What is behind ambiguity aversion? An experiment disentangling model uncertainty and risk aversion

Transcription

1 What is behind ambiguity aversion? An experiment disentangling model uncertainty and risk aversion Loïc Berger 2 Valentina Bosetti 3 Fondazione Eni Enrico Mattei (FEEM) and 2 Federal Planning Bureau (FPB) 3 Department of Economics, Bocconi University January 28, 206

2 Introduction Motivation () Uncertainty is ubiquitous. Deep uncertainty/ambiguity is more and more recognized to play a central role in decision making processes Partly because of some recent "catastrophic" events Economic uncertainty: nancial crisis Technological uncertainty: Fukushima Partly because of a growing awareness about Environmental uncertainty: climate change Demographic uncertainty: longevity / mortality risk 2 / 37

3 Introduction Motivation (2) What is ambiguity/deep uncertainty/knightian uncertainty? "all kind of situations in which a decision maker does not have sucient information to quantify through a single probability distribution the stochastic nature of the problem she is facing" (Cerreia-Vioglio et al., 203) 3 / 37

4 Introduction Motivation (3) All uncertainty relevant for decision making is ultimately subjective Yet, in applications (especially with data) it is convenient to distinguish between aleatory and epistemic uncertainty 4 / 37

5 Introduction Motivation (3) All uncertainty relevant for decision making is ultimately subjective Yet, in applications (especially with data) it is convenient to distinguish between aleatory and epistemic uncertainty - Aleatory uncertainty (=risk): examples: games chance (roulette, coin, dice), measurement errors deals with variability in data (because of inherent randomness, measurement errors, omitted minor explanatory variables) characterizes data generating processes (DGP) (i.e. probability models) probability is a measure of randomness/variability 4 / 37

6 Introduction Motivation (3) All uncertainty relevant for decision making is ultimately subjective Yet, in applications (especially with data) it is convenient to distinguish between aleatory and epistemic uncertainty - Aleatory uncertainty (=risk): examples: games chance (roulette, coin, dice), measurement errors deals with variability in data (because of inherent randomness, measurement errors, omitted minor explanatory variables) characterizes data generating processes (DGP) (i.e. probability models) probability is a measure of randomness/variability - Epistemic uncertainty (=model uncertainty): deals with the truth of propositions: "the composition of the urn is 50 red and 50 black balls" "the parameter that characterizes the DGP has value x" probability is a measure of degree of belief 4 / 37

7 Introduction Motivation (3) All uncertainty relevant for decision making is ultimately subjective Yet, in applications (especially with data) it is convenient to distinguish between aleatory and epistemic uncertainty - Aleatory uncertainty (=risk): examples: games chance (roulette, coin, dice), measurement errors deals with variability in data (because of inherent randomness, measurement errors, omitted minor explanatory variables) characterizes data generating processes (DGP) (i.e. probability models) probability is a measure of randomness/variability - Epistemic uncertainty (=model uncertainty): deals with the truth of propositions: "the composition of the urn is 50 red and 50 black balls" "the parameter that characterizes the DGP has value x" probability is a measure of degree of belief Deep uncertainty/ambiguity: combination of both aleatory and epistemic uncertainty 4 / 37

8 Introduction Research questions From Ellsberg (96), we know that people are generally ambiguity averse but... What is the source of ambiguity non neutrality? (normative vs. descriptive) 2 What is the degree of ambiguity aversion? What is the degree of model uncertainty aversion (in comparison to risk aversion)? 3 What are the underlying properties of the ambiguity aversion function? (CAAA?, DAAA?) 5 / 37

9 Introduction Why is it important for us? Deep uncertainty plays an important role in the economic and policy science related to climate change. 6 / 37

10 Introduction Why is it important for us? Deep uncertainty plays an important role in the economic and policy science related to climate change. These uncertainties arise from: the underlying science of climate (and the extreme complexity of the climate system) our inability to perfectly capture the way our socio-economic system would respond, mitigate and adapt to climate change 6 / 37

