A Quantum-like Model of Selection Behavior
Masanari Asano, Irina Basieva, Andrei Khrennikov, Masanori Ohya, Yoshiharu Tanaka
aa r X i v : . [ q -f i n . E C ] A p r A Quantum-like Model of Selection Behavior
Masanari AsanoLiberal Arts Division, National Institute of Technology, Tokuyama CollegeGakuendai, Shunan, Yamaguchi 745-8585 JapanIrina Basieva and Andrei KhrennikovInternational Center for Mathematical Modelingin Physics and Cognitive SciencesLinnaeus University, V¨axj¨o, SwedenMasanori Ohya and Yoshiharu TanakaDepartment of Information Sciences, Tokyo University of ScienceYamasaki 2641, Noda-shi, Chiba, 278-8510 JapanMay 25, 2017
Abstract
In this paper, we introduce a new model of selection behavior underrisk that describes an essential cognitive process for comparing valuesof objects and making a selection decision. This model is constructedby the quantum-like approach that employs the state representationspecific to quantum theory, which has the mathematical frameworkbeyond the classical probability theory. We show that our quantumapproach can clearly explain the famous examples of anomalies forthe expected utility theory, the Ellsberg paradox, the Machina para-dox and the disparity between WTA and WTP. Further, we pointout that our model mathematically specifies the characteristics of theprobability weighting function and the value function, which are basicconcepts in the prospect theory. Introduction
Many studies on selection behavior have been done mainly in economics andpsychology. In economics, the expected utility theory ([Neumann & Morgenstern(1953)])is traditionally used to discuss selection behaviors under risk, which are re-garded as normative and rational from the view of the probability theory.In psychology, anomalies for the expected utility theory have been verifiedthrough a large number of experimental tests.Among the two disciplines, behavioral economics based on the prospecttheory ([Kahneman & Tversky(1979), Tversky & Kahneman(1992)]) has beendeveloped. The prospect theory is categorized into subjective expected util-ity (SEU) approach that tries to explain the anomalies by simulating thedecision maker who makes a choice for maximizing the value of SEU. TheSEU in the prospect theory is defined by the probability weighting func-tion and the value function . The probability weighting function representsthe psychological tendency to overestimate small probabilities and underes-timate large ones. The value function represents the tendency that a lossgives a greater feeling of pain compared to the joy given by an equivalentgain. (The amount of loss or gain is measured from reference point whoseposition fluctuates situationally.) However, the development of experimen-tal economics brought the finds of anomalies that cannot be captured inthe prospect theory or its gentle modification. For examples, the anomaliesshown by [Ert & Erev(2013)],[Thaler & Johnson(1990)], [Payne(2015)] and[Birnbaum(2008)] are well known and they are difficult to explain.In recent years, trying to find a theory/model that can explain all anoma-lies is a major topic in behavioral economics and, many researchers competefor developing descriptive decision-making model .In this paper, we propose a new decision-making model that is not a gen-tle modification of SEU. It is a model designed in the quantum-like approach ,where the state representation specific to the quantum mechanics is em-ployed. Quantum mechanics is originally established for the description of Actually, Erev et al. [Ert & Erev(2013)] proposed and organized the fair competitionof model (
From Anomalies to Forecasts: Choice Prediction Competition for Decisionsunder Risk and Ambiguity (CPC2015)). The detail of this competition is reported in thewebsite http://departments.agri.huji.ac.il/cpc2015. The problem of the state interpretation is one of the most complicated problemsof quantum foundations. The present situation is characterized by a huge diversity ofinterpretations [(2009a)]. One of them is the information interpretation which was strongly Our main interest is not just to make a model that fits experimentaldata of anomalies, but to offer the foundation for the new theory of expectedtheory, that is, to describe human behavior using non-classical probabilities.We believe that in order to develop the quantum-like approach further tobecome an established theory, we need to investigate in a deep way how thequantum-like approach compares to prospect theory. In this paper, we willshow that the characters of the probability weighting function and the valuefunction can be realized mathematically in a part of our quantum-like model.Further, we will show that our model can explain several anomalies includingnon-classical ones. These results suggest that our quantum-like model hasthe potential to be a mile stone toward the development of model to coverall known and yet unknown anomalies.In Sec. 2, we mathematically define a cognitive process essential to makea preference. Firstly, the decision maker in our model is aware of the exis-tence of objects to be compared. The structure of awareness is representedby density operator(matrix) , which is the most general state representationin quantum mechanics. We call it the comparison state . The decision makersecondly compares the values (utilities) of objects quantitatively. This func- supported by the recent quantum information revolution. We shall use this interpretation.Thus a quantum-like mental state represents information available to decision makers andstructured in the special (quantum-like) way by their brains. The quantum-like approach works very well to model a variety of behavioral effects,e.g., the order effect or disjunction and conjunction effects. However, it is not clear whetherthe mathematical formalism of quantum theory can serve to model all possible behavioralphenomena, see Khrennikov et al. [(2014)] and Boyer-Kassem et al. [(2016a)], [(2016b)]for the recent analysis of this problem. evaluation function . A selection decision is de-rived from the above comparison state and evaluation function. In Sec. 2.3,we discuss the modeling of selection between two lotteries. The main prob-lem is how to embed the probability distributions into the comparison state.Especially, the comparison state designed in Sec. 2.3.3 is important, becauseit is closely related with the crucial concept in quantum theory: The statetransition , which occurs when a physical value is measured on a system, isthe basic assumption in quantum theory. A physical state before measure-ment is generally represented with the form called quantum superposition that is clearly distinguished from a statistical description as obtained aftermeasurement. If the comparison state at the stage before drawing the lotsis represented with quantum superposition, the character of the probabilityweighting function can be explained, see Fig. 1 in Sec. 2.3.4.It is also crucial that our model describes the cognitive process that isnever explained in the expected utility theory. For example, for the lotterythat pays $100 or $1 according to a probability distribution, a decision makermight fear to get $1 by missing the chance of $100. Then, the difference ofthese potential outcomes will affect his/her evaluation of risk . Such a processis to be experienced before drawing the lottery but never after drawing,and actually, its effect is represented in the comparison state with quantumsuperposition, as the parameter called degree of evaluation of risk (DER).We expect, the evaluation of risk is an essential cause of anomaly. In Sec.3,this point will be shown clearly, where we simulate the famous anomalies inselection behavior under ambiguity ; Ellsberg paradox ([Ellsberg(1961)]) andthe
Machina paradox ([Machina(2009)]) . Ellsberg paradox, see Table. 1 isthe first example that shows the anomaly due to the ambiguity aversion ,which lets one to prefer the known risk to the unknown risk. The Machinaparadox, see Table. 3 points out an existence of anomaly that is impossible tobe explained even in the popular models of ambiguity aversion. Our analysesare summarized in the diagrams of Fig. 2 and Fig. 3 in Sec. 3.In Sec. 4, we discuss the determination of cash equivalent (CE), which isan amount of money whose value is indifferent from a given lot. The CE isrelated to the willingness to accept (WTA) and willingness to pay (WTP)([Horowitz et al.(2003)]), each of which is interpreted as cash determined in There exists another quantum-like approach trying to solve the two paradoxes, see([Aerts et al.(2012)]), in which the effect of DER is not assumed. > WTP is generally observed in experimental tests, and this disparityis an important topic in economics. As seen in Fig. 4, CE is defined asthe function of the degree of evaluation of risk (DER). Therefore, we canexplain the disparity from DER whose value is to be changed depending onthe decision maker’s situation. Note that in such a situation dependency isconsistent with the character of value function in the prospect theory.
