The Distorted Theory of Rank-Dependent Expected Utility *

Transcription

1 ANNALS OF ECONOMICS AND FINANCE -, (0) The Distorted Theory of Rank-Dependent Expected Utility * Hui Huang Faculty of Business Administration, University of Regina Regina, Saskatchewan, S4S 0A, Canada h34huang@uwaterloo.ca and Shunming Zhang China Financial Policy Research Center, Renmin University of China Beijing, 0087, P.R.China szhang@ruc.edu.cn This paper re-examines the rank-dependent expected utility theory. Firstly, we follow Quiggin s assumption (Quiggin 98) to deduce the rank-dependent expected utility formula over lotteries and hence extend it to the case of general random variables. Secondly, we utilize the distortion function which reflects decision-makers beliefs to propose a distorted independence axiom and then to prove the representation theorem of rank-dependent expected utility. Finally, we make direct use of the distorted independence axiom to explain the Allais paradox and the common ratio effect. Key Words: Expected utility; Rank-dependent expected utility; Distortion function; Distorted independence axiom; The Allais paradox; The Common ratio effect. JEL Classification Number: D8.. INTRODUCTION It is well known that the independence axiom (IA), the key behavioral assumption of the expected utility (EU) theory, is often violated in practice in experimental studies. Amongst other theories, the anticipated utility (AU) theory, which is also known as rank-dependent expected utility * We are grateful financial supports from National Natural Science Foundation of China ( ) and National Social Sciences of China (07AJL00). We would like to thank Leigh Roberts and Matthew Ryan, seminar participants at Victoria University of Wellington and University of Auckland for their helpful comments and suggestions /0 All rights of reproduction in any form reserved.

2 34 HUI HUANG AND SHUNMING ZHANG RDEU (Quiggin 98, Segal 989, and Quiggin and Wakker 994), has successfully resolved this issue. There exist several axiomatic systems for this theory. The weak certainty equivalent substitution axiom in Quiggin (98) and Quiggin and Wakker (994) implies that the weights function maps to. However, such an assumption takes a lot of power out of this theory. Segal (989) utilizes a measure approach to axiomatize RDEU theory. He proposes a projection IA to graphically compare two cumulative distribution functions (or lotteries), which lacks normative appeal. Here we propose a new axiomatic foundation to RDEU. In our opinion, the weights function in RDEU reflects decision-makers beliefs and their attitude to risks, and therefore can be treated as exogenous. Based on the weights function, we propose a distorted independence axiom (DIA) and establish a new axiomatic system to build the distorted theory of RDEU. This paper also shows how DIA can be used directly to determine the specific forms of weights function under which some examples violating IA would no longer be paradoxical. We first follow the assumption in Quiggin (98) and Quiggin and Wakker (994) that the transformation of cumulative distribution functions (CDFs) is continuous on the whole probability distribution. Applying the reduction of lottery dimensions we prove the RDEU formula over lotteries and show that it also holds for general (continuous) random variables. Quiggin s assumption is a breakthrough in successfully extending EU to RDEU. The essence of the von Neumann - Morgenstern EU theory is a set of restrictions imposed on the preference relations over lotteries that allows their representation by a mathematical expectation of a real function on the set of outcomes. One main aspect of this theory is the specific functional form of the representation, namely, the linearity in probabilities. The EU hypothesis is widely used in various disciplines. However, it sometimes fails to explain some counterexamples. Quiggin (98) successfully resolves this issue by proposing that probability weights of every prospect are derived from the entire original probability distribution. He tries a special case of three outcomes and writes the general form of his RDEU formula. Segal (987a) claims that this formula can be extended to any (one-stage) lottery. In this paper we prove Quiggin s RDEU formula over prospects by using the continuity of utility function (von Neumann - Morgenstern) and further generalize this formula for arbitrary random variables. For the case of general random variables we employ Helly theorem to prove the RDEU formula. Then we re-axiomatize the RDEU theory along the line of Quiggin (98), Chew (985) and Segal (989), using the methods in Fishburn (98) and Yaari (987). In the RDEU formula the transformation of CDFs or the deci- Later Chew (985) removes this restriction.

3 THE DISTORTED THEORY OF RANK-DEPENDENT 35 sion weights function reflects decision-makers beliefs. It distorts prospects probability distributions to reflect a decision maker s own evaluation of probabilities. Therefore we call it the distortion function. We can interpret the distortion function as the decision-maker s attitude to risk when choosing among lotteries. In this paper we assume that this distortion function is exogenous. We observe that the distortion function is also a CDF and can be represented by a random variable. Therefore, the distorted CDF also corresponds to a random variable, which is a compound of the inverse of the CDF of a risky prospect and the random variable corresponding to the distortion function. From this approach we provide our DIA, and hence prove the representation theorem of RDEU by modifying EU. The format of the distorted independence axiom (DIA) is analogous to IA, but in DIA we use a mixture operator instead of the conventional addition. The independence, instead of being hypothesized for convex combinations formed along the CDFs, is postulated for convex combinations formed along the distorted CDFs. Therefore, while IA is applied to the family of CDFs, DIA is applied to the distorted CDFs. We can use DIA directly to rationalize the most famous paradoxes in uncertainty theory such as the Allais paradox (or the common consequences effect) and the common ratio effect (or the certainty effect) without resorting to the RDEU formula as in Segal (987a). We use the distortion function to transform the unit triangle in Machina (987) to obtain triangles under the framework of DIA. In the new unit triangles, indifference curves keep parallel but positions of prospects change such that lines of compared prospect pairs may not parallel. We show that the compared prospects form parallelograms in the transformed unit triangle if and only if DIA reduces to IA. In this case the inconsistency in these paradoxes would arise. Furthermore, we are able to show that in the transformed unit triangle under DIA, when the distortion function takes specific forms such that the lines of compared prospect pairs fan in, the behavioral patterns in these examples may be rational. Our approach here fundamentally departs from that of Machina (987). In his diagrams, prospects are fixed and form parallelograms while indifference curves fan out. The RDEU theory has received several axiomatizations. In Quiggin (98) and Quiggin and Wakker (994), a preference relation satisfies a set of axioms including the key weak certainty equivalent substitution axiom if and only if it has an expected utility with rank-dependent probabilities where the probability transformation function maps to. As Chew, Karni and Safra (987), Röell (987) and Segal (987a) suggest, risk aversion holds in this theory if and only if von Neumann-Morgendtern utility function is concave and the weights function is convex. Assuming that the weights function maps to takes much power out of the theory. Chew (985) shows that the latter restriction is not necessary. Segal (989)

4 36 HUI HUANG AND SHUNMING ZHANG presents another set of axioms to prove RDEU by a measure representation approach. His projection IA is of a simple mathematical form which, in our opinion, is lack of interpretations in terms of behavioral foundations. Yaari (987) also suggests an expected utility theory with rank-dependent probabilities, but with the roles of payment and probability reversed. He cites Fishburn s (98) five axioms in EU and replaces IA with the dual IA. Our paper presents a more appealing axiomatic system for RDEU by replacing the dual IA in Yaari (987) with our DIA. To some extent, Yaari s theory can be treated as a special case of our distorted theory of RDEU. The difference is that in RDEU the utility function is endogenous and the distortion function is exogenous while in Yaari the endogeneity of these two is reversed. Our paper further differs from Yaari (987) in that all random variables in Yaari s model take values in the unit interval so that the inverse of a CDF is still a CDF, but in our model random variables take values from any (closed) interval. We use the distortion function which is also a CDF to distort the CDF of a risky prospect. There are other papers on RDEU. Chew, Karni and Safra (987) and Karni (987) study the risk aversion in expected utility theory with rankdependent probabilities and state-dependent preferences. The RDEU approach can be used not only to explain the examples with uncertainty such as the Allais paradox and the common ratio effect, but also to interpret the Ellsberg paradox (Segal 987b). Furthermore, Karni and Schmeidler (99) summarize the utility theory with uncertainty. On the other hand, RDEU can be used to explain ambiguity aversion, as Miao(004, 009) and Zou (006). This paper is composed of five sections. In section we formally generalize the RDEU formula from Quiggin (98) and Quiggin and Wakker (994). In section 3 we propose an axiomatic system with DIA and prove the representation theorem of RDEU. Section 4 explains the Allais paradox and the common ratio effect by directly using DIA, in addition to using the RDEU formula. Section 5 concludes this paper.. RANK-DEPENDENT EXPECTED UTILITY FORMULA This section outlines the RDEU theory, which represents decision makers preferences using mathematical expectations of a utility function with respect to a transformation of probabilities on a set of outcomes. The transformation function can be found by induction, and generally is not a linear function of the CDF. Each component of the transformation is a function of the whole probability distribution of the prospect and does not depend upon the winning probability of this prize only. For the case of discrete random variables we show that, for any natural number N =,,, the N-th component is an increment of the transformation of the sum of

5 THE DISTORTED THEORY OF RANK-DEPENDENT 37 the winning probabilities for the smallest N outcomes. Further, we can extend the EDRU formula to a more general one by using a convergence theorem of CDFs. Consider a compact interval [m, M] of monetary prizes. Let L be the set of lotteries (probability measures) over [m, M] and L 0 in L be the set of lotteries with finite support. For each X L, its CDF of X is defined by F X (x) = P {X x} for x [m, M]. Now we build the RDEU theory on L 0. For any natural number N =,,, denote X N = (x N, p N ; ; x N N, pn N ) as a prospect, which yields x N n dollars with probability p N n for n =,, N, where x N x N x N N xn N in [m, M], pn n 0 for n =,, N, and N n= pn n =. The RDEU function is defined to be RDEU(X N ) = N Hn N (p N,, p N N)U(x N n ) () n= where Hn N : [0, ] N [0, ] is a continuous function for n =,, N with N n= HN n (p N,, p N N ) =, and U is a continuous and increasing von Neumann - Morgenstern utility function. (H N,, HN N ) is a transformation of the probability distribution and produces a new probability distribution. Quiggin (98) assumes that, for n =,, N, Hn N (p N,, p N N ) is a function of (p N,, p N N ). Under the environment of the RDEU theory, Hn N (p N,, p N N ) is a function of (pn,, p N n ) for n =,, N, which is proved in this section by induction. For any N =,,, X N = (x N, p N ; ; x N N, pn N ) with N n= pn n =, we have H N (p N,, p N N) = g(p N ) () ( n ) ( n ) Hn N (p N,, p N N) = g p N n g p N n, for n =,, N(3) n = n = where g(p) = H (p, p) for p [0, ]. Thus we have the RDEU formula in L 0, which is an assertion in Quggin (98). Theorem (Quiggin 98). In the rank-dependent expected utility function (), the behavior of Hn N (p N,, p N N ) on arbitrary probability distributions is fully determined by the values of g as in () and (3). The CDF F X satisfies the following three conditions: [] F X is non-decreasing; [] F X (m ) = 0 and F X (M) = ; and [3] F X is right-continuous. If function F satisfies the three conditions, then there exists a random variable X such that its CDF F X is equal to F.

6 38 HUI HUANG AND SHUNMING ZHANG From this theorem, the RDEU function in L 0 is, for X N = (x N, p N ; ; x N N, pn N ), [ ( N n ) ( n )] RDEU(X N ) = g(p N )U(x N ) + g p N n g p N n U(x N n ). (4) n= From () and (3), we have Hn N (p N,, p N N ) is the increment of the transformation of the sum of the first n winning probabilities. Expression (4) can be re-written in a general form as RDEU(X N ) = U(x)dg(F X N (x)). n = n = From the proof of Theorem in the Appendix, we summarize the property of function g as follows. Proposition. The function g is a continuous and increasing function with g(0) = 0 and g() =. Furthermore, on L we can also prove the rank-dependent expected utility function by using a convergence theorem of CDFs. Theorem. The rank-dependent expected utility function in L is RDEU(X) = U(x)dg(F X (x)). (5) The RDEU formula describes a class of models of decision making under risk in which risks are represented by CDFs, and preference relations on risks are represented by mathematical expectations of a utility function with respect to a transformation of probabilities on a set of outcomes. The distinguishing characteristic of these models is that the transformed probability of an outcome depends on the rank of the outcomes in the induced preference ordering on the set of outcomes. When the function g reduces to the identity, the RDEU formula reduces to the EU one. However, EU does not depend on the rank of the outcomes. 3. A REPRESENTATION THEOREM OF RANK-DEPENDENT EXPECTED UTILITY In this section we axiomatize the RDEU theory, following the axiomatic systems of Fishburn (98) and Yaari (987). We present our five axioms and then prove the representation theorem. As we know, the existence of von Neumann and Morgenstern EU is e- quivalent to three axioms: the preference relation axiom, the independence

