Quantum Markov Model for Data from Shafir

Open Systems & Information Dynamics Vol. 16, No. 4 (2009) 371–385 c World Scientific Publishing Company °

Quantum Markov Model for Data from Shafir-Tversky Experiments in Cognitive Psychology Luigi Accardi Mathematical Center Vito Volterra, Department of Mathematics University of Rome II, Italy e-mail: [email protected]

Andrei Khrennikov International Center for Mathematical Modeling in Physics and Cognitive Sciences University of V¨ axj¨ o, S-35195, Sweden e-mail: [email protected]

Masanori Ohya Department of Information Science Tokyo University of Science Noda-city, Chiba, 278-8510, Japan e-mail: [email protected]

(Received: April 5, 2009) Abstract. We analyze, from the point of view of quantum probability, statistical data from two interesting experiments, done by Shafir and Tversky [1, 2] in the domain of cognitive psychology. These are gambling experiments of Prisoner Dilemma type. They have important consequences for economics, especially for the justification of the Savage “Sure Thing Principle” [3] (implying that agents of the market behave rationally). Data from these experiments were astonishing, both from the viewpoint of cognitive psychology and economics and probability theory. Players behaved irrationally. Moreover, all attempts to generate these data by using classical Markov model were unsuccessful. In this note we show (by inventing a simple statistical test — generalized detailed balance condition) that these data are non-Kolmogorovian. We also show that it is neither quantum (i.e., it cannot be described by Dirac-von Neumann model). We proceed towards a quantum Markov model for these data.

1.

Introduction

The development of probability theory can be compared with development of geometry [5, 6]: from a single model, implicitly supposed to be the only possi-

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ble one, to the emergence of a variety of different models and the understanding of the impossibility to describe everything by just one (even very successful) model. In geometry the very slow transition was from the Euclidean model towards the non-Euclidean models (invented by Lobachevsky, Gauss and others). In probability theory it was from the Kolmogorov measuretheoretic modela to a special non-Kolmogorovian model — the quantum probabilistic model based on Dirac–von Neumann formalism.b We mention here recent publication [7] devoted to fundamental questions of applications of non-Kolmogorovian models, in particular, quantum-like models in economy, cognitive science, and genetics; see also works [8 – 16] on contextual approach to probability, classical as well as quantum, with applications outside of QM. In general, there are no reasons to believe that quantum probabilistic model is the only possible deviation from the Kolmogorov model. One may try to find non-Kolmogorovian probabilistic data even outside quantum physics, in other words, there may exist nonclassical probabilistic data which need not be quantum. A few years ago, a group of neurophysiologists and psychologists at the University of Bari, Italy, performed an experiment designed by one of the authors of this paper, see [10 – 12], and obtained non-Kolmogorovian probabilistic data, see Conte et al. [16 – 18]. Although this specially designed experiment confirmed (at least preliminarly) that non-Kolmogorovian data can appear not only in quantum physics, it would be extremely interesting to find such data in some well-known experiments, e.g., in psychology — experiments which were known as inducing paradoxical consequences. In the domain of cognitive psychology, it was well known that data collected in gambling experiments of the Prisoner Dilemma (PD) type [19], see Shafir and Tversky [1, 2] and also Croson [20], cannot be generated by a natural (classical) Markov model, see Busemeyer et al. [21]. In the latter papers an attempt was taken to use quantum probabilistic description. In this paper, we present the rigorous proofs that: a) the data from the Shafir and Tversky [1, 2] experiments cannot be embedded in a Kolmogorov modelc ; b) these data are neither quantum — they cannot be described by probabilities given by Dirac–von Neumann’s formalism. a The model, since the early 30’s, has been the dominating one in many domains of science: classical statistical physics, economy, biology, psychology. b We remark that, in geometry, non-Euclidean models were invented for purely mathematical reasons, but in probability theory the first non-Kolmogorov model, the quantum one, was invented by demand of physics and it emerged even before the first publication defining the Kolmogorov model. c In particular, this explains the reasons of the failure of the attempts to generate this data by using Markov models.

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Finally, the authors proceed towards the description of data from Shafir and Tversky [1, 2] experiments by a quantum Markov model [22, 23, 24]. 2. 2.1.

