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 Email: accardi@volterra.mat.uniroma2.it

Andrei Khrennikov

International Center for Mathematical Modeling

in Physics and Cognitive Sciences University of Växjö, S-35195, Sweden Email: Andrei.Khrennikov@vxu.se

Masanori Ohya

Department of Information Science

Tokyo University of Science Noda-city, Chiba, 278-8510, Japan

Email: ohya@rs.noda.tus.ac.jp

January 12, 2012

Abstract

We analyze from the viewpoint of quantum probability statistical data from two interesting experiments, done by Shafir and Tversky [?], [?], 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” [?] (implying that agents of the market behave rationally). Data from these experiments was 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 quantum Markov model for these data.

1 Introduction

The development of probability theory can be compared with development of geometry [?], [?]: from a single model, implicitly supposed to be the only possible 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,....). In probability theory it was from the Kolmogorov measure-theoretic model^{1} to a special non-Kolmogorovian model – the quantum probabilistic model based on Dirac-von Neumann formalism. ^{2} We mention recent publication [?] devoted to fundamental questions of applications of non-Kolmogorovian models, in particular, quantum-like models in in economy, cognitive science, and genetics; see also works [?]–[?] 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 non classical probabilistic data which need not to 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 [?]–[?], and obtained non-Kolmogorovian probabilistic data, see Conte et al. [?] – [?]. Although this specially designed experiment confirmed (at least preliminary) 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 paradoxic consequences.

1which, since the early 1930’s, has been dominating in many domains of science: classical statistical physics, economy, biology, psychology.

2We remark that, in geometry, non-Euclidean models were invented by purely mathematical reasons, but in probability 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.

In the domain of cognitive psychology, it was well known that data collected in gambling experiments of the Prisoner Dilemma (PD) type [?], see Shafir and Tversky [?], [?] and also Croson [?], cannot be generated by a natural (classical) Markov model, see Busemeyer et al. [?]–[?]. In the latