A Proposal to Extend Expected Utility in a Quantum Probabilistic Framework
aa r X i v : . [ q -f i n . E C ] D ec A Proposal to Extend Expected Utility in a Quantum ProbabilisticFramework
Diederik Aerts ∗ Emmanuel Haven † Sandro Sozzo ‡ Abstract
Expected utility theory (EUT) is widely used in economic theory. However, its subjective probabilityformulation, first elaborated by Savage, is linked to Ellsberg-like paradoxes and ambiguity aversion. Thishas led various scholars to work out non-Bayesian extensions of EUT which cope with its paradoxesand incorporate attitudes toward ambiguity. A variant of the Ellsberg paradox, recently proposedby Mark Machina and confirmed experimentally, challenges existing non-Bayesian models of decision-making under uncertainty. Relying on a decade of research which has successfully applied the formalismof quantum theory to model cognitive entities and fallacies of human reasoning, we put forward a non-Bayesian extension of EUT in which subjective probabilities are represented by quantum probabilities,while the preference relation between acts depends on the state of the situation that is the object of thedecision. We show that the benefits of using the quantum theoretical framework enables the modeling ofthe Ellsberg and Machina paradoxes, as the representation of ambiguity and behavioral attitudes towardit. The theoretical framework presented here is a first step toward the development of a ‘state-dependentnon-Bayesian extension of EUT’ and it has potential applications in economic modeling.
Keywords:
Expected utility theory; Ellsberg paradox; Machina paradox; quantum probability; quantummodeling.
Economic theory crucially rests on the so-called ‘Bayesian paradigm’: any source of uncertainty can be prob-abilistically quantified and the ensuing probability theory satisfies the axioms of Kolmogorov (Kolmogorov,1933). Von Neumann and Morgenstern successfully applied the Bayesian paradigm when elaborating anaxiomatic form of the expected utility theory (EUT), which is the predominant model of decision-makingunder uncertainty (von Neumann & Morgenstern, 1944).It is however well known that, for many events of interest, one cannot define an objective agreed-uponprobability. This led Savage to extend the von Neumann and Morgenstern ‘objective EUT’ within theBayesian paradigm (Savage, 1954). The traditional line of reasoning is that, in the absence of objectiveprobabilities, the decision-maker forms her/his own subjective probabilities, and takes decisions on the basisof these subjective probabilities. This is the basic tenet of subjective EUT: individuals take decisions as ifthey maximized expected utility with respect to a Kolmogorovian probability measure which is interpretedas their subjective probability. ∗ Center Leo Apostel for Interdisciplinary Studies, Krijgskundestraat 33, 1160 Brussels (Belgium). Email address: [email protected] † School of Business and Institute IQSCS, University Road, LE1 7RH Leicester (United Kingdom). Email address: [email protected] ‡ School of Business and Institute IQSCS, University Road, LE1 7RH Leicester (United Kingdom). Email address: [email protected] free from anyphysical connotation or interpretation , can be used to model human choices in the presence of ambiguity. Inthis respect, we apply here the lesson we have learned from quantum physics. A measurement process givesrise to an interaction between the microscopic quantum entity that is measured and a measuring apparatus,and thus the latter acts as a measurement context for the former. This contextual interaction is non-controllable (which is formalized by the Heisenberg uncertainty principle), and determines an intrinsicallyprobabilistic change of the state of the measured entity. Then, a measurement outcome is actualized amonga set of possible outcomes, as a consequence of this contextual interaction. Whenever one formalizes thestatistical frequencies of repeated experiments to derive probabilities, one discovers that this kind of ‘purepotentiality’ and ‘contextuality’ cannot be formalized in a single Kolmogorov probability space: they indeedrequire quantum probability (Aerts, 2009).The quantum measurement lesson was firstly applied with success to the representation of conceptualentities. We introduced the notion of ‘state of a concept’ to model the type of interaction that occurswhen people are asked about membership, or typicality, of specific exemplars with respect to pairs ofconcepts and their combinations, e.g., conjunction, disjunction and negation. We applied the quantum-conceptual approach to solve various difficulties of the Kolmogorovian model of probability in conceptualcombinations. Then, we extended the approach to more complex situations, showing that genuine quantumeffects, i.e. ‘superposition’, ‘interference’, ‘entanglement’, etc., systematically occur in the combination ofnatural concepts (Aerts, 2009; Aerts, Broekaert, Gabora & Sozzo, 2013; Aerts, Gabora & Sozzo, 2013;Aerts, Sozzo & Veloz, 2015; Sozzo, 2015).The quantum-conceptual approach supports a growing research that uses the mathematical formalism2f quantum theory to model complex cognitive processes, like probability and similarity judgment, per-ception, knowledge representation and decision-making. More specifically, quantum models have showntheir effectiveness in the explanation of the so-called ‘fallacies of human reasoning’, like ‘conjunctive anddisjunctive fallacies’, ‘disjunction effects’, ‘question order effects’, etc. (see, e.g., Busemeyer & Bruza, 2012;Busemeyer, Pothos, Franco & Trueblood, 2011; Haven & Khrennikov, 2013; Khrennikov, 2010; Pothos &Busemeyer, 2013; Wang, Solloway, Shiffrin & Busemeyer, 2014; Khrennikov, 2015). This novel researchprogramme has now become a valid alternative to the Kahneman and Tversky theory of individual heuris-tics and biases in providing an explanation for the observed fallacies (Tversky & Kahneman, 1974; 1983;1992).What about the above paradoxes of EUT? We have recently applied the quantum conceptual approachto decision-making processes. In a human decision-making process the interaction between the object ofthe decision and the decision-maker is of a cognitive nature rather than of a physical nature like it is thecase when quantum models are applied in physics. The result of this interaction leads to the decisionitself, which is actualized among the possible alternatives. Again, the probabilistic model formalizingthe ambiguity occurring in the decision-making interaction cannot generally be Kolmogorovian: it shouldbe quantum probability. In particular, the notion of state of the cognitive entity that is the object ofthe decision, the ‘decision-making (DM) entity’, incorporates the notion of ambiguity and generates the(non-Kolmogorovian) quantum probability distribution modeling subjective probabilities. In addition,the change of state of the DM entity after the interaction incorporates attitudes toward ambiguity, e.g.,‘ambiguity aversion’, or ‘ambiguity attraction’, or even other attitudes, depending on the overall ‘conceptuallandscape surrounding the DM entity’. We also show that preferences between acts are not absolutenotions, but they depend on the state of the DM entity and on how the state of the DM changes as aconsequence of the interaction with the decision-maker. This suggests associating the DM entity with afamily of subjective probability distributions, parametrized by the states of the DM entity itself. We thusput forward a ‘state-dependent extension of EUT’, of which the present formalism is a first step.We may say that the use of quantum probabilities helps clarifying how subjective probabilities areformed. We indeed suggest that the kind of uncertainty that occurs in an ambiguity situation is similarto the kind of uncertainty that occurs in quantum theory as an effect of superposition. It is importantto mention that the quantum theoretical approach allows reduction of ambiguity to risk, as subjectiveprobabilities can be estimated from quantum probabilities. However, this is a context-dependent riskwhich cannot be modeled by Kolmogorovian probability. Rather, it is a ‘contextual risk’ that is modeledin the mathematical formalism of quantum theory.We put forward a non-Bayesian generalization of the Bayesian paradigm, where human decision-makingunder uncertainty is modeled in the mathematical formalism of quantum theory, quantum states incor-porate the uncertainty that is present in situations of ambiguity, and state transformations incorporatepeople’s attitude towards ambiguity. This quantum-conceptual approach was recently employed with suc-cess to model Ellsberg- and Machina-like preferences in the corresponding paradox situations, and tofaithfully represent data collected in experimental tests on the ‘three-color Ellsberg urn’ and the ‘Machinareflection example’.Let us summarize the content of this paper as follows.In Section 3 we give an overview of the technical details of the mathematical formalism of quantumtheory that are needed to model cognitive entities, like ‘concepts’, ‘conceptual combinations’, or more com-plex ‘DM entities’. We then apply in Section 3 the quantum conceptual approach to model the ‘disjunctioneffect’, which entails a violation of one of the axioms of subjective EUT, the ‘sure thing principle’ (Tversky& Shafir, 1992). Successively, we introduce in Section 4 the foundations of subjective EUT and brieflyreview its non-Bayesian extensions. We explicitly present the Ellsberg paradox in the ‘three-color example’and the Machina paradox in the ‘reflection example’. In Section 5 we instead work out a general quantum3heoretical framework to represent events, states, measurements, subjective probabilities, acts and deci-sions, which we apply to model the Ellsberg paradox (Section 5.1) and the Machina paradox (5.2). InSections 5.1 and 5.2 we also show that the quantum theoretical framework enables a faithful representationof concrete experiments on the Ellbserg three-color example and the Machina reflection example, respec-tively. We finally discuss in Section 6 some potential applications of the quantum theoretical approach tomodel ambiguity and ambiguity aversion in economics and finance.Before proceeding further, we would like to premise that the mathematical framework presented inSection 5 is not derived from a representation theorem. It rather emerges from heuristic operationalconsiderations on events, measurements, outcomes and probabilities and their representation within theformalism of quantum theory. And, indeed, it is able to reconstruct, via the quantum state of the DMentity, the subjective probability distributions that are actualized in concrete experiments as a consequenceof a particular attitude toward ambiguity.
We illustrate in this section how the mathematical formalism of quantum theory can be applied to modelsituations outside the microscopic quantum world, more specifically, in the representation of cognitiveentities. This formalism will be applied to conceptual entities in Section 3 and to decision-making (DM)entities in Section 5. We avoid in our presentation superfluous technicalities, but aim to be synthetic andrigorous at the same time.When the quantum mechanical formalism is applied for modeling purposes, each considered entity – inour case a cognitive entity – is associated with a complex Hilbert space H , that is, a vector space over thefield C of complex numbers, equipped with an inner product h·|·i that maps two vectors h A | and | B i ontoa complex number h A | B i . We denote vectors by using the bra-ket notation introduced by Paul AdrienDirac, one of the pioneers of quantum theory (Dirac, 1958). Vectors can be ‘kets’, denoted by | A i , | B i , or‘bras’, denoted by h A | , h B | . The inner product between the ket vectors | A i and | B i , or the bra-vectors h A | and h B | , is realized by juxtaposing the bra vector h A | and the ket vector | B i , and h A | B i is also called a‘bra-ket’, and it satisfies the following properties:(i) h A | A i ≥ h A | B i = h B | A i ∗ , where h B | A i ∗ is the complex conjugate of h A | B i ;(iii) h A | ( z | B i + t | C i ) = z h A | B i + t h A | C i , for z, t ∈ C , where the sum vector z | B i + t | C i is called a‘superposition’ of vectors | B i and | C i in the quantum jargon.From (ii) and (iii) follows that the inner product h·|·i is linear in the ket and anti-linear in the bra, i.e.( z h A | + t h B | ) | C i = z ∗ h A | C i + t ∗ h B | C i .We recall that the ‘absolute value’ of a complex number is defined as the square root of the product ofthis complex number times its complex conjugate, that is, | z | = √ z ∗ z . Moreover, a complex number z caneither be decomposed into its cartesian form z = x + iy , or into its polar form z = | z | e iθ = | z | (cos θ + i sin θ ).As a consequence, we have |h A | B i| = p h A | B ih B | A i . We define the ‘length’ of a ket (bra) vector | A i ( h A | )as ||| A i|| = ||h A ||| = p h A | A i . A vector of unitary length is called a ‘unit vector’. We say that the ketvectors | A i and | B i are ‘orthogonal’ and write | A i ⊥ | B i if h A | B i = 0.We have now introduced the necessary mathematics to state the first modeling rule of quantum theory,as follows. First quantum modeling rule:
A state A of an entity – in our case a cognitive entity – modeled by quantumtheory is represented by a ket vector | A i with length 1, that is h A | A i = 1.An orthogonal projection M is a linear operator on the Hilbert space, that is, a mapping M : H →H , | A i 7→ M | A i which is Hermitian and idempotent. The latter means that, for every | A i , | B i ∈ H and4 , t ∈ C , we have:(i) M ( z | A i + t | B i ) = zM | A i + tM | B i (linearity);(ii) h A | M | B i = h B | M | A i ∗ (hermiticity);(iii) M · M = M (idempotency).The identity operator maps each vector onto itself and is a trivial orthogonal projection operator.We say that two orthogonal projections M k and M l are orthogonal operators if each vector contained inthe range M k ( H ) is orthogonal to each vector contained in the range M l ( H ), and we write M k ⊥ M l , inthis case. The orthogonality of the projection operators M k and M l can also be expressed by M k M l = 0,where 0 is the null operator. A set of orthogonal projection operators { M k | k = 1 , . . . , n } is called a‘spectral family’ if all projectors are mutually orthogonal, that is, M k ⊥ M l for k = l , and their sum is theidentity, that is, P nk =1 M k = . A spectral family { M k | k = 1 , . . . , n } identifies an Hermitian operatorˆ O = P ni =1 o k M k , where o k is called ‘eigenvalue of ˆ O ’, i.e. is a solution of the equation ˆ O | o i = o k | o i – thenon-null vectors satisfying this equation are called ‘eigenvectors of ˆ O ’.The above definitions give us the necessary mathematics to state the second modeling rule of quantumtheory, as follows. Second quantum modeling rule:
A measurable quantity Q of an entity – in our case a cognitive entity –modeled by quantum theory, and having a set of possible real values { q , . . . , q n } is represented by a spectralfamily { M k | k = 1 , . . . , n } , equivalently, by the Hermitian operator ˆ Q = P nk =1 q k M k , in the following way.If the conceptual entity is in a state represented by the vector | A i , then the probability of obtaining thevalue q k in a measurement of the measurable quantity Q is h A | M k | A i = || M k | A i|| . This formula is calledthe ‘Born rule’ in the quantum jargon. Moreover, if the value q k is actually obtained in the measurement,then the initial state is changed into a state represented by the vector | A k i = M k | A i|| M k | A i|| (1)This change of state is called ‘collapse’ in the quantum jargon.Let us now come to the formalization of quantum probability. A major structural difference betweenclassical probability theory, which satisfies the axioms of Kolmogorov, and quantum probability theory,which is non-Kolmogorovian, relies on the fact that the former is defined on a Boolean σ -algebra ofevents, whilst the latter is defined on a more general algebraic structure. More specifically, let us denoteby L ( H ) the set of all orthogonal projection operators over the complex Hilbert space H . L ( H ) hasthe algebraic properties of a complete orthocomplemented lattice, but L ( H ) is not distributive, hence L ( H ) does not form a σ -algebra. A ‘generalized probability measure’ over L ( H ) is a function µ : M ∈ L ( H ) µ ( M ) ∈ [0 , µ ( ) = 1, and µ ( P ∞ k =1 M k ) = P ∞ k =1 µ ( M k ), for any countable sequence { M k ∈ L ( H ) | k = 1 , , . . . } of mutually orthogonal projection operators. The elements of L ( H ) are saidto represent ‘events’, in this framework. Referring to the definitions above, the event “a measurement ofthe quantity Q gives the outcome q k ” is represented by the orthogonal projection operator M k .The Born rule establishes a connection between states and generalized probability measures, as follows.Given a state of a cognitive entity represented by the vector | A i ∈ H with length 1, it is possible toassociate | A i with a generalized probability measure µ A over L ( H ), such that, for every M ∈ L ( H ), µ A ( M ) = h A | M | A i . The generalized probability measure µ A is a ‘quantum probability measure’ over L ( H ). Interestingly enough, if the dimension of the Hilbert space is greater than 3, all generalizedprobability measures over L ( H ) can be written as functions µ A ( M ) = h A | M | A i , for some unit vector | A i ∈ H (Gleason theorem; Gleason, 1957).The quantum theoretical modeling above can be extended by adding further quantum rules to modelcompound cognitive entities and more general classes of measurements on cognitive entities. However, thepresent definitions and results are sufficient to attain the results in this paper.5 The quantum cognition lesson
In 2002 Daniel Kahneman was awarded the Nobel Prize in Economic Science for “having integrated insightsfrom psychological research into economic science, especially concerning human judgement and decision-making under uncertainty”.Researchers have historically formalized human cognition and behavior by using set theoretical struc-tures, mainly, Boolean logic and a probability theory that satisfies the axioms of Kolmogorov (Kolmogorov,1933) (‘Kolmogorovian probability’). These are also called ‘classical structures’, as they were originallyemployed in classical physics, and later extended to economics, finance, statistics, psychology, etc. As men-tioned in Section 1, this conception has particularly flowed in economics into EUT to form the so-called‘Bayesian paradigm’ and, with it, the roots of ‘rational behavior’.However, since the beginning of the previous century, researchers have known that the probabilitytheory axiomatized by Kolmogorov is not the only way to talk about probabilities and uncertainty. Indeed,quantum probability formalizes uncertainty into the microscopic realm (see, e.g., (Pitowsky, 1989)). A hugeamount of literature has discussed the conceptual differences between classical and quantum probability.Leaving aside these differences, that are still object of scientific debate, we can certainly pin point thestructural differences between classical and quantum probabilistic theories. The former probability indeedrests a normalized probability measure on a σ -algebra of events represented by sets and set theoreticaloperations. The latter probability rests instead on a more abstract generalized measure on a lattice oforthogonal projection operators over a complex Hilbert space (see Section 2).What has become evident in the last three decades is that accumulating paradoxical findings in cognitivepsychology suggest that this classical conception of logic and probability theory is also fundamentallyproblematical. Puzzling cognitive phenomena have been identified, revealing the so-called ‘fallacies ofhuman reasoning’, which can be roughly divided in two main classes, as follows.(i) ‘Probability judgment errors’. People estimate the conjunction event ‘ A and B ’ (disjunction event‘ A or B ’) as more (less) likely than the events A or/and B separately, which entails a violation of themonotonicity law of Kolmogorovian probability.