Half-full or half-empty? A model of decision making under risk

Journal of Mathematical Psychology 68–69 (2015) 1–6

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Journal of Mathematical Psychology journal homepage: www.elsevier.com/locate/jmp

Half-full or half-empty? A model of decision making under risk Marisa Cenci a , Massimiliano Corradini a , Alberto Feduzi b,c,d , Andrea Gheno a,∗ a

Department of Business Studies, University of Rome III, Via Silvio D’Amico 77, 00145 Roma, Italy

b

Department of Financial and Management Studies, SOAS University of London, Thornhaugh Street, Russell Square, London WC1H 0XG, UK

c

Department of Economics, University of Rome III, Via Silvio D’Amico 77, 00145 Roma, Italy

d

Judge Business School, University of Cambridge, Trumpington Street, Cambridge CB2 1AG, UK

highlights • • • •

A new descriptive model of decision-making under risk. Retains much of the intuitive appeal of the expected value model. Explains various paradoxes in decision-making under risk. Relies on only two parameters that have clear behavioral interpretations.

article

info

Article history: Received 24 March 2015 Received in revised form 16 June 2015

Keywords: Decision under risk Decision-making paradoxes Optimism/pessimism Expected value criterion Prospect theory

abstract We propose a descriptive model of decision making under risk, inspired by the ‘‘half-full, half-empty’’ glass metaphor, that explains well-known paradoxes identified by Allais (1953), Kahneman and Tversky (1979), and Birnbaum (2008). The model is intuitive in that it is closely related to the expected value criterion and its parameters have a clear behavioral interpretation, and parsimonious in that it provides an approach to modeling behavior based on only two parameters. © 2015 Elsevier Inc. All rights reserved.

1. Introduction The development of the literature on decision-making under risk has been driven by the discovery of paradoxes afflicting existing theories (Fox, Erner, & Walters, in press, Luce, Ng, Marley, & Aczél, 2008, Wu, Zhang, & Gonzalez, 2004). Old paradoxes such as the St. Petersburg paradox (Bernoulli, 1738) and the Allais (1953) paradox and its variations (Kahneman & Tversky, 1979) challenge the descriptive power of the expected value (EV) criterion and expected utility (EU) theory. It is for this reason that, over the last 60 years or so, decision theorists have sought a theory of choice able to provide a satisfactory description of decision-makers’ behavior in risky situations. While numerous theories have been proposed (reviewed by Fox et al., in press, Starmer, 2000, Wu et al., 2004), Kahneman and

∗

Corresponding author. E-mail address: [email protected] (A. Gheno).

http://dx.doi.org/10.1016/j.jmp.2015.06.006 0022-2496/© 2015 Elsevier Inc. All rights reserved.

Tversky’s (1979) original prospect theory and Tversky and Kahneman’s (1992) new prospect theory have become the leading descriptive framework for modeling decision making under risk (Camerer, 1998, Fox & Poldrack, 2014) and have inspired a large body of theoretical and empirical work.1 However, despite this apparent success, various studies have questioned the underlying assumptions and applicability of prospect theory (PT) on the basis of empirical violations it seems unable to accommodate.2 These violations include forms of description invariance, context independence, internality, gain–loss separability, coalescing, and prob-

1 See, for instance, Abdellaoui (2000, 2002), Camerer (1992), Gonzalez and Wu (1999), Karni and Safra (1987), Luce (2000, 2001), Machina (1982), Prelec (1998), Quiggin (1993), Schmeidler (1989), Starmer and Sugden (1989), Tversky and Wakker (1995), Wakker (1994, 1996, 2001), Yaari (1987). 2 See, for instance, Baltussen, Post, and van Vliet (2006), Brandstätter, Gigerenzer, and Hertwig (2006), Hertwig, Barron, Weber, and Erev (2004), Humphrey (1995), Lopes and Oden (1999), Marley and Luce (2005), Neilson and Stowe (2002), Payne (2005), Starmer (2000), Starmer and Sugden (1993), Wu and Markle (2008), Wu, Zhang, and Abdelloui (2005).

