- Email: [email protected]
Comment on Cenci et al. (2015): “Half-full or half-empty? A model of decision making under risk”
Journal of Mathematical Psychology 81 (2017) 110–113
Contents lists available at ScienceDirect
Journal of Mathematical Psychology journal homepage: www.elsevier.com/locate/jmp
Comment
Comment on Cenci et al. (2015): ‘‘Half-full or half-empty? A model of decision making under risk’’ Marc Oliver Rieger University of Trier, Chair of Banking and Finance, 54286 Trier, Germany
highlights • • • • •
We revisit the decision model introduced by Cenci et al. (2015). We show that their model corresponds to an extension of prospect theory. We also show its relation to Disappointment Aversion. The model is also related to standard risk measures in finance. These results give new insights into their model, its validity and its applicability.
article
info
Article history: Received 13 December 2016 Received in revised form 26 September 2017 Available online 2 November 2017 Keywords: Decisions under risk Prospect theory Disappointment aversion Probability weighting Loss aversion Risk measure
a b s t r a c t In this comment, we demonstrate that the decision model proposed by Cenci et al. (2015) can be reformulated as an extension of normalized Prospect Theory and is also related to Disappoint Aversion. These reformulations allow us to understand better the novel ideas in this model and why its choice of the weighting function is in a certain sense natural. Moreover, we point out an interesting connection of the model to risk measures in finance. © 2017 Elsevier Inc. All rights reserved.
1. The model of Cenci, Corradini, Feduzi, and Gheno (2015)
Second, they modify this model by introducing weights for the probabilities pi . The weighted probabilities are defined by
Cenci et al. (2015) introduce a new, parsimonious model for decisions under risk that is able to capture some of the most important observations of actual behavior. Their model consists of two key ideas: First, they introduce different decision weights for outcomes above and below the expected value. For a lottery X with finitely many non-negative outcomes X = (pi : x1 , p2 : x2 , . . . , pn : xn ) and µ = E[X ], this leads to the evaluation function1
p
j=1
µq := Eq [X ] :=
n ∑
w(pi , (p1 , . . . , pn ), q)xi .
All in all, their complete evaluation function becomes (1)
where λ ∈ [0, 1] describes the relative decision weights of above and below average outcomes.
https://doi.org/10.1016/j.jmp.2017.09.007 0022-2496/© 2017 Elsevier Inc. All rights reserved.
,
i=1
Hλ (X ) = µ + 2 λE [max(0, X − µ)]
DOI of original article: http://dx.doi.org/10.1016/j.jmp.2015.06.006. E-mail address: [email protected]. 1 This is Eq. (3) in Cenci et al. (2015).
q
pj
where q is a weighting parameter. For instance, the expected value is subsequently calculated using the weighted probabilities, i.e.
(
) + (1 − λ)E [min(0, X − µ)] ,
q
w(pi , (p1 , . . . , pn ), q) := ∑n i
(
H(X ) = µq + 2 λEq [max(0, X − µ)]
) + (1 − λ)Eq [min(0, X − µ)] .
(2)
As the authors point out in Remark 2, the resulting model can be described as H(X ) = Eq [u(x)]
M.O. Rieger / Journal of Mathematical Psychology 81 (2017) 110–113
with u(X ) = X + (2λ − 1)|X − µq | being ‘‘a peculiar utility function that is lottery dependent being a function of µq ’’ (Cenci et al., 2015). In this comment, we want to clarify the relations of this model to a number of similar decision theories. We show in particular, that the model can be reformulated as extension of a variant of Prospect Theory (compare Kahneman & Tversky, 1979) which had been introduced by Karmarkar (1978). We will also show that the model can be reformulated as risk–return trade-off with a behavioral risk measure—a somehow surprising connection to financial mathematics. Before that, however, we explore the connection to ReferenceDependent Preferences in the sense of Koszegi and Rabin (2006) and to Disappointment Aversion as introduced by Gul (1991). 2. Connection to reference-dependent preferences Koszegi and Rabin (2006) suggest a model where reference points are chosen endogenously2 : We consider a lottery p with finitely many, ordered outcomes x1 , . . . , xn and associated probabilities pi . With the help of a utility function u and an increasing gain–loss function g we evaluate the lottery p with respect to a lottery p′ (our expectations) with the function U(p, p′ ) =
∑
u(xi )pi +
i
∑∑ i
g(u(xi ) − u(xj ))pi p′j .
j
In the next step, we consider an equilibrium where the expectations coincide with the actual lottery, providing us with the ultimate valuation formula V (p) = U(p, p). At first glance this looks similar to the model by Cenci et al. (2015) in that the outcomes are evaluated with respect to a quantity that in itself depends on the lottery, but there are important differences: First, there is no obvious probability weighting.—In the words of the authors: ‘‘Despite the clear evidence that peoples evaluation of prospects is not linear in probabilities, our model simplifies things by assuming preferences are linear.’’ Koszegi and Rabin (2006, page 1137, footnote 2).3 Second, the gain–loss utility compares the potential outcomes not to one fixed value (the expected value in Cenci et al. (2015)), but with all possible values simultaneously: ‘‘the sense of gain or loss from a given consumption outcome derives from comparing it with all outcomes possible under the reference lottery’’ (Koszegi & Rabin, 2006, page 1137). As Masatlioglu and Raymond (2016) point out, this may lead to violation of first order stochastic dominance. 3. Connection to disappointment aversion The model by Cenci et al. (2015) has an interesting relation to disappointment aversion (Gul, 1991). This is mentioned in Remark 5 of Cenci et al. (2015), but we want to elaborate on this relation in more detail: Gul (1991) introduced a decision model that is, in a certain sense, the most parsimonious extension of Expected Utility Theory that achieves to cover the famous Allais paradox (Allais, 1953). We briefly summarize the model in the following (willingly ignoring some more subtle difficulties for the sake of an easier exposition).
We consider a lottery p with finitely many, ordered outcomes x1 , . . . , xn and associated probabilities pi . Denote the certainty equivalent of this lottery by x. Then we denote the part of the lottery p with outcomes less than x by r˜ , we denote the part of the lottery p with outcomes larger than x by q˜ , and we denote the probability to have an outcome less than x by a ∈ [0, 1].4 Thus p = aq˜ + (1 − a)r˜ . The probabilities of the outcomes of q˜ and r˜ will accordingly be q˜ i = pi /a (for all i such that xi < x) and r˜i = pi /(1 − a) (for all i such that xi > x). Gul (1991) calls this decomposition (a, q˜ , r˜ ) of p an ‘‘elation disappointment decomposition’’ (EDD). He then proves that under certain natural axioms there exist a function u : R → R and a constant β ∈ (−1, ∞) such that we can evaluate the lottery p with EDD (a, q˜ , r˜ ) simply by computing V (p) = γ (a)
some cases be reformulated into a Rank-Dependent Utility model (Quiggin, 1982) with quadratic utility function and a certain probability weighting function.
∑
u(xi )q˜ i + (1 − γ (a))
∑
u(xi )r˜i ,
where
γ (a) :=
a 1 + (1 − a)β
, for a ∈ [0, 1].
It is important to notice that this defines V only indirectly, as Gul (1991, page 673) points out: ‘‘To see that V (p) is not explicitly defined note that the certainty equivalence of p needs to be known in order to determine an EDD of p’’. While this is different to the model of Cenci et al. (2015), there is an interesting connection: If we assume u(x) = x and also change the model of Disappointment Aversion in that the initial EDD is not made according to the certainty equivalence x of p, but according to the expected value µ of p, we can evaluate the lottery p with the modified function V˜ as follows: V˜ (p) = γ (a)
=
∑
u(xi )q˜ i + (1 − γ (a))
∑
u(xi )r˜i
a 1 E [min(0, X − µ)] 1 + (1 − a)β a ( ) a 1 + 1− E [max(0, X − µ)] . 1 + (1 − a)β 1 − a
A short computation leads to V˜ (p) =
1 1 + β − aβ
(µ + β E [max(0, X − µ)]) .
This resembles the evaluation function (1), but the coefficients differ: in particular, V˜ depends on a, the probability to be disappointed (i.e. to fall short of the expected value), while this probability itself does not play a role in the model by Cenci et al. (2015). 4. Reformulation in an extended prospect theory framework Let us now look at the connection of the model to the variant of Prospect Theory (compare Kahneman and Tversky, 1979) introduced by Karmarkar (1978). We will show that λ is related to the concept of loss aversion in Prospect Theory and how the particular probability weighting applied by Cenci et al. (2015) follows naturally from the framework of Karmarkar (1978). We start from Karmarkar’s Prospect Theory model that has been reformulated in modern notation in Rieger and Wang (2008): For a lottery X with outcomes xi , associated with probabilities pi and a reference point RF , we define the evaluation function PT (X ) :=
2 We omit some details in this exposition. Please consider the original source for further information. 3 Surprisingly, Masatlioglu and Raymond (2016) show that the model can in
111
n ∑ i=1
w (pi ) ∑n v (xi − RF ), j=1 w (pj )
4 Here one subtlety is that x might coincide to one of the x or that a = 0 or i a = 1. We refer to the original paper on how to cover these special cases with a more general definition.
