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 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.

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 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.

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

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)

Einhorn & Hogarth 1985 ( ). 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

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)

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

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

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 ) > < Under probabilistic sophistication with a (non)expected utility model:

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

> 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 ) > < Under probabilistic sophistication with a (non)expected utility model: 13 two models, depending on source reconsidered.

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

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

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

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 ` 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 d = 0.11 Fig.b. Insensitivity index a: 0; pessimism index b: 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: d = 0 1 p c = 0 0 Theory continued: (Chateauneuf, Eichberger, & Grant 2005 ; Kilka & Weber 2001; Tversky & Fox 1995)

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

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.

Median choice-based probabilities (real incentives) Real data over 1900  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 11 22 3 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.

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

23 Results for uncertainty ("ambiguity?")

Figure 8.3. Probability transformations for participant 2 Fig. a. Raw data and linear interpolation 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 participant 2; a = 0.78, b = * Fig. a. Raw data and linear interpolation. * Figure 8.4. Probability transformations for Paris temperature participant 48; a = 0.21, b = 0.25 Between-person comparisons

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 Under Bayesian EU we’d know all now. NonEU: need some more graphs; we have them! 26

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

28 Subject 2, p = decision weight expectation certainty equivalent uncertainty premium risk premium ambiguity premium –3099 Subject 48, p = Subject 2, p = Subject 48, p = –39 –4034 – Between-person comparisons; Paris temperature

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

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