There are two lots, say A and B . If you chose A , you will get outcome x i ( i = 1 · · · n ) with probability P i . If you chose B , you will get outcome x i withprobability Q i . All of the outcomes are different from each other. Which lotdo you select? When a decision maker decides the preference A ≻ B or B ≻ A in thissituation, he/she will recognize the following three points.1. What objects exist,2.
Which pairs of objects are to be compared,3.
How comparisons are evaluated.Here, we have to emphasize that “objects” mean “events” that will be expe-rienced in the future. He/she can simulate the experience that he draws thelot A ( B ) and get the outcome x i . Let us represent such an event by ( A, x i )or ( B, x i ). Further, we assume that the decision maker sets the utilities of( A, x i ) and ( B, x i ) by u ( x i ) ≡ u i . (The utility of event depends on only out-come.) Here, u ( x ) is a utility function of outcome x . Under this assumption,comparing an arbitrary pair of objects (events) becomes possible. Lastly,the decision maker evaluates various comparisons for making the preference A ≻ B or B ≻ A . If von Neumann-Morgenstern (VNM) utility theorem isapplied to this third point, the method consistent with VNM axioms (Com-pleteness, Transitivity, Independence, Continuity) will be used. VNM axiomsare given for the relation of utilities like u ≻ v and the operation using prob-ability like pu + (1 − p ) v . Therefore, these encourage the decision makerto estimate the expected utilities, E A = P ni P i u i and E B = P ni Q i u i , andemploy its difference as a sort of a criterion for making the preference.However, the fact of anomalies shows that many people seem to violateVNM axioms in their selection behaviors. For this problem, we have to men-5ion the interpretation of probability used in the axioms. In the book “Theoryof Games and Economic Behavior”([Neumann & Morgenstern(1953)]), VNMstressed the well founded interpretation of probability as frequency , but nota subjective concept of probability , which is often visualized. What is the subjective concept?
It is not the frequency of event countedthrough a large number of trial experiments, rather it is the weight of aware-ness assigned for unmeasured event. The reason that many people oftenvisualize the latter impression might be simple: It is difficult for them tosimulate a large number of trial experiments in their minds, even if theyknow the meaning of frequency probability. We believe, in a realistic selec-tion behavior using the subjective probability, a “natural” operation exists,which is different from the form of pu + (1 − p ) v , and in this section, we de-scribe it by using the framework of quantum theory. It is a significant pointthat quantum theory mathematically defines a state before measurement.Here, one can interpret the state as a subjective recognition for uncertain(unmeasured) events, if the measurement is regarded as an acquisition ofsubjective experience. Such an interpretation of quantum theory is called“Qbism”, see the papers of [Fuchs & Schack(2013)]. (Note, QBism neverdeny the standard interpretation in quantum physics, rather, includes it.) InSec.2.1 and Sec.2.2, the three points introduced in the above are explainedin the state representation based on QBism, and in Sec.2.3, a criterion forselection is defined as a “natural” operation with potential to violate VNMaxioms. Firstly, the decision maker is aware of existences of objects. Secondly, he/sheis aware of various pairs of objects to be compared. As mentioned in the be-ginning of section, the “objects” we consider here are the “events” individ-ual and incompatible each other. The decision maker preliminarily assignshis/her awarenesses for these events. In our model, its distribution is math-ematically represented in the form of a density operator , which is the generalstate representation in quantum theory. In a sense, this representation playsthe role of prior probability distribution that is different from statistical prob-ability distribution.The density operator ρ we use here is defined as a N × N complex matrix6n a finite dimensional complex vector space H = C N , which is described as ρ = N X k,l ρ kl | k ih l | = N X k,l ρ kl t kl . (1)The notation | k i ( h k | ), which is called ket vector (bra vector), means a columnvector (row vector) with the k -th component is 1 , and other components arezero. Thus, {| k i} ( k = 1 · · · N ) is a set of vectors that are normalized andorthogonal to each other; h k | l i = δ kl . ( h k | l i means the inner product of twovectors.) Hereafter, we call the set {| k i} a complete orthonormal system (CONS) in H . h k | ρ | l i = ρ kl ∈ C is ( k, l )-component of the matrix ρ . Thenotation t kl denotes the operator | k ih l | , which acts as follows: t kl | h i = δ lh | k i .In general, a density operator satisfies the following conditions:1. ρ kl = ρ ∗ lk ; the self-adjoint is satisfied.2. tr ( ρ ) = 1 ; the diagonal sum is one.3. ∀ x ∈ H , h x | ρ | x i ≥ ρ ∗ lk means the complex conjugate of ρ lk .)We divide this matrix into the diagonal part and non-diagonal part; ρ = ρ d + ρ nd (2)where ρ d = N X k ρ kk t kk = N X k ρ kk E k ρ nd = X k
Let us introduce a most simple comparison state with N = 2. The decisionmaker is aware of two objects and recognizes them as one pair to be compared.Then, the following 2 × H = C is considered as a form ofcomparison state. ρ = (cid:18) α αβ e i θ αβ e − i θ β (cid:19) = ( α E + β E ) + αβC ↔ (3) α and β are real numbers satisfying α + β = 1. The weights α , β and αβ are assigned for the cognitive processes E , and C ↔ . In principle, thevalues of α and β are to be determined subjectively by the decision maker.In the case of ( α, β ) = (0 ,
1) or (1 , αβ is zero. This implies that thedecision maker recognizes the pair of objects, if and only if he/she is awareof the existence of both objects. Also, the condition of ( α, β ) = ( √ , √ )implies that the decision maker pays attention to the existence of them, in a fair way . The decision maker we postulate has a function to compare an arbitrarilypair which is chosen from N objects. It is the quantitative evaluation defined The form of Eq. (3) is rewritten as ρ = | ψ ih ψ | , | ψ i = α e i θ | i + β | i = (cid:18) α e i θ β (cid:19) . Such representation, which is called pure state in quantum theory, is used in the generalconstruction of comparison state, see Sec. 2.3.
8y the following map. ϕ D : S ( H ) R , ϕ D ( · ) := tr ( D · ) . (4) S ( H ) denotes the set of all of comparison states (density matrices) on H = C N . The operator D is defined by a self-adjoint operator , which is alsowritten as a N × N complex matrix; D = N X k,l F ( k, l ) | k ih l | = N X k,l F ( k, l ) t kl (5) F ( k, l ) ∈ C is ( k, l )-component of the matrix. From the self-adjoint property, F ( k, l ) ∗ = F ( l, k ) is satisfied. The objects k and l are events independent andincompatible each other. (The orthogonality h k | l i = δ k,l implies this fact.)However, the decision maker correlates them in his/her cognitive operationof comparison. The correlation, that is, how the decision maker sees thedifference between k and l is encoded in the term of F ( k, l ) = h k | D | l i . Wegive its form as F ( k, l ) = ( u k − u l )e i φ kl , (6)where u k is the utility that the decision maker feels for the object- k . Here-after, we call ϕ D the evaluation function . Simple Example of Evaluation
Let us explain the use of evaluation function ϕ D ( ρ ) in the selection betweentwo objects: The comparison state ρ is given in the form of Eq. (3): ρ = 12 (cid:18) i θ e − i θ (cid:19) . where the fairness ( α, β ) = ( √ , √ ) is assumed. From the definition ofEq. (6), the operator D is given as 2 × D = (cid:18) u − u )e i φ ( u − u )e − i φ (cid:19) . In the formalism of quantum theory, a measurable physical quantity is given by aself-adjoint operator like D , and the value of tr ( Dρ ) is defined as the expectation value of D for the state ρ . Dρ = u − u (cid:18) e i( φ − θ ) e i φ e − i φ e − i( φ − θ ) (cid:19) , the evaluation ϕ D ( ρ ) = tr ( Dρ ) = cos Θ ( u − u ) , Θ = φ − θ is obtained. The parameter Θ determines the direction for evaluation . Forexample, in the case of cos Θ >
0, the above ϕ D ( ρ ) is used in the followingway. If ϕ D ( ρ ) ≥ , then (cid:23) , If ϕ D ( ρ ) ≤ , then (cid:23) . In the case of cos Θ < If ϕ D ( ρ ) ≤ , then (cid:23) , If ϕ D ( ρ ) ≥ , then (cid:23) . Let us remind the situation in the beginning of section: If a decision makerdraws the lot A ( B ), he/she will get the outcome x i ( i = 1 , · · · , n ) with theprobability P i ( Q i ). We defined the objects in this situation as the events(experiences), { ( A, x ) , · · · , ( A, x n ) , ( B, x ) , · · · , ( B, x n ) } . (For example, ( A, x k ) denotes the event (experience) that he/she draws thelot A and gets the outcome x k .) In our modeling, these are to be representedby vectors in Hilbert space. Firstly, let us introduce the tensor product ofspaces , H = H Lot ⊗ H
Outcome ; H Lot is the two dimensional space spannedby the two vectors | A i and | B i , and H Outcome is the n dimensional spacespanned by {| x i i , i = 1 , · · · , n } . Here, h A | B i = h B | A i = 0 , h x i | x j i = δ i,j areassumed, that is, {| A i ⊗ | x i , · · · , | A i ⊗ | x n i , | B i ⊗ | x i , · · · , | B i ⊗ | x n i} . is a CONS of H = C n . These vectors correspond to the above objects.Hereafter, {| A i ⊗ | x i i} and {| B i ⊗ | x i i} are replaced by {| i i , i = 1 , · · · , n } and {| ¯ i i = | n + i i , i = 1 , · · · , n } : We call the events (experiences) ( A, x i )10nd ( B, x i ), object- i and object-¯ i . (The utility of object- i or object-¯ i is givenby u ( x i ) = u i .)The comparison state ρ can be defined in H = C N with N = 2 n . Then,the statistical information of { P i } and { Q i } will be embedded in its struc-ture. In this subsection, the two types of ρ are designed under differentinterpretations, and in each case, the evaluation ϕ D ( ρ ) is used as the crite-rion for selection between A and B . After you draw the two lots, you will recognize the experiences ( A, x i ) and ( B, x j ) with the probability P i × Q j . This context is explained in the following comparison state. ρ C = n X i,j =1 P i Q j | ψ ij ih ψ ij | ∈ S ( H ) , (7)where | ψ ij i is the normalized vector with the form of | ψ ij i = 1 √ | A i ⊗ | x i i + | B i ⊗ | x j i ) = 1 √ | i i + | ¯ j i ) , where ¯ j = n + j . The state ρ C has the form of so-called mixed state . Notethat | ψ ij ih ψ ij | in ρ C is rewritten as | ψ ij ih ψ ij | = 12 ( E i + E ¯ j ) + 12 C i ↔ ¯ j , which is the same with the state of Eq. (3) with ( α, β ) = ( √ , √ ): Thedecision maker at | ψ ij ih ψ ij | is aware of the existence of objects denoted by i ,¯ j and recognizes them as one pair. This is interpreted as the cognitive statethat is experienced after drawing the two lots. The evaluation of the comparison state ρ C is calculated as ϕ D ( ρ C ) = n X i,j =1 P i Q j ϕ D ( | ψ ij ih ψ ij | ) = n X i,j =1 P i Q j cos Θ i ¯ j ( u i − u j ) . (8) Generally, a mixed state is defined as ρ = P Mk α k | ψ k ih ψ k | , where { α k } is a probabilitydistribution satisfying P Mk α k = 1, and { | ψ k i ∈ C N } are N -dimensional vectors whosenorms are one. D of Eq. (6). The value of ϕ D ( ρ C ) is the statisticalexpectation of { ϕ D ( | ψ ij ih ψ ij | ) } . Especially, if all of { Θ i ¯ j } have the samevalue Θ, it is proportional to the difference of expected utilities; ϕ D ( ρ C ) = cos Θ( n X i =1 P i u i − n X j =1 Q j u j ) ∝ E A − E B . (9)As result, our model can realize the criterion based on the expected utilitytheory as one example. The classical comparison state ρ C of Eq. (7) has the form as the statisticaldescription of the cognitive states {| ψ ij ih ψ ij |} , which are experienced afterdrawing the lots. In this subsection, we discuss another comparison statethat describes cognitive processes before drawing . We start this discussion byregarding the action of drawing of lot as a sort of measurement . Acquisition ofinformation through a measurement resolves uncertainty , which an observerholds before measurement. In other word, by the measurement, awarenessof uncertainty is transited to the one of definite information. We believethat such transition is described in the mathematical formalism of quantumtheory, which have discussed state changes by measurement, generally. Inour description, awareness of uncertainties on the two lots is represented bythe following two density operators. σ A = | ψ A ih ψ A | , σ B = | ψ B ih ψ B | , (10)where | ψ A i and | ψ B i are defined as normalized vectors in H = C N =2 n . Ifthe decision maker draws the lot A and knows the result of event ( A, x i )(object- i ), for example, the uncertainty symbolized by σ A is transited to thedefinite cognition | i ih i | . According to quantum theory, the state transitionfrom σ A to | i ih i | is represented by M i σ A M ∗ i tr ( M i σ A M ∗ i ) = | i ih i | , where M i ( i = 1 , · · · , N ) is called measurement operator and defined by M i = | i ih i | . Further, the statistical description obtained after measurements { M i } is defined by n X i =1 M i σ A M ∗ i = n X i =1 tr ( M i σ A M ∗ i ) | i ih i | = n X i =1 | h i | ψ A i | | i ih i | . tr ( M i σ A M ∗ i ) corresponds to the probability of measure-ment of result i : tr ( M i σ A M ∗ i ) = | h i | ψ A i | = P i . Thus, the vector | ψ A i in σ A is written as | ψ A i = n X i =1 p P i e i θ A i | i i . (11)This form is called quantum superposition . Similarly, | ψ B i in σ B has the formof | ψ B i = n X i =1 p Q i e i θ B¯ i | ¯i i . (12)As seen in these results, the probability distributions { P i } and { Q j } areembedded as elements determining the directions of | ψ A,B i .With using the above | ψ A,B i , we define the following comparison state: ρ = | Ψ ih Ψ | , | Ψ i = 1 √ | ψ A i + | ψ B i ) . (13)Note, this ρ is rewritten as ρ = 12 ( σ A + σ B ) + 12 C A ↔ B . (14)where C A ↔ B = | ψ A ih ψ B | + | ψ B ih ψ A | . The form of Eq (14) clearly explains the basic cognitive processes for theselection between A and B at the stage before measurement: the term of ( σ A + σ B ) in Eq. (14) implies that the decision maker is aware of the existenceof two lots. The term of C A ↔ B implies that he/she recognizes the two lotsas one pair to be compared. Let us discuss the evaluation of the non-classical comparison state ρ ; ϕ D ( ρ ) = ϕ D (cid:18) σ A (cid:19) + ϕ D (cid:18) σ B (cid:19) + ϕ D (cid:18) C A ↔ B (cid:19) . (15)13ere, we use the evaluation function ϕ D with D of Eq. (6). The first andsecond terms ϕ D (cid:0) σ A (cid:1) and ϕ D (cid:0) σ B (cid:1) are calculated as ϕ D (cid:18) σ A (cid:19) = n X i,j =1 p P i P j cos Θ ij ( u i − u j ) ϕ D (cid:18) σ B (cid:19) = n X i,j =1 p Q i Q j cos Θ ¯ i ¯ j ( u i − u j ) , (16)where Θ ij = φ ij − θ Ai + θ Aj and Θ ¯ i ¯ j = φ ¯ i ¯ j − θ B ¯ i + θ B ¯ j . We call these terms the evaluations of risks . For example, if the outcome in the lot A is $100 withprobability P and $10 with probability 1 − P , a decision maker might feel therisk by seeing the difference between $100 and $10, and he/she will estimatethe highest risk when P = 1 /
2. In this way, a decision maker will evaluatethe risk from the differences of outcomes and probabilities. As seen in theform of Eq. (16), the degrees of evaluations of risks are specified by the valuesof { cos Θ ij } and { cos Θ ¯ i ¯ j } . Hereafter, we call them DERs. The evaluationof risk seems to be experienced in realistic selection behavior. However, inthe classical criterion ϕ D ( ρ C ), this empirical fact is ignored, see Sec. 2.3.2.The third term ϕ D (cid:0) C A ↔ B (cid:1) in Eq. (15) is calculated as ϕ D (cid:18) C A ↔ B (cid:19) = n X i,j =1 p P i Q j cos Θ i ¯ j ( u i − u j ) , (17)where Θ i ¯ j = φ i ¯ j − θ Ai + θ B ¯ j . This term corresponds to the classical criterion ϕ D ( ρ C ) = P i,j P i Q j cos Θ i ¯ j ( u i − u j ) of Eq. (8). It should be noted here thatthe weights of awareness in Eq. (17) are given by square roots of productsof probabilities. In the case of Θ i ¯ j = Θ, the classical criterion realizes theselection based on the expected utility theory, see Eq. (9). In the same case,Eq. (17) is written as ϕ D (cid:18) C A ↔ B (cid:19) = cos Θ " n X i,j =1 p P i Q j u i − n X i,j =1 p P i Q j u j . Here, we consider the following value: ϕ D (cid:0) C A ↔ B (cid:1)P k,l √ P k Q l = cos Θ " n X i =1 √ P i P k √ P k u i − n X j =1 p Q j P l Q l u j
14 cos Θ " n X i =1 ˜ P i u i − n X j =1 ˜ Q j u j . (18)Note, P ni =1 ˜ P i = 1 and P nj =1 ˜ Q j = 1. From the above equation, one can seethat the evaluation ϕ D (cid:0) C A ↔ B (cid:1) is essentially equivalent to the calculationof expected utilities, P i ˜ P i u i and P j ˜ Q j u j . We call { ˜ P i } and { ˜ Q i } , whichare different from the statistical probabilities, subjective probabilities . Thesubjective probability ˜ P i is rewritten as˜ P i = √ P i √ P i + (cid:16)P k = i q P k − P i (cid:17) √ − P i = √ P i √ P i + ξ √ − P i . (19)The values of { P k = i − P i } in the above equation mean the probability distributionof results except for i . The range of ξ = P k = i q P k − P i is 1 ≤ ξ ≤ √ m − w ξ ( x ) = √ x √ x + ξ √ − x (0 ≤ x ≤ . (20)Figure.1 shows the behavior of w ξ ( x ) at m = 3 and ξ = √ m −
1. Onecan find that w ξ ( x ) realizes the character of probability weighting function ,which explains the experimental fact that small (statistical) probabilities areoverestimated, and large probabilities are underestimated, subjectively. Inthe prospect theory, the probability weighting function is the important con-cept to explain the violation of independence axiom in VNM theory. Actu-ally, from phenomenological discussions, various weighting functions has beenproposed([Prelec(1998), Rieger& Wang(2006), Tversky & Kahneman(1992)]).Especially, we note the form of two-parameter weighting function, w λ,δ ( x ) = δx λ δx λ + (1 − x ) λ , which was discussed in [Gonzalez & Wu (1999)]. The parameters λ and δ control the curvature and elevation of function, respectively. This phe-nomenological function with λ = 1 / δ = 1 /ξ is realized in the form ofEq. (20), which is derived from the state representation by density operator.15igure 1: The behavior of w ξ ( x ) at m = 3 and ξ = √ m − ϕ D ( ρ )had the two characteristics:1. ϕ D ( ρ ) includes the evaluation of risk.2. ϕ D ( ρ ) explains the property of probability weighting function.These are important points needed for the description of realistic selectionbehaviors. Actually, in the below sections, ϕ D ( ρ ) is used to describe severalfamous decision-making phenomena, which are impossible to be explained inthe framework of the standard expected utility theory. In this section, we consider selection behavior under unknown risk whoseprobability distribution is not informed. Selection behavior under ambigu-ity was first discussed by Ellsberg: He pointed out that many real decisionmakers prefer known risks to unknown risks, and such a tendency called16able 1: Example of Ellsberg Paradox.30balls 60ballsR W Y f $100 0 0 f f $100 0 $100 f ambiguity aversion cannot be explained in the standard expected utility the-ory. To solve this paradox, several mathematical models have been proposed.However, Machina recently presented the example of selection behavior underambiguity that cannot be explained even in the popular models of ambigu-ity aversion. In Sec.3.1, we discuss the solution of Ellsberg paradox in ourmodel, and in Sec.3.2, we show that our model also can solve the Machinaparadox in the same manner. The Ellsberg Paradox is explained in the following sentence:
An urn contains balls. balls are red ( R ). The other balls are eitherwhite ( W ) or yellow ( Y ), but its ratio is unknown. Consider the four lotteriesshown in Table.1, in each of which, if you draw one ball from the urn, youmight get $100 . Note, all of the lotteries consist of the event of getting $100 and the one ofgetting nothing. The table.2 shows the probabilities that are assigned for theevents, { ( f k , , ( f k , , k = 1 , , , } . The parameter α (0 ≤ α ≤
1) in thetable specifies the unknown ratio of white and yellow balls, and it is includedin the probabilities of ( f k , f k ,
0) with k = 2 ,
3. One can see thatthe lots f and f have ambiguities on the probabilities of results. Ellsbergpredicted that many real decision makers prefer f to f and f to f , thatis, ( f ≻ f ) ∧ ( f ≻ f ). This tendency of ambiguity aversion is inconsistentwith the standard manner according to the expected utility theory, in which, f ≻ f if and only if f ≻ f . The aim in this subsection is to simulatethe selection behavior in the above situation by using the model in Sec. 2.Firstly, let us consider the comparison state given for the selection between17able 2: Probabilities of events ( f k , f k , f k , f k , f / / f α/ − α ) / f (3 − α ) / α/ f / / f and f . We use the form of non-classical state of Eq. (13); ρ f − f = | Ψ ih Ψ | , | Ψ i = 1 √ | ψ f i + | ψ f i ) , (21)where | ψ f i = r | f i ⊗ | i + r | f i ⊗ | i = r | i + r | i , | ψ f i = r α | f i ⊗ | i + r − α | f i ⊗ | i = r α | i + r − α | i . (22)Here, {| i , | i , | i , | i} is a CONS on H = C , which corresponds to the setof events in the lots f and f . Note, ρ f − f is rewritten as ρ f − f = 12 σ f + 12 σ f + 12 C f ↔ f , with σ f k = | ψ f k ih ψ f k | and C f ↔ f = | ψ f ih ψ f | + | ψ f ih ψ f | .The operator D in the evaluation function ϕ D is defined in the form ofEq. (6); D = X i,j F ( i, j ) | i ih j | , (23)where F ( i, j ) ∗ = F ( j, i ). If ( i, j ) = (( f k , , ( f k ′ , F ( i, j ) = ( u ($100) − u (0))e i φ ij and if ( i, j ) = (( f k , , ( f k ′ , f k , , ( f k ′ , F ( i, j ) = 0.In the above definitions, the criterion ϕ D ( ρ f − f ) is calculated as ϕ D ( ρ f − f ) = ϕ D (cid:18) σ f (cid:19) + ϕ D (cid:18) σ f (cid:19) + ϕ D (cid:18) C f ↔ f (cid:19) , D (cid:18) σ f (cid:19) = √
23 cos Θ δu, (24) ϕ D (cid:18) σ f (cid:19) = p α (3 − α )3 cos Θ δu, (25) ϕ D (cid:18) C f ↔ f (cid:19) = (cid:18) √ − α + 2 √ α (cid:19) δu. (26)Here δu = u ($100) − u (0) > ϕ D (cid:0) C f ↔ f (cid:1) consists of √ − α cos Θ δu and √ α cos Θ δu , which are the terms of comparison between ( f , f , f ,