2

M. Cenci et al. / Journal of Mathematical Psychology 68–69 (2015) 1–6

ability outcome independence (Fox et al., in press). PT is consequently at once one of the most confirmed and falsified decision making models currently out there (Wakker, 2010, p. 351). Although there is a debate about whether some of these studies provide a definite falsification of PT,3 Birnbaum (2008) has recently produced a systematic empirical study that has received considerable attention in the decision making literature (Fox et al., in press, Luce et al., 2008, Wakker, 2010). In particular, he presents 11 new paradoxes that lead him to conclude that PT cannot ‘‘be retained as a descriptive model of decision making’’ (Birnbaum, 2008, p. 464). Various ‘‘configural weight models’’ have been devised that accommodate some of these paradoxes, including the rank-affected multiplicative weights (RAM) model (Birnbaum, 1997), the transfer of attention exchange (TAX) model (Birnbaum & Chavez, 1997, Birnbaum & Stegner, 1979), and the gains decomposition utility (GDU) model (Luce, 2000, Marley & Luce, 2001, Marley & Luce, 2005). However, while these models fit the data better than PT in many experiments (Birnbaum, 2008, Fox et al., in press, Wakker, 2010) and are more flexible as they usually have more free parameters, they are less parsimonious (Wu et al., 2004, p. 408) and ‘‘do not have the particularly tractable form of prospect theory, with psychological interpretations for its parameters. . . ’’ (Wakker, 2010, p. 351). The present paper accordingly aims to provide an intuitive and parsimonious descriptive model of decision making under risk that can explain well-known ‘‘old and new’’ paradoxes at one stroke. The intuition underlying the model can be easily grasped by way of the ‘‘glass half-full, glass half-empty’’ metaphor, which suggests that different individuals may evaluate the same situation in different ways, and will typically be divided between those who focus on positive aspects of the situation (the optimists) and those who focus on negative aspects (the pessimists). In the same spirit, we suggest that a decision maker’s (DM) assessment of a choice situation may depend on the way she values the difference between the outcomes above and below the mean value of the lottery in question. The lottery’s evaluation (and consequently the DM’s risk attitude) depends only on two parameters: her degree of optimism/pessimism λ, and the probability distortion parameter q that determines her decision weights. We will show that λ and q can be calibrated such that old and new paradoxes are explained. When compared to other models of decision-making recently proposed to accommodate Birnbaum’s (2008) paradoxes, our model, which we call the Half-Full/Half-Empty (henceforth HFHE) model, has two important advantages. The first is that it is highly intuitive, both for being closely related to the EV criterion and for its parameters having a clear and widely used behavioral interpretation (see, for instance, Arrow & Hurwicz, 1972, Hurwicz, 1951, Kahneman & Tversky, 1979, Tversky & Wakker, 1995). The second is that it is highly parsimonious in that it provides a very simple way of modeling behavior based on only two parameters. We begin by introducing the HFHE model and discussing its main properties. To facilitate the presentation, we develop the model with reference to the Allais (1953) paradox and explain how this paradox can be accommodated. We then show how the HFHE model can provide a solution to additional paradoxes presented in Birnbaum (2008) and Kahneman and Tversky (1979). We close with a short conclusion. 2. The HFHE model The glass half-full/half-empty adage suggests that, when observing a glass that contains 50% of wine, optimists focus on the

3 See, for instance, Baucells and Heukamp (2006), Birnbaum and McIntosh (1996), Camerer and Ho (1994), Fox and Hadar (2006), Rieger and Wang (2008), Wakker (2003), Wu and Gonzalez (1996), Wu et al. (2004).

Table 1 The Allais (1953) choice problems. Problem 1

Problem 2

A

B

C

D

(100 million, 1)

(500 million, 0.10) (100 million, 0.89) (0, 0.01)

(100 million, 0.11) (0, 0.89)

(500 million, 0.10) (0, 0.90)

half-full part and therefore see the glass half-full, while pessimists focus on the half-empty part and therefore see the glass halfempty. This is the intuition we pursue here, namely that the DM’s assessment of a given lottery depends on the way she evaluates the difference between the outcomes above and below the expected value of the lottery. Too see this, consider a lottery X with finitely many nonnegative outcomes X = (p1 : x1 , p2 : x2 , . . . , pn : xn ) with xi ≥ 0 for i = 1, . . . , n. We can express the EV representing function for the lottery X as: HEV (X ) = E [X ] = µ + E [X − µ]

(1)

where µ = E [X ] is the expected value of X . Since y = (y)+ + (y)− , where (y)+ = max {y, 0} and (y)− = min {y, 0}, Eq. (1) can be rewritten as: HEV (X ) = µ + 2



1 2

   1   E (X − µ)+ + E (X − µ)− .

(2)

2

The EV  representing  function  HEV in Eq. (2) assigns an equal weight to E (X − µ)+ and E (X − µ)− . However, in line with the half-full/half-empty glass metaphor, suppose that the DM assigns different weights according to her degree of optimism/pessimism (see, for instance, Arrow & Hurwicz, 1972, Hurwicz, 1951). In particular, suppose that an optimist who sees the glass half-full, overweighs the ‘‘large’’ outcomes (xi > µ) relative to the ‘‘small’’ ones (xi < µ), while a pessimist, who sees the glass half-empty, does the opposite. To take into account this behavioral feature, we introduce into the model a parameter λ representing the DM’s degree of optimism/pessimism, and propose the following simple generalization of the EV representing function in Eq. (2): Hλ (X ) = µ + 2 λE (X − µ)+ + (1 − λ) E (X − µ)−







with 0 ≤ λ ≤ 1.



 (3)

If the DM is an optimist, < λ ≤ 1 and she overweighs     E (X − µ)+ and underweights E (X − µ)− . If the DM is a pessimist, 0 ≤ λ < 12 and the opposite occurs. If the DM is neither an optimist nor a pessimist, λ = 12 and the EV representing function HEV in Eq. (2) is recovered. 1 2

Example 1. Explanation of the Allais paradox. Allais (1953) introduced the most famous counter-example to EU theory showing that people overweight certain outcomes relative to probable ones (the so-called certainty effect). The representing function Hλ that we have introduced in Eq. (3) is very simple, yet able to describe the original Allais (1953) paradox represented in Table 1. The notation used is (outcome, corresponding probability) and monetary outcomes are in French francs. When facing Problem 1 and Problem 2, according to EU theory, a DM should choose either A and C or B and D. The paradox arises because when presented with a choice between A and B most DMs select A, while when presented with a choice between C and D, most DMs select D. A DM characterized by the representing function in Eq. (3) prefers A to B when 0 ≤ λ < 0.23 and prefers D to C when 0.22 < λ ≤ 1. Hence for 0.22 < λ < 0.23 the representing function Hλ is able to explain the Allais (1953) original paradox. 

M. Cenci et al. / Journal of Mathematical Psychology 68–69 (2015) 1–6

Fig. 1. Weighting function w (pi , p, q) for a binary lottery with q < 1.

Fig. 2. Weighting function w (pi , p, q) for a binary lottery with q > 1.

However, even if the representing function in Eq. (3) is able to explain the Allais (1953) paradox, it does not allow that the DM may also distort the lottery’s probabilities, something widely recognized in the literature (see, for instance, Fehr-Duda & Epper, 2012, Kahneman & Tversky, 1979, Tversky & Wakker, 1995). In order to model such distortion, we replace the probabilities pi with q the following simple transformation pi → cpi where c = n 1 q j=1 pj

so that the transformed probabilities are normalized to sum to 1 q

p

and the corresponding weighting function is w (pi , p, q) = n i

q j=1 pj

with p = {p1 , p2 , . . . , pn }.4 The weighting function w (pi , p, q) for a binary lottery X = {(x1 , p) ; (x2 , 1 − p)} is depicted in Figs. 1 and 2, respectively for q < 1 and q > 1. In particular, if q < 1, probabilities of the more likely outcomes decrease and those of the less likely outcomes increase and the weighting function has the typical inverse S-shaped form. If q > 1, the opposite occurs and the weighting function has an S-shaped form. If q = 1 there is no distortion. Considering the probability distortion, we obtain the representing function H that characterizes the HFHE model from Eq. (3):



H (X ) = µq + 2 λEq



where Eq [X ] =

n

i =1

 

X − µq +

+ (1 − λ) Eq



 

X − µq −

(4)

w (pi , p, q) xi = µq .