112
M.O. Rieger / Journal of Mathematical Psychology 81 (2017) 110–113
where w is a probability weighting function and v is a value function. The choices of w and v are not fixed, but they have to satisfy a number of natural conditions. (We refer to the aforementioned papers for details.) Let us now choose as reference point the expected value of the lottery. This choice is not typical: usually, one chooses the current wealth level or – less frequently – the lowest or highest outcome or the current wealth level plus interest. Nevertheless, it is perfectly possible to choose the expected value as reference point, so we define RF = E[X ]. For the time being, we use the weighting function w (p) = p, i.e. we neglect probability weighting altogether. The simplest meaningful value function that has been used in the literature (particularly frequently in behavioral finance) is a piecewise affine function
v˜ (x) =
x, if x ≥ 0, λ′ x, if x < 0,
{
where λ′ is called loss aversion. It is usually denoted by the parameter λ. In order not to mix it up with the parameter in Cenci et al. (2015), we denote it by λ′ . We modify v˜ by adding (λ′ + 1)/2 times the reference point RF = E[X ] to it and define
λ′ + 1
v (x) = v˜ (x) +
E[X ]. 2 This is the only deviation that we take from the classical choice of a value function. It is, of course, a substantial extension of classical Prospect Theory using the aforementioned ideas (Gul, 1991; Koszegi & Rabin, 2006): in the original Prospect Theory, the value function does not depend on the lottery! We will show now that there is a close connection between λ and λ′ , since the above formulation of Prospect Theory reduces in this case to Hλ . To prove this, we first calculate (for the moment assuming w (p) = p): PT (X ) :=
n ∑
w(pi ) ∑n v (xi − E[X ]) j=1 w (pj )
i=1 n
=
∑
pi v (xi − E[X ])
∑
pi
( ) λ′ + 1 λ′ (xi − E[X ]) + E[X ] 2
i with xi
∑
+
pi
λ +1 ′
( (xi − E[X ]) +
i with xi ≥E[X ]
λ +1
2
) E[X ]
′
=
2
µw :=
n ∑ i=1
E[X ] + E [max(0, X − E[X ])]
+ λ E [min(0, X − E[X ])] . ′
Substituting λ := 1/(λ′ + 1), multiplying the PT-value with 2/(λ′ + 1) = 2λ (which does not affect the resulting preferences as it is a positive constant), and abbreviating µ := E[X ], we arrive at 2λ · PT (X ) = µ + 2 λE [max(0, X − µ)]
(
+ (1 − λ)E [min(0, X − µ)] = Hλ [X ].
)
This shows that the new model (without probability weighting yet!) can be reformulated as an extension of Prospect Theory. This extension requires the reference point and the value function to depend on the expected value of the lottery which is certainly unusual and an innovative idea. We now take a look at the probability weighting with a probability weighting function w . The reference point and the additional
w(pi ) ∑n xi . j=1 w (pj )
Redoing the above computation leads to 2λPT (X ) = µw + 2 λE [max(0, X − µw )]
(
) + (1 − λ)E [min(0, X − µw )] . For the particular choice of w (p) = pq , we arrive at 2λPT (X ) = H(X ) which proves the correspondence of the new theory to the Prospect Theory framework also in the case with probability weighting. We might wonder, however, whether this particular form of the probability weighting function is admissible, since it does not show the familiar S-shape of Prospect Theory weighting functions like the one introduced in Tversky and Kahneman (1992). This seems to be a strong concern, but we have to recall that this is the ‘‘normalized’’ formulation by Karmarkar, and a functional choice like w (p) = pq will still lead to overweighting of small and underweighting of large probabilities, ∑n since the ‘‘effective’’ weights include the normalizing term 1/( j=1 w (pj )), so that only ‘‘below average’’ probabilities are effectively over weighted, while ‘‘above average’’ probabilities are indeed under weighted.5 But even more so, the functional form w (p) = pq is not only possible, but also in a certain sense even the most natural choice: it has been shown by Rieger and Wang (2008) that in the limit of a continuous probability distribution all (reasonable) probability weighting functions in this model converge to the weighting function w (p) = pq . Thus, one might argue that using this function in the case of finitely many outcomes is also canonical. Rieger and Wang (2008) also provide us with a way to compare the probability weight q with the standard probability weighting parameter γ of the classical weighting function
wγ (p) =
i=1
=
term of the value function are now chosen as the weighted expected value
pγ (pγ
+ (1 − p)γ )1/γ
from Tversky and Kahneman (1992). Indeed, Rieger and Wang (2008) prove that for sufficiently small probabilities q = γ , since wγ (p) in the Prospect Theory utility becomes pγ as p → 0. This allows us to compare the estimates (q and λ) from Cenci et al. (2015) to standard estimates (γ and λ′ ), e.g. from Cenci et al. (2015) and Tversky and Kahneman (1992) obtain q = 0.48 and λ = 0.305. This would correspond to γ = 0.48 and λ′ = (1 − λ)/λ = 2.279, while Tversky and Kahneman (1992) obtained for their variant of Prospect Theory (cumulative Prospect Theory) values of γ = 0.61 and λ′ = 2.25. While the loss aversion parameter is (within any reasonable error bar) identical, there is some difference in the probability weighting parameter—not too surprising, given that the probabilities in the experimental data were way above zero, and therefore the different functional form should indeed make some difference. The measured value, however, even fits well to some other previous measurements. Kilka and Weber (2001), e.g., estimated γ ∈ [0.30, 0.51] in their experimental data. 5. Relations to risk measures in finance There is another interesting correspondence of the model, namely to risk measures in finance: starting with the work of Markowitz (1952), decisions in portfolio selection have been using the expected return and a risk measure as sole criteria, where the most classical risk measure is the variance (or, equivalently, 5 Another example where this probability weighting function occurred is Handa (1977).