(ii) ‘Decision-making errors’. People prefer action A over action B if they know that an event E occurs,and also if they know that E does not occur, but they prefer B over A if they do not know whether E occurs or not, which entails a violation of the total law of Kolmogorovian probability.Fallacies of type (i) include the ‘conjunction’ and the ‘disjunction fallacy’, ‘non-monotonic reasoning’and ‘over/under-extension effects’ in membership judgements in conceptual combinations, and violations ofdistance axioms in similarity judgments. Fallacies of type (ii) include the ‘disjunction effect’ and violationsof the axioms of EUT (see, e.g, Busemeyer & Bruza (2012)), which we will extensively discuss in Section4. While a well established proposal of solution comes from the research programme on ‘individual heuris-tics and bias’ developed by Kahneman and Tversky (1979, 1983, 1992), an alternative proposal has recentlygrown which uses the mathematical formalism of quantum theory to model the observed deviations fromclassicality in human reasoning. In particular, quantum probabilistic models have shown distinctive ad-vantages over Bayesian models in representing experimental data on the fallacies above, also allowing tomake predictions and to discover new non-classical effects (see, e.g., the monographs Khrennikov, 2010;Busemeyer & Bruza, 2012; Haven & Khrennikov, 2013).The origins of our quantum theoretical approach can be traced back to the studies on the structuralconnections between cognitive and microphysical entities, namely, their behavior with respect to ‘con-textuality’ and ‘pure potentiality’. We recognized that any decision process involves a ‘transition frompotential to actual’ in which an outcome is actualized from a set of possible outcomes as a consequenceof a contextual interaction (of a cognitive nature) between the decision-maker and the cognitive situation6hat is the object of the decision. Hence, human decision processes exhibit deep analogies with what occursin a quantum measurement process, where the measurement context (of a physical nature) influences themeasured quantum particle in a non-deterministic way. Quantum probability - which is able to formalizethis ‘contextually driven actualization of potential’, rather than classical probability, which only formalizes alack of knowledge about actuality - can conceptually and mathematically cope with this situation underlyingboth quantum and cognitive realms (Aerts, 2009).The above analysis was the starting point for the development of a quantum theoretical perspectiveto represent conceptual entities and their combinations – conjunctions, disjunctions and negations (Aerts,2009; Aerts, Gabora & Sozzo, 2013; Aerts, Broekaert, Gabora & Sozzo, 2013; Sozzo, 2015; Aerts, Sozzo &Veloz, 2015). We modeled different sets of experimental data that exhibited deviations from classical logicaland Kolmogorovian structures, also identifying new non-classical mechanisms and extending the approachto more complex decision-making situations (Sozzo, 2015; Aerts & Sozzo, 2016). A systematic treatment ofthese results would lead us beyond the scope of the present paper. Hence, we limit ourselves to summarizehere a quantum theoretical modeling of the disjunction effect as an example of a long standing cognitivepuzzle. The solution offered here is paradigmatic, because the disjunction effect entails a violation of the‘sure thing principle’, exactly like the paradoxes of subjective EUT mentioned in Section 1.Savage stated the sure thing principle by means of the following story.“A businessman contemplates buying a certain piece of property. He considers the outcome of the nextpresidential election relevant. So, to clarify the matter to himself, he asks whether he would buy if heknew that the Democratic candidate were going to win, and decides that he would. Similarly, he considerswhether he would buy if he knew that the Republican candidate were going to win, and again finds thathe would. Seeing that he would buy in either event, he decides that he should buy, even though he doesnot know which event obtains, or will obtain, as we would ordinarily say.” (Savage, 1954).Tversky and Shafir tested the sure thing principle in an experiment where they presented a group ofstudents with a ‘two-stage gamble’, that is, a gamble which can be played twice (Tversky & Shafir, 1992;Busemeyer & Bruza, 2012). At each stage the decision consisted in whether or not playing a gamble thathas an equal chance of winning, say $200, or losing, say $100. The key result is based on the decision forthe second bet, after finishing the first bet. The experiment included three situations: (i) the students wereinformed that they had already won the first gamble; (ii) the students were informed that they had lostthe first gamble; (iii) the students did not know the outcome of the first gamble. Tversky and Shafir foundthat 69%, i.e. the majority, of the students who knew they had won the first gamble chose to play again,59%, i.e. the majority, of the students who knew they had lost the first gamble, chose to play again; butonly 36% of the students who did not know whether they had won or lost chose to play again (equivalently,64%, i.e. the majority, decided not to play in the second gamble).The two-stage gamble experiment violates Savage’s sure thing principle: students generally prefer toplay again if they know they won, and they also prefer to play again if they know they lost, but theygenerally prefer not to play again when they do not know whether they won or lost. More generally, theexperiment performed by Tversky and Shafir violates the total law of Kolmogorovian probability. If wedenote by µ ( P ) the total probability that a student decides to play again without knowing whether he/shehas won or lost in the first gamble, by µ ( W ) and µ ( L ) the probability that the student wins or loses,respectively, by µ ( P | W ) the conditional probability that the student decides to play again when he/sheknows he/she has won, and by µ ( P | L ) the conditional probability that the student decides to play againwhen he/she knows he/she has lost, then it is not possible to find any value of µ ( W ) and µ ( L ) = 1 − µ ( W )such that µ ( P | W ) = 0 .
69 and µ ( P | L ) = 0 . p ( P ) = 0 .
36 and the law of total probability µ ( P ) = µ ( W ) µ ( P | W ) + µ ( L ) µ ( P | L ) (2)is satisfied. This violation of the laws of Kolmogorovian probability is called the ‘disjunction effect’.7n equivalent formulation of the disjunction effect is known as the ‘Hawaii problem’, and it is againdue to Tversky and Shafir (1992). Consider the following situations.‘Disjunctive version’. Imagine that you have just taken a tough qualifying examination. It is the end ofthe fall quarter, you feel tired and run-down, and you are not sure that you passed the exam. In case youfailed you have to take the exam again in a couple of months after the Christmas holidays. You now havean opportunity to buy a very attractive 5-day Christmas vacation package to Hawaii at an exceptionallylow price. The special offer expires tomorrow, while the exam grade will not be available until the followingday. Would you: x buy the vacation package; y not buy the vacation package; z pay a $5 non-refundablefee in order to retain the rights to buy the vacation package at the same exceptional price the day aftertomorrow after you find out whether or not you passed the exam?‘Pass/fail version’. Imagine that you have just taken a tough qualifying examination. It is the endof the fall quarter, you feel tired and run-down, and you find out that you passed the exam (failed theexam. You will have to take it again in a couple of months after the Christmas holidays). You now havean opportunity to buy a very attractive 5-day Christmas vacation package to Hawaii at an exceptionallylow price. The special offer expires tomorrow. Would you: x buy the vacation package; y not buy thevacation package: z pay a $5 non-refundable fee in order to retain the rights to buy the vacation packageat the same exceptional price the day after tomorrow.In the Hawaii experiment, Tversky and Shafir experienced the same pattern of the two-stage gamblesituation. Indeed, more than half of the subjects chose option x (buy the vacation package) if they knewthe outcome of the exam (54% in the pass condition and 57% in the fail condition), whereas only 32% choseoption x (buy the vacation package) if they did not know the outcome of the exam. The Hawaii problemclearly shows a violation of the sure thing principle: subjects generally prefer option x (buy the vacationpackage) when they know that they passed the exam, and they also prefer x when they know that theyfailed the exam, but they refuse x (or prefer z ) when they don’t know whether they passed or failed theexam. Moreover, as in the two-stage gamble experiment, the Hawaii problem also violates the total law ofKomogorovian probability.A seemingly plausible explanation is that the origin of the violation of the sure thing principle in thedisjunction effect is ‘uncertainty aversion’, that is, people prefer to buy the vacation package in both caseswhere they have certainty about the outcome of the exam, while they refuse to buy the package when theydo not yet know whether they passed or failed the exam and hence lack this certainty. We will come backto this when studying ‘ambiguity aversion’ in Section 4.We now work out a quantum theoretical model for these two experiments, where the above mentioneddeviation is described in terms of genuine quantum effects.Let us firstly consider the Hawaii problem and denote by A the conceptual situation in which theparticipant has passed the exam, and by B the conceptual situation in which the participant has failed theexam. The disjunction of both conceptual situations, denoted by ‘ A or B ’, is the conceptual situation inwhich the participant ‘has passed or failed the exam’. The participant has to make a decision whether tobuy the vacation package – positive outcome, or not to buy it – negative outcome.We introduce the notion of state of a conceptual entity, as in Section 2. Thus, each conceptual situationabove is described by a defined state and represented by a unit vector of a complex Hilbert space. Moreexplicitly, we represent A by the unit vector | A i and B by the unit vector | B i in a complex Hilbert space.We assume that | A i and | B i are orthogonal, that is, h A | B i = 0, and represent the disjunction ‘ A or B ’by means of the normalized superposition state vector √ ( | A i + | B i ). The decision to be made is ‘to buythe vacation package’ or ‘not to buy the vacation package’. This decision is represented by an orthogonalprojection operator M of the Hilbert space H in our modeling scheme. The probability of the outcome‘yes’, i.e. ‘buy the package’, in the ‘pass’ situation, i.e. state vector | A i , is 0.54, and we denote it by µ ( A ) = 0 .