Remark 1. Compact form. Since



X − µq − 





X − µq + it follows that Eq X − µq − Hence an equivalent form for Eq. (4) is:





H (X ) = µq + 2 (2λ − 1) Eq





 

X − µq + .

  X − µq −    = −Eq X − µq + . =

(5)

Eq. (5) can also be nicely expressed in terms of mean absolute deviation (MAD). Since (X )− = X − (X )+ and |X | = (X )+ − (X )− it follows that |X | = (X )+ − X + (X )+ that can be rewritten as: X − µq + =





    X − µq  + X − µq 2

.

3

(6)

4 This particular weighting function has been chosen for its simplicity and appropriateness for the present paper. Obviously, alternative weighting functions may also be used.

Substituting Eq. (6) into Eq. (5) yields: H (X ) = µq + (2λ − 1) Eq X − µq 





(7)

  where Eq X − µq  is the MAD of lottery X . Remark 2. Utility function interpretation. The representing function H can also be expressed as: H (X ) = Eq [u(X )]

(8)

  where u(X ) = X + (2λ − 1) X − µq  is a peculiar utility function that is lottery dependent being a function of µq .  Remark 3. Risk attitude. In reference to a lottery X , we define a risk averse DM as an individual who prefers a monetary amount equal to the expected value of the lottery rather than the lottery itself. In other words, given a lottery X , the DM is risk-averse if H (X ) < µ, risk-loving if H (X ) > µ, or risk-neutral if H (X ) = µ.5  Example 2. Risk attitude graphic representation. The DM’s risk attitude can also be described graphically with respect to the range of the parameters (q, λ). An example of risk attitude obtained by varying the parameters (q, λ) is depicted in Fig. 3 for the lottery X = {(100, 0.3) ; (10, 0.7)}. The line represents riskneutrality (H (X ) = µ), the region above the line is characterized by risk-love (H (X ) > µ), and the region below by risk-aversion (H (X ) < µ).  Remark 4. Stochastic dominance. The HFHE model is not a Rank Dependent Utility (RDU) model and violates the property of stochastic dominance (see, for instance, Wakker, 2010 Exercise 6.7.1). However, whereas this property is desirable from a prescriptive viewpoint, a good descriptive model should be able to accommodate empirically observed violations of stochastic dominance (Tversky & Kahneman, 1986, Weber & Camerer, 1987). The HFHE model performs well in this regard as it allows some violations of stochastic dominance found empirically (see, for instance, Appendix Table A.2, Problems 2, 3.1, 3.2, 3.3, 4) if an appropriate set of parameters is considered. In contrast to RDU models, it is thus able to accommodate all the old and new paradoxes presented here.  Remark 5. A related theory. Gul (1991) presents a theory of disappointment aversion that is consistent with the Allais paradox

5 From Eq. (4) it follows that H (µ) = µ.

4

M. Cenci et al. / Journal of Mathematical Psychology 68–69 (2015) 1–6 Table 2 Solutions to old and new paradoxes. Kahneman and Tversky’s (1979) choice problems are denoted by KT and Birnbaum’s (2008) by B. ‘‘Solution not available’’ is denoted by NA. q

0.42 0.44 0.46 0.48 0.50 0.52 0.54

KT 1–4, 7, 8

B 1.1–6.2

B7.1–8.2

B10.1–10.5

B13.1–14.2

B15.1–17.2

All

λ

λ

λ

λ

λ

λ

λ

(0.24, 0.29) (0.25, 0.31) (0.25, 0.32) (0.25, 0.33) (0.25, 0.34) (0.25, 0.35) (0.25, 0.36)