11 Introduction Why is it important for us? Deep uncertainty plays an important role in the economic and policy science related to climate change. These uncertainties arise from: the underlying science of climate (and the extreme complexity of the climate system) our inability to perfectly capture the way our socio-economic system would respond, mitigate and adapt to climate change in many situations, the probabilistic model is neither explicitly given, nor can be perfectly approximated or inferred with the available data and current scientic methods 6 / 37

12 Introduction Examples () Heal and Millner (203) IPCC (203) 7 / 37

13 Introduction Examples (2) Probability of climate catastrophe? (Berger, Emmerling, Tavoni; 205) Probability of AMOC collapse (%) Exp Exp 2 Exp 3 Exp 4 Exp 6 Exp 7 Exp 9 Exp 0 Exps 8,, Temperature increase in 200 (K) / 37

14 Roadmap. Experimental procedures 2. Theoretical predictions 3. Results 4. Conclusion 9 / 37

15 . Experimental procedure. Experimental procedure 0 / 37

16 . Experimental procedure The choices situations The choices situations Urn : (risk) The number or red and black balls is perfectly known / 37

17 . Experimental procedure The choices situations The choices situations Urn 2: (compound risk) heads tails EURO or? The number of red and black balls is determined by ipping a fair coin in the air 00 black 00 red 2 / 37

18 . Experimental procedure The choices situations The choices situations Task elow Urn 3: (model uncertainty) Expert!! Expert!2! Only!Red! balls!! Only!Black! balls!! The number of red, black and the total number of balls in the urn are unknown, but information is provided by two "experts", each giving her own assessment of the composition of the urn RD 3 / 37

19 . Experimental procedure The choices situations The choices situations Urn 4: (Ambiguity à la Ellsberg) The total number of balls is given, but the exact composition of the urn is unknown 4 / 37

20 . Experimental procedure The choices situations The choices situations Urn : Urn 2: heads tails Consider the two Urns A and B illustrated below EURO or? Task Urn 3: Urn 4: Expert!! Only!Red! balls!! Expert!2! Only!Black! balls!! 00 black 00 red Type P(r) Proba. Pres. Risk /2 objective simple Urn A The number of Red and Black balls is determined as follows. A fair coin is being tossed: if the result is "Heads", the urn contains only 00 Black balls; if the result is "Tails", the urn contains only 00 Red balls Compound risk /2 ( if "tails"; 0 if "heads") objective compound Urn B The total number of balls and the composition of the urn are both unknown, but in order to help you, two experts have provided their assessment of the urn composition. These experts are the best we could find for this situation. They are both experienced and both have excellent track record. Model uncertainty unknown Expert says there are only Red and no Black balls. (Exp: ; Exp2: 0) Expert 2 says there are only Black and no Red balls. Remember that subjective these experts assessments are given to help you, but it does not mean the experts are correct. The experts do their best but they may be wrong. compound A ball will be drawn from each urn. You are asked to choose a colour (Red or Black), and to place a bet on the colour of the ball drawn from each urn. If your bet is correct, you win 5 euros. If your bet is incorrect, nothing happens. Ambiguity unknown subjective? 5 / 37

21 . Experimental procedure The RLP task Random lotery pairs (RLP) Urns, 2, 3 and 4 presented 2-by-2 Randomized sequence In each decision, subjects have to:. place a bet on the color (Red or Black) drawn 2. decide which of the two urns presented to place their bet (or whether indierent) If bet is correct: win e5, otherwise nothing happens 6 / 37

22 . Experimental procedure Literature review Literature review In general: Urn Urn 4: Ellsberg (96),... (Trautmann and van Kuilen, 204) Ambiguity aversion 7 / 37

23 . Experimental procedure Literature review Literature review In general: Urn Urn 4: Ellsberg (96),... (Trautmann and van Kuilen, 204) Ambiguity aversion Urn Urn 2: Halevy (2007), Abdellaoui et al. (205), Dean and Ortoleva (205), Armentier and Treich (205), Harrison et al. (205) Compound lottery aversion 7 / 37