54. The probability of the outcome ‘yes’, i.e. buy the package, in the ‘fail’ situation, i.e. state8ector | B i , is 0.57, and we denote it by µ ( B ) = 0 .
57. The probability of the outcome ‘yes’, i.e. buythe package, in the ‘pass or fail’ situation, i.e. state vector √ ( | A i + | B i ), is 0.32, and we denote it by µ ( A or B ) = 0 . µ ( A ) = h A | M | A i (3) µ ( B ) = h B | M | B i (4) µ ( A or B ) = 12 ( h A | + h B | ) M ( | A i + | B i ) (5)By applying the linearity of the Hilbert space and the hermiticity of M , that is, h B | M | A i ∗ = h A | M | B i ,we then get µ ( A or B ) = 12 ( h A | M | A i + h A | M | B i + h B | M | A i + h B | M | B i )= µ ( A ) + µ ( B )2 + Re( h A | M | B i ) (6)where Re( h A | M | B i ) is the real part of the complex number h A | M | B i , i.e. the typical interference term ofquantum theory. Its presence allows to produce a deviation from the average value ( µ ( A ) + µ ( B )), whichwould be the outcome in the absence of interference. Note that, also in this disjunction effect situation,we have applied two key quantum features, namely, ‘superposition’, in taking √ ( | A i + | B i ) to represent‘ A or B ’, and ‘interference’, as the effect appearing in (6).Our quantum model can be realized in the three-dimensional complex Hilbert space C (Aerts, 2009;Sozzo, 2015), as follows. Let us distinguish two cases:(i) if µ ( A ) + µ ( B ) ≤
1, we put a ( A ) = 1 − µ ( A ) , b ( B ) = 1 − µ ( B ) and γ = π ;(ii) if µ ( A ) + µ ( B ) >
1, we put a ( A ) = µ ( A ), b ( B ) = µ ( B ) and γ = 0.Moreover, we choose | A i = ( p a ( A ) , , p − a ( A )) (7) | B i = ( e i ( β + γ ) (cid:16)q (1 − a ( A ))(1 − b ( B )) a ( A ) , q a ( A )+ b ( B ) − a ( A ) , − p − b ( B ) (cid:17) if a ( A ) = 0 e iβ (0 , ,
0) if a ( A ) = 0 (8) β = ( arccos (cid:16) µ ( A or B ) − µ ( A ) − µ ( B )2 √ (1 − a ( A ))(1 − b ( B )) (cid:17) if a ( A ) = 1 , b ( B ) = 1arbitrary if a ( A ) = 1 or b ( B ) = 1 (9)If µ ( A ) + µ ( B ) ≤
1, we take M to project orthogonally onto the subspace of C spanned by the vector(0 , , µ ( A ) + µ ( B ) >
1, we take M to project orthogonally onto the subspace of C spanned by thevectors (1 , ,
0) and (0 , , µ ( A ) , µ ( B ) and µ ( A or B ). In particular, the interference term in (6)is given by Re( h A | M | B i ) = p (1 − a ( A ))(1 − b ( B )) cos β (10)where β is the ‘interference angle for the disjunction’.Equations (6) and (10) can be used to represent the Hawaii problem situation. If we set µ ( A ) = 0 . µ ( B ) = 0 .
57 and µ ( A or B ) = 0 .
32, and observe that µ ( A ) + µ ( B ) = 1 . >
1, then we have a ( A ) = 0 . b ( B ) = 0 .
57 and γ = 0. After making the calculations of (7), (8) and (9), we obtain | A i = (0 . , , . | B i = e i . ◦ (0 . , . , − .
66) and we take M to project onto the subspace of C spanned by the vectors(1 , ,
0) and (0 , , µ ( A ) = 0 . µ ( B ) = 0 .
59 and µ ( A or B ) = 0 .
36, hence µ ( A ) + µ ( B ) = 1 . > a ( A ) = 0 . b ( B ) = 0 .
59 and γ = 0. Equations (6)and (10) can be solved for β = 141 . ◦ . In addition, (7), (8) and (9) can be solved for | A i = (0 . , , . | B i = e i . ◦ (0 . , . , − .
64) and M ( C ) is the subspace of C spanned by vectors (1 , ,
0) and (0 , , Researchers in probability theory distinguish between probabilities that are known, or knowable, e.g.,from past data, which they call ‘objective probabilities’, and probabilities that are not known nor can bededuced, calculated or estimated in an objective way. For this reason, Knight introduced the term ‘risk’to designate situations that can be described by objective probabilities, and ‘uncertainty’ to designatesituations that cannot be described by objective probabilities (Knight, 1921). The Bayesian paradigmmentioned in Section 1 minimizes this distinction by introducing the notion of ‘subjective probability’:even when the probabilities are not known, people may form their own ‘beliefs’, or ‘priors’, thus reducingproblems of decision under ambiguity to problems of decision under risk. Within the Bayesian paradigm,people then update their beliefs according to the Bayes rule of Kolmogorovian probability.The predominant model of choice under risk, i.e. in the presence of objective probabilities, is theEUT elaborated by von Neumann and Morgenstern (von Neumann & Morgenstern, 1944), which we call‘objective EUT’. Von Neumann and Morgenstern presented a set of axioms allowing to uniquely representhuman preferences over ‘lotteries’ by maximization of the expected utility functional. The reasonabilityof their axioms is so widely accepted that these axioms constitute the ‘normative counterpart of rationalbehavior’. However, the ‘Allais paradox’ revealed that simple situations exist where decision-makers violatesome axioms of objective EUT (Allais, 1953). In addition, the von Neumann-Morgenstern framework doesnot apply to problems where objective probabilities are not given. For this reason, Savage extended EUTto subjective probabilities within the Bayesian paradigm above: decision-makers behave as if they hadsubjective probabilities with respect to which they maximize expected utility (Savage, 1954).We illustrate in the following the main definitions and results of ‘subjective EUT’. While other elegantformulations of subjective EUT have been widely used in the literature, like the ‘Anscombe-Aumann’(Anscombe & Aumann, 1963), or the ‘Fishburn’ (Fishburn, 1970) formulation, we prefer presenting theSavage original formulation, as it will be naturally extended in Section 5. As in Section 2, we try to berigorous, without however dwelling on mathematical technicalities.Our basic mathematical framework requires a set S = { . . . , s, . . . } of states of nature. A Boolean σ -algebra of subsets of S is denoted by A ⊆ P ( S ) ( P ( S ) is the power set of S ), while the elements of We prefer using the the term ‘ambiguity’ when referring to situations involving unknown probabilities, as done in manytextbooks and papers on the topic. f $100 $0 $0 f $0 $0 $100 f $100 $100 $0 f $0 $100 $100 Table 1.