M.O. Rieger / Journal of Mathematical Psychology 81 (2017) 110–113
113
the volatility) and the second frequently used one is the value at risk. Both concepts have been shown to have deficiencies: while value at risk does not satisfy certain natural assumptions on diversification (Artzner, Delbaen, Eber, & Heath, 1999), variance is symmetric and thus sees ‘‘risk’’ equally in small chance losses as in small chance gains. Alternative concepts like semi-variance that consider only bad outcomes have been suggested in the literature. More generally, one can consider one-sided moment-based risk measures defined by Rp := E |min(0, X − E[X ])|p
[
]
for p ≥ 1 (Fischer, 2003). In the case p = 2, we obtain the usual semi-variance, while for p = 1, we get R := E |min(0, X − E[X ])| = −E [min(0, X − E[X ])] .
[
]
(3)
The simplest decision model based on expected return in combination with the risk measure (3) is given by Fig. 1. Relations of the decision model by Cenci et al. (2015) to selected other decision models. Further details in the main text.
U(X ) := µ − α R = µ + α E[min(0, X − µ)], where α > 0 is a risk aversion parameter and µ = E(X ). By choosing
α :=
2 − 4λ 2λ + 1
,
(4)
we obtain (2λ + 1)U(X ) = (2λ + 1)µ + (2 − 4λ)E[min(0, X − µ)]
= µ + 2λE[max(0, X − µ)] + 2λE[min(0, X − µ)] + (2 − 4λ)E[min(0, X − µ)] ( = µ + 2 λE[max(0, X − µ)] ) + (1 − λ)E[min(0, X − µ)] = Hλ (X ). Moreover, we note that risk aversion in the sense of risk measures, i.e. α > 0, corresponds to λ ∈ (0, 0.5), as can be seen from (4). This situation is defined as ‘‘pessimism’’ in the model by Cenci et al. (2015). Thus, the model of Cenci et al. (2015) in the case without probability weighting can be expressed as a simple risk–return trade-off where the risk is measured by the risk measure (3). This connection is certainly surprising and novel, since it brings together models for risk of two very different academic fields. In the case with probability weighting, we could interpret the model of Cenci et al. (2015) as a risk–return trade-off after adjusting all probabilities involved with the corresponding weighting function. This interpretation, therefore, describes the new model as a ‘‘behavioral risk measure’’ in connection with a probability weighted return. 6. Conclusion We have found that the new decision model by Cenci et al. (2015) can be either understood as a simplification of Disappointment Aversion or an extension of Prospect Theory (see Fig. 1). Its novel feature is essentially the combination of a new reference point choice (the expected value of the lottery) with normalized Prospect Theory where the value function depends on the expected value of the lottery. In this way, the theory can reduce a lot of the usual complexity of Prospect Theory (convex–concave structure of the value function, precise shape of the probability weighting function) to a more parsimonious model. The above derivations demonstrate in particular that the choice of the weighting function is in a certain sense natural.
We also derived an alternative interpretation of the new decision model as a behavioral (i.e. probability weighted) variant of a classical risk–return trade off as it is standard in mathematical finance. We hope that our results help to put the new theory into a framework and can help to understand why it works well to describe several features of behavioral decision making under risk. This new framework might also lead to other variants of the model, e.g. by changing the risk measure. Acknowledgments The author is grateful to Artem Dyachenko, Frank Seifried and to the anonymous referees for their valuable input. References Allais, M. (1953). Le comportement de l’homme rationnel devant le risque: critique des postulats et axiomes de l’ćole amŕicaine. Econometrica, 21(4), 503–546. Artzner, P., Delbaen, F., Eber, J.-M., & Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9(3), 203–228. Cenci, M., Corradini, M., Feduzi, A., & Gheno, A. (2015). Half-full or half-empty? a model of decision making under risk. Journal of Mathematical Psychology, 68, 1–6. Fischer, T. (2003). Risk capital allocation by coherent risk measures based on onesided moments. Insurance: Mathematics & Economics, 32(1), 135–146. Gul, F. (1991). A theory of disappoinment aversion. Econometrica, 59(3), 667–686. Handa, J. (1977). Risk, probabilities, and a new theory of cardinal utility. Journal of Political Economy, 85(1), 97–122. Kahneman, D., & Tversky, A. (1979). Prospect Theory: An analysis of decision under risk. Econometrica, 47, 263–291. Karmarkar, U. S. (1978). Subjectively weighted utility: A descriptive extension of the expected utility model. Organizational Behavior and Human Performance, 21, 61–72. Kilka, M., & Weber, M. (2001). What determines the shape of the probability weighting function under uncertainty? Management Science, 47(12), 1712–1726. Koszegi, B., & Rabin, M. (2006). A model of reference-dependent preferences. The Quarterly Journal of Economics, CXXI(4), 1133–1165. Markowitz, H. M. (1952). Portfolio selection. The Journal of Finance, 7(1), 77–91. Masatlioglu, Y., & Raymond, C. (2016). A behavioral analysis of stochastic reference dependence. American Economic Review, 106, 2760–2782. Quiggin, J. (1982). A theory of anticipated utility. Journal of Economic Behavior & Organization, 3(4), 323–343. Rieger, M. O., & Wang, M. (2008). Prospect Theory for continuous distributions. Journal of Risk and Uncertainty, 36, 83–102. Tversky, A., & Kahneman, D. (1992). Advances in Prospect Theory: Cumulative representation of uncertainty. Journal of Risk and Uncertainty, 5, 297–323.
Contents lists available at ScienceDirect
Journal of Mathematical Psychology journal homepage: www.elsevier.com/locate/jmp
Comment
Comment on Cenci et al. (2015): ‘‘Half-full or half-empty? A model of decision making under risk’’ Marc Oliver Rieger University of Trier, Chair of Banking and Finance, 54286 Trier, Germany
highlights • • • • •
We revisit the decision model introduced by Cenci et al. (2015). We show that their model corresponds to an extension of prospect theory. We also show its relation to Disappointment Aversion. The model is also related to standard risk measures in finance. These results give new insights into their model, its validity and its applicability.
article
info
Article history: Received 13 December 2016 Received in revised form 26 September 2017 Available online 2 November 2017 Keywords: Decisions under risk Prospect theory Disappointment aversion Probability weighting Loss aversion Risk measure
a b s t r a c t In this comment, we demonstrate that the decision model proposed by Cenci et al. (2015) can be reformulated as an extension of normalized Prospect Theory and is also related to Disappoint Aversion. These reformulations allow us to understand better the novel ideas in this model and why its choice of the weighting function is in a certain sense natural. Moreover, we point out an interesting connection of the model to risk measures in finance. © 2017 Elsevier Inc. All rights reserved.
1. The model of Cenci, Corradini, Feduzi, and Gheno (2015)
Second, they modify this model by introducing weights for the probabilities pi . The weighted probabilities are defined by
Cenci et al. (2015) introduce a new, parsimonious model for decisions under risk that is able to capture some of the most important observations of actual behavior. Their model consists of two key ideas: First, they introduce different decision weights for outcomes above and below the expected value. For a lottery X with finitely many non-negative outcomes X = (pi : x1 , p2 : x2 , . . . , pn : xn ) and µ = E[X ], this leads to the evaluation function1
p
j=1
µq := Eq [X ] :=
n ∑
w(pi , (p1 , . . . , pn ), q)xi .
All in all, their complete evaluation function becomes (1)
where λ ∈ [0, 1] describes the relative decision weights of above and below average outcomes.
https://doi.org/10.1016/j.jmp.2017.09.007 0022-2496/© 2017 Elsevier Inc. All rights reserved.
,
i=1
Hλ (X ) = µ + 2 λE [max(0, X − µ)]
DOI of original article: http://dx.doi.org/10.1016/j.jmp.2015.06.006. E-mail address: [email protected]. 1 This is Eq. (3) in Cenci et al. (2015).
q
pj
where q is a weighting parameter. For instance, the expected value is subsequently calculated using the weighted probabilities, i.e.
(
) + (1 − λ)E [min(0, X − µ)] ,
q
w(pi , (p1 , . . . , pn ), q) := ∑n i
(
H(X ) = µq + 2 λEq [max(0, X − µ)]
) + (1 − λ)Eq [min(0, X − µ)] .
(2)
As the authors point out in Remark 2, the resulting model can be described as H(X ) = Eq [u(x)]
M.O. Rieger / Journal of Mathematical Psychology 81 (2017) 110–113
with u(X ) = X + (2λ − 1)|X − µq | being ‘‘a peculiar utility function that is lottery dependent being a function of µq ’’ (Cenci et al., 2015). In this comment, we want to clarify the relations of this model to a number of similar decision theories. We show in particular, that the model can be reformulated as extension of a variant of Prospect Theory (compare Kahneman & Tversky, 1979) which had been introduced by Karmarkar (1978). We will also show that the model can be reformulated as risk–return trade-off with a behavioral risk measure—a somehow surprising connection to financial mathematics. Before that, however, we explore the connection to ReferenceDependent Preferences in the sense of Koszegi and Rabin (2006) and to Disappointment Aversion as introduced by Gul (1991). 2. Connection to reference-dependent preferences Koszegi and Rabin (2006) suggest a model where reference points are chosen endogenously2 : We consider a lottery p with finitely many, ordered outcomes x1 , . . . , xn and associated probabilities pi . With the help of a utility function u and an increasing gain–loss function g we evaluate the lottery p with respect to a lottery p′ (our expectations) with the function U(p, p′ ) =
∑
u(xi )pi +
i
∑∑ i
g(u(xi ) − u(xj ))pi p′j .
j
In the next step, we consider an equilibrium where the expectations coincide with the actual lottery, providing us with the ultimate valuation formula V (p) = U(p, p). At first glance this looks similar to the model by Cenci et al. (2015) in that the outcomes are evaluated with respect to a quantity that in itself depends on the lottery, but there are important differences: First, there is no obvious probability weighting.—In the words of the authors: ‘‘Despite the clear evidence that peoples evaluation of prospects is not linear in probabilities, our model simplifies things by assuming preferences are linear.’’ Koszegi and Rabin (2006, page 1137, footnote 2).3 Second, the gain–loss utility compares the potential outcomes not to one fixed value (the expected value in Cenci et al. (2015)), but with all possible values simultaneously: ‘‘the sense of gain or loss from a given consumption outcome derives from comparing it with all outcomes possible under the reference lottery’’ (Koszegi & Rabin, 2006, page 1137). As Masatlioglu and Raymond (2016) point out, this may lead to violation of first order stochastic dominance. 3. Connection to disappointment aversion The model by Cenci et al. (2015) has an interesting relation to disappointment aversion (Gul, 1991). This is mentioned in Remark 5 of Cenci et al. (2015), but we want to elaborate on this relation in more detail: Gul (1991) introduced a decision model that is, in a certain sense, the most parsimonious extension of Expected Utility Theory that achieves to cover the famous Allais paradox (Allais, 1953). We briefly summarize the model in the following (willingly ignoring some more subtle difficulties for the sake of an easier exposition).