7 THE DISTORTED THEORY OF RANK-DEPENDENT 39 (substitution) axiom, and the Archimedean axiom. The representation of linearity in probabilities in EU is a direct consequence of the restriction on preference relations known as IA. Fishburn (98) proves the representation theorem of EU from an axiomatic system with five axioms which has been widely used. The five axioms are the neutrality axiom, the complete weak order axiom, the continuity (with respect to L - convergence) axiom, the monotonicity (with respect to first-order stochastic dominance) axiom, and the independence axiom. IA is the key behavioral assumption of EU, which is often violated in experimental studies. Yaari (987) cites Fishburn s (98) five axioms and replaces IA with the dual IA, and hence establishes the dual theory of choice under risk. At the core of the dual theory is the dual IA. Yaari (987) develops an expected utility theory with rank-dependent probabilities (EURDP) with the roles of payments and probabilities reversed. In this paper, we replace the dual IA in Yaari (987) with our DIA. We then use this system of five axioms to prove the representation theorem of RDEU. We first describe the five axioms in Yaari (987) and the representation theorem of EU. A strict preference relation is assumed to be defined on L. Let the symbols and stand for preference relation and indifference relation, respectively. The following axiom suggests itself: [Axiom A - Neutrality]: Let X and X belong to L with respective CDFs F X and F X. If F X = F X then X X. Denote F to be a family of CDFs by F = {F : [m, M] [0, ] F is a CDF}. Define on F by F F if and only if X X for which F = F X and F = F X. [Axiom A - Weak Order]: is asymmetric and negatively transitive. [Axiom A3 - Continuity with respect to L -Convergence]: Let F, F, F, F belong to F ; assume that F F. Then there exists an ε > 0 such that F F < ε and F F < ε imply F F, where is the L -norm F = F (x) dx. [Axiom A4 - Monotonicity with respect to First-Order Stochastic Dominance]: If F X (x) F X (x) for all x [m, M], then F X F X. [Axiom A5EU - Independence (Substitution)]: If F, F and F belong to F and α is a real number satisfying 0 < α <, then F F implies αf + ( α)f αf + ( α)f. By using the five axioms, Yaari (987) proves the following representation theorem of EU, which is a modification of Fishburn (98). Theorem 3. A preference relation satisfies Axioms A - A4 and A5EU if and only if there exists a continuous and non-decreasing real function u, defined on [m, M], such that, for all X and X belonging to L, X X E[u(X )] > E[u(X )].

8 40 HUI HUANG AND SHUNMING ZHANG Moreover, the function u, which is unique up to a positive transformation, can be selected in such a way that, for all x [m, M], u(x) solves the preference equation (m, u(x); M, u(x)) (x, ). From Theorem 3, the expected utility is given by EU(X) = E[u(X)] = u(x)df X (x). We now present our DIA and prove the representation theorem of RDEU. The distorted theory of choice under risk is obtained when IA (Axiom A5EU) of EU is replaced. We postulate independence for convex combinations that are formed along the distorted CDFs instead of for convex combinations formed along the CDFs. The best way to achieve that is to consider an appropriately defined distortion of CDFs. In RDEU (Quiggin 98 and Segal 989), the rank-dependent expected utility value is RDEU(X) = u(x)dg(f X (x)) = u(x)d[g F X ](x) which is in Section. The function g(p) = H (p, p) for each p [0, ] defines the behavior of (H, H ) on the pair (p, p) in Theorem. Then g : [0, ] [0, ] is a transformation to change probability distributions. As we have explained in above section, we can treat the function g as exogenous; and we call it the distortion function. Suppose that g satisfies some conditions such that g F X is a CDF, then we can represent the RDEU value in the form of mathematical expectations. Thus the representation theorem of RDEU can be checked by using Theorem 3 of EU. We now consider the corresponding axiom for independence. We assume that the distortion function g satisfies Proposition. Then the function g is onto and invertible. The inverse of the function g, g : [0, ] [0, ], also satisfies Proposition. In addition, We assume that the function g : [0, ] [0, ] and its inverse g : [0, ] [0, ] satisfy Lipschitz conditions. From Proposition, the function g : [0, ] [0, ] satisfies all the conditions of CDFs. Then we can consider it as a CDF. Therefore there exists a random variable ξ on [0, ] such that g(p) = P {ξ p}. From now on we always use ξ as the random variable with the CDF g.

9 THE DISTORTED THEORY OF RANK-DEPENDENT 4 The inverse F : [0, ] [m, M] of a CDF F : [m, M] [0, ] is given by: F (p) = sup{x [m, M] F (x) p}. Proposition. Let X L be a random variable, then F X (X) follows a uniform distribution on [0, ]. If the random variable θ follows a uniform distribution on [0, ], for any CDF F, F (θ) follows the CDF F. If the random variable θ follows the uniform distribution on [0, ], the random variable ξ on [0, ] can be chosen from Proposition as ξ = g (θ). In fact, P {ξ p} = P {g (θ) p} = P {θ g(p)} = g(p). For any CDF F, we consider the compound function g F of the two CDFs g and F, which is defined by [g F ](x) = g(f (x)). It is clear that g F satisfies all the conditions of CDFs. Then g F is a CDF, and we call it a distorted CDF of CDF F. For any distorted CDF g F, we have that [g F ](x) = g(f (x)) = P {ξ F (x)} = P {F (ξ) x} = F F (ξ)(x) which is the CDF of random variable F (ξ) = F (g (θ)) = [F g ](θ) = [g F ] (θ). Thus the compound function g F : [m, M] [0, ] is also a CDF. Hence g F = F F (ξ). We denote the set of such distorted CDFs as F = {g F F F F } = {F F (ξ) F F F }. We can simply write F = g(f ). From the property of Proposition we have F = F. A mixture operation for distorted CDFs in F may now be defined as follows: if g F and g F belong to F and if 0 α, then α[g F ] ( α)[g F ] F is given by α[g F ] ( α)[g F ] g [αf + ( α)f ].

10 4 HUI HUANG AND SHUNMING ZHANG Equivalently, if F F (ξ) and F F (ξ) belong to F and if 0 α, then αf F (ξ) ( α)f F (ξ) F is given by αf F (ξ) ( α)f F (ξ) F [αf +( α)f ] (ξ). For some 0 α, α[g F ] ( α)[g F ] = αf F (ξ) ( α)f F (ξ) is called a harmonic convex combination of F and F. With the operation, the set F of all distorted CDFs becomes a mixture space. Returning to the preference relation, we are now in a position to state the axiom that gives rise to the distorted theory of choice under risk: [Axiom A5 - Distorted Independence]: If g F, g F and g F belong to F and α is a real number satisfying 0 < α <, then g F g F implies α[g F ] ( α)[g F ] α[g F ] ( α)[g F ]. Equivalently, this axiom can be written as [Axiom A5 - Distorted Independence]: If F F (ξ), F F (ξ) and F F (ξ) belong to F and α is a real number satisfying 0 < α <, then F F (ξ) F F (ξ) implies αf F (ξ) ( α)f F (ξ) αf F (ξ) ( α)f F (ξ). For any distorted CDF F in F, there exists a CDF F in F such that F = g F, then F (x) = [g F ](x) = g(f (x)) and F (x) = g (F (x)) = [g F ](x) for x [m, M], hence F = g F. A mixture operation for CDFs in F can be defined as follows: if F and F belong to F and if 0 α, then αf ( α)f F is given by αf ( α)f g {α[g F ] + ( α)[g F ]}. Then we can write DIA in a simple form. [Axiom A5 - Distorted Independence]: If F, F and F belong to F and α is a real number satisfying 0 < α <, then F F implies αf ( α)f αf ( α)f. From the above axioms, we have the following representation theorem of RDEU by using Theorem 3. Theorem 4. Assume that the distortion function g and its inverse g satisfy Lipschitz conditions. A preference relation satisfies Axioms A - A5 if and only if there exists a continuous and non-decreasing real function u, defined on [m, M], such that, for all X and X belonging to L, X X u(x)dg(f X (x)) > u(x)dg(f X (x)). (6)