0) and ( f , f , f ,
0) will contribute to increase the preferenceof f , and the difference of utilities of ( f ,
0) and ( f , ≥ ≤ ϕ D (cid:0) σ f (cid:1) and ϕ D (cid:0) σ f (cid:1) represent the evaluations of risks in f and f .For the decision maker who has the tendency to dislike (like) risks, the risk in f will become a cause to decrease (increase) the preference of f , and the onein f will be a cause to increase (decrease) it. Thus, the two degrees of evalua-tions of risks (DERs) given by cos Θ and cos Θ satisfy cos Θ cos Θ ≤ ϕ D (cid:0) σ f (cid:1) has the unknown parameter α , seeEq. (25), that is, it is the evaluation of risk with ambiguity. According toEllsberg’s prediction, generally, the unknown risk affects the selection be-havior more strongly than the known risk. This essence is represented in thecondition | cos Θ | < | cos Θ | in our model. To simplify the discussion, letus set the parameters { Θ ij } as satisfyingcos Θ = 1 , cos Θ = − , cos Θ = 0 , cos Θ = λ. Then, ϕ D ( ρ f − f ) = √ − α − √ α λ p α (3 − α )3 ! δu. (27)The parameter λ is DER for the lot f with ambiguity; if λ > < f ≻ f if ϕ D ( ρ f − f ) > f ≻ f if ϕ D ( ρ f − f ) <
0. In the same manner, we canconsider the criterion for the selection between f and f that is given by ϕ D ( ρ f − f ) = √ α − √ − α λ p α (3 − α )3 ! δu. (28)19igure 2: α − λ phase diagram of selection in Ellseberg Paradox, in which,the parameter α specifies the unknown ratio and λ is DER for the lots f and f . Ellsberg’s prediction (( f ≻ f ) ∧ ( f ≻ f )) is realized in the rangeof λ > ρ f − f is defined on the Hilbert space H ′ = C ,which is different from H . This criterion is used as f ≻ f if ϕ D ( ρ f − f ) > f ≻ f if ϕ D ( ρ f − f ) <
0. The parameter λ in the third term is DER forthe lot f with ambiguity.The criterions shown in Eqs. (27) and (28) are regarded as the functions ofvariables α and λ : ϕ D ( ρ f − f ) := S f − f ( α, λ ) and ϕ D ( ρ f − f ) := S f − f ( α, λ ).Thus, the decision maker’s selection depends on the values of ( α, λ ). SeeFig. 2 that shows α − λ phase diagram giving ( f ≻ f ) ∧ ( f ≻ f ), ( f ≻ f ) ∧ ( f ≻ f ), ( f ≻ f ) ∧ ( f ≻ f ) or ( f ≻ f ) ∧ ( f ≻ f ).Let us see the line of λ = 0 in this diagram: The preference ( f ≻ f ) ∧ ( f ≻ f ) is obtained when α < /
2, and ( f ≻ f ) ∧ ( f ≻ f )when α < /
2. This is consistent with the result derived from the standardexpected utility theory. One can see that the preference ( f ≻ f ) ∧ ( f ≻ f ),which was predicted by Ellsberg, appears in the range of λ >
0. Then, thedecision maker has the tendency to dislike the unknown risks in f and f and embeds this evaluation into his (her) selection behavior. Especially, at α ≈ /
2, the preference ( f ≻ f ) ∧ ( f ≻ f ) is realized even if the positive20able 3: Example of the Machina Paradox.50balls 51ballsR B W Y f
202 202 101 101 f
202 101 202 101 f
303 202 101 0 f
303 101 202 0 λ is very small. In general, α and λ are quantities that are fluctuated ordetermined subjectively by the decision maker. However, we predict thatmany people stay at α ≈ / Machina presented the following situation as an example of selection underambiguity []:
An urn contains balls. The balls are either red ( R ) or black( B ).The balls are either white (W) or yellow (Y). These ratios are unknown.Consider the four lotteries shown in Table. 3, where you will get outcomeswhose utilities are 303, 202 , 101 or 0, if you draw one ball from the urn. The lots f and f consist of ( f k , a ) and ( f k , a ) with a = 101. The lots f and f consist of ( f k , a ), ( f k , a ), ( f k , a ) and ( f k , p = 50 / q = 1 − p = 51 / α, β (0 ≤ α, β ≤
1) specify the unknown ratio of R and B and the one of W and Y . One can see that only the lot f is unambiguous,and other lots have ambiguity on the probabilities of events. Machina pointedout that this example falsifies the popular models of ambiguity aversionincluding Choquet expected utility([Schmeidler(1989)]), maxmin expectedutility([Gilboa & Schmeidler(1989)]), variational preferences([Maccheroni et al.(2006)]), α -maxmin([Ghirardato et al.(2004)]) and the smooth model of ambiguityaversion([Klibanoff et al.(2005)]). According to these models, f ≻ f if and21able 4: Probabilities of events ( f k , a ), ( f k , a ), ( f k , a ), ( f k , f k , a ) ( f k , a ) f p qf pα + qβ − ( pα + qβ )( f k , a ) ( f k , a ) ( f k , a ) ( f k , f pα p (1 − α ) qβ q (1 − β ) f pα qβ p (1 − α ) q (1 − β )only if f ≻ f . Since f is unambiguous, the preference f ≻ f can be in-terpreted as the result of the ambiguity aversion. However, both of f and f include ambiguities. Note, the lotteries f and f have a slight advantage dueto the 51th ball that may yield 202. In the choice between f and f , there isa tradeoff between the advantage offered by f and the unambiguity offeredby f . On the other hand, such trade-off seems to be less in the selection of f or f . From this perspective, ( f ≻ f ) ∧ ( f ≻ f ) can not be denied as arealistic selection.With using the manner discussed in Sec.2.1, we analyze the selectionbehavior in this situation. Criterion for the selection between f and f isdesigned as ϕ D ( ρ f − f ) = ϕ D (cid:18) σ f (cid:19) + ϕ D (cid:18) σ f (cid:19) + ϕ D (cid:18) C f ↔ f (cid:19) , (29)where ϕ D (cid:18) σ f (cid:19) = 0 , (30) ϕ D (cid:18) σ f (cid:19) = p ( pα + qβ )(1 − ( pα + qβ )) λa, (31) ϕ D (cid:18) C f ↔ f (cid:19) = (cid:16)p p (1 − ( pα + qβ )) − p q ( pα + qβ ) (cid:17) a. (32)We assume, the DER for the lot f with the known risk is zero, see Eq. (30),and the DER for the lot f with the unknown risk is λ , see Eq. (31). All ofthe comparisons of utilities are embedded in the term of Eq. (32). We define22his criterion as a function of α , β and λ ; ϕ D ( ρ f − f ) := S f − f ( α, β, λ ).If S f − f ( α, β, λ ) > S f − f ( α, β, λ ) < f ≻ f ( f ≻ f ). Next, thecriterion for the selection between f and f is designed as ϕ D ( ρ f − f ) = ϕ D (cid:18) σ f (cid:19) + ϕ D (cid:18) σ f (cid:19) + ϕ D (cid:18) C f ↔ f (cid:19) , (33)where ϕ D (cid:18) σ f (cid:19) = − (cid:16) p p α (1 − α ) + q p β (1 − β ) (cid:17) λa, (34) ϕ D (cid:18) σ f (cid:19) = (cid:16) p p α (1 − α ) + q p β (1 − β ) (cid:17) λa, (35) ϕ D (cid:18) C f ↔ f (cid:19) = (cid:16)p pqαβ + q p β (1 − β ) + qβ (cid:17) a − (cid:16)p pq (1 − α )(1 − β ) + p p α (1 − α ) + p (1 − α ) (cid:17) a. (36)Note, the DERs for f and f with the unknown risks are also specified bythe parameter λ , see Eqs. (34) and (35). If ϕ D ( ρ f − f ) := S f − f ( α, β, λ ) > S f − f ( α, β, λ ) < f ≻ f ( f ≻ f ). Figure. 3 shows α − λ phasediagram in the case of α = β . Let us see the line of λ = 0 in this figure. If α > / f ≻ f ) ∧ ( f ≻ f ) is realized. As mentionedin the previous section, the decision maker will stay at α = β ≈ /
2, unlesssome information on the unknown ratios is given. Since 1 / > / f and f ; in a sense, the difference between 1 / / f and f , which comes from the beliefthat the 51th ball might be W . When λ becomes large, ( f ≻ f ) ∧ ( f ≻ f )and ( f ≻ f ) ∧ ( f ≻ f ) are appearing. As mentioned previously, thepositive λ implies the tendency of ambiguity aversion which decreases thevalues of f , f and f with ambiguity. As seen in the diagram of Fig. 3,the value of λ needed to invert f ≻ f to f ≻ f is not so large, but λ toinvert f ≻ f to f ≻ f has the higher value. The suppression of inversionfrom f ≻ f to f ≻ f is due to the competition between the evaluations ofunknown risks offered by the both lots, see Eqs. (34) and (35). As a result,our model explains the existence of selection ( f ≻ f ) ∧ ( f ≻ f ), which hasnot been explained in the popular models of ambiguity aversion.23igure 3: α − λ Phase Diagram of Selection in the Machina Paradox
In this section, we consider cash equivalent (CE) for lot with risk that meansamount of money whose utility is indifferent with the utility of receiving lot: u CE = u LOT . This relation can be interpreted as a result of comparison between the twoalternatives, the selection of definite cash and the one of lot. Thus, thedecision of CE can be discussed in the framework of our comparison model.For example, let us consider a comparison between a cash x and a lot whoseoutcome is y ( >
0) with probability p or nothing with q = 1 − p . Then, thecomparison state is given as ρ = | Ψ ih Ψ | , | Ψ i = 1 √ | ψ lot i + 1 √ | ψ cash i , (37)where | ψ lot i = √ p | L i ⊗ | y i + √ q | L i ⊗ | i = √ p | i + √ q | i , | ψ cash i = | C i ⊗ | x i = | i . (38)24igure 4: Behavior of cash equivalent of lot.The comparison state ρ is defined in H = C . With the same manner in theprevious sections, we design the following criterion. ϕ D ( ρ ) = √ p ( u y − u x ) − √ qu x − λ √ pqu y . (39) u x and u y are the utilities of the outcomes x and y , and u = 0 is assumed.The parameter λ is the DER for the lot. If ϕ D ( ρ ) = 0, the lot and x areindifferent for the decision maker. We define the cash x in this case as theCE of the lot; u CE = u x = √ p (1 − λ √ q ) u y √ p + √ q . (40)Figure. 4 shows the behavior of u CE /u y for the parameters of p and λ (0 < p, λ < p , u CE is smaller than u y , whichis the utility of the highest outcome in lot, and its value is decreased as λ becomes large. Note, the decision maker at positive λ dislikes the risk of lot.The CE defined in Eq. (40) will be closely related to willingness to accept (WTA) and willingness to pay (WTP) for the lot, because, each of them isalso defined as cash equivalent to the lot. WTA and WTP are conceptually25istinguished in the difference situations that require the determination ofCE. WTA is the minimum cash which the decision maker accepts to sell thelot. WTP is the maximum cash which the decision maker pays for gettingthe lot. In the determination of WTA, the decision maker is endowed withthe lot. On the other hand, in the case of WTP, the decision maker does notget the lot yet. In our model, this apparent difference of position is reflectedon the value of λ in Eq. (40). The seller abandons the right of drawing thelot, therefore, he/she might not fear the underlying risk in the lot. The buyerwill be aware of the risk more strongly, because he/she might draw the lot.From such a perspective, it is expected that seller’s λ will be smaller thanthe one the buyer has. As seen in Fig. 4, if λ < λ ′ , CE ( λ ) > CE ( λ ′ )is always satisfied. Actually, WTA > WTP is the fact that has been con-firmed in various experimental