We consider a lottery p with finitely many, ordered outcomes x1 , . . . , xn and associated probabilities pi . Denote the certainty equivalent of this lottery by x. Then we denote the part of the lottery p with outcomes less than x by r˜ , we denote the part of the lottery p with outcomes larger than x by q˜ , and we denote the probability to have an outcome less than x by a ∈ [0, 1].4 Thus p = aq˜ + (1 − a)r˜ . The probabilities of the outcomes of q˜ and r˜ will accordingly be q˜ i = pi /a (for all i such that xi < x) and r˜i = pi /(1 − a) (for all i such that xi > x). Gul (1991) calls this decomposition (a, q˜ , r˜ ) of p an ‘‘elation disappointment decomposition’’ (EDD). He then proves that under certain natural axioms there exist a function u : R → R and a constant β ∈ (−1, ∞) such that we can evaluate the lottery p with EDD (a, q˜ , r˜ ) simply by computing V (p) = γ (a)
some cases be reformulated into a Rank-Dependent Utility model (Quiggin, 1982) with quadratic utility function and a certain probability weighting function.
∑
u(xi )q˜ i + (1 − γ (a))
∑
u(xi )r˜i ,
where
γ (a) :=
a 1 + (1 − a)β
, for a ∈ [0, 1].
It is important to notice that this defines V only indirectly, as Gul (1991, page 673) points out: ‘‘To see that V (p) is not explicitly defined note that the certainty equivalence of p needs to be known in order to determine an EDD of p’’. While this is different to the model of Cenci et al. (2015), there is an interesting connection: If we assume u(x) = x and also change the model of Disappointment Aversion in that the initial EDD is not made according to the certainty equivalence x of p, but according to the expected value µ of p, we can evaluate the lottery p with the modified function V˜ as follows: V˜ (p) = γ (a)
=
∑
u(xi )q˜ i + (1 − γ (a))
∑
u(xi )r˜i
a 1 E [min(0, X − µ)] 1 + (1 − a)β a ( ) a 1 + 1− E [max(0, X − µ)] . 1 + (1 − a)β 1 − a
A short computation leads to V˜ (p) =
1 1 + β − aβ
(µ + β E [max(0, X − µ)]) .
This resembles the evaluation function (1), but the coefficients differ: in particular, V˜ depends on a, the probability to be disappointed (i.e. to fall short of the expected value), while this probability itself does not play a role in the model by Cenci et al. (2015). 4. Reformulation in an extended prospect theory framework Let us now look at the connection of the model to the variant of Prospect Theory (compare Kahneman and Tversky, 1979) introduced by Karmarkar (1978). We will show that λ is related to the concept of loss aversion in Prospect Theory and how the particular probability weighting applied by Cenci et al. (2015) follows naturally from the framework of Karmarkar (1978). We start from Karmarkar’s Prospect Theory model that has been reformulated in modern notation in Rieger and Wang (2008): For a lottery X with outcomes xi , associated with probabilities pi and a reference point RF , we define the evaluation function PT (X ) :=
2 We omit some details in this exposition. Please consider the original source for further information. 3 Surprisingly, Masatlioglu and Raymond (2016) show that the model can in
111
n ∑ i=1
w (pi ) ∑n v (xi − RF ), j=1 w (pj )
4 Here one subtlety is that x might coincide to one of the x or that a = 0 or i a = 1. We refer to the original paper on how to cover these special cases with a more general definition.