11 THE DISTORTED THEORY OF RANK-DEPENDENT 43 Moreover, the function u, which is unique up to a positive transformation, can be selected in such a way that, for all x [m, M], u(x) solves the preference equation g F (m, u(x);m,u(x)) g F (x,). (7) We first note that g satisfying Lipschitz condition is not required for the proof of the sufficient condition for (6). We also note that from Theorem 4, the rank-dependent expected utility is given by RDEU(X) = = u(x)dg(f X (x)) = u(x)d[g F X ](x) u(x)df F (ξ)(x) = E[u(FX (ξ))] (8) X which is presented in Quiggin (98) and Segal (987a and 989). Chew, Karni and Safra (987) state that the distortion function g is Lipschitz continuous on [0, ] if and only if the RDEU functional in (8) is weakly Gateaux differentiable on F. If all random variables take values in the unit interval [0, ], Yaari (987) proposes the dual IA as follows: If F X, F X and F X in F are pairwise comonotonic and α is a real number satisfying 0 < α <, then F X F X implies αf X ( α)f X αf X ( α)f X where αf X ( α)f X F αx +( α)x. Using Axioms A - A4 and his dual IA, he then proves his EURDP (Yaari s Theorem of Dual Theory) in which the utility function in payments is linear. The dual utility is given by DU(X) = f 0 ( F X (x))dx = xdf 0 ( F X (x)). [0,] [0,] Define g 0 : [0, ] [0, ] by g 0 (p) = f 0 ( p), then we have DU(X) = [0,] xdg 0 (F X (x)). If we take the utility function u in Theorem 4 to be linear, then RDEU degenerates into Yaari s dual utility. Yaari s dual utility can be considered as a special case of Quiggin s AU/RDEU. In Theorem 4 of RDEU, the utility function u is endogenous and the distortion function g is exogenous,

12 44 HUI HUANG AND SHUNMING ZHANG while, in Yaari s Theorem of Dual Theory, the weights function f 0 (and hence g 0 ) is endogenous but the utility function u is exogenous and linear. Yaari (987) considers random variables assuming values in the unit interval [0, ] (that is, [m, M] = [0, ]); then the inverse of a CDF is also a CDF. However, for a general interval [m, M] [0, ], the inverse of a CDF is not a CDF. Therefore we introduce the distortion function such that the distorted CDF is a CDF. As we have seen earlier, the distorted function is a CDF and there exists a random variable following this distribution; hence we can find the random variable associated with the distorted CDF. This is what leads us to obtain the distorted theory of rank-dependent expected utility. To unravel the paradoxes in next section, we need to use the specific forms of the RDEU formula. From Theorems and 4, the rank-dependent expected utility value of random variable X L is RDEU(X) = M u(x)dg(f X (x)) = u(m) g(f X (x))du(x). m We define a decision-weights function f : [0, ] [0, ] by f(p) = g( p); then it also satisfies f(0) = 0 and f() =. Therefore the RDEU functional is given by M RDEU(X) = u(x)df( F X (x)) = u(m)+ f( F X (x))du(x). m When we consider a simple lottery X = (x, p ; ; x N, p N ), the RDEU functional is [ ( N n ) ( n )] RDEU(X) = g(p )u(x ) + g p n g p n u(x n ) and RDEU(X) = = u(x N ) N n= [ f = u(x ) + N g n= ( N n =n p n n= ( n ) ( N N f n= n =n n = f p n n = p n ) ( N ) n = [u(x n ) u(x n )]. (9) n =n+ p n )] u(x n ) + f(p N )u(x N ) [u(x n ) u(x n )]. (0)

13 THE DISTORTED THEORY OF RANK-DEPENDENT 45 The expressions (9) (0) are more concise formulas of the RDEU theory for discrete random variables. 4. DIRECT APPLICATIONS OF DISTORTED INDEPENDENCE AXIOM In this section, we show the significance of DIA. We use DIA directly to rationalize two famous examples the Allais paradox and the common ratio effect. The two examples are inconsistent with IA and EU, but may agree with RDEU, which can be checked by applying RDEU formulas (9) (0) (Segal 987a and 989). Using DIA directly, we can determine the 4 Direct Applications of Distorted Independence Axiom functional forms of the distortion function such that the two examples are paradoxical. In this Furthermore, section, we show the wesignificance look beyond of DIA. these We use functional DIA directly toforms rationalize andtwo are able to obtain conditions under which the two examples accord with DIA. famous examples the Allais paradox and the common ratio effect. The two examples are In order to explain the roles played by IA and DIA for the two examples, inconsistent with IA and EU, but may agree with RDEU, which can be checked by applying we will use isosceles right triangles in Machina (987). As he demonstrates RDEU formulas (3.4) - (3.7) (Segal 987a and 989). Using DIA directly, we can determine in his well-known article, the set of all prospects over the fixed outcome the functional forms of the distortion function such that the two examples are paradoxical. levels 0 < x < y can be represented by the set of all probability triples of the Furthermore, we look beyond these functional forms and are able to obtain conditions under form (p 0, p x, p y ) where p 0 = P {X = 0}, p x = P {X = x}, p y = P {X = y}, which the two examples accord with DIA. and p 0 + p x + p y =. Graphically, this set of gambles can be represented in two dimensions, In order to explainin the (proles 0, pplayed y ) plane by IA(Figure and DIA for ), thesince two examples, the third we willdimension, use isosceles p x, isright implicit triangles in Machina the graph (987). byasphe x demonstrates = p 0 in his p y well-known. Thenarticle, the indifference the set of all curvesprospects underover EUtheinfixed the outcome triangle levels 0 diagram < x < y canare be represented straightby lines the setwith of all probability the same slope, triples which of the illustrates form (p 0, p x, the p y ) where property p 0 = P {X of linearity = 0}, p x = P in{x probabilities. = x}, p y = P {X = Attitude y}, and to riskp 0 +p can x +palso y =. be Graphically, illustrated this set of ingambles the unit can betriangle representedwhere in two dimensions, relatively in (p 0 steep, p y ) utilityplane indifference (Figure 4.), curves since the third represent dimension, risk p x, aversion is implicit inand the graph relatively by p x = flat p 0 utility p y. indifference curves represent risk loving. Then the indifference curves under EU in the triangle diagram are straight lines with the same slope, which illustrates the property of linearity in probabilities. Attitude to risk can also be illustrated in the unit triangle where relatively steep utility indifference curves represent risk FIG. aversion. Indifference and relativelycurves flat utility under indifference EU in curves the triangle represent diagram. risk loving. Solid lines are indifference curves and dotted lines are iso-expected value lines. p y Increasing Preference 0 p 0 p y Increasing Preference 0 p 0 Figure 4.: Indifference curves under EU in the triangle diagram. Solid lines are indifference curves and dotted lines are iso-expected value lines. 3