Statistical Tests of Rational Behaviour

The Prisoner’s Dilemma

In game theory, PD is a type of non-zero-sum game in which anyone of two players can cooperate with or defect (i.e. betray) the other one. In this game, as in all game theory, the only concern of each individual player (prisoner) is maximizing his/her own payoff, without any concern for the other player’s payoff. In the classical form of this game, cooperating is strictly dominated by defecting, so that the only possible equilibrium for the game is for all players to defect. In simpler terms, no matter what the other player does, one player will always gain a greater payoff by playing defect. Since in any situation playing defect is more beneficial than cooperating, all rational players will play defect. The classical PD is as follows: Two suspects, A and B, are arrested by the police which, having insufficient evidence for a conviction and having separated both prisoners, visits each of them to offer the same deal: if one testifies for the prosecution against the other and the other remains silent, the betrayer goes free and the silent accomplice receives the full 10-year sentence. If both stay silent, both prisoners are sentenced to only six months in jail for a minor charge. If each betrays the other, each receives a two-year sentence. Each prisoner must make the choice of whether to betray the other or to remain silent. However, neither prisoner knows for sure what choice the other prisoner will make. How should the prisoners behave? The dilemma arises when one assumes that both prisoners only care about minimizing their own jail terms. Each prisoner has two options: to cooperate with his accomplice and stay quiet, or to defect from their implied pact and betray his accomplice in return for a lighter sentence. The outcome of each choice depends on the choice of the accomplice, but each prisoner must choose without knowing what his accomplice has chosen. In deciding what to do in strategic situations, it is normally important to predict what others will do. This is not the case here. If you knew the other prisoner would stay silent, your best move would be to betray as you then walk free instead of receiving the minor sentence. If you knew the other prisoner would betray, your best move would still be to betray, as you receive a lesser sentence than by silence. Betraying is the dominant strategy. The other prisoner reasons similarly, and therefore also chooses to betray. Yet by both defecting they get a lower payoff than they would get by staying silent. So rational, selfinterested play results in each prisoner being worse off than if they had stayed silent, see e.g. Wikipedia — “Prisoner’s dilemma.”

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This is the principle of rational behaviour which is basic for the rational choice theory — the dominant theoretical paradigm in microeconomics. It is also central to modern political science and is used by scholars in other disciplines such as sociology. However, Shafir and Tversky [1] found that players frequently behave irrationally. 2.2.

Gambling experiment

Shafir and Tversky also performed a simpler version of PD’s gamble experiment, see [2]. They proposed to test rational behaviour for the following gambling experiment. In this experiment, one is presented with two possible plays of a gamble that is equally likely to win 200 USD or lose 100 USD. You are instructed that the first play has been completed, and now one is faced with the possibility of another play. Here a gambling device, e.g., roulette, plays the role of one of the players, say B; A is a real player, his actions are A = +, to play the second game, A = −, not. 3.

Experimental Evidence of Nonclassical Statistics in Cognitive Psychology

3.1.

General description of experiments

The general context of these experiments is that of game theory. There are two players: Alice (A) and Bob (B). A can choose among nA possible behaviours: a1 , . . . , anA B can choose among nB possible behaviours: b1 , . . . , bnB Events [A = aα ] ≡ A chooses the behaviour aα [B = bβ ] ≡ B chooses the behaviour bβ C ≡ A knows nothing about the choice of B, B knows nothing about the choice of A and both A and B have this information. The experimental data are the conditional probabilities: • P (A = aα | B = bβ ) — probability that A chooses the behaviour aα knowing that B has chosen the behaviour bβ , • P (B = bβ | A = aα ) — probability that B chooses the behaviour bβ knowing that A has chosen the behaviour aα ,

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• P (A = aα | C) — probability that A chooses the behaviour aα given that A knows nothing about the choice of B and conversely. Analogue interpretation can be used for P (B = bβ |C). 3.2.