(0.13, 0.49) (0.14, 0.48) (0.15, 0.46) (0.21, 0.45) (0.26, 0.43) (0.31, 0.41) (0.35, 0.38)

(0.21, 0.37) (0.22, 0.36) (0.22, 0.36) (0.22, 0.35) (0.23, 0.34) (0.23, 0.34) (0.24, 0.33)

(0.18, 0.35) (0.19, 0.35) (0.19, 0.35) (0.20, 0.34) (0.20, 0.34) (0.21, 0.34) (0.21, 0.34)

(0.31, 0.32) (0.30, 0.32) (0.29, 0.32) (0.28, 0.32) (0.27, 0.32) (0.26, 0.32) (0.25, 0.32)

(0.10, 0.37) (0.07, 0.38) (0.04, 0.38) (0.00, 0.38) (0.00, 0.38) (0.00, 0.38) (0.00, 0.38)

NA (0.30, 0.31) (0.29, 0.32) (0.28, 0.32) (0.27, 0.32) (0.31, 0.32) NA

inverse S-shaped form that also characterizes PT. Moreover, since 0 ≤ λ < 12 , a representative DM is pessimist and sees the glass half-empty. 4. Conclusion

Fig. 3. Risk attitude for the lottery X = {(100, 0.3) ; (10, 0.7)}.

and includes EU theory as a special case. Gul’s axiomatic model requires a utility function and an additional parameter that permits overweighting of all outcomes above the certainty equivalent of a lottery. Our descriptive model is similar in spirit but does not require a utility function and considers the expected value instead of the certainty equivalent.  3. Explaining old and new paradoxes In this section we show the results of the application of the HFHE model to the one-stage choice problems of Birnbaum (2008) and Kahneman and Tversky (1979) with non-negative monetary outcomes. In other words, we calibrate our model’s parameters q and λ of Eq. (4) in order to explain the modal behavior of the DM in these paradoxes. For the interested reader, the choice problems studied are reported in the Appendix. The way we calibrate the HFHE model is straightforward: for any given value of q, we find the range of values λ compatible with the modal choice according to the representing function H in Eq. (4). Table 2 shows, for any relevant q value, the corresponding interval of λ values consistent with the modal choices reported in Birnbaum (2008) and Kahneman and Tversky (1979). The last column of Table 2 is crucial: it reports, for each q value considered, the interval of λ values that is able to explain all problems simultaneously. In particular it can be inferred that all the paradoxes are explained for q = [0.48 ± 0.02] and λ = (0.305 ± 0.015), and that therefore a representative DM can be characterized by q = 0.48 and λ = 0.305. Since q < 1 (i.e. probabilities of the more likely outcomes are underweighted whilst those of the less likely outcomes are overweighted), the weighting function has the typical

While it has been shown that some recent configural weight models fit some of the empirical data better than prospect theory does, the latter still remains the dominant descriptive framework for decision making under risk. The search for a simple, intuitive, and parsimonious descriptive decision-making model that is able to accommodate old and new paradoxes is therefore still ongoing. Our paper aims to contribute to this search by showing that it is possible to provide a solution to well-known old and new paradoxes through a decision model inspired by the ‘‘half-full, half-empty’’ glass metaphor. The HFHE model suggests that a DM assesses the value of a lottery by evaluating the difference between the outcomes above and below its mean value. This assessment ultimately depends on her degree of optimism/pessimism and her decision weights. Relative to models recently proposed to accommodate Birnbaum’s (2008) new paradoxes, the HFHE model is intuitive insofar as it is closely related to the EV criterion and its parameters have a well-known behavioral interpretation. It is also parsimonious, in that it provides a very simple way to model behavior based on only two parameters. Acknowledgments For discussion and comments on earlier versions of this paper we are grateful to Jochen Runde, Stefan Scholtes, and Peter Wakker. Appendix. Kahneman and Tversky’s and Birnbaum’s choice problems In this Appendix we report the one-stage choice problems of Birnbaum (2008) and Kahneman and Tversky (1979) considered in Section 3 of our paper, i.e. those with non-negative monetary outcomes. Kahneman and Tversky (1979) introduce 14 problems in order to show that EU theory axioms are systematically violated. Problems 1–4, 7, 8 deal with the violation of the EU substitution axiom (also known as the independence axiom). In particular, Problems 1–4 are based on Allais (1953) examples and Problems 7 and 8 are instead characterized by the same probabilities ratio. The pairs of prospects that form each problem appear in Table A.1. The notation used is (outcome, corresponding probability) and monetary outcomes are in Israeli pounds. The percentage of choice of each prospect is in square brackets. Birnbaum (2008) presents 11 problems (new paradoxes) that our model is able to accommodate. In Table A.2 we report those related to violations of coalescing, stochastic dominance, and upper

M. Cenci et al. / Journal of Mathematical Psychology 68–69 (2015) 1–6 Table A.1 Kahneman and Tversky (1979): substitution axiom. A

B

Problem 1 (2500, 0.33) (2400, 0.66) (0, 0.01) [18]

(2400, 1)

[82]

D

Problem 7.1

(2500, 0.33) (2400, 0.66) (0, 0.01) [83]

(2400, 1)

[17]

Problem 4 (3000, 1) [80]

Problem 7 (6000, 0.45) [14]

Table A.3 Birnbaum (2008): upper (Problems 7.1 and 7.2) and lower cumulative independence (Problems 8.1 and 8.2), and dissection of Allais paradox (Problems 10.1–10.5).

Problem 2

Problem 3 (4000, 0.8) [20]

C

(4000, 0.20) [65]

(6000, 0.001) [73]

Problem 7.2

M (110, 0.8) (44, 0.1) (40, 0.1) [30]

N (110, 0.8) (98, 0.1) (10, 0.1) [70]

Problem 8.1 (3000, 0.25) [35]

Problem 8 (3000, 0.9) [86]

(3000, 0.002) [27]

Table A.2 Birnbaum (2008): coalescing (Problems 1.1 and 1.2), stochastic dominance (Problems 2, 3.1, 3.2, 3.3, 4), and upper tail independence (Problems 5.1, 5.2, 6.1, 6.2).

A (100, 0.85) (50, 0.1) (50, 0.05) [38]

Problem 1.2

Q (96, 0.05) (12, 0.05) (3, 0.9) [38]

R (52, 0.05) (48, 0.05) (3, 0.9) [62]

B (100, 0.85) (100, 0.1) (7, 0.05) [62]

Problem 2, 3.1 J (96, 0.85) (90, 0.05) (12, 0.1)

[27]

[73]

Problem 3.3 J′ (90, 0.85) (80, 0.05) (10, 0.1) [57]

(110, 0.8) (44, 0.1) (40, 0.1) [33]

[26]

M (96, 0.85) (96, 0.05) (14, 0.05) (12, 0.05) [94]

N (96, 0.85) (90, 0.05) (12, 0.05) (12, 0.05) [6]

K (96, 0.9) (14, 0.05) (12, 0.05) [65]

L (96, 0.25) (90, 0.05) (12, 0.7) [35]

Problem 5.2 t (92, 0.48) (0, 0.52) [34]

u (97, 0.43) (68, 0.07) (0, 0.5) [38]

x (110, 0.8) (96, 0.1) (10, 0.1) [67]

y (96, 0.8) (44, 0.1) (40, 0.1) [67]

Problem 6.1

w

[74]

Problem 4

Problem 5.1 s (92, 0.43) (68, 0.07) (0, 0.5) [66]

B′ (100, 0.95) (7, 0.05)

Problem 3.2

I (96, 0.9) (14, 0.05) (12, 0.05)

I′ (97, 0.9) (15, 0.05) (13, 0.05) [43]

