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) Decision and Uncertainty, Rotterdam, the Netherlands, April 11 '07 Topic: Uncertainty/Ambiguity. We make it operational; measuring, predicting, quantifying completely. Only gains today.

Good starting point for uncertainty: Risk. Allais: nonEU. Theories: 1.OPT (Kahneman &Tversky '79) 2.RDU (Quiggin '81) 3.Betweenness (Dekel, Chew '83, '86) 4.Quadratic utility (Chew, Epstein, Segal, '91) 5.Disappointment aversion (Gul '91) 6.Regret theory (Loomes, Sugden, Bell '82) 7.Skew-symmetric utility (Fishburn '87) 8.PT (Tversky & Kahneman '92) 2

All nonEU theories popular today: x  y; 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.

Models for risk came to a stop in 1990s. After Gilboa (1987) & Schmeidler (1989), Gilboa & Schmeidler (1989), and Tversky & Kahneman (1992): Now we turn to uncertainty/ambiguity (unknown probabilities). Still Allais  nonEU. Theories: 1.Rank-dependent utility (CEU; Gilboa 1987; Schmeidler 1989); 2.Multiple priors (Wald 1950; Gilboa & Schmeidler 1989); 3.PT (Tversky & Kahneman 1992); 4.Endogeneous definitions of ambiguity …; 5.Smooth model (Klibanoff, Marinacci, Mukerji, 2005); 6.Variational model (Maccheroni, Marinacci, Rustichini, 2006). Economic models mostly normative. We: descriptive. 5

All popular static nonEU theories (except variational): x  y; xEy  W(E)U(x) + ( 1–W(E) ) U(y). (Ghirardato & Marinacci 2001; Luce 1991; Miyamoto 1988) EU: W is probability; for n states of nature, need n–1 assessments. General nonEU: need 2 n – 2 assessments. Pffff. No more graphs. Not "one more graph." Pffff. 6

Machina & Schmeidler (1992), probabilistic sophistication: x  y; xEy  w ( P(E) ) U(x) + ( 1–w ( P(E) ) ) U(y). This is doable. Relative to EU: one more graph! However, …. Ellsberg! 7

Common preferences between gambles for $100: (R k : $100)  (R u : $100) (B k : $100)  (B u : $100) > 8 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. Not "one more graph." 9 (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: 10 two models, depending on source reconsidered.

x  y; xEy  w S ( P(E) ) U(x) + ( 1– w S ( P(E) ) ) U(y). w S : source-dependent probability transformation. S: "Uniform" source. Ellsberg: w u (0.5) < w k (0.5) u k unknown known (Choice-based) probabilities can be maintained! Not one more graph. But a few more graphs. 11

Data: 12

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

14 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! 15

16 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):

17 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; (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. 18

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