24 . Experimental procedure Literature review Literature review In general: Urn Urn 4: Ellsberg (96),... (Trautmann and van Kuilen, 204) Ambiguity aversion Urn Urn 2: Halevy (2007), Abdellaoui et al. (205), Dean and Ortoleva (205), Armentier and Treich (205), Harrison et al. (205) Compound lottery aversion Attitudes to ambiguity and compound lottery are tightly associated 7 / 37

25 2. Theoretical predictions A theory of choice under uncertainty 2. Theoretical predictions 8 / 37

26 2. Theoretical predictions A theory of choice under uncertainty A theory of choice under uncertainty Two layers uncertainty (Cerreia-Vioglio et al., 203; Marinacci, 205) Distinction between risk (aleatory uncertainty) and model uncertainty (epistemic uncertainty) Possible alternative models P θ (r) (i.e. compositions of urns) belong to a set M taken as given Single prior probability measure over models: q θ The DM chooses the act that maximizes his utility: U(f i ) = E θ (v u ) P θ (c)u (f i (c)), i {,..., 4} c {r i,b i } 9 / 37

27 2. Theoretical predictions A theory of choice under uncertainty A theory of choice under uncertainty Two layers uncertainty (Cerreia-Vioglio et al., 203; Marinacci, 205) Distinction between risk (aleatory uncertainty) and model uncertainty (epistemic uncertainty) Possible alternative models P θ (r) (i.e. compositions of urns) belong to a set M taken as given Single prior probability measure over models: q θ The DM chooses the act that maximizes his utility: U(f i ) = E θ (v u ) P θ (c)u (f i (c)), i {,..., 4} }{{} φ c {r i,b i } 9 / 37

28 2. Theoretical predictions A theory of choice under uncertainty In particular: Risk (Urn ): M = {P(r) = /2} is a singleton U(f ) = c {r,b } 2 u (f (c)). () 20 / 37

29 2. Theoretical predictions A theory of choice under uncertainty In particular: Risk (Urn ): M = {P(r) = /2} is a singleton U(f ) = c {r,b } Compound risk (Urn 2): M = {P(r) =, P(r) = 0} U(f 2 ) = Eθ(u u ) P θ (c) u (f 2 (c)) = 2 u (f (c)). () c {r 2,b 2} c {r 2,b 2} 2 u (f 2(c)) (2) 20 / 37

30 2. Theoretical predictions A theory of choice under uncertainty In particular: Risk (Urn ): M = {P(r) = /2} is a singleton U(f ) = c {r,b } Compound risk (Urn 2): M = {P(r) =, P(r) = 0} U(f 2 ) = Eθ(u u ) P θ (c) u (f 2 (c)) = 2 u (f (c)). () c {r 2,b 2} c {r 2,b 2} Model uncertainty (Urn 3): M = {P(r) =, P(r) = 0} U d (f 3 ) = c {r 3,b 3} 2 u (f 2(c)) (2) 2 v( f 3 (c) ), (3) 20 / 37

31 2. Theoretical predictions A theory of choice under uncertainty In particular: Risk (Urn ): M = {P(r) = /2} is a singleton U(f ) = c {r,b } Compound risk (Urn 2): M = {P(r) =, P(r) = 0} U(f 2 ) = Eθ(u u ) P θ (c) u (f 2 (c)) = c {r 2,b 2} 2 u (f (c)). () c {r 2,b 2} Model uncertainty (Urn 3): M = {P(r) =, P(r) = 0} U d (f 3 ) = c {r 3,b 3} Ellsberg (Urn 4): M = {P θ (r) : P(r) = θ N U(f 4 ) = 0 0 θ= (v u ) 2 u (f 2(c)) (2) 2 v( f 3 (c) ), (3) c {r 4,b 4} for θ = {,..., 0}} P θ (c) u (f 4 (c)). (4) 20 / 37

32 2. Theoretical predictions Predictions Predictions Let Ci be the CE for urn i, and C 0 be the sure amount that corresponds to the expected gain obtained in the uncertain situation, then: H: C 0 C and C 0 C 3 Risk and model uncertainty aversion 2 / 37

