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In economics, probabilities usually unknown; inflation next year, strategy opponent, …

Using Prospect Theory to Study Unknown Probabilities ("Ambiguity") by Peter P. Wakker , Econ. Dept., Erasmus Univ. Rotterdam (joint with Mohammed Abdellaoui & Aurélien Baillon) 6 th Tilburg Symposium on Psychology and Economics , August 31 '07. In economics, probabilities usually unknown;

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In economics, probabilities usually unknown; inflation next year, strategy opponent, …

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  1. Using Prospect Theory to Study Unknown Probabilities ("Ambiguity")by Peter P. Wakker, Econ. Dept.,Erasmus Univ. Rotterdam(joint with Mohammed Abdellaoui & Aurélien Baillon)6th Tilburg Symposium on Psychology and Economics, August 31 '07 In economics, probabilities usually unknown; inflation next year, strategy opponent, … Long time not studied, simply cause no models. Recently, models: multiple priors & Choquet-expected utility, Gilboa & Schmeidler 1989; robust control, Hansen & Sargent 2001; Marinacci; &: prospect theory 1992!

  2. 2 None operational quantitatively (yet) toobserve data … analyze … predict. We make prospect theory operational by introducing source functions. We give first quantitative assessment of ambiguous behavior with risk premiums etc.

  3. 3 1. Introduction First some risk (known probabilities). Many nonEU theories exist; virtually all amount to: x  y  0; xpy  w(p)U(x) + (1–w(p))U(y); Relative to EU:one more graph … more work; but can live with it.

  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; Van de Kuilen & Wakker (MS 2nd round) 4

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

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

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

  8. 8 Einhorn & Hogarth 1985 They have nice graphs: go to pdf file of Hogarth & Einhorn (1990, Management Science 36, p. 785/786).Problem of the x-axis … Our source functions will solve that within revealed preference (no introspection needed).

  9. 9 2. Theory Only binary acts with gains. All popular static nonEU theories (except "variational theory"): 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.

  10. 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!W.r.t. Bayesian, still "one more graph." However,

  11. Known urnk Unknown urnu ? 20–? 20 R&B in unknown proportion 10 R 10 B < > + + 1 1 11 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(Bu) > Under probabilistic sophistication with a (non)expected utility model:

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

  13. 13 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(Bu) + + 1 1 > Under probabilistic sophistication with a (non)expected utility model: < > models, depending on source two reconsidered.

  14. 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.

  15. 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

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

  17. 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!?).

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

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

  20. 20 Figure 6.1. Decomposition of the universal event E = S b0 b1 E E a1/2 b0 b1 E E E E a1/4 a1/2 a3/4 b0 b1 E E a1/4 a3/8 a1/2 a3/4 a7/8 a1/8 a5/8 b0 b1 The italicized numbers and events in the bottom row were not elicited. E E E E E E Method for measuring choice-based probabilities

  21. Figure 7.1. Probability distributions for CAC40 Figure 7.2. Probability distributions for Paris temperature 1.0 1.0 Real data over the year 2006 Real data over 19002006 0.8 0.8 Median choice-based probabilities (real incentives) Median choice-based probabilities (hypothetical choice) 0.6 0.6 Median choice-based probabilities (hypothetical choice) Median choice-based probabilities (real incentives) 0.4 0.4 0.2 0.2 0.0 0.0 30 25 35 2 1 0 1 20 3 2 3 15 10 21 Results for choice-based probabilities Uniformity confirmed 5 out of 6 cases.

  22. Figure 7.3. Cumulative distribution of powers 1 Real Hypothetical 0.5 0 2 3 1 0 Method for measuring utility 22 Certainty-equivalents of 50-50 prospects. Fit power utility with w(0.5) as extra unknown. Results for utility

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

  24. 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 Many economists, erroneously, take this symmetric weighting function as unambiguous or ambiguity-neutral. 0 0.25 0.75 1 0.125 0.50 0.875 0 Fig. a. Raw data and linear interpolation. 24 Within-person comparisonsthrough source functions

  25. 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. Between-person comparisonsthrough source functions 25

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

  27. 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 27 Within-person comparisons (to me big novelty of Ellsberg):

  28. 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 28 Between-person comparisons; Paris temperature

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

  30. 30 The end

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