A′ (100, 0.85) (50, 0.15)

v (97, 0.43) (92, 0.05) (0, 0.52) [62]

Problem 6.2

F (40, 0.2) (2, 0.8)

[62]

[38]

Problem 10.3

[33]

K (98, 0.8) (98, 0.1) (2, 0.1) [57]

[42]

S (96, 0.05) (12, 0.95)

T (52, 0.1) (12, 0.9)

[74]

[26]

G (98, 0.1) (2, 0.1) (2, 0.8) [36]

H (40, 0.1) (40, 0.1) (2, 0.8) [64] Problem 10.5

L (98, 0.8) (40, 0.1) (40, 0.1) [43]

M (98, 0.9) (2, 0.1)

N (98, 0.8) (40, 0.2)

[22]

[78]

Table A.4 Birnbaum (2008): restricted branch independency (Problems 13.1 and 13.2), 4-distribution independence (Problems 14.1 and 14.2), 3-lower distribution independence (Problems 15.1 and 15.2), 3–2 lower distribution independence (Problems 16.1 and 16.2), and 3-upper distribution independence (Problems 17.1 and 17.2). Problem 13.1 S (44, 0.25) (40, 0.25) (5, 0.5) [60]

Problem 13.2 R (98, 0.25) (10, 0.25) (5, 0.5) [40]

Problem 14.1 S (110, 0.01) (49, 0.2) (45, 0.2) (4,0.59) [66]

S (58, 0.2) (56, 0.2) (2, 0.6) [76]

S′ (111, 0.5) (44, 0.25) (40, 0.25) [38] S′ (110, 0.59) (49, 0.2) (45, 0.2) (4, 0.01) [49]

R′ (110, 0.01) (97, 0.2) (11, 0.2) (4,0.59) [51]

Problem 15.2 R (96, 0.2) (4, 0.2) (2, 0.6) [24]

S2 (58, 0.45) (56, 0.45) (2, 0.1) [81]

S0 (44, 0.5) (40, 0.5)

R0 (96, 0.5) (4, 0.5)

[69]

[31]

S (44, 0.48) (40, 0.48) (2, 0.4) [66]

R′ (110, 0.8) (96, 0.1) (10, 0.1) [56]

S2′ (100, 0.1) (44, 0.45) (40, 0.45) [67]

R2 (96, 0.45) (4, 0.45) (2, 0.1) [19]

Problem 16.2

Problem 17.1 S′ (110, 0.8) (44, 0.1) (40, 0.1) [44]

R′ (111, 0.5) (98, 0.25) (10, 0.25) [62]

Problem 14.2 R (110, 0.01) (97, 0.2) (11, 0.2) (4,0.59) [34]

Problem 16.1

tail independence. From Table A.2 onwards, monetary outcomes are in US dollars. Upper cumulative independence and lower cumulative independence are studied in Problems 7.1–8.2. A joint analysis of branch independence and coalescing is performed through Problems 10.1–10.5 (dissection of Allais paradox). These problems are in Table A.3. Problems 9.1–9.3 are not considered because the size of the outcomes is in millions instead of tens of US dollars, and therefore their solutions are not comparable with the others (see also the TAX model’s performance reported in Table 2 of Birnbaum, 2008). Problems 13.1–17.2 in Table A.4 are instead aimed at testing restricted branch independency, 4-distribution independence, 3lower distribution independence, 3–2 lower distribution independence, and 3-upper distribution independence.

[58]

Problem 10.4 J (40, 0.1) (40, 0.8) (40, 0.1) [54]

Problem 15.1 z (96, 0.9) (10, 0.1)

P (98, 0.9) (10, 0.1)

Problem 10.2

E (98, 0.1) (2, 0.9)

I (98, 0.1) (40, 0.8) (2, 0.1) [46]

O (98, 0.8) (40, 0.2)

Problem 8.2

Problem 10.1

Problem 1.1

5

R (96, 0.48) (4, 0.48) (2, 0.4) [34]

Problem 17.2 R2′ (100, 0.1) (96, 0.45) (4, 0.45) [33]

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