tests. Also, this discussion is closely relatedto preference reversal phenomena ([Tversky & Thaler(1990)]). It is the fol-lowing decision making: There are two lots, one of which, say A , has a highchance of winning a small prize, and another, say B , has a lower chance ofwinning a larger. According to experimental tests, many people prefer A to B ( A ≻ B ) but, they estimate WTA of B larger than the one of A . This factseems to show the violation of transitivity axiom: In the situation of sellingthe lots, there exist a cash X satisfying the preference as B ≻ X ≻ A , inspite of A ≻ B in the simple comparison of the two lots. The essence of pref-erence reversal is almost same with the disparity between WTA and WTP:In the situation of selling, they will estimate WTA with using λ , which issmaller than λ ′ expected to be used in the simple comparison between A and B . Here, we mention the popular interpretations on the disparity betweenWTA and WTP: In the standard economic theory, the income effect hasbeen thought to be a cause of the disparity. WTP is generally constrainedby the present income, but WTA is not. However, the existence of disparityis found in many experimental tests, in which, income effects are expected tobe small([Carmon & Ariely(2000), Kahneman et al.(1990)]). On the otherhand, in the prospect theory, the disparity is explained from the difference ofreference point in the value function . The value function evaluates gain or loss We remark the real situation is even more complicated: in standard economics thereare typically considered two effects - the income effect and substitution effect.
26f outcome from reference point that represents a decision maker’s situation.The decision maker endowed with the lot and the one not endowed are positedat different reference points. From these points, the compensation of loss oflot (WTA) and the cost of gain of lot (WTP) are estimated respectively.Generally, in the value function, the degree of depression of utility by lossbecomes larger than the one of enhancement of utility by gain. Therefore,WTA > WTP is realized.In our model, the existence of the parameter λ (DER) explains the dispar-ity between WTA and WTP, and it is a psychological factor, which fluctuatesdepending on the decision maker’s situation. This point is consisting withthe concept of value function in the prospect theory, where the referencepoint fluctuates. Furthermore, the evaluation of loss or gain in the terms ofprospect theory is clearly represented as the function of λ . The decision maker simulated in our model performs various comparisonsof objects to make a criterion of selection. As discussed in Sec. 2, all ofthe comparisons are embedded into the comparison state ρ , and the crite-rion is defined by the function ϕ ( ρ ). This quantum-like model has potentialto explain realistic selection behaviors. Actually, it clearly explains the be-havior of probability weighting function, see Fig. 1. In the prospect theory,the probability weighting function is assumed in a phenomenological senseand used to explain the violation of independence axiom in VNM theory.Further, our model describes comparisons, which are realistic operations buthave been ignored in the expected utility theory. The evaluation of suchcomparison is reflected in the criterion ϕ ( ρ ), as the degree of evaluation ofrisk (DER). As discussed in Sec. 3, DER plays the important role to solveEllsberg paradox and Machina paradox, see the diagrams of Figs. 2 and 3where the parameters α and λ specify the unknown ratio and DER respec-tively. Here, we predicted that many people would pay fair attentions to theunknown probabilities and dislike the risk, that is, α ≈ / λ >
0. Thedecision maker at this position has the tendency of ambiguity aversion andcan realize the choice that Machina pointed out. Our solution is crucial, be-cause the Machina paradox is difficult to be solved in the popular models ofambiguity aversion. (Our solution might be closely related to the discussionin the paper of [Baillon et al.(2011)], in which, models of decision making are27iscussed to account for the Machina paradox.) In Sec. 4, the disparity be-tween WTA(willingness to accept) and WTP(willingness to pay) is explained.Here, we assumed that DER is the sensitive quantity varying depending onthe situation of “accept” or “pay”, see Fig. 4. This property is closely re-lated to the preference reversal phenomenon that shows anomaly violatingthe transitivity axiom in VNM theory. More generally, DER might specifypersonalities including habits, experiences and cognitive abilities. This pointwill be very important from the view of experimental psychology.Lastly, let us point out that this general modeling is applicable to variousdecision making phenomena. For example, it can address more complicatedcase such that a large number of alternatives exist. Naturally, a real decisionmaker cannot pay attention to all alternatives, and such a situation will bereflected on the form of comparison state. The problem on limited atten-tion has been often discussed in the economic literatures, see the papers of[Manzini & Mariotti(2014)] and [Masatlioglu et al. (2012)].
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