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M.O. Rieger / Journal of Mathematical Psychology 81 (2017) 110–113
where w is a probability weighting function and v is a value function. The choices of w and v are not fixed, but they have to satisfy a number of natural conditions. (We refer to the aforementioned papers for details.) Let us now choose as reference point the expected value of the lottery. This choice is not typical: usually, one chooses the current wealth level or – less frequently – the lowest or highest outcome or the current wealth level plus interest. Nevertheless, it is perfectly possible to choose the expected value as reference point, so we define RF = E[X ]. For the time being, we use the weighting function w (p) = p, i.e. we neglect probability weighting altogether. The simplest meaningful value function that has been used in the literature (particularly frequently in behavioral finance) is a piecewise affine function
v˜ (x) =
x, if x ≥ 0, λ′ x, if x < 0,
{
where λ′ is called loss aversion. It is usually denoted by the parameter λ. In order not to mix it up with the parameter in Cenci et al. (2015), we denote it by λ′ . We modify v˜ by adding (λ′ + 1)/2 times the reference point RF = E[X ] to it and define
λ′ + 1
v (x) = v˜ (x) +
E[X ]. 2 This is the only deviation that we take from the classical choice of a value function. It is, of course, a substantial extension of classical Prospect Theory using the aforementioned ideas (Gul, 1991; Koszegi & Rabin, 2006): in the original Prospect Theory, the value function does not depend on the lottery! We will show now that there is a close connection between λ and λ′ , since the above formulation of Prospect Theory reduces in this case to Hλ . To prove this, we first calculate (for the moment assuming w (p) = p): PT (X ) :=
n ∑
w(pi ) ∑n v (xi − E[X ]) j=1 w (pj )
i=1 n
=
∑
pi v (xi − E[X ])
∑
pi
( ) λ′ + 1 λ′ (xi − E[X ]) + E[X ] 2
i with xi
∑
+
pi
λ +1 ′
( (xi − E[X ]) +
i with xi ≥E[X ]
λ +1
2
) E[X ]
′
=
2
µw :=
n ∑ i=1
E[X ] + E [max(0, X − E[X ])]
+ λ E [min(0, X − E[X ])] . ′
Substituting λ := 1/(λ′ + 1), multiplying the PT-value with 2/(λ′ + 1) = 2λ (which does not affect the resulting preferences as it is a positive constant), and abbreviating µ := E[X ], we arrive at 2λ · PT (X ) = µ + 2 λE [max(0, X − µ)]
(
+ (1 − λ)E [min(0, X − µ)] = Hλ [X ].
)
This shows that the new model (without probability weighting yet!) can be reformulated as an extension of Prospect Theory. This extension requires the reference point and the value function to depend on the expected value of the lottery which is certainly unusual and an innovative idea. We now take a look at the probability weighting with a probability weighting function w . The reference point and the additional
w(pi ) ∑n xi . j=1 w (pj )
Redoing the above computation leads to 2λPT (X ) = µw + 2 λE [max(0, X − µw )]
(
) + (1 − λ)E [min(0, X − µw )] . For the particular choice of w (p) = pq , we arrive at 2λPT (X ) = H(X ) which proves the correspondence of the new theory to the Prospect Theory framework also in the case with probability weighting. We might wonder, however, whether this particular form of the probability weighting function is admissible, since it does not show the familiar S-shape of Prospect Theory weighting functions like the one introduced in Tversky and Kahneman (1992). This seems to be a strong concern, but we have to recall that this is the ‘‘normalized’’ formulation by Karmarkar, and a functional choice like w (p) = pq will still lead to overweighting of small and underweighting of large probabilities, ∑n since the ‘‘effective’’ weights include the normalizing term 1/( j=1 w (pj )), so that only ‘‘below average’’ probabilities are effectively over weighted, while ‘‘above average’’ probabilities are indeed under weighted.5 But even more so, the functional form w (p) = pq is not only possible, but also in a certain sense even the most natural choice: it has been shown by Rieger and Wang (2008) that in the limit of a continuous probability distribution all (reasonable) probability weighting functions in this model converge to the weighting function w (p) = pq . Thus, one might argue that using this function in the case of finitely many outcomes is also canonical. Rieger and Wang (2008) also provide us with a way to compare the probability weight q with the standard probability weighting parameter γ of the classical weighting function
wγ (p) =
i=1
=
term of the value function are now chosen as the weighted expected value
pγ (pγ
+ (1 − p)γ )1/γ
from Tversky and Kahneman (1992). Indeed, Rieger and Wang (2008) prove that for sufficiently small probabilities q = γ , since wγ (p) in the Prospect Theory utility becomes pγ as p → 0. This allows us to compare the estimates (q and λ) from Cenci et al. (2015) to standard estimates (γ and λ′ ), e.g. from Cenci et al. (2015) and Tversky and Kahneman (1992) obtain q = 0.48 and λ = 0.305. This would correspond to γ = 0.48 and λ′ = (1 − λ)/λ = 2.279, while Tversky and Kahneman (1992) obtained for their variant of Prospect Theory (cumulative Prospect Theory) values of γ = 0.61 and λ′ = 2.25. While the loss aversion parameter is (within any reasonable error bar) identical, there is some difference in the probability weighting parameter—not too surprising, given that the probabilities in the experimental data were way above zero, and therefore the different functional form should indeed make some difference. The measured value, however, even fits well to some other previous measurements. Kilka and Weber (2001), e.g., estimated γ ∈ [0.30, 0.51] in their experimental data. 5. Relations to risk measures in finance There is another interesting correspondence of the model, namely to risk measures in finance: starting with the work of Markowitz (1952), decisions in portfolio selection have been using the expected return and a risk measure as sole criteria, where the most classical risk measure is the variance (or, equivalently, 5 Another example where this probability weighting function occurred is Handa (1977).
