000 177836424 135762217 5879 7697–3662 Within-person comparisons (to me big novelty of Ellsberg):

28. 28 Subject 2, p = 0.125 decision weight expectation certainty equivalent uncertainty premium risk premium ambiguity premium 0.350.67 500035000 12133 159742732 5717 10257–3099 Subject 48, p = 0.125 Subject 2, p = 0.875 Subject 48, p = 0.875 500035000 0.080.52 2268 654 9663 –39 –4034 –7133 2078 9624 19026 25376 Between-person comparisons; Paris temperature

29. Conclusion: By (1) recognizing importance of uniform sources and source-dependent probability transformations; (2) carrying out quantitative measurements of (a) probabilities (subjective), (b) utilities, (c) uncertainty attitudes (the graphs), all in empirically realistic and tractable manner, we make ambiguity completely operational at a quantitative level. 29

30. The end 30