33 2. Theoretical predictions Predictions Predictions Let Ci be the CE for urn i, and C 0 be the sure amount that corresponds to the expected gain obtained in the uncertain situation, then: H: C 0 C and C 0 C 3 Risk and model uncertainty aversion H2: C = C 2 C 3 C C 4 (i.e. v u φ φ 0) v u Reduction of compound risk, stronger aversion to model uncertainty than to risk, ambiguity aversion 2 / 37

34 2. Theoretical predictions Predictions Predictions Let Ci be the CE for urn i, and C 0 be the sure amount that corresponds to the expected gain obtained in the uncertain situation, then: H: C 0 C and C 0 C 3 Risk and model uncertainty aversion H2: C = C 2 C 3 C C 4 (i.e. v H3: H4: u φ φ 0) v u Reduction of compound risk, stronger aversion to model uncertainty than to risk, ambiguity aversion [ ] [ w u (w) u (w) 0 and w u (w) [ ] w v (w) v (w) 0 and w Decreasing absolute risk aversion (DARA) Increasing relative risk aversion (IRRA) [ ] ] u (w) w 0 [ ] v (w) v (w) w 0 U φ (U) φ (U) 0 Decreasing absolute ambiguity aversion (DAAA) 2 / 37

35 3. General results 3. General results 22 / 37

36 3. General results Implementation 89 subjects (Bocconi students) 2 sessions of +/- 75 min (instructions, training session, 9 tasks, questionnaire, payment) 9 tasks: RLP task, 2 CE tasks, 6 MPL tasks Random incentive system Average gain: e8.5 (min e5, max e40, median e9) 23 / 37

37 3. General results Implementation 89 subjects (Bocconi students) 2 sessions of +/- 75 min (instructions, training session, 9 tasks, questionnaire, payment) 9 tasks: RLP task, 2 CE tasks, 6 MPL tasks Random incentive system Average gain: e8.5 (min e5, max e40, median e9) Robustness round: 9 subjects at COP2 (negotiators, NGOs,...) 23 / 37

38 3. General results RQ Table : Association between ambiguity neutrality, ROCL and ROCU RQ: What is the source of ambiguity non neutrality? Table 2: Association between ambiguity neutrality, ROCL and ROCU C =C2 C =C3 C =C4 No Yes No Yes Total No Count Yes Count Total Note: Chi-square test: Chi-square test: 8.9e / 37

39 3. General results RQ Table : Association between ambiguity neutrality, ROCL and ROCU RQ: What is the source of ambiguity non neutrality? Table 2: Association between ambiguity neutrality, ROCL and ROCU C =C2 C =C3 C =C4 No Yes No Yes Total No Count Yes Count Total Note: Chi-square test: Chi-square test: 8.9e-09 79% (50/89) are ambiguity non neutral 24 / 37

40 3. General results RQ Table : Association between ambiguity neutrality, ROCL and ROCU RQ: What is the source of ambiguity non neutrality? Table 2: Association between ambiguity neutrality, ROCL and ROCU C =C2 C =C3 C =C4 No Yes No Yes Total No Count Yes Count Total Note: Chi-square test: Chi-square test: 8.9e-09 79% (50/89) are ambiguity non neutral 7% (34/89) of subjects reduce compound risks 24 / 37

41 3. General results RQ Table : Association between ambiguity neutrality, ROCL and ROCU RQ: What is the source of ambiguity non neutrality? Table 2: Association between ambiguity neutrality, ROCL and ROCU C =C2 C =C3 C =C4 No Yes No Yes Total No Count Yes Count Total Note: Chi-square test: Chi-square test: 8.9e-09 79% (50/89) are ambiguity non neutral 7% (34/89) of subjects reduce compound risks 69% (30/89) do not reduce compound uncertainty with subjective probabilities 24 / 37

42 3. General results RQ Table : Association between ambiguity neutrality, ROCL and ROCU RQ: What is the source of ambiguity non neutrality? Table 2: Association between ambiguity neutrality, ROCL and ROCU C =C2 C =C3 C =C4 No Yes No Yes Total No Count Yes Count Total Note: Chi-square test: Chi-square test: 8.9e-09 79% (50/89) are ambiguity non neutral 7% (34/89) of subjects reduce compound risks 69% (30/89) do not reduce compound uncertainty with subjective probabilities Association between ambiguity and reduction of compound risk but / 37