M.O. Rieger / Journal of Mathematical Psychology 81 (2017) 110–113
113
the volatility) and the second frequently used one is the value at risk. Both concepts have been shown to have deficiencies: while value at risk does not satisfy certain natural assumptions on diversification (Artzner, Delbaen, Eber, & Heath, 1999), variance is symmetric and thus sees ‘‘risk’’ equally in small chance losses as in small chance gains. Alternative concepts like semi-variance that consider only bad outcomes have been suggested in the literature. More generally, one can consider one-sided moment-based risk measures defined by Rp := E |min(0, X − E[X ])|p
[
]
for p ≥ 1 (Fischer, 2003). In the case p = 2, we obtain the usual semi-variance, while for p = 1, we get R := E |min(0, X − E[X ])| = −E [min(0, X − E[X ])] .
[
]
(3)
The simplest decision model based on expected return in combination with the risk measure (3) is given by Fig. 1. Relations of the decision model by Cenci et al. (2015) to selected other decision models. Further details in the main text.
U(X ) := µ − α R = µ + α E[min(0, X − µ)], where α > 0 is a risk aversion parameter and µ = E(X ). By choosing
α :=
2 − 4λ 2λ + 1
,
(4)
we obtain (2λ + 1)U(X ) = (2λ + 1)µ + (2 − 4λ)E[min(0, X − µ)]
= µ + 2λE[max(0, X − µ)] + 2λE[min(0, X − µ)] + (2 − 4λ)E[min(0, X − µ)] ( = µ + 2 λE[max(0, X − µ)] ) + (1 − λ)E[min(0, X − µ)] = Hλ (X ). Moreover, we note that risk aversion in the sense of risk measures, i.e. α > 0, corresponds to λ ∈ (0, 0.5), as can be seen from (4). This situation is defined as ‘‘pessimism’’ in the model by Cenci et al. (2015). Thus, the model of Cenci et al. (2015) in the case without probability weighting can be expressed as a simple risk–return trade-off where the risk is measured by the risk measure (3). This connection is certainly surprising and novel, since it brings together models for risk of two very different academic fields. In the case with probability weighting, we could interpret the model of Cenci et al. (2015) as a risk–return trade-off after adjusting all probabilities involved with the corresponding weighting function. This interpretation, therefore, describes the new model as a ‘‘behavioral risk measure’’ in connection with a probability weighted return. 6. Conclusion We have found that the new decision model by Cenci et al. (2015) can be either understood as a simplification of Disappointment Aversion or an extension of Prospect Theory (see Fig. 1). Its novel feature is essentially the combination of a new reference point choice (the expected value of the lottery) with normalized Prospect Theory where the value function depends on the expected value of the lottery. In this way, the theory can reduce a lot of the usual complexity of Prospect Theory (convex–concave structure of the value function, precise shape of the probability weighting function) to a more parsimonious model. The above derivations demonstrate in particular that the choice of the weighting function is in a certain sense natural.
We also derived an alternative interpretation of the new decision model as a behavioral (i.e. probability weighted) variant of a classical risk–return trade off as it is standard in mathematical finance. We hope that our results help to put the new theory into a framework and can help to understand why it works well to describe several features of behavioral decision making under risk. This new framework might also lead to other variants of the model, e.g. by changing the risk measure. Acknowledgments The author is grateful to Artem Dyachenko, Frank Seifried and to the anonymous referees for their valuable input. References Allais, M. (1953). Le comportement de l’homme rationnel devant le risque: critique des postulats et axiomes de l’ćole amŕicaine. Econometrica, 21(4), 503–546. Artzner, P., Delbaen, F., Eber, J.-M., & Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9(3), 203–228. Cenci, M., Corradini, M., Feduzi, A., & Gheno, A. (2015). Half-full or half-empty? a model of decision making under risk. Journal of Mathematical Psychology, 68, 1–6. Fischer, T. (2003). Risk capital allocation by coherent risk measures based on onesided moments. Insurance: Mathematics & Economics, 32(1), 135–146. Gul, F. (1991). A theory of disappoinment aversion. Econometrica, 59(3), 667–686. Handa, J. (1977). Risk, probabilities, and a new theory of cardinal utility. Journal of Political Economy, 85(1), 97–122. Kahneman, D., & Tversky, A. (1979). Prospect Theory: An analysis of decision under risk. Econometrica, 47, 263–291. Karmarkar, U. S. (1978). Subjectively weighted utility: A descriptive extension of the expected utility model. Organizational Behavior and Human Performance, 21, 61–72. Kilka, M., & Weber, M. (2001). What determines the shape of the probability weighting function under uncertainty? Management Science, 47(12), 1712–1726. Koszegi, B., & Rabin, M. (2006). A model of reference-dependent preferences. The Quarterly Journal of Economics, CXXI(4), 1133–1165. Markowitz, H. M. (1952). Portfolio selection. The Journal of Finance, 7(1), 77–91. Masatlioglu, Y., & Raymond, C. (2016). A behavioral analysis of stochastic reference dependence. American Economic Review, 106, 2760–2782. Quiggin, J. (1982). A theory of anticipated utility. Journal of Economic Behavior & Organization, 3(4), 323–343. Rieger, M. O., & Wang, M. (2008). Prospect Theory for continuous distributions. Journal of Risk and Uncertainty, 36, 83–102. Tversky, A., & Kahneman, D. (1992). Advances in Prospect Theory: Cumulative representation of uncertainty. Journal of Risk and Uncertainty, 5, 297–323.