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Ambiguity Made Operational by Peter P. Wakker, Econ. Dept., Erasmus Univ. Rotterdam (joint with Mohammed Abdellaoui & Aurélien Baillon) ESA, Tucson, Oct. 19 '07 Keynes 1921 & Knight 1921: probabilities unknown. Long time not studied, simply because no models. Only in 1989: Choquet-expected utility & multiple priors (Gilboa & Schmeidler); 1992: Prospect theory (Tversky & Kahneman); other theories: Robust control (Hansen & Sargent 2001); Smooth (Klibanoff, Marinacci & Mukerji 2005); Variational (Maccheroni, Marinacci, Rustichini 2006).
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These theories not implemented quantitatively. Empirical studies are as yet qualitative. We make ambiguity quantitatively operational by introducing source functions. We get exact ambiguity premium etc. 2
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1. Introduction First some words on risk (so no ambiguity yet). Most descriptive (nonEU) theories: x y 0; xpy w(p)U(x) + ( 1–w(p) ) U(y); Relative to EU: one more graph … 3
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4 inverse-S, (likelihood insensitivity) p w expected utility pessimism extreme inverse-S ("fifty-fifty") prevailing finding pessimistic "fifty-fifty" Common graphs found:
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Now to Uncertainty (unknown probabilities); x-axis has events. So, no nice graphs … 5
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2. Theory Most ambiguity theories: x y 0; xEy W(E)U(x) + ( 1–W(E) ) U(y). For rich state space, such as continuum, general W is too complex. Machina & Schmeidler (1992) considered tractable special case. 6
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Machina & Schmeidler (1992), probabilistic sophistication (in a "global" sense): W(. ) = w(P(. )) for a probability P. So, xEy w ( P(E) ) U(x) + ( 1–w ( P(E) ) ) U(y). Then still get nice x-axis for uncertainty. W.r.t. Bayesian, still "one more graph." Tractable and OK. However, 7
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Ellsberg (1961) 2-color paradox: 2 different urns with irreconcileable prob.beliefs; no global probabilistic sophistication; no p's exist. No x-axis … Life remains difficult … Now to our operationalization of ambiguity (we will get back an x-axis and probabilistic sophistication but in a "local" sense): 8
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Theory for our operationalization of ambiguity: 9 For x y 0: xEy w S ( P(E) ) U(x) + ( 1–w S ( P(E) ) ) U(y). Like Machina & Schmeidler, but with S added. Explanation: Not all events alike. Divide events into different sources S. Source S is set of events referring to same "mechanism generating uncertainty"; Define uniform sources: its events have uniform degree of ambiguity = within such a source we have probabilistic sophistication ("local" prob. soph.); w S is the source function. We get back P & x-axis & nice graphs!
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Wait a minute … One graph for every source!? Many graphs!? More complex than Machina-Schmeidler!? Yes it is. Ambiguity more complex than risk or Machina-Schmeidler "global" probabilistic sophistication. No free lunch … Still, major simplification wrt general W or general set of priors. 10
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3. Rubber Meets the Road: An Experiment Data: 11 4 sources: 1.CAC40; 2.Paris temperature; 3."Foreign" temperature; 4.Risk.
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12 Method for measuring choice-based probabilities: 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. repeated bisection into equally likely events of probability 2 –n.
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13 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 11 22 33 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.
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Certainty-equivalents of 50-50 prospects. Fit power utility with w(0.5) as extra unknown. 14 0 Hypothetical Real 1 23 0 1 0.5 Figure 7.3. Cumulative distribution of powers Method for measuring utility Results for utility
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15 Results for uncertainty ("ambiguity?")
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16 These were within-person comparisons 0.125 0 0 Source functions 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 choice-based probabilities wSwS (source function weights)
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17 participant 2; a = 0.78, b = 0.69 0 * Fig. a. Raw data and linear interpolation. * Source functions 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:
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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: no homebias. NonEU: need source function; we have them! 18
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19 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:
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20 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
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Conclusion: By carrying out quantitative measurements of (a) probabilities (subjective), (b) utilities, 21 we make ambiguity operational at a quantitative level. Bayesian (c) source functions (uncertainty attitudes), non-Bayesian (ambiguity)
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The end 22
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23 Many advanced theories; CEU (Gilboa 1987; Schmeidler 1989) PT (Tversky & Kahneman 1992) Multiple priors (Gilboa & Schmeidler 1989) Endogenous definitions (Epstein, Zhang, Kopylov, Ghirardato, Marinacci) Smooth (KMM; Nau) Variational model (Maccheroni, Marinacci, Rustichini) Biseparable (Ghirardato & Marinacci 2001) Choice-based but intractable Nice graphs!? x-axis-problem: choice- less probabilities We connect Many tractable empirical studies; Curley & Yates 1985 Fox & Tversky 1995 Kilka & We- ber 2001 Cabantous 2005 di Mauro & Maffioletti 2005 Einhorn & Hogarth 1985 (over 400 citations) "Anchor probabilities" on x-axis (c.f. Hansen & Sargent). Others take midpoint of probability interval.
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24 Results for measuring ambiguity attitudes
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25 0 0 * * * * * * * Figure 8.3. Probability transformations for participant 2 Fig. a. Raw data and linear interpolation. 0.25 0.125 0.875 0.75 1 0.50 0.125 0.875 Fig. b. Best-fitting (exp( ( ln(p)) )) . 0.25 0.75 1 0.50 0.125 0.8750.2500.500.751 0.1250.875 0.25 0 0.50 0.751 CAC40; a = 0.80; b = 0.30 risk: a = 0.42, b = 0.13 Paris temperature; a = 0.78, b = 0.12 foreign temperature; a = 0.75, b = 0.55 CAC40; = 0.15; = 1.14 risk: = 0.47, = 1.06 Paris temperature; = 0.17, = 0.89 foreign temperature; = 0.21, = 1.68
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