43 3. General results RQ Table : Association between ambiguity neutrality, ROCL and ROCU RQ: What is the source of ambiguity non neutrality? Table 2: Association between ambiguity neutrality, ROCL and ROCU C =C2 C =C3 C =C4 No Yes No Yes Total No Count Yes Count Total Note: Chi-square test: Chi-square test: 8.9e-09 79% (50/89) are ambiguity non neutral 7% (34/89) of subjects reduce compound risks 69% (30/89) do not reduce compound uncertainty with subjective probabilities Association between ambiguity and reduction of compound risk but... Stronger association between ambiguity and reduction of compound uncertainty COP2 24 / 37

44 3. General results RQ RQ: What is the source of ambiguity non neutrality? Logistic regressions: Dependent variable: Ambiguity neutrality (C = C 4) Regressor: - reduction of compound risk (C = C 2) - reduction of compound uncertainty (C = C 3) Table : Characteristics of Ambiguity Neutrality: Logistic Regressions Odds Ratio Standard Error Lower 95% Upper 95% Confidence Interval Confidence Interval C =C C =C Notes: :89observations, p value < / 37

45 3. General results RQ RQ: What is the source of ambiguity non neutrality? Logistic regressions: Dependent variable: Ambiguity neutrality (C = C 4) Regressor: - reduction of compound risk (C = C 2) - reduction of compound uncertainty (C = C 3) Table : Characteristics of Ambiguity Neutrality: Logistic Regressions Odds Ratio Standard Error Lower 95% Upper 95% Confidence Interval Confidence Interval C =C C =C Notes: :89observations, p value < 0.00 Probability of C = C 4 is 2% 25 / 37

46 3. General results RQ RQ: What is the source of ambiguity non neutrality? Logistic regressions: Dependent variable: Ambiguity neutrality (C = C 4) Regressor: - reduction of compound risk (C = C 2) - reduction of compound uncertainty (C = C 3) Table : Characteristics of Ambiguity Neutrality: Logistic Regressions Odds Ratio Standard Error Lower 95% Upper 95% Confidence Interval Confidence Interval C =C C =C Notes: :89observations, p value < 0.00 Probability of C = C 4 is 2% it increases to 46% if C = C 3, and decreases to 9% if C C 3 25 / 37

47 3. General results RQ RQ: What is the source of ambiguity non neutrality? Logistic regressions: Dependent variable: Ambiguity neutrality (C = C 4) Regressor: - reduction of compound risk (C = C 2) - reduction of compound uncertainty (C = C 3) Table : Characteristics of Ambiguity Neutrality: Logistic Regressions Odds Ratio Standard Error Lower 95% Upper 95% Confidence Interval Confidence Interval C =C C =C Notes: :89observations, p value < 0.00 Probability of C = C 4 is 2% it increases to 46% if C = C 3, and decreases to 9% if C C 3 it increases to 3% if C = C 2, and decreases to 2% if C C 2 COP ML 25 / 37

48 3. General results RQ RQ: What is the source of ambiguity non neutrality? Result : Attitudes to ambiguity and compond uncertainty are closely related (association much stronger when subjective probabilities than when objective ones) 26 / 37

49 3. General results RQ2 RQ2: What is the degree of ambiguity aversion? (a) How do risk and model uncertainty aversion dier? 2 certainty equivalent (CE) tasks: risk and model uncertainty Sequence of 0 decisions Decision Option A Option A Option B Option B P(Red) outcome Red P(Black) outcome Black outcome ! 50% 25 50% 4 50% 25 50% 4 50% 25 50% 4 50% 25 50% 4 50% 25 50% 4 50% 25 50% 4 50% 25 50% 4 50% 25 50% 4 50% 25 50% 4 50% 25 50% / 37