14 46 HUI HUANG AND SHUNMING ZHANG 4.. The Allais Paradox [The Allais Paradox]: Consider the following two problems: Problem : Choose between 4. The Allais Paradox A = (0, p ε; x, p+ε) and A = (0, q ε; x, ε; y, q); [The Allais Paradox]: Consider the following two problems: Problem : Choose between Problem : Choose between A = (0, p ε; x, p+ε) and A = (0, q ε; x, ε; y, q); A 3 = (0, p; x, p) and A 4 = (0, q; y, q) Problem : Choose between where 0 < x < y, 0 < q < p <, and 0 < ε p. Most people prefer A A 3 = (0, p; x, p) and A 4 = (0, q; y, q) to A and A 4 to A 3 (Allais 953). However, they are not consistent with IA orwhere EU. 0 Allais < x < y, (953) 0 < q < takes p <, and the0 parameter < ε p. Most values people asprefer x = A$m, to A y and = A$5m, 4 p = 0., to A 3 (Allais q = 953). 0.0, However, and ε they = are not consistent In Kahneman with IA or EU. and Allais Tversky (953) takes(979), the x = 400, y = 500, p = 0.34, q = 0.33, and ε = parameter values as x = $m, y = $5m, p = 0., q = 0.0, and ε = In Kahneman and Tversky (979), x = 400, y = 500, p = 0.34, q = 0.33, and ε = FIG.. The Allais Paradox and the Independence Axiom p + ε < p + ε = Y Y C C p y p y A A 4 A A 4 D D D C D X A C A 3 Z X A A 3 Z p 0 p 0 Figure 4.: The Allais Paradox and the Independence Axiom EU implies that A A and A 3 A 4 are equivalent. Under RDEU, A A EUand implies A 4 that A 3 are A compatible and A 3 A 4 are if and equivalent. only if Under therdeu, distortion A Afunction and g is concave A 4 A 3 are (Segal compatible 987a). if and only Under if theia, distortion A function A isg only is concave compatible (Segal 987a). with A 3 Under A 4, which IA, A we A use is only thecompatible unit triangle with A 3 toaillustrate. 4, which we use Later the unit wetriangle will apply to DIA to illustrate. the unit Later triangle we will apply to resolve DIA to the the unit paradox. triangle to resolve Figure theparadox. shows Figure us the4.four prospects A, A, A 3 and A 4 in the plane (p 0, p y ), where A A A 3 A 4 and shows us the four prospects A, A, A 3 and A 4 in the plane (p 0, p y ), where A A A 3 A 4 and A A 3 A A 4. Slope of A A = Slope of A 3 A 4 = q q. We can find two pairs of prospects p q p q to represent the original four prospects A, A, A 3, and A 4. Figure 4. reports how to take A A 3 A A 4. Slope of A A = Slope of A 3 A 4 =. We can find two pairs of prospects to represent the original four prospects A, A, A 3, and A 4. Figure reports how to take the 4 two pairs of new prospects. First, we take point C to be the intersection of line XA and line ZY and point C to be the point on the line XZ such that C C A A. Then we take D to be X (the origin) and D to be Z. The two pairs of prospects are defined

15 THE DISTORTED THEORY OF RANK-DEPENDENT 47 ( as C = 0, p ε ) ( ε ; x, p and C = 0, q ε ) ε ε ; y, q, D = (x, ) ε and D = (0, ). Then A = ( ε)c + εd and A = ( ε)c + εd, A 3 = ( ε)c + εd and A 4 = ( ε)c + εd. In addition, F A = ( ε)f C + εf D and F A = ( ε)f C + εf D, F A3 = ( ε)f C + εf D and F A4 = ( ε)f C + εf D. From IA, F C F C implies F A F A and F A3 F A4. Then A A is only compatible with A 3 A Under DIA, the distortion function g is the identity if and only if A A A 3 A 4, and in this case A A and A 3 A 4 hold simultaneously Now we explain this paradox by directly using DIA. In Figure 3, we define four CDFs F A, F A, F A 3 and F A 4 in F as F A = g F A, F A = g F A, F A 3 = g F A3 and F A 4 = g F A4. That is, A A A 3 A 4 = (0, g ( p ε); x, g ( p ε)), = (0, g ( q ε); x, g ( q) g ( q ε); y, g ( q)), = (0, g ( p); x, g ( p)), = (0, g ( q); y, g ( q)). We take point C to be the intersection of line X A and line Z Y and point C to be the point on the line X Z such that C C A A. Then the two prospects can be expressed as C = C = ( 0, ( 0, g ( p ε) g ( q)+g ( q ε) g ( q ε) g ( q)+g ( q ε) ; y, g ( p ε) ; x, g ( q)+g ( q ε) ). g ( q) g ( q)+g ( q ε) Then A = ( ε )C + ε D and A = ( ε )C + ε D where ε = g ( q) g ( q ε). In addition, F A = ( ε )F C + ε F D and F A = ( ε )F C + ε F D. Then F A = g F A = g [( ε )F C + ε F D ] = ( ε )[g F C ] ε [g F D ]; F A = g F A = g [( ε )F C + ε F D ] = ( ε )[g F C ] ε [g F D ]. From DIA, g F C g F C implies F A F A. As we know A A 3 A A 4, then D A 4 D C = D A D C = D A D C = ε. If D A 3 D C = ε, then C C A 3 A 4, and hence A 3 = ( ε )C + ε D ) ;