General Statement of the Problem

Given the experimental data P (B|A) P (A|B) PA PB

≡ ≡ ≡ ≡

(P (B = bβ |A = aα )) (P (A = aα |B = bβ )) (P (A = aα |C)) (P (B = bβ |C))

(1) (2) (3) (4)

decide if they admit a Kolmogorov model. The manifold of K-data (i.e., data which can be described by a Kolmogorov model) is determined by the full statistical invariant, see [25] for details. However, it is difficult to compute it (just like in geometry!). The full characterization of the manifold of Kolmogorovian probabilistic data is a hard problem. Sometimes it is more practical to find some good (i.e., easily applicable and not too narrow) necessary conditions: if they are not satisfied we are sure the K-model does not exist. In this case we are sure that it is meaningless to try to describe such sort of data by classical Kolmogorov model. For example, it is meaningless to try to simulate such data by using classical Markov model. The 2-slit inequality, see [25], and the Bell inequality are of this form as well as interference conditions considered in [10 – 12]. 3.3.

Simple necessary conditions I

Necessary conditions for the Kolmogorovianity of the given statistical data (also sufficient for two observables) are that the marginal probabilities (3), (4) and the transition probabilities (1), (2) satisfy the following compatibility condition (which essentially amounts to Bayes’s theorem): X P (B = bβ |C)P (A = aα |B = bβ ) = P (A = aα |C) , (5) β

X

P (A = aα |C)P (B = bβ |A = aα ) = P (B = bβ |C) .

(6)

α

Replacing the value of P (A = aα | C) in (6) by the left-hand side of (5), one finds the necessary condition: X P (B = bβ |C)P (A = aα |B = bβ )P (B = bβ |A = aα ) = P (B = bβ |C) β,α

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and the one obtained by exchanging the roles of (A, B), (α, β). These are equivalent to the fixed point equations, PB P (A|B)P (B|A) = PB , and the equation obtained exchanging the roles of (A, B), i.e., PA P (B|A)P (A|B) = PA . 3.4.

Simple necessary conditions II

Necessary conditions, even simpler than those discussed in Sect. 3.3, are deduced from the following remark: P (Aα ∩ Bβ ) := P (Bβ |C)P (Aα |Bβ ) , P (Aα ∩ Bβ ) = P (Aα |C)P (Bβ |Aα ) , which implies the so-called generalized detailed balance condition, P (Bβ |C)P (Aα |Bβ ) = P (Aα |C)P (Bβ |Aα ) which is a necessary condition for the existence of the K-model even if the fixed point conditions (5) or (6) are satisfied. The usual detailed balance condition is recovered for A = B. It is known that, in classical statistical mechanics, this condition characterizes Gibbs equilibrium states. 3.5.

Tversky and Shafir gambling experiment

We start with the simpler experiment discussed in Sect. 2.2, see Tversky and Shafir [2], then we will consider the PD experiment (discussed in Sect. 2.1, see Shafir and Tversky [1]). In this experiment, B is a computer and its choices are determined by a random generator with probability 1/2, i.e. P (B1 |C) = P (B2 |C) =

1 . 2

A and C are human and • the reaction of A to the choice of B, i.e. PA|B , • the reaction of A to the choice of C, i.e. PA|C , are found in experiments. They are given by: µ ¶ µ ¶ P (A1 |B1 ) P (A1 |B2 ) 0.16 0.03 PA|B := = P (A2 |B1 ) P (A2 |B2 ) 0.84 0.97 µ ¶ 0.63 PA = P (A = · | C) = . 0.37

(7)

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Notice that, in this experiment, the knowledge of PB|A is not required. A necessary condition for the existence of a Kolmogorov model is the validity of the Bayes rule, P (A1 |B1 )P (B1 |C) + P (A1 |B2 )P (B2 |C) = P (A1 |C) , which, given (7), becomes 1 [P (A1 |B1 ) + P (A1 |B2 )] = P (A1 |C) . 2 However, in our case 1 [P (A1 |B1 ) + P (A1 |B2 )] = (0.5)[0.16 + 0.03] = (0.5)(0.19) 2 = 0.95 6= 0.63 = P (A1 |C) . 3.6.