50 uncertain situations and respectively). Each interval is defined by the highest outcome for 3. General results CE tasks which the uncertain situation is preferred and the lowest outcome which is preferred to the uncertain situation. The results from the table confirms our predictions: subjects are on average ready to pay a CE tasks Table 7: Descriptive Statistics Mean Median Mode SD Min Max Obs C [.92 ; 3.22] [3 ; 4] [4 ; 5] [2.35 ;.98] a [6 ; 8] [5 ; 8] 69 C3 [0.25 ;.80] [0 ; 2] [8 ; 0] [3.6 ; 2.82] a < 4 > a The first (second) number corresponds to the standard deviation of the lower (upper) bound / 37

51 uncertain situations and respectively). Each interval is defined by the highest outcome for 3. General results CE tasks tightly which the associated, uncertain while situation we cannot is preferred reject the andindependence the lowest outcome hypothesis which between is preferred the attitudes to the uncertain towards compound situation. The lottery results and from ambiguity the table (p-value=0.9), confirms ouror predictions: between the subjects attitudes aretowards on average compound ready tolottery pay a and model uncertainty (p-value=0.83). The CE tasks then enable us to obtain Table a 7: direct Descriptive measure Statistics of the strength of model uncertainty aversion CE tasks relative to risk aversion. Figure displays the proportions of safe choices expressed by preference for the Mean Median Mode SD Min Max Obs sure amount for each of the the ten decisions between either Urn or Urn 2 and a list of sure amounts C [.92 ; 3.22] [3 ; 4] [4 ; 5] [2.35 ;.98] a [6 ; 8] [5 ; 8] 69 in descending order (see the description of the CE tasks above). C3 [0.25 ;.80] [0 ; 2] [8 ; 0] [3.6 ; 2.82] a < 4 > a The first (second) number 00 corresponds to the standard deviation of the lower (upper) bound. risk model uncertainty 90 prediction neutrality Proportion of safe choices (%) Decision 28 / 37

52 3. General results CE tasks RQ2: What is the degree of ambiguity aversion? (a) How do risk and model uncertainty aversion dier? Result 2: (needs to be conrmed) Subjects are both risk and model uncertainty averse. Model uncertainty aversion is stronger than risk aversion. 29 / 37

53 4. Structural estimations 4. Structural estimations 30 / 37

54 4. Structural estimations Double MPL tasks Double MPL tasks Decision tables Risky tasks à la Holt and Laury (2002) Model uncertain tasks For each task: 2 options (A and B) 0 decisions made switching point identied 3 / 37

55 4. Structural estimations Double MPL tasks RQ2: What is the degree of ambiguity aversion? Expo-Power function (Saha, 993): U(x) = exp ( ax r ) DARA and IRRA when a 0 and r 0 Special cases: CRRA when a = 0: U(x) = x r CARA when r = 0: U(x) = exp( ax) a a 32 / 37

56 4. Structural estimations Double MPL tasks RQ2: What is the degree of ambiguity aversion? (b) What are the degrees of risk and model uncetainty aversion? Table : Estimates of risk, model uncertainty and ambiguity preferences u v a r noise parameter CRRA EP CRRA EP CRAA (0.0025) (0.0542) (0.09) (0.093) (0.020) (0.0542) (0.026) ( ) ( ) ( ) ( ) (0.0023) Observations Loglikelihood Notes: Luce error specification is used in the estimation. Standard errors in parentheses. The EP risk specification is used to estimate v p value < 0.05, p value < 0.0, p value < 0.00 Model uncertainty aversion stronger than risk aversion CE task only. When the EP specification is considered, we estimate r u =0.35 and a u = EP specication seems better: DARA and IRRA implies IRRA. While the focus or our analysis is on comparing these estimates with the ones the model uncertainty function v, we note that their absolute magnitudes are consistent wit obtained by Holt and Laury (2002); Andersen et al. (2008). 43 We however recognize 33 / that 37 t