16 48 HUI HUANG AND SHUNMING ZHANG FIG. 3. The Allais Paradox and the Distorted Independence Axiom EU Theory Y Y RDEU Theory C C p y p y A A 4 A A 4 D D D D X A C A 3 Z X A C A 3 Z p 0 p 0 Figure 4.3: The Allais Paradox and the Distorted Independence Axiom and A 4 = ( ε )C F A 3 = ( ε + ε D. In addition, F A )F C ε F D and F A 4 = ( ε )F C + ε F D. Then 3 = ( ε )F C + ε F D and = ( ε )F C + ε F D. Then F A 4 F A3 = g F A 3 = g [( ε )F C + ε F D ] = ( ε )[g F C ] ε [g F D ]; F A3 = g F A 3 = g [( ε )F C + ε F D ] = ( ε )[g F C ] ε [g F D ]; F A4 = g F A 4 = g [( ε )F C + ε F D ] = ( ε )[g F C ] ε [g F D ]. F A4 = g F A 4 = g [( ε )F C + ε F D ] = ( ε )[g F C ] ε [g F D ]. From DIA, g F C g F C implies F A3 F A4. From DIA, g F C g F C implies F A3 F A4. The condition D A 3 D C = ε implies The condition D A 3 D C = ε implies g ( p) g = g ( q)+g ( q ε). ( p ε) g ( q)+g ( q ε) g ( p) g = g ( q)+g ( q ε). ( p ε) g ( q)+g ( q ε) Therefore, we have, for any 0 < q < p < and 0 < ε p, g ( q) g ( q ε) = g ( p) g ( p ε). (4.) Then g ( p) g ( p ε) is independent of p. For any 0 < p < and 0 < ε p, Therefore, we have, for any 0 < q < p < and 0 < ε p, g ( p) g ( p ε) = R(ε). (4.) g ( q) g ( q ε) = g ( p) g ( p ε). () Differentiating (4.) with respect to p, we then have [g ] ( p) = [g ] ( p ε). That is, Then[g g ] (( p) is a constant g ( p ε) and thus gis independent is linear. Since gof (0) p. = For 0 and any g 0 () < = p, < then and 0 < ε p, g (p) = p and g(p) = p. Therefore the distortion function g is the identity. In this case A A is only compatible with A 3 A 4. g ( p) g ( p ε) = R(ε). () 6 Differentiating () with respect to p, we then have [g ] ( p) = [g ] ( p ε). That is, [g ] ( p) is a constant and thus g is linear. Since g (0) = 0 and g () =, then g (p) = p and g(p) = p. Therefore the distortion function g is the identity. In this case A A is only compatible with A 3 A 4.

17 THE DISTORTED THEORY OF RANK-DEPENDENT 49 In summary, we find the condition for the distortion function such that A A C C A 3 A 4 and hence g ( q) g ( q ε) g ( p ε) = Slope of A A = Slope of A 3 A 4 = g ( q) g ( q) g ( p). Then we have () and the distortion function g is the identity. In this case DIA reduces to IA and the RDEU formula reduces to the EU one, and In summary, we find the condition for the distortion function such that A we have A A and A 3 A 4 hold simultaneously. A C C A 3 A 4 and hence g ( q) g ( q ε) g ( p ε) = Slope of A A = Slope of A 3 A 4 = g ( q) g ( q) g ( p) A Closer Look NowThen we we take have a(4.) closer and the look distortion at thefunction Allaisg paradox is the identity. byinusing this case DIAinreduces the unit to triangle. IA and In the RDEU left formula diagram reduces of to Figure the EU3, one, the andindifference we have A A curves and A 3 under A 4 holdeu are linear; simultaneously. and in the right diagram of Figure 3 which is the transformed triangle, the indifference curves are also linear. Based on Figure 3, we can 4.. A Closer Look use DIA directly to explain the Allais paradox. As we have shown above, Now we take a closer look at the Allais paradox by using DIA in the unit triangle. In the left Slope of A A = g ( q) g ( q ε) g ( p ε) ; diagram of Figure 4.3, the indifference curves under EU are linear; and in the right diagram of Figure 4.3 which is the transformed triangle, the indifference curves are also linear. Based on Figure 4.3, we can use DIA directly to explain the Allais paradox. As we have shown above, Slope of A 3 A 4 = g ( q) g ( q) g ( p). Slope of A A = g ( q) g ( q ε) g ( p ε) ; Slope of A g ( q) 3 FIG. 4. Rationalization of the A 4 = Allais Paradox g ( q) by g DIA. ( p) A. 0, A, and A correspond to the cases that g is the identity, g is concave, and g is convex, respectively. p y EU Theory A A 4 0 A A 3 p 0 p y A A 0 RDEU Theory A 4 A 0 4 A A 4 0A A 0 A A 3 A 0 3 A 3 p 0 Figure 4.4: Rationalization of the Allais Paradox by DIA. A 0, A, and A correspond to the Thecases distortion that g the function identity, g gis concave, is the and identity g is convex, if and respectively. only if Slope of A A = Slope of A 3 A 4 if and only if A A A 3 A 4 if and only if A A and The distortion function g is the identity if and only if Slope of A A = Slope of A 3 A 4 if and only if A A A 3 A 4 if and only if A A and A 3 A 4. In the right-hand-side of Figure 4.4 we 7

18 50 HUI HUANG AND SHUNMING ZHANG A 3 A 4. In the right-hand-side of Figure 4 we use superscript 0 to replace in A to denote for this case. As we know, under RDEU, A A and A 4 A 3 are compatible if and only if g is concave. Does this result hold under DIA? We discuss the two cases where g is not the identity as follows. Note [g ] (g(p)) = g (p) [g (p)] 3. [] The distortion function g is concave if and only if the weights function g is convex if and only if g ( p) g ( p ε) < g ( q) g ( q ε) if and only if g ( q ε) g ( p ε) < g ( q) g ( p) if and only if Slope of A A > Slope of A 3 A 4. In the right-hand-side of Figure 4 we use superscript to replace in A. In this case, it is possible that A A and A 4 A 3 are compatible. [] The distortion function g is convex if and only if the weights function g is concave if and only if g ( p) g ( p ε) > g ( q) g ( q ε) if and only if g ( q ε) g ( p ε) > g ( q) g ( p) if and only if Slope of A A < Slope of A 3 A 4. In the right-hand-side of Figure 4 we use superscript to replace in A. In this case, A A and A 3 A 4 are compatible. We summarize the above results from the view of DIA as follows.. The distortion function g is the identity if and only if A A A 3 A 4, then A A and A 3 A 4 hold simultaneously.. The distortion function g is concave if and only if Slope of A A > Slope of A 3 A 4, and it is possible that A A and A 4 A 3 are compatible. 3. The distortion function g is convex if and only if Slope of A A < Slope of A 3 A 4, then A A and A 3 A 4 are compatible. 4.. The Common Ratio Effect [The Common Ratio Effect]: Consider the following two problems: Problem : Choose between A = (0, p; x, p) and A = (0, q; y, q); Problem : Choose between A 3 = (0, λp; x, λp) and A 4 = (0, λq; y, λq) where 0 < x < y, 0 < q < p, and 0 < λ <. Most people prefer A to A and A 4 to A 3 (MacCrimmon and Larsson 979). However, they are not consistent with IA or EU. MacCrimmon and Larsson (979) take the parameter values as x = $m, y = $5m, p =.00, q = 0.80, and λ = 0.05.