Shafir and Tversky PD experiment

In the experiment nA = nB = 2 and µ ¶ µ ¶ P (B1 |A1 ) P (B1 |A2 ) 0.16 0.03 PB|A := = . P (A2 |A1 ) P (B2 |A2 ) 0.84 0.97 We postulate that A and B have similar psychology: ¶ µ 0.63 PA = PB = P ( · | C) = 0.37 µ ¶ µ ¶ P (A1 |B1 ) P (A1 |B2 ) P (B1 |A1 ) P (B1 |A2 ) PA|B := = = PB|A . P (A2 |B1 ) P (A2 |B2 ) P (B2 |A1 ) P (B2 |A2 ) We have seen that a necessary condition for the existence of the K-model is the generalized detailed balance condition, P (B1 |C)P (A2 |B1 ) = P (A2 |C)P (B1 |A2 ) . However, (0.63) · (0.84) = 0.5292 6= (0.37) · (0.003) = 0.00111 . 3.7.

Alternative system of equations based on quantum Markov techniques

As we have already seen, data from experiments performed in [1, 2] cannot be embedded in a Kolmogorov probability space. The next natural question is whether it can be embedded in Dirac–von Neumann’s (pure) quantum model. One can easily see that neither this is the case because the matrices of the

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transition probabilities are not doubly stochastic. Thus one should look for some generalisation of Dirac–von Neumann pure model. Notice that Dirac– von Neumann nonpure model, i.e. based on non-rank-one density matrices, is applicable only to compatible observables, which would bring us back to the Kolmogorov case. Another interesting possibility for the quantum model is provided by the quantum Markov model, see [22, 23, 24] which goes beyond the Dirac–von Neumann model because von Neumann introduced the notion of density matrix, which is the quantum analogue of a probability density, but not the quantum analogue of the conditional probability density. This was introduced in [23] and is a more subtle notion because it is an operator generalisation of the basic intuition that quantum theory deals with amplitudes rather than directly with probabilities. Since, intuitively, amplitudes are square roots of probabilities, given a classical Markov transition matrix (pij ) with state space {1, . . . , d} (d ∈ N) a natural way to associate to it a conditional density amplitude is to define X√ K = pij eii ⊗ ejj ∈ Md ⊗ Md , i,j

where Md denotes the d × d complex matrices, (ei ) an orthonormal basis of √ Cd , (eij ) the associated system of matrix units (eij ehk = δj,h eik ), and pij are arbitrary (usually complex) square roots of pij . Any conditional density amplitude determines a linear lifting (see [24]) E ∗ : S(Md ) → S(Md ⊗ Md ) from density matrices on Md to density matrices on Md ⊗ Md or, dually, a transition expectation, i.e., a completely positive, identity preserving linear map E : Md ⊗ Md → Md defined by

E(x) = Tr2 (K ∗ xK) ;

x ∈ Md ⊗ Md ,

(8)

where Tr2 denotes the partial trace over the second factor (Tr2 (a ⊗ b) := aTr(b)) and Tr denotes the non-normalized trace on Md . More explicitly, if X, Y ∈ Md one has E(X ⊗ Y ) = Tr2 (K ∗ (X ⊗ Y )K) ³ ´ XX√ √ = pij ∗ phk Tr2 eij Xehk ⊗ eji Y ekk i,j h,k

XX√ √ = pij ∗ phk eii Xehk δjk Tr(Y ejj ) . i,j h,k

Therefore, using the identities ejj Y ejj = hej , Y ej iejj , eii eαβ ehh = δix δβh eih , eii Xehh = hei , Xeh ieih ,

Quantum Markov Model in Cognitive Psychology

one obtains E(X ⊗ Y ) =

X√ √ pij ∗ phj hei , Xeh ieih hej , Y ej i .