The payoff matrix for the Ellsberg three-color thought experiment. A denote events. A probability measure µ : A ⊆ P ( S ) −→ [0 ,
1] over A is such that, for every E ∈ A , µ ( E ) = Z E dµ ( s ) (11)Then, we denote by X the set of consequences. For our purposes, it is sufficient that the elements x ∈ X are monetary payoffs, so that x belongs to the real line ℜ . A decision-maker is assumed to have preferencesover acts. An act is a function f : s ∈ S f ( s ) ∈ X , and we denote by F the set of all acts. Then,one assumes a weak order (i.e. reflexive, symmetric and transitive) relation % on the Cartesian product F × F , and one introduces a utility function u : X −→ ℜ .Savage proved that, if one assumes that the preference relation % satisfies a number of axioms (includingthe sure thing principle), then, a probability measure µ : A ⊆ P ( S ) −→ [0 ,
1] and a function u : X −→ ℜ exist such that, for every f, g ∈ F , f % g iff Z S u ( f ( s )) dµ ( s ) ≥ Z S u ( g ( s )) dµ ( s ) (12)The integrals in (12) express the expected utility functionals W ( f ) and W ( g ) associated with the acts f and g , respectively. The probability measure µ is interpreted as a subjective probability measure andrepresents the decision-maker’s beliefs, while u is a utility function and represents the decision-maker’staste. In addition, µ is unique and u is uniquely defined up to a positive affine transformation (see,e.g., Etner, Jeleva & Tallon, 2012; Gilboa & Marinacci, 2013; Karni, 2014). As in the von Neumann-Morgenstern formulation, the concavity of u is a measure of the decision-maker’s risk aversion (for givenbeliefs). Deriving both probability and utility from observed choices, Savage was able to give both anormative and descriptive status to his subjective EUT. It is however important, at this stage, to stressthat the probability distribution µ is unique and satisfies the axioms of Kolmogorovian probability theory,in agreement with the Bayesian paradigm.Subjective EUT has been widely applied to economics and finance. However, in many economic prob-lems of interest it is not clear how one should define probabilities and, if the latter cannot be defined ina satisfactory way, how people form beliefs. In addition, in a seminal 1961 paper, Ellsberg predicted inhis ‘two-color example’ that people do not always choose by maximizing their subjective expected util-ity, but they generally prefer acts over events with known (or objective) probabilities to acts over eventswith unknown (or subjective) probabilities, a phenomenon called ‘ambiguity aversion’ (Ellsberg, 1961).Interestingly enough, the explanation proposed by Ellsberg for the observed pattern closely resembles the‘uncertainty aversion’ proposed by Tversky and Shafir for the disjunction effect (see Section 3).The thought experiment where ambiguity aversion manifestly clashes with subjective EUT is the ‘three-color example’.Consider one urn with 30 red balls and 60 balls that are either yellow or black in unknown proportion.One ball will be drawn at random from the urn. Then, free of charge, a person is asked to bet on one of theacts f , f , f and f defined in Table 1. Ellsberg suggested that, when asked to rank these acts, most ofthe persons will prefer f over f and f over f . On the other hand, acts f and f are ‘unambiguous’, as11hey are associated with events over known probabilities, while acts f and f are ‘ambiguous’, as they areassociated with events over unknown probabilities. The conclusion is that people prefer the unambiguousact over its ambiguous counterpart. There is a huge experimental evidence that decision-makers actuallyshow this ambiguity aversion (see, e.g., the extensive review by Machina and Siniscalchi (2014)). Only theexperiment by Slovic and Tversky (1974) indicated that ‘ambiguity attraction’ was at play, while morerecent experiments identify other behavioral mechanisms as primary (Binmore, Stewart & Voorhoeve,2012).Neither ambiguity aversion nor ambiguity attraction can be explained within subjective EUT, as theyviolate the sure thing principle, according to which, preferences should be independent of the commonoutcome. For example, in the three-color urn, preferences should not depend on whether the commonevent “a yellow ball is drawn” pays off $0 or $100. More technically, subjective EUT predicts ‘consistencyof decision-makers’ preferences’, that is, f is preferred to f if and only if f is preferred to f . A simplecalculation shows that this is impossible. Indeed, if we denote by p R , p Y and p B the subjective probabilitythat a red ball, a yellow ball, a black ball, respectively, is drawn (with p R = 1 / − ( p Y + p B )), then theexpected utilities W ( f i ), i = 1 , , ,
4, are such that W ( f ) > W ( f ) if and only if ( p R − p B )( u (100) − u (0)) > W ( f ) > W ( f ). Hence, no assignment of the subjective probabilities p R , p Y and p B reproduces a preference with W ( f ) > W ( f ) and W ( f ) > W ( f ).In the last forty years various extensions of subjective EUT have been elaborated, mainly in an ax-iomatic form, to cope with the Ellsberg paradox and ambiguity aversion. These proposals weaken someof the axioms of subjective EUT, e.g., the sure thing principle, taking a non-Bayesian direction. Withoutpretending to be exhaustive (extensive reviews can be found in, e.g., Gilboa & Marinacci (2013), Etner,Jelena & Tallon (2012) and Machina & Siniscalchi (2014)), we can roughly group these alternative decisionmodels as follows.(i) ‘Type-I models’ include ‘Choquet expected utility’ (Schmeidler, 1989) and ‘cumulative prospecttheory’ (Kahneman & Tversky, 1979; 1992). These models assume ‘non-additive capacities’, rather thanKolmogorovian probabilities, to represent beliefs. In particular, Choquet expected utility introduces rank-dependent axioms.(ii) ‘Type-II models’ include ‘max-min expected utility’ (Gilboa & Schmeidler, 1989) and ‘ α -max minexpected utility‘ (Ghirardato, Maccheroni & Marinacci, 2004). These models rest on the insight that, inthe absence of relevant information, asking for precise subjective beliefs is too demanding. Hence, thesemodels assume a set of probability distributions, or ‘multiple priors’, underlying actual decisions.(iii) ‘Type-III models’ include ‘variational preference’ (Maccheroni, Marinacci & Rustichini, 2006) and‘robust control’ (Hansen & Sargent, 2001). These models assume that the decision-maker has a benchmarkprobability distribution in mind, but he/she is not ‘completely confident’ about it.(iv) ‘Type-IV models’ include ‘smooth ambiguity preferences’ (Klibanoff, Marinacci & Mukerij, 2005).These models still assume that the decision-maker has a set of priors, but in a concrete decision, he/shealso comes up with a prior over this set of priors, or ‘second order belief’.The proposals above successfully cope with the Allais and Ellsberg paradoxes and, though some of themhave been criticized (Epstein, 1999), they have manifold applications in economic and financial modeling.More important, they depart from the assumption that only Kolmogorovian probabilities can representsubjective beliefs, but they, more or less explicitly, assume non-Kolmogorovian probability distributions.In 2009 Mark Machina proposed new thought experiments, which seriously challenge existing results onEUT, the ‘50:51 example’ and the ‘reflection example’ (Machina, 2009). Like Machina and other scholarshave proved, the decision models above are incompatible with the choices expected in these two examples(Machina, 2009; Baillon, l’Haridon & Placido, 2011). In particular, the reflection example questions anaxiom of Choquet expected utility, the so-called ‘tail separability’, exactly as the Ellsberg three-colorexample questions the sure thing principle of subjective EUT.12ct Red Yellow Black Green f $0 $50 $25 $25 f $0 $25 $50 $25 f $25 $50 $25 $0 f $25 $25 $50 $0 Table 2.
The payoff matrix for the Machina reflection example with lower tail shifts.Act Red Yellow Black Green f $50 $50 $25 $75 f $50 $25 $50 $75 f $75 $50 $25 $50 f $75 $25 $50 $50 Table 3.
The payoff matrix for the Machina reflection example with upper tail shifts.We present here two versions of the reflection example, as in l’Haridon & Placido (2010).
Reflection example with lower tail shifts . Consider one urn with 20 balls, 10 are either red or yellowin unknown proportion, 10 are either black or green in unknown proportion. One ball will be drawn atrandom from the urn. Then, free of charge, a person is asked to bet on one of the acts f , f , f and f defined in Table 2. Machina introduced the notion of ‘informational symmetry’, namely, the events “thedrawn ball is red or yellow” and “the drawn ball is black or green” have known and equal probability and,further, the ambiguity about the distribution of colors is similar in the two events. In an informationalsymmetry scenario, people should prefer act f over act f and act f over act f , or they should preferact f over act f and act f over act f . On the other hand, let us introduce the utilities u (0), u (25) and u (50), the subjective probabilities p R , p Y , p B and p G , and calculate the expected utilities W ( f i ) of the acts f i , i = 1 , , ,
4. Then, we have that preferences should be consistent according to subjective EUT, namely, W ( f ) > W ( f ) if and only if ( u (50) − u (25))( p Y − p B ) > W ( f ) > W ( f ). The interestingaspect of this example is that Choquet expected utility predicts similar consistency requirements on thebasis of tail separability.An experiment by L’Haridon and Placido (2010) confirms the Machina preference f % f and f % f ,consistently with informational symmetry. We will illustrate in detail the experiment in Section 5.2, wherewe will represent it within our general quantum theoretical modeling. Reflection example with upper tail shifts . Consider one urn with 20 balls, 10 are either red or yellowin unknown proportion, 10 are either black or green in unknown proportion. One ball will be drawn atrandom from the urn. Then, free of charge, a person is asked to bet on one of the acts f , f , f and f defined in Table 3. According to Machina’s informational symmetry, people should again prefer act f over act f and act f over act f , or they should prefer act f over act f and act f over act f . Onthe other hand, let us introduce the utilities u (25), u (50) and u (75), the subjective probabilities p R , p Y , p B and p G , and calculate the expected utilities W ( f i ) of the acts f i , i = 1 , , ,
4. Then, we have thatpreferences should again be consistent according to subjective EUT, namely, W ( f ) > W ( f ) if and onlyif ( u (50) − u (25))( p Y − p B ) > W ( f ) > W ( f ). One shows that tail separability of Choquetexpected utility leads to a similar prediction.The experiment by L’Haridon and Placido (2010) confirms again the Machina preference f % f and f % f , consistently with informational symmetry, and its results will be reviewed in Section 5.2.The theoretical and experimental arguments above strongly require a novel approach to ambiguity. Asmentioned in Section 1, a possible way out of these difficulties is assuming non-Kolmogorovian probability13istributions to represent subjective beliefs. Following our considerations in Section 3, we believe thatquantum probability distribution is a proper candidate to represent the uncertainty surrounding decision-making situations. Inspired by the cognitive approach in Section 3, we have recently worked out a quantum theoreticalframework to model the Ellsberg and Machina paradox situations (Aerts, Sozzo & Tapia, 2012; 2014). Wehave also successfully represented data collected on the three-color Ellsberg (Aerts, Sozzo & Tapia, 2014)and Machina reflection (L’Haridon & Placido, 2010) experiments within the quantum framework (Aerts &Sozzo, 2016). In this section we generalize these results by presenting a unified perspective to model events,states, subjective probabilities, acts, preferences and decisions within the quantum mechanical formalism.This can be considered as a first step toward the elaboration of ‘state dependent EUT’ where subjectiveprobabilities are represented by quantum probabilities, and attitudes toward ambiguity are incorporatedinto the states of the cognitive entities.We begin with the introduction of some basic notions and definitions that are needed to operationallydescribe the cognitive aspects of decision-making under uncertainty, following the approach in Section 3.(i) The cognitive situation that is the object of the decision identifies a DM entity in a defined state p v . We denote by Σ DM the set of all possible states of the DM entity. This state has a cognitive nature,hence it should be distinguished from a physical state or a state of nature (see Section 4). The cognitivestate mathematically captures aspects of ambiguity.(ii) There is a contextual interaction of a cognitive, not physical , nature between the decision-makerand the DM entity. This contextual interaction determines a change of the state of the DM entity. Theway in which this change occurs depends on subjective attitudes toward ambiguity (ambiguity aversion,ambiguity attraction, etc.).(iii) Events correspond to measurements that can be performed on the DM entity. For each state p v of the DM entity, each event E is associated with a probability µ v ( E ) that the event occurs when the DMentity is in the state p v .(iv) The DM entity, its states, events, probabilities and the decision-making process are modeled byusing the mathematical formalism of quantum theory in Section 2. In particular, the state of the DMentity identifies a single quantum probability distribution via the Born rule. We interpret this quantumprobability distribution, which is generally non-Kolmogorovian, as the subjective probability distributionassociated with the specific DM process.Let us start with the simple case where the set S of states of nature is finite and let { E , E , . . . , E n } be a set of mutually exclusive and exhaustive elementary events, which form a partition of S . We denoteby X the set of consequences, and suppose that X contains monetary outcomes, for the sake of simplicity.An act is defined as a function f : S −→ X , and we denote the set of acts by F . If the act f mapsthe elementary event E i into the outcome x i ∈ ℜ , then we can equivalently define f by the 2n-tuple( E , x ; . . . ; E n , x n ). We assume that a utility function u : X −→ ℜ exists over the set of consequenceswhich incorporates individual preferences toward risk.We refer to the mathematics introduced in Section 2. The DM entity is associated with a Hilbertspace H over the field C of complex numbers. Since n is the number of elementary events, the space H can be chosen to be isomorphic to the Hilbert space C n of n-ples of complex numbers. We thus denoteby {| α i , | α i , . . . , | α n i} the canonical orthonormal (ON) basis of C n , that is, | α i = (1 , , . . . | α n i = (0 , , . . . n ). The event E i is then represented by the orthogonal projection operator P i = | α i ih α i | , i ∈ { , . . . , n } . 14or every state p v ∈ Σ DM of the DM entity, represented by the unit vector | v i = P ni =1 h α i | v i| α i i ∈ C n ,the quantum probability distribution µ v : P ∈ L ( C n ) µ v ( P ) ∈ [0 ,
1] (13)( L ( C n ) is the lattice of all orthogonal projection operators over the complex Hilbert space C n ) inducedby the Born rule, associates the probability that the event E , represented by the orthogonal projectionoperator P , occurs when the DM entity is in the state p v . Thus, in particular, µ v ( E i ) = h v | P i | v i = |h α i | v i| (14)for every i ∈ { , . . . , n } .Suppose that, when the decision-maker is presented with a choice between the acts f and g , the DMentity is in the initial state p v . This state is interpreted as the state of the DM entity when no cognitivecontext is present. As the decision-maker starts pondering between f and g , this mental action can bedescribed as a cognitive context interacting with the DM entity and changing its state. The type of statechange directly depends on the decision-maker’s attitude toward ambiguity. More precisely, if the DMentity is in the initial state p v and the decision-maker is asked to choose between the acts f and g , a givenattitude toward ambiguity, say ambiguity aversion, will determine a given change of state of the DM entityto a state p v , leading the decision-maker to prefer, say f to g . But, a different attitude toward ambiguitywill determine a different change of state of the DM entity to a state p w , leading the decision-maker toprefer g to f . In this way, different attitudes toward ambiguity are formalized by different changes of stateinducing different subjective probabilities.The considerations above suggest associating the DM entity with a ‘family of subjective probabilitydistributions’ { µ v : L ( C n ) −→ [0 , | p v ∈ Σ DM } , represented by quantum probabilities and parametrizedby the state of the DM entity. In a concrete choice between two acts we will derive the exact state, hencethe specific subjective probability distribution that represents the actual choice.Let us now come to the representation of acts. The act f = ( E , x ; . . . ; E n , x n ) is represented by theHermitian operator ˆ F = n X i =1 u ( x i ) P i = n X i =1 u ( x i ) | α i ih α i | (15)For every p v ∈ Σ DM , we introduce the functional ‘expected utility in the state p v ’ W v : F −→ ℜ as follows.For every f ∈ F , W v ( f ) = h v | ˆ F | v i = h v | (cid:16) n X i =1 u ( x i ) P i (cid:17) | v i = n X i =1 u ( x i ) h v | P i | v i = n X i =1 u ( x i ) |h α i | v i| = n X i =1 u ( x i ) µ v ( E i ) (16)because of (14) and (15). Equation (16) generalizes the expected utility formula of subjective EUT. Aswe can see, expected utility explicitly depends on the state p v of the DM entity. This means that, for twoacts f and g , two states p v and p w may exist such that W v ( f ) > W v ( g ), but W w ( f ) < W w ( g ), dependingon subjective attitudes toward ambiguity. This suggests introducing a state-dependent preference relation % v on the set of acts F , as follows.For every f, g ∈ F , p v ∈ Σ DM , f % v g iff W v ( f ) ≥ W v ( g ) (17)It follows that the DM entity incorporates the presence of ambiguity, as the quantum probability dis-tribution representing subjective probabilities depends on the state of the DM entity. Furthermore, theway in which the state of the DM entity changes in the interaction with the decision-maker incorporatespeople’s attitude toward ambiguity, as it determines the state-dependent preference relation % v . The state-dependence enables the ‘inversion of preferences’ observed in the Ellsberg and Machina paradox situations,as will become evident from the next sections. 15 .1 Application to the Ellsberg three-color urn Let us now consider the Ellsberg three-color example and represent it by using the formalism in Section 5.The DM entity, which we call the ‘Ellsberg entity’, is a cognitive representation of the urn with 30 redballs and 60 yellow and black balls in unknown proportion. Let us point out explicitly that the ‘Ellsbergentity’ is ‘not’ the physical entity of the urn with 30 red balls and 60 yellow and black balls. We canalso easily understand that it cannot be, because for the physical entity the proportion of black balls andyellow balls ‘is’ always determined. Thus, if we specify in our description of the Ellsberg situation that thisproportion is unknown we bring in explicitly a cognitive element, which makes the Ellsberg entity, i.e. theentity experimented upon, a cognitive entity and no longer a physical entity. One can wonder whether thiscognitive aspect which render the Ellsberg entity from a physical entity into a cognitive entity is not justa subjective element which can be recuperated in the notion of subjective probability. It is not, at leastnot if subjective probabilities are meant to describe the subjectiveness pertained to the person performingthe Ellsberg test. Indeed, the specification of the proportion of black balls and yellow balls to be unknownis presented each time again to each different participant when the test is performed, and hence cannotbe attributed to the subjectivity of one individual test participant. It is objectively part of the beginningexperimental situation that each person is confronted with in an experiment, and this is the reason itpertains to the cognitive representation of the Ellsberg physical entity which is indeed the ‘object’ of thetest. The above reasoning clarifies why we introduce the notion of ‘state’ to describe it. Hence, the state p v of the Ellsberg entity exactly describes the cognitive situation of the urn with 30 red balls and 60 yellowand black balls in unknown proportion. However, many states of the Ellsberg entity are possible if we,for example, also specify something more than just ‘unknown proportion’ about yellow and black balls, inwhich case the cognitive situation is described by a different state. We represent each state p v by the unitvector | v i of the complex Hilbert space C over complex numbers. We denote by (1 , , , ,
0) and(0 , ,
1) the unit vectors of the canonical basis of C .Let us now consider the elementary, exhaustive and mutually exclusive events E R : “a red ball is drawn”, E Y : “a yellow ball is drawn”, and E B : “a black ball is drawn”. They define a ‘color measurement’ on theEllsberg entity. This color measurement has three outcomes, corresponding to the three colors red, yellowand black, and it is represented by a Hermitian operator with eigenvectors | R i = (1 , , | Y i = (0 , , | B i = (0 , ,
1) or, equivalently, by the spectral family { P i = | i ih i | | i = R, Y, B } . In other terms, theevent E i is represented by the orthogonal projection operator P i = | i ih i | , i = R, Y, B . In the canonicalbasis of C , a state p v of the Ellsberg entity is represented by the unit vector | v i = ρ R e iθ R | R i + ρ Y e iθ Y | Y i + ρ B e iθ B | B i = ( ρ R e iθ R , ρ Y e iθ Y , ρ B e iθ B ) (18)By using the Born rule of quantum probability, the probability µ v ( E i ) of drawing a ball of color i , i = R, Y, B , when the Ellsberg entity is in a state p v , is given by µ v ( E i ) = h v | P i | v i = |h i | v i| = ρ i (19)We have ρ R = 1 /
3, as the urn contains 30 red balls. Therefore, a state p v of the Ellsberg entity isrepresented by the unit vector | v i = ( 1 √ e iθ R , ρ Y e iθ Y , r − ρ Y e iθ B ) (20) As mentioned in Section 5, the choice of C depends on the fact that there are three mutually exclusive and exhaustiveevents in the three-color example – the generalization to the Ellsberg n-color example is straightforward. p RY and p RB of the Ellsberg entity. The states p RY and p RB arerepresented by the unit vectors | v RY i = ( 1 √ e iθ R , r e iθ Y ,
0) (21)and | v RB i = ( 1 √ e iθ R , , r e iθ B ) (22)and describe the cognitive situation “there are no black balls” and “there are no yellow balls”, respectively.As the Ellsberg entity is a cognitive entity, ‘cognitive contexts’ have an influence on its state, and ingeneral will make a specific state change into another state. This is what happens in the cognitive realmin analogy with the physical realm, where a physical context will in general change the physical state of aphysical entity. Hence, whenever a decision-maker is asked to ponder between the choice of f and f , thepondering itself, before a choice is made, is a cognitive context, and hence it changes in general the state ofthe Ellsberg entity. Similarly, whenever a decision-maker is asked to ponder between the choice of f and f , also this introduces a cognitive context, before the choice is made – and a context which in general isdifferent from the one introduced by pondering about the choice between f and f – which will in generalchange the state of the Ellsberg entity.Let us now introduce a state p describing the situation where no cognitive context is present. Thisis the initial state of the Ellsberg entity, and symmetry reasons suggest to represent it by the unit vector | v i = √ (1 , , f and f will make the state p of theEllsberg entity change to a state p w that is generally different from the state p w in which the Ellsbergentity changes from p when pondering about a choice between f and f . In particular, a ‘highly ambiguityaverse’ decision-maker will be, as a consequence of her/his pondering in the choice between f and f ,confronted with the Ellsberg entity which changes from the initial state p to a state p w that is very closeto the state p RY represented in (21). Analogously, a ‘highly ambiguity averse’ decision-maker will be, asa consequence of her/his pondering in the choice between f and f , confronted with the Ellsberg entitywhich changes from the state p to a state p w that is very close to the state p RB represented in (22).Let us then come to the representation of the acts f , f , f and f in Table 1, Section 4. For a givenutility function u , to be estimated from empirical data, we respectively associate f , f , f and f with theHermitian operators ˆ F = u (100) P R + u (0) P Y + u (0) P B (23)ˆ F = u (0) P R + u (0) P Y + u (100) P B (24)ˆ F = u (100) P R + u (100) P Y + u (0) P B (25)ˆ F = u (0) P R + u (100) P Y + u (100) P B (26)The corresponding expected utilities in a state p v of the Ellsberg entity are W v ( f ) = h v | ˆ F | v i = 13 u (100) + 23 u (0) (27) W v ( f ) = h v | ˆ F | v i = ( 13 + ρ Y ) u (0) + ( 23 − ρ Y ) u (100) (28) W v ( f ) = h v | ˆ F | v i = ( 13 + ρ Y ) u (100) + ( 23 − ρ Y ) u (0) (29) W v ( f ) = h v | ˆ F | v i = 13 u (0) + 23 u (100) (30)We observe that W v ( f ) and W v ( f ) do not depend on the state p v , hence they are ambiguity-free, i.e.independent of the state, while W v ( f ) and W v ( f ) do depend on p v . This means that it is possible to find17 state p w , e.g., the state represented in (21), such that W w ( f ) > W w ( f ), and a state p w , e.g., thestate represented in (22), such that W w ( f ) > W w ( f ). These two states reproduce Ellsberg preferences,in agreement with an ambiguity aversion attitude.We repeated the Ellsberg three-color experiment, asking 57 persons, chosen among our colleagues andfriends, to rank the four acts in Table 1, Section 4 (Aerts, Sozzo & Tapia, 2014). We found that 34participants preferred acts f and f , 12 participants preferred acts f and f , 7 participants preferred acts f and f , and 6 participants preferred acts f and f . This makes the weights with preference of acts f over act f to be 0.68 against 0.32, and the weights with preference of act f over act f to be 0.69against 0.31. Hence, 46 participants over 57 chose f and f or the inversion f and f , for an ‘inversionpercentage’ of 78%, thus confirming the typical behavior observed in the Ellsberg three-color example.A quantum model for these data can be constructed by finding two orthogonal states p w and p w ,represented by the unit vectors | w i and | w i , respectively, such that h w | ˆ F − ˆ F | w i = 0 .
68 (31) h w | ˆ F − ˆ F | w i = 0 .
69 (32)where ˆ F i , i = 1 , , ,
4, are defined in (23)–(26). In the canonical basis of C , the solution is | w i = ( 1 √ , . e i ◦ , . e i . ◦ ) (33) | w i = ( 1 √ , . e i ◦ , . e i . ◦ ) (34)as we have proved in Aerts & Sozzo (2016). The states p w and p w represented in (33) and (34) identify thesubjective probability distributions µ w and µ w , respectively, reproducing the ambiguity aversion patternof the experiment, for an utility value u (100) − u (0) ≈ .
4. But, generally speaking, preferences do dependon the state p v of the Ellsberg entity. This completes the construction of a quantum model for the Ellsbergthree-color example, which follows the prescriptions in Section 5. In this section we represent the ‘Machina reflection example’ within the formalism. We start from thereflection example with lower tail shifts. The DM entity, which we call the ‘Machina entity’, is the urnwith 10 red or yellow balls and 10 black or green balls, in both cases in unknown proportion. A possiblestate p v of the Machina entity has a cognitive nature and is represented by the unit vector | v i of the complexHilbert space C over complex numbers. We denote by (1 , , , , , , , , ,
0) and (0 , , ,
1) theunit vectors of the canonical basis of C .Let us consider the elementary, exhaustive and mutually exclusive events E R : “a red ball is drawn”, E Y :“a yellow ball is drawn”, E B : “a black ball is drawn”, and E G : “a green ball is drawn”. They define a ‘colormeasurement’ that can be performed on the Machina entity. This color measurement has four outcomescorresponding to the four colors red, yellow, black and green, and it is represented by a Hermitian operatorwith eigenvectors | R i = (1 , , , | Y i = (0 , , , | B i = (0 , , ,
0) and | G i = (0 , , ,
1) or, equivalently,by the spectral family { P i = | i ih i | | i = R, Y, B, G } . Hence, the event E i is represented by the orthogonalprojection operator P i , i = R, Y, B, G . In the canonical basis of C , a state p v of the Machina entity isrepresented by the unit vector | v i = ρ R e iθ R | R i + ρ Y e iθ Y | Y i + ρ B e iθ B | B i + ρ G e iθ G | G i = ( ρ R e iθ R , ρ Y e iθ Y , ρ B e iθ B , ρ G e iθ G ) (35)By using quantum probabilistic rules, the probability µ v ( E i ) of drawing a ball of color i , i = R, Y, B, G ,when the Machina entity is in a state p v is given by µ v ( E i ) = h v | P i | v i = |h i | v i| = ρ i (36)18he reflection with lower tail shifts situation requires that ρ R + ρ Y = 1 / ρ B + ρ G . Therefore, a state p v of the Machina entity is represented by the unit vector | v i = ( ρ R e iθ R , r − ρ R e iθ Y , ρ B e iθ B , r − ρ B e iθ G ) (37)As in the Ellsberg case, let us suppose that the initial state p of the Machina entity completely reflectsthe symmetry between the different colors. Thus, p is represented by the unit vector | v i = (1 , , , p to a generally different state p w , represented by the unit vector | w i , as in (37). In this framework, thepondering about a choice between f and f will make the initial state p of the Machina entity changeto a state p w that is generally different from the state p w in which the Machina entity changes from theinitial state p in a pondering about the choice between f and f .Let us now come to the representation of the acts f , f , f and f in Table 2, Section 4. For a givenutility function u , to be estimated from empirical data, we respectively associate f , f , f and f with theHermitian operators ˆ F = u (0) P R + u (50) P Y + u (25) P B + u (25) P G (38)ˆ F = u (0) P R + u (25) P Y + u (50) P B + u (25) P G (39)ˆ F = u (25) P R + u (50) P Y + u (25) P B + u (0) P G (40)ˆ F = u (25) P R + u (25) P Y + u (50) P B + u ((0) P G (41)The corresponding expected utilities in a state p v are W v ( f ) = h v | ˆ F | v i = u (0) ρ R + u (50) ρ Y + 12 u (25) (42) W v ( f ) = h v | ˆ F | v i = u (0) ρ R + u (25) ρ Y + u (50) ρ B + u (25) ρ G (43) W v ( f ) = h v | ˆ F | v i = u (25) ρ R + u (50) ρ Y + u (25) ρ B + u (0) ρ G (44) W v ( f ) = h v | ˆ F | v i = 12 u (25) + u (50) ρ B + u (0) ρ G (45)All expected utilities depend on the state p v , thus it is possible to find a state p w such that W w ( f ) >W w ( f ) ( W w ( f ) > W w ( f )), and a state p w such that W w ( f ) > W w ( f ) ( W w ( f ) > W w ( f )).Indeed, let us consider the state p Y G describing the cognitive situation where there are no red balls andno black balls. This state is represented by the unit vector | v Y G i = (0 , r e iθ Y , , r e iθ G ) (46)Then, let us consider the state p RB describing the cognitive situation where there are no yellow balls andno green balls. This state is represented by the unit vector | v RB i = ( r e iθ R , , r e iθ B ,
0) (47)By using (42), (43), (44) and (45), we have W Y G ( f ) = W RB ( f ) and W Y G ( f ) = W RB ( f ). Therefore, thestates p Y G and p RB perfectly reproduce informational symmetry in the Machina reflection example withlower tail shifts. 19et us now apply the quantum model for the Machina paradox situation to represent the data collectedin L’Haridon & Placido (2010) on the reflection example with lower tail shifts. These authors asked94 students to rank the four acts in Table 2, Section 4. The students’ response was that 11 studentspreferred acts f and f , 44 students preferred acts f and f , 15 students preferred acts f and f , and 24students preferred acts f and f . This entails that 68 students over 94 reversed their preferences, for aninversion percentage of 72%, thus violating subjective EUT and in agreement with Machina’s expectations(L’Haridon & Placido, 2010). Equivalently, a rate of 0.59 preferred act f over act f , and a rate of 0.63preferred act f over f . As we have seen in Section 4, this result is problematical also from the point ofview of Choquet expected utility.A quantum mechanical model for the experimental data above can be constructed by finding twoorthogonal states p w and p w , represented by the unit vectors | w i and | w i , respectively, such that h w | ˆ F − ˆ F | w i = 0 .