19 THE DISTORTED THEORY OF RANK-DEPENDENT 5 In Kahneman and Tversky (979), x = 3000, y = 4000, p =.00, q = 0.80, and λ = 0.5. EU implies that A A and A 3 A 4 are equivalent. Under RDEU A A and A 4 A 3 are compatible if and only if the elasticity of the q = 0.80, and λ = In Kahneman and Tversky (979), x = 3000, y = 4000, p =.00, weights function f is increasing (Segal 987a). Under IA, F A F q = 0.80, and λ = 0.5. A implies F A3 F A4, which we illustrate in the unit triangle. Later we will EU implies that A A and A 3 A 4 are equivalent. Under RDEU A A and apply DIA to the unit triangle to resolve the common ratio effect. Figure 5 A 4 A 3 are compatible if and only if the elasticity of the weights function f is increasing shows us the four prospects A, A, A 3 and A 4 in the plane (p 0, p y ), where (Segal 987a). Under IA, F A F A implies F A3 F A4, which we A A A 3 A 4 (Slope of A A = Slope of A 3 A 4 = q illustrate in the unit triangle. Later we will apply DIA to the unit triangle to resolve the common). ratio Define effect. D Figure = (0, 4.5). p q then Ashows 3 = us λathe +( λ)d four prospectsand A, A, A 4 = 3 and λaa +( λ)d, 4 in the plane (p F 0, p A3 y ), = where λf A A +( λ)f A A 3 A 4 D q p q ). Define D = (0, ). then A 3 = λa + ( λ)d and A 4 = λa +( λ)d, F A3 = λf A +( λ)f D and F A4 = λf A +( λ)f D. From IA, F A F A and F(Slope A4 = ofλf A A = A + Slope ( λ)f of A 3 A D 4. = From IA, F A F A implies F A3 F A4. implies F A3 F A4. FIG. 5. The Common Ratio Effect and the Independence Axiom p y Y X A A p < p = Y p 0 A 4 A 3 D Z p y X A A p 0 A 3 Figure 4.5: The Common Ratio Effect and the Independence Axiom A 4 D Z Under Under DIA, the weights function f isf ofisconstant constant elasticity elasticity if and only if and if only if A A A A A 3 A 4,, andin in this this case case A A and A 3 and A 4 hold A 3 simultaneously A 4 hold simultaneously Now we explain this example by using DIA. Define two new prospects to be A = (0, g ( p); Now we explain this x, g ( p)) and A = example (0, by using DIA. Define two new prospects g ( q); y, g ( q)) such that their CDFs F A and F A to be A in F = (0, g satisfy F A ( p); x, = g F A and F A g ( p)) and A = g F A, as illustrated in Figure = (0, g ( q); 4.6. By DIA, F A y, F A if g ( q)) such that their CDFs F and only if g F A g F A implies, for any A α and F [0, ] and A 0 in F satisfy F z x, α[g F A ] ( α)[g G A = g F z ] A and Fα[g F A = g F A ] ( α)[g G A, as z illustrated in Figure ]. That is, g [αf A + ( α)g z 6. By DIA, ] g [αf A + ( α)g z F]. A F A if and only if g F We take D A such that F D = g F D, then D g F A implies, for any α [0, ] and 0 = D = (0, ). For 0 z x, define G z z x, in F α[g F A ] ( α)[g G z ] α[g F A ] ( α)[g G z ]. That is, g [αf A + ( α)g z ] g [αf A + ( α)g z ]. 9 We take D such that F D = g F D, then D = D = (0, ). For 0 z x, define G z in F to be a CDF for a degenerate distribution

20 5 HUI HUANG AND SHUNMING ZHANG which assigns the value z with probability. Then the random variable is δ z = (z, ) and hence G 0 = F D = F D. to be a CDF for a degenerate distribution which assigns the value z with probability. Then the random variable is δ z = (z, ) and hence G 0 = F D = F D. FIG. 6. The Common Ratio Effect and the Distorted Independence Axiom p y Y X A EU A p 0 A 4 A 3 D Z Y RDEU A p y A 4 D X A A 3 Z p 0 Figure 4.6: The Common Ratio Effect and the Distorted Independence Axiom We now find α [0, ] such that F A3 = g [αf A + ( α)f D ] and We now find α [0, ] such that F A3 = g [αf A + ( α)f D ] and F A4 = g [αf A + F A4 = g [αf A +( α)f D ]. Define A ( α)f D ]. Define A 3 and A 4 as F A = αf 3 A + 3 and A ( α)f 4 as F A D and F 3 = αf A A = αf 4 A + ( α)f +( α)f D D, then and F A 4 = αf A + ( α)f D, then F A3 = g F A 3 and F A4 = g F A F A3 = g F A 3 and F A4 = g F A 4 (See Figure 4.6). In this case, 3 A 4 A A and A 4 (See 3 A 4. Figure 6). In this case, A 3 A 4 A A and A 3 A 4. Thus for any 0 < q < p < and 0 < λ <, λp = g(αg ( p) + ( α)) λp = g(αg ( p) + ( α)) λq = g(αg ( q) + ( α)). λq = g(αg ( q) + ( α)). Thus for any 0 < q < p < and 0 < λ <, α = g ( λp) g ( p) = g ( λq) g ( q). (4.3) α = g ( λp) g ( p) = g ( λq) g ( q). (3) Then g ( λp) g ( p) is independent of p. For any 0 < p < and 0 < λ <, g ( λp) g ( p) = R(λ) [0, ]. That is, g ( λp) = R(λ)[ g ( p)] (4.4) Then g ( λp) g is independent of p. For any 0 < p < and 0 < λ < ( p) with R() =. Differentiating (4.4) with respect to p and λ, we then have, g ( λp) g = R(λ) [0, ]. That is, ( p) λ[g ] ( λp) = R(λ)[g ] ( p) p[g ] ( λp) = R (λ)[ g ( p)] g ( λp) = R(λ)[ g ( p)] (4) with R() =. Differentiating (4) with respect to p and λ, we then have λ[g ] ( λp) = R(λ)[g ] ( p) p[g ] ( λp) = R (λ)[ g ( p)] 0