379

(9)

i,j,h

Notice that, if X, Y ∈ Md are diagonal in the (ei )-basis, then this formula reduces to the usual formula for the conditional expectation in classical probability, ´ X³ X X E(X ⊗ Y ) = pij Xii eii Yjj = Xii pij Yjj eii = XP (Y ) . i

i,j

Given any density matrix in Md , ρ0 =

X

j

ρα0 ,γ 0 eα0 ,γ 0 ,

α0 ,γ 0

the pair {ρ0 , E} uniquely determines the joint n-point correlations, ϕ (A0 ⊗ A1 ⊗ . . . An ) := Tr (ρ0 E(A0 ⊗ E(A1 ⊗ E(A2 ⊗ . . . E(An ⊗ 1) . . .)))) , where A0 , A1 , . . . , An ∈ Md . We are interested in a time horizon with two instants: initial and final. The corresponding joint probabilities are in this case Trρ0 E(B ⊗ E(A ⊗ 1)) . Using (9) we obtain E(A ⊗ 1) =

X√ √ pij ∗ phj hei , Aeh ieih . i,j,h

Therefore, E(B ⊗ E(A ⊗ 1)) =

X√ √ pij ∗ phj hei , Aeh iE(B ⊗ eih ) i,j,h

X√ X√ √ √ = pij ∗ phj hei , Aeh i pαβ ∗ pγβ heα , Beγ ieαγ heβ , eih eβ i α,β,γ

i,j,h

X √ √ √ √ = pij ∗ pij hei , Aei iheα , Beγ ieαγ pαi ∗ pγi . i,j,α,γ

By using the fact that

P

Trρ0 E(B ⊗ E(A ⊗ 1)) =

j

pij = 1, for any i, we obtain X √ √ hei , Aei iheα , Beγ i pαi ∗ pγi ρα0 γ 0 T reα0 γ 0 eαγ

i,α,γ,α0 ,γ 0

and, using the identity Treα0 γ 0 eαγ = δγ 0 α δα0 γ

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and the notations hei , Aei i =: Aii ,

heα , Beγ i =: Bαγ ,

one finally obtains Trρ0 E(B ⊗ E(A ⊗ 1)) = =

X

X

√ √ hei , Aei iheα , Beγ i pαi ∗ pγi ργα

i,α,γ

X √ ∗√ X √ √ Aii pαi pαi ) Aii Bαγ pαi ∗ pγi ργα = ργα Bαγ ( α,γ

i,αγ

=:

X

i

ραγ Bαγ L(A)αγ .

α,γ

P P If the observables A, B have the form B = bj |bj ihbj | and A = aj |aj ihaj | the marginal probabilities for B are obtained by putting A = 1, i.e., X√ √ Tr (ρ0 E(|b1 ihb1 | ⊗ 1)) = pij ∗ phj hei , b1 ihb1 , eh iρhi i,j,h

X

=

ρhi hei , b1 ihb1 , eh i

X√ √ pij ∗ phj , j

h,i

where we used the identity X X Trρeih = ρα,γ Treαγ eih = ραγ δαi δαh = ρhi . α,γ

α,γ

Now, we specialise to the case d = 2 so that ρ0 has the form ρ11 = p ,

ρ22 = 1 − p ,

ρ12 = ρ21 = z ,

for some p ∈ [0, 1] and z ∈ C. We will look for solutions with z = 0. We remark that the density matrix can always be diagonalised in some basis. So, we work with this basis. This gives the equation p|he1 , b1 i|2 + (1 − p)|he2 , b1 i|2 = P (B = b1 |C) . A similar calculation for the marginal probabilities for A gives the equation X X Trρ0 E(1 ⊗ E(|a1 iha1 |)) = ραα |ha1 , ei i|2 pαi = P (A = a1 |C) . α

i

Expanding the eigenvectors of A and B in the (ei )-basis we obtain, for some pA , pB ∈ [0, 1]: X X pp1i |ha1 , ei i|2 + (1 − p)p2i |ha1 , ei i|2 = P (A = a1 |C) , i

i

Quantum Markov Model in Cognitive Psychology

p √ |a1 i =: pA e1 + 1 − pA e2 , p √ |b1 i =: p B e1 + 1 − p B e2 ,

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and therefore the above equations become ppB + (1 − p)(1 − pB ) = P (B = b1 |C) , pp11 pA + pp12 (1 − pA ) + (1 − p)p21 pA + (1 − p)p22 (1 − pA ) = P (A = a1 |C) , or ppB − (1 − p)pB + (1 − p) = (2p − 1)pB + (1 − p) = P (B = b1 |C) , p(p11 − p12 )pA + pp12 + (1 − p)pA (p21 − p22 ) + (1 − p)p22 = P (A = a1 |C) . Recalling that the empirical values, found by Tversky and Shafir, are P (B = b1 |C) = 0.5 , P (A = a1 |C) = 0.63 , the first equation becomes equivalent to (2p − 1)pB + (1 − p) =

