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Good starting point for uncertainty: Risk. Allais: nonEU. Theories: OPT (Kahneman &Tversky '79)

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Good starting point for uncertainty: Risk. Allais: nonEU. Theories: OPT (Kahneman &Tversky '79)

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  1. Combining Bayesian Beliefs and Willingness to Bet to Analyze Attitudes towards Uncertaintyby Peter P. Wakker, Econ. Dept.,Erasmus Univ. Rotterdam(joint with Mohammed Abdellaoui & Aurélien Baillon) Decision and Uncertainty, Rotterdam, the Netherlands, April 11 '07Topic: Uncertainty/Ambiguity. We make it operational; measuring, predicting, quantifyingcompletely. Only gains today.

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

  3. 3 All nonEU theories popular today: x  y; xpy  w(p)U(x) + (1–w(p))U(y); Relative to EU:one more graph …

  4. w inverse-S, (likelihood insensitivity) extreme inverse-S ("fifty-fifty") expected utility motivational pessimism prevailing finding pessimistic "fifty-fifty" cognitive p Abdellaoui (2000); Bleichrodt & Pinto (2000); Gonzalez & Wu 1999; Tversky & Fox, 1997. 4

  5. 5 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: Rank-dependent utility (CEU; Gilboa 1987; Schmeidler 1989); Multiple priors (Wald 1950; Gilboa & Schmeidler 1989); PT (Tversky & Kahneman 1992); Endogeneous definitions of ambiguity …; Smooth model (Klibanoff, Marinacci, Mukerji, 2005); Variational model (Maccheroni, Marinacci, Rustichini, 2006). Economic models mostly normative. We: descriptive.

  6. 6 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 2n – 2 assessments. Pffff. No more graphs. Not "one more graph." Pffff.

  7. 7 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!

  8. Known urnk Unknown urnu ? 20–? 20 R&B in unknown proportion 10 R 10 B < > + + 1 1 8 Ellsberg paradox. Two urns with 20 balls. Ball drawn randomly from each. Events: Rk: Ball from known urn is red. Bk, Ru, Bu are similar. Common preferences between gambles for $100: (Rk: $100)  (Ru: $100) (Bk: $100)  (Bu: $100)  P(Rk) > P(Ru)  P(Bk) > P(Bk) > Under probabilistic sophistication with a (non)expected utility model:

  9. 9 Ellsberg: There cannot exist probabilities in any sense. Not "one more graph." (Or so it seems?)

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

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

  12. 12 Data:

  13. Figure 8.3. Probability transformations for participant 2 risk: a=0.42, b=0.13 1 0.875 0.75 Paris temperature; a=0.78, b=0.12 0.50 0.25 foreign temperature; a=0.75, b=0.55 0.125 0 0.25 0.75 1 0.125 0.50 0.875 0 Fig. a. Raw data and linear interpolation. 13 Within-person comparisons

  14. Figure 8.4. Probability transformations for Paris temperature 1 0.875 0.75 participant 2; a=0.78, b=0.69 0.50 0.25 participant 48; a=0.21, b=0.25 0.125 * 0 0 0.25 0.50 0.75 0.875 1 0.125 * Fig. a. Raw data and linear interpolation. 14 Between-person comparisons

  15. 15 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) = x0.88. Under Bayesian EU we’d know all now. NonEU: need some more graphs; we have them!

  16. Foreign temperature Paris temperature decision weight 0.49 0.20 expectation 20000 20000 certainty equivalent 17783 6424 uncertainty premium 2217 13576 risk premium 5879 5879 ambiguity premium –3662 7697 16 Within-person comparisons (to me big novelty of Ellsberg):

  17. Subject 48, p = 0.875 Subject 2, p = 0.875 Subject 48, p = 0.125 Subject 2, p = 0.125 0.08 0.52 decision weight 0.35 0.67 expectation 5000 5000 35000 35000 25376 2268 19026 certainty equivalent 12133 uncertainty premium 2732 15974 9624 –7133 risk premium 5717 2078 –39 –4034 ambiguity premium –3099 10257 9663 654 17 Between-person comparisons; Paris temperature

  18. 18 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.

  19. 19 The end

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