57 4. Structural estimations Risk and Model uncertainty aversion RQ2-3: Characterizing ambiguity attitude Characterizing risk and model uncertainty attitudes model uncertainty risk.5 model uncertainty risk.8 Absolute aversion.6.4 Relative aversion Monetary outcome (euros) Monetary outcome (euros) Figure 2: Absolute (left) and relative (right) risk and model uncertainty aversion using EP estimates (95% confidence in grey). This important result confirms one of our central hypothesis that we started observing in the RLP task which is, that subjects are more averse to model uncertainty than to risk. The di erence with 34 / the 37

58 4. Structural estimations Ambiguity aversion RQ2-3: Characterizing ambiguity attitude Easy to compute the degree of ambiguity aversion Remember φ v u.5 Absolute Ambiguity Aversion Relative Ambiguity Aversion Expected Utility levels Expected Utility levels Figure 3: Absolute (left) and relative (right) ambiguity aversion obtained with EP function estimates What we directly observe is that, while the degree of absolute ambiguity is clearly decreasing, the degree of relative ambiguity aversion seems to be fairly constant on the domain considered. It should here be noticed that the domain of the ambiguity function is not the monetary outcome domain 35 / 37

59 4. Structural estimations Ambiguity aversion RQ2-3: Characterizing ambiguity attitude Easy to compute the degree of ambiguity aversion Remember φ v u.5 Absolute Ambiguity Aversion Relative Ambiguity Aversion Expected Utility levels Expected Utility levels Figure 3: Absolute (left) and relative (right) ambiguity aversion obtained with EP function estimates Result 3: What we directly observe is that, while the degree of absolute ambiguity is clearly decreasing, the Decreasing absolute ambiguity aversion (DAAA) and constant relative here be noticed that the domain of the ambiguity function ambiguity aversion (CRAA) + degree of relative ambiguity aversion seems to be fairly constant on the domain considered. It should is not the monetary outcome domain 35 / 37

60 4. Conclusion 4. Conclusion 36 / 37

61 4. Conclusion 4. Conclusion Objective: characterize ambiguity aversion 37 / 37

62 4. Conclusion 4. Conclusion Objective: characterize ambiguity aversion Individuals exhibit dierent attitudes to dierent types of uncertainty 37 / 37

63 4. Conclusion 4. Conclusion Objective: characterize ambiguity aversion Individuals exhibit dierent attitudes to dierent types of uncertainty When the setup is simple enough: subjects reduce compound risk (objective probabilities) subjects do not reduce compound uncertainty (subjective probabilities) 37 / 37

64 4. Conclusion 4. Conclusion Objective: characterize ambiguity aversion Individuals exhibit dierent attitudes to dierent types of uncertainty When the setup is simple enough: subjects reduce compound risk (objective probabilities) subjects do not reduce compound uncertainty (subjective probabilities) Subjects are more averse to model uncertainty than to risk strongly correlated with ambiguity aversion 37 / 37

65 4. Conclusion 4. Conclusion Objective: characterize ambiguity aversion Individuals exhibit dierent attitudes to dierent types of uncertainty When the setup is simple enough: subjects reduce compound risk (objective probabilities) subjects do not reduce compound uncertainty (subjective probabilities) Subjects are more averse to model uncertainty than to risk strongly correlated with ambiguity aversion Evidence of decreasing absolute ambiguity aversion (DAAA) 37 / 37

66 4. Conclusion Thank you RISICO ERC / 37

67 Miscellaneous Art from the lab / 37

68 Miscellaneous Art from the lab / 37

69 Appendix The randomness device The randomness device Ideally, absent of experts' information: perfect ignorance innite number of balls? Infeasible... Randomness device: total number of balls in Urn 3 itself unknown, and comprised between and 00 Total number of potential objectives models: 3045 (= cardinality of Farey sequence of order 00) of order 00. Moreover, even in the case in which all the possible total numbers of balls, and all their possible compositions are assumed to be equally probable (assumptions that would be supported by a symmetry of ignorance argument), the possible models are not weighted equally. To see this, consider Table 2 below, which presents the sets of potential models when the maximum number of balls N is known to be (first row), 2 (second row), up to 8 (in the last row). N Set of possible models: M N = {P (r)} M N Table 2: Sets of models and their corresponding cardinality when the maximum number of balls in the urn is N M 00 = {P(r) {F 00 }} {P(r) [0, ]} back 37 / 37