59 (48) h w | ˆ F − ˆ F | w i = 0 .
63 (49)where ˆ F i , i = 1 , , ,
4, are defined in (38)–(41). In the canonical basis of C , the solution is | w i = (0 , . e i . ◦ , . e i ◦ , . e i . ◦ ) (50) | w i = (0 . e i . ◦ , . e i . ◦ , . e i . ◦ , . e i . ◦ ) (51)as we have proved in Aerts & Sozzo (2016). The states p w and p w represented in (50) and (51) identifythe subjective probability distributions µ w and µ w , respectively, reproducing the experimental pattern,for an utility value u (50) − u (25) ≈ . p v of the Machina entity. This completes the construction of a quantum model for the Machina reflectionexample with lower tail shifts, which follows the prescriptions in Section 5.L’Haridon and Placido (2010) also tested the reflection example with upper tail shifts. The authorsasked the same 94 students to rank the four acts in Table 3, Section 4. The students’ response was that8 students preferred acts f and f , 47 students preferred acts f and f , 6 students preferred acts f and f , and 33 students preferred acts f and f . This entails that 47 + 33 = 80 students over 94 reversedtheir preferences, for a ratio of 0.85, thus violating subjective EUT and in agreement with Machina’sexpectations. Equivalently, a rate of 0.59 preferred act f over act f , and a rate of 0.56 preferred act f over f .A quantum model for these experimental data can be constructed by following along the lines above.We have proved in Aerts & Sozzo (2016) that the states p w and p w reproducing the data collected in theMachina reflection example with upper tail shifts are represented by the unit vectors | w i = (0 . e i . ◦ , . e i . ◦ , . e i . ◦ , . e i . ◦ ) (52) | w i = (0 . e i . ◦ , , . e i . ◦ , . e i . ◦ ) (53)for an utility value u (50) − u (25) = 1 . .3 Possible extensions and a definition of ambiguity We conclude the presentation of the quantum theoretical framework with some technical remarks.The quantum theoretical model for human preferences can be extended to the case in which the set ofstates of nature S has a continuous cardinality, as follows.We introduce, in addition to S , the set of consequences X , the set of acts F = { f : S −→ X } , theutility function u : X −→ ℜ . Further, let H be the Hilbert space representing states of the DM entity, andlet {| α i} be an orthonormal basis of H , so that h α | α ′ i = δ ( α − α ′ ), where δ ( · ) is the δ -Dirac distribution.For every state p v ∈ Σ DM describing the DM entity, the represented vector | v i can be written as | v i = Z ℜ h α | v i| α i dα = Z ℜ c ( α ) | α i dα (54)An event E is represented by the orthogonal projection operator P E = R E | α ih α | dα . Hence, the subjectiveprobability that the event E occurs when the DM entity is in the state p v is µ v ( E ) = h v | (cid:16) Z E | α ih α | dα (cid:17) | v i = Z E |h α | v i| dα = Z ℜ d || P α | v i|| (55)where the integral is intended in the Lebesgue sense.The act f is instead represented by the Hermitian operatorˆ F = Z ℜ u ( f ( α )) | α ih α | dα (56)Hence, the expected utility of the act f in the state p v is W v ( f ) = h v | ˆ F | v i = Z ℜ u ( f ( α )) | c ( α ) | dα = Z ℜ u ( f ( α )) d || P α | v i|| (57)The right side of (57) is also useful when one does not specifies the cardinality of the set of states. It isindeed sufficient to require that the integral is intended in the Lebesgue-Stieltjes sense.The treatment of the Ellsberg and Machina paradox situations enables providing a general definitionof ambiguity within the quantum theoretical framework, as follows.We say that ‘an event E is unambiguous’ if the subjective probability µ v ( E ) does not depend on thestate p v of the DM entity. In the Ellsberg three-color example, the event E R : “a red ball will be drawn” isindeed unambiguous in the sense specified here. Indeed, for every state p v of the Ellsberg entity, representedby the unit vector | v i ∈ C , µ v ( E R ) = .Finally, we say that ‘an act f is unambiguous’ if the expected utility W v ( f ) does not depend on thestate p v of the DM entity. Again in the Ellsberg three-color example, the act f is indeed unambiguousin the sense specified above, since, for every state p v of the Ellsberg entity, represented by the unit vector | v i ∈ C , W v ( f ) = ( u (0) + u (100)). We have worked out in this paper a theoretical framework to represent preferences and decisions underuncertainty. This framework uses the mathematical formalism of quantum theory. In it, subjective proba-bilities are represented by families of quantum probability distributions, parametrized by the state of theDM entity under investigation. The interaction with the overall cognitive landscape, which includes thedecision-maker’s pondering about two acts, provokes a change of state of the DM entity, which enablesmodeling of the ambiguity aversion affecting Ellsberg- and Machina-like preferences. However, the present21heoretical framework is flexible enough to reproduce different experimental patterns arising from differentattitudes toward ambiguity. We have also stressed that the present approach allows modeling of beliefs,but in a generally non-Bayesian setting where a kind of ‘contextual risk’ is present. Finally, the presentapproach suggests the development of a quantum-based subjective EUT with state-dependent preferencesbetween acts.In our opinion, this quantum theoretical framework can be successfully used to represent ambiguity-laden situations outside pure decision theory, in particular in strategic management decisions and somelong standing economic puzzles. We conclude this paper with some hints in this direction.Experiments have revealed a mixture of ambiguity aversion and ambiguity attraction in decisions underuncertainty. In this respect, managers typically compare the performance of an investment with a bench-mark, or targeted performance, e.g., the return on investment (ROI), or the internal rate of return (IRR).A ‘gain’ is realized when the performance is above the benchmark, a ‘loss’ is realized when the performanceis below the benchmark. For example, risk occurs in a situation where the probability that the ROI ofa given investment is above the benchmark is x per cent. Ambiguity occurs instead in a situation wherethe probability that the ROI of the investment is above the benchmark is between ( x − ∆) and ( x + ∆).Viscusi and Chesson (1999) identified a ‘fear effect’, as well as a ‘hope effect’, in their experimental study.More precisely, they found that, as the probability of a loss increases, managers become less ambiguityaverse, reaching a ‘crossover point’ at which they become ambiguity seeking, which indicates a shift froma fear to a hope effect. Viceversa, as the probability of a gain increases, managers become less ambiguityseeking, reaching a ‘crossover point’ at which they become ambiguity averse, which indicates this timea shift from a hope to a fear effect. This result was confirmed by Ho, Keller and Keltyka (2002). Thequantum theoretical framework can reproduce this dual behavior by reconstructing the quantum states,hence the subjective probability distributions underlying the observed behavior, as we have done in thethree color Ellsberg urn and the Machina reflection example (Aerts & Sozzo, 2016).Coming to economics and finance, Epstein and Miao (2003) have put forward that ambiguity aversionmay explain the ‘home bias puzzle’ in international finance: people prefer to trade stocks of their owncountry rather than foreign stocks. This is in principle compatible with the quantum theoretical approach,where ambiguity aversion can be explained in terms of the overall cognitive landscape surrounding thedecision-making situation. Further, Hansen, Sargent and Wang (2002) have shown that ambiguity aversionmodels lead to market prices that are closer to the empirical prices than the prices predicted by EUT, whichmay explain the ‘equity premium puzzle’. Again, this behavior can in principle be reproduced within atheoretical framework where ambiguity aversion depends on the way the cognitive landscape influences theDM entity.We believe that the quantum theoretical approach presented here can be naturally applied to thefascinating problems above, and we plan to dedicate future research to this. References
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