1 , 2

which gives, independently of p ³ 1´ 1 1 pB = p − = . 2 2p − 1 2 It is not especially surprising, since here the B-player is computer which produces wins and loses with equal probabilities. It is natural for the Aplayer (who is really a player) simply to incorporate this probabilities in the quantum-like representation of the B-strategy. This representation is given by vectors |bj i, j = 1, 2, which are created by B in his “mental Hilbert space” (which is used by him to represent the gambling situation). Thus the simplest representation √ |b1 i = (e1 + e2 )/ 2 , the second vector can be chosen as any normalised vector which is orthogonal to |b1 i, say √ |b2 i = (e1 − e2 )/ 2 . Of course, mathematically the situation is essentially more general. The vector representation of the state of B can involve complex phases, √ √ |b1 i = (e1 + eiθB e2 )/ 2 , |b2 i = (e1 − eiθB e2 )/ 2 . In a more complicated gambling such that the B-player is not a computer, the A-player may really use the additional degree of freedom, namely, the phase

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θ, to represent the B-strategy. However, in the present gambling experiment it seems that A can be fine with purely real eigenvectors. The second equation becomes equivalent to pA (p(p11 − p12 ) + (1 − p)(p21 − p22 )) + pp12 + (1 − p)p22 = 0.63 ⇔ pA (p(0.16 − 0.84) + (1 − p)(0.03 − 0.97)) + p 0.84 + (1 − p)0.97 = 0.63 ⇔ −pA p 0.68 − pA (1 − p)0.94 + p 0.84 + (1 − p)0.97 = 0.63 ⇔ p(0.84 − pA 0.68) + (1 − p)(0.97 − pA 0.94) = 0.63 ⇔p =

0.63 − (0.97 − pA 0.94) . 0.84 − pA 0.68

Thus any pA ∈ [0, 1] such that 0 ≤

0.63 − (0.97 − pA 0.94) ≤ 1 0.84 − pA 0.68

(10)

will give a unique p. The two inequalities (10) are equivalent to 0 ≤ 0.63 − 0.97 + pA 0.94 , 0.63 − 0.97 + pA 0.94 ≤ 0.84 − pA 0.68 . The first is equivalent to 0.34 ≤ pA 0.94 ⇔

34 ≤ pA . 94

The second is equivalent to pA (0.94 + 0.68) ≤ 0.84 + 0.34 ⇔ pA 1.62 ≤ 1.18 ⇔ pA ≤ Since

118 . 162

34 118 ≈ 0.36 < ≈ 0.73 94 162 118 any pA in the nonempty interval [ 34 94 , 162 ] will give a solution. 34 If pA = 94 , then p = 0. Thus in the density matrix ρ11 = 0 and ρ22 = 1, nondiagonal elements equal to zero. It is the pure state ρ = e2 ⊗ e2 . It is a “pure state of mind.” In the same way in the case pA = 118 162 we get ρ = e1 ⊗e1 . We point out to the crucial difference of the quantum Markov model from the conventional Dirac–von Neumann model based on Born’s rule, where P (A = +1 | C) ≡ Pρ (A = +1) = Trρ(|a1 i ⊗ ha1 |) and P (B = +1 | C) ≡ Pρ (B = +1) = Trρ(|b1 i ⊗ hb1 |) .

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In the mind of the A-player vectors |aj i and |bj i are created. One can speculate that they provide a kind of “internal mental representation” of the context C. However, this internal representation does not determine completely real probabilities of answers. The latter arise in the process of interaction of the internal mental representation with previous experience which is encoded in probabilities pij and, finally, in the lifting operator E ? which is determined by the operator K. Of course, it is just the first reflection to interpret moving from the conventional quantum model to the quantum Markov model: interference of the internal mental representation of context C with representations of contexts corresponding to fixed strategies, say Caj = (A = aj ) and Cbj = (B = bj ). The latter are based on previous experience. We hope that psychologists may come with more justified motivations of the shift from the conventional quantum model to the quantum Markov model. We would not be surprised if possible psychological interpretations would be different from our interpretation. 4.