70 The main experiment presented in the Appendix body of therq paper is not immune to imperfections and biases. We conducted a robustness round of the experiment at the COP2, held in Paris in December 205. RQ: Robustness round COP2 Participants originated from and represented more than 00 countries and most were climate negotiators. In individual in-person interviews, we prompted respondents who volunteered for the study for their prior probability distribution over possible 200 temperature increases as the result of current INDCs, Table A.: Association between ambiguity neutrality, ROCL and ROCU (robustness round) C =C2 C =C3 C =C4 No Yes No Yes Total No Yes Total Count Expected (4.76%) (27.47%) (60.44%) (8.79%) (69.23%) Count Expected (6.59%) (24.8%) (9.89%) (20.88%) (30.77%) (48.35%) (5.65%) (70.33%) (29.67%) (00%) Notes: Relative frequencies in parentheses. Chi-square test: 6.e-4 Chi-square test:.e-07 These resuls of the logistic regression for the probability of being ambiguity neutral in the robustness back round are reported in Table A.2. As in the original experiment, keep in mind that the ambiguity non neutral subjects are overrepresented in the sample under study. The results are then very similar to that of Table 6 presented in the body of the paper. In particular, exhibiting an identical attitude towards objective and subjective probabilities enables to statistically significantly predict ambiguity neutrality. The odds of being ambiguity neutral when C =C3 corresponds to 4.5 the odds when C 6= C3 (p-value 37 / 37

71 Appendix RQ Robustness round Logistic regressions: Dependent variable: Ambiguity neutrality (C = C 4) Regressors: - reduction of compound risk (C = C 2) - reduction of compound uncertainty (C = C 3) Table A.2: Characteristics of Ambiguity Neutrality: Logistic Regressions (robustness round) Odds Ratio Standard Error Lower 95% Upper 95% Confidence Interval Confidence Interval C =C2 C =C (.793) (.74) (0.497) (6.472) (0.67 ) (6.790) (3.066) (37.38) Notes: :Logisticregressions.Adjustedresultsinparentheses.Dependentvariable:Ambiguityneutrality(C =C4). 9 observations. p value < 0.05, p value < 0.0, p value < 0.00 Probability of C = C 4 is 3% it increases to 70% if C = C 3, and decreases to 4% if C C 3 back it increases to 47% if C = C 2, and decreases to 4% if C C 2 37 / 37

72 Appendix RQ Multinomial logistic Figure B.: Adjusted predictions of model uncertainty and compound risk atitudes C3 C2 0.8 C3 C2 P(C>C4) P(C=C4) P(C<C4) C3 C2 0 0 < C = C > C < C = C > C < C = C > C C3 C2 0.8 C3 C2 P(C>C4) P(C=C4) P(C<C4) C3 C2 0 0 < C = C > C < C = C > C < C = C > C back Notes: First row represents results of the main experiment (N=89), second row represents results of robustness round (N=9). Bars represent 95% confidence levels 37 / 37

73 therefore means, that we let the ambiguityappendix aversion function Ambiguity be: aversion r RQ2-3: Characterizing ambiguity exp a (U) attitude Direct estimation of φ: (U) = where U represents the expected utility for a given model : U ˆp u(ō)+( a, (6) ˆp )u(o), and u is defined as in equation (4). The results are provided in the last column of Table (0). In this case, the coe of the EP formulation is not significant at the 5% significance level (p should rather be of the CRRA type. cient a value = 0.062), and the function In the second column of Table (0), we provide the estimates CRAA EP a (0.9655) r (0.026) (0.0452) noise parameter (0.0023) (0.0084) Observations Loglikelihood Standard errors in parentheses p value < 0.05, p value < 0.0, p value < 0.00 Table 0: Estimates of model uncertainty aversion parameters using Luce error specification (a u =0.0294, r u =0.35) back obtained using this particular specification. As can be observed, the constant relative ambiguity aversion / 37