21 THE DISTORTED THEORY OF RANK-DEPENDENT 53 Thus λr (λ) = R ()R(λ) and hence R(λ) = λ R () (5) When p approaches unity in (4), we have the limit as g ( λ) = R(λ). It follows, from (5), g ( λ) = λ R () and g( λ R () ) = λ. Therefore g(p) = ( p) R (). Therefore, under DIA, A A A 3 A 4 if and only if the distortion function g satisfies g(p) = ( p) γ where γ > 0, and in this case both A A and A 3 A 4 hold together. The value of α in (3) is chosen such that A A A 3 A 4 and hence g ( q) g ( q) g ( p) = Slope of A A = Slope of A 3 A 4 = g ( λq) g ( λq) g ( λp). We can then conclude that the distortion function g is of the form g(p) = ( p) γ. Note that if the form of the distortion function is g(p) = ( p) γ with γ > 0, then the function f(p) defined in Section 3 becomes p γ, and hence the elasticity of f, which is defined as p f (p), equals to f(p) γ. Conversely, if p f (p) f(p) = γ, then f(p) = pγ and g(p) = ( p) γ. Therefore under DIA, both A A and A 3 A 4 hold if and only if the elasticity of the weights function f is a positive constant A Closer Look The common ratio effect example is also regarded as irrational behavior under IA and EU. However, the inconsistent result holds when the distortion function g satisfies g(p) = ( p) γ where γ > 0 under DIA (which is IA when γ = ). Under RDEU, the paradox can disappear when the corresponding weights function has an increasing elasticity. As for the explanation for the Allais paradox, we turn to the unit triangle to explain the common ratio effect by directly using DIA, which is

22 54 HUI HUANG AND SHUNMING ZHANG illustrated in Figure 6. As we know, Slope of A A = g ( q) g ( q) g ( p) ; Slope of A 3 A 4 = g ( λq) g ( λq) g ( λp). Define a function h : [0, ] [0, ] by h(p) = g ( p), then Slope of A A = h(q) h(p) h(q) = h(p) h(q) ; Slope of A 3 A 4 = h(λq) h(λp) h(λq) = h(λp) h(λq). Now we can find the relation between the function f(p) = g( p) and h(p) = g ( p) as follows: h(p) = g ( p) if and only if g (p) = h( p) if and only if p = g (g(p)) = h( g(p)) if and only if p = h( g(p)) if and only if h ( p) = g(p) if and only if h (p) = g( p) = f(p) if and only if h(f(p)) = p. Since h(f(p)) = p implies h (f(p))f (p) = and f (p) = h (h (p)), the elasticity of f at p is p f (p) f(p) = p h (p)h (h (p)) = h(h (p)) h (p)h (h (p)) = h (p) h (h (p)) h(h (p)) which is the inverse of the elasticity of h at h (p). As we know, the elasticity of the weights function f is a positive constant, p f (p) f(p) = γ, if and only if f(p) = pγ if and only if h(p) = p γ if and only if Slope of A A = = Slope of A 3 A 4 if and only if A A A 3 A 4 if [ p q ] γ and only if A A and A 3 A 4. In the right-hand-side of Figure 7 we use superscript 0 to replace in A to denote for this case. As we know, under RDEU, A A and A 4 A 3 are compatible if and only if the elasticity of the weights function f is a positive constant. Does this result hold under DIA? [] The elasticity of the weights function f is increasing if and only if the elasticity of the weights function h is decreasing if and only if h(λp) h(p) > h(λq) h(q) if and only if h(λp) h(λq) > h(p) h(q) if and only if Slope of A A > Slope of A 3 A 4.

23 Now we can find the relation between the function f(p) = g( p) and h(p) = g ( p) as follows: h(p) = g ( p) if and only if g (p) = h( p) if and only if p = g (g(p)) = h( g(p)) if and only if p = h( g(p)) if and only if h ( p) = g(p) if and only if h (p) = g( p) = f(p) if and only if h(f(p)) = p. is Since h(f(p)) = p implies h (f(p))f (p) = and f (p) = h (h, the elasticity of f at p (p)) THE DISTORTED THEORY OF RANK-DEPENDENT 55 p f (p) f(p) = p h (p)h (h (p)) = h(h (p)) h (p)h (h (p)) = h (p) h (h (p)) h(h FIG. 7. Rationalization of the Common Ratio Effect by the distorted (p)) independence axiom. which A 0, is Athe, and inverse A of correspond the elasticitytooftheh at cases h (p). that the elasticity of the weights function f is constant, increasing, and decreasing, respectively. p y EU A A 4 0 A A 3 p 0 p y A 0 A A 0 RDEU A 0 A A A 4 A 3 p 0 A 0 3 A 0 4 A 4 A 3 Figure 4.7: Rationalization of the Common Ratio Effect by the distorted independence axiom. In the right-hand-side of Figure 7 we use superscript to replace in A. A 0, A, and A correspond to the cases that the elasticity of the weights function f is constant, Thus, the elasticity of the weights function f is increasing if and only if increasing, and decreasing, respectively. Slope of A A > Slope of A 3 A 4. In this case, it is possible that A A and A 4 A 3 are compatible. As we know, the elasticity of the weights function f is a positive constant, p f (p) = γ, if and [] The elasticity of the weights function f is decreasing f(p) if and only if the elasticity of the weights function h is increasing p γ only if f(p) = p γ if and only if h(p) = p γ if and only if Slope of A A = [ ] = Slope of A 3 A 4 q if and only if h(λp) if and only if A A A 3 A 4 if and only if A A and A 3 A 4. In the right-hand-side of Figure h(p) < h(λq) h(λp) if and only if 4.7 we use h(q) superscript 0 to replace h(λq) < h(p) in A to denote h(q) if and only if Slope of A A < for this case. As we know, under RDEU, SlopeA of A 3and A 4. A In 4 A the 3 are right-hand-side compatible if only of if Figure the elasticity 7 we of the use weights superscript function f is ato replace positive inconstant. A. Thus, Does this theresult elasticity hold under of DIA? the weights function f is decreasing if and only if Slope of A A < Slope of A 3 A 4. In this case, A A and A 3 A 4 are compatible. We summarize the above results from the view of DIA as follows.. The weights function f satisfies f(p) = p γ where γ > 0 if and only if A A A 3 A 4. In this case both A A and A 3 A 4 hold together.. The weights function f is of increasing elasticity if and only if Slope of A A > Slope of A 3 A 4, and it is possible that A A and A 4 A 3 are compatible. 3. The weights function f is of decreasing elasticity if and only if Slope of A A < Slope of A 3 A 4, then A A and A 3 A 4 are compatible. Machina (987) uses the unit triangle to explain the Allais paradox and the common ratio effect. In his diagrams, prospects are fixed and form parallelograms, but indifference curves fan out. In this paper, we examine