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).

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

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

4 inverse-S, (likelihood insensitivity) p w expected utility pessimism extreme inverse-S ("fifty-fifty") prevailing finding pessimistic "fifty-fifty" Common graphs found:

Now to Uncertainty (unknown probabilities); x-axis has events. So, no nice graphs … 5

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

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

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

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!

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

3. Rubber Meets the Road: An Experiment Data: 11 4 sources: 1.CAC40; 2.Paris temperature; 3."Foreign" temperature; 4.Risk.

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.

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

15 Results for uncertainty ("ambiguity?")

16 These were within-person comparisons Source functions 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 choice-based probabilities wSwS (source function weights)

17 participant 2; a = 0.78, b = * Fig. a. Raw data and linear interpolation. * Source functions 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: no homebias. NonEU: need source function; we have them! 18

19 Paris temperature Foreign temperature decision weight expectation certainty equivalent uncertainty premium risk premium ambiguity premium –3662 Within-person comparisons:

20 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 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)

The end 22

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.

24 Results for measuring ambiguity attitudes

* * * * * * * Figure 8.3. Probability transformations for participant 2 Fig. a. Raw data and linear interpolation Fig. b. Best-fitting (exp(  (  ln(p))  ))  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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