Discussion

As it was pointed out, the experimental data obtained by Shafir and Tversky contradict the postulate on the rational behaviour of players. Since such gambling is typical for economic and especially financial “games”, one should conclude that traders of the market do not behave rationally. This result definitely contradicts the postulate on rationality of agents of economics, e.g., in the form of Savage “Sure Thing Principle”, [3]. We would like to comment on this situation. As we have seen, violation of rationality of actions is related to non-Kolmogorovianity of data induced by actions. Thus, it is better to speak about violation of Kolmogorovian rationality. Moreover, we can associate with each probability model, say PM, its own type of rationality, e.g., the standard (Dirac–von Neumann) quantum probability describes “quantum rationality”. Agents of the market can establish such rules of behaviour (based on mixture of economics and psychology) that their actions will be described by some PM. In this case, they behave rationally, but with respect to this model (created in their minds on the basis of context), i.e., PM-rationally. At the same time they can be K-irrational. Finally, we remark that players of PD-type games participated in Shafir and Tversky experiments behave irrationally with respect to both the classical Kolmogorov model and the quantum Dirac–von Neumann model. However, as we proved, they are rational with respect to the quantum Markov model. One of the interesting open problems is to study other databases induced by cognitive psychology and to check the possibility to describe them by a quantum Markov model. Our conjecture is that behaviour of cognitive

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systems can be completely described by this model in the same way as behaviours of classical and quantum particles were described by Kolmogorov and Dirac–von Neumann models, respectively. Unfortunately, all known experiments are based on two-step games and they can generate only two-step quantum Markov chains. Our conjecture on quantum Markov description of cognitive phenomena can be completely checked only by designing and performing new experiments. They should involve in general n steps and they should be described (by our conjecture) by Markov chains of length n. The present paper is devoted to theoretical study of existing data and we do not try to design here concrete experiments. We can just mention (as the first step) a variant of the Prisoner’s Dilemma game with three players. In this paper we did not try to describe how the “black box” (the brain) producing “irrational data” works. One of the possibilities is to try to use the adaptive dynamics approach for a possible description of the internal processes in this black box [4]. Acknowledgment This project was partially supported by Centro V. Volterra, Universit´ a di Roma Torvergata (visit of A. Yu. Khrennikov to this center, June 2008), by International Center for Mathematical Modelling, University of V¨axj¨ o (visits of L. Accardi and M. Ohya, August 2008) and QBIC Center, Tokyo University of Science (visits of A. Yu. Khrennikov and L. Accardi, March 2009). Bibliography [1] E. Shafir and A. Tversky, Thinking through uncertainty: nonconsequential reasoning and choice, Cognitive Psychology 24, 449 (1992). [2] A. Tversky and E. Shafir, The disjunction effect in choice under uncertainty, Psychological Science 3, 305 (1992). [3] L. J. Savage, The foundations of statistics, Wiley and Sons, New York, 1954. [4] L. Accardi and M. Ohya, A stochastic limit approach to the SAT problem, Open Sys. Information Dyn. 11, 219 (2004). [5] L. Accardi, Foundations of Quantum Mechanics: a quantum probabilistic approach, in: The Nature of Quantum Paradoxes, G. Tarozzi, A. van der Merwe, eds., Reidel, pp. 257–323, 1988; Preprint Dipartimento Di Matematica, Universit´ a di Roma Torvergata, 1986. [6] A. Yu. Khrennikov, Interpretations of Probability, VSP Int. Sc. Publishers, Utrecht/Tokyo, 1999; De Gruyter, Berlin (2009), second edition (completed). [7] L. Accardi, A. Yu. Khrennikov, and M. Ohya, The problem of quantum-like representation in economy, cognitive science, and genetics, in: Quantum Bio-Informatics-2, L. Accardi, W. Freudenberg, M. Ohya, eds., pp. 1–8, World Scientific, Singapore, 2008. [8] A. Yu. Khrennikov, Linear representations of probabilistic transformations induced by context transitions, J. Phys. A: Math. Gen. 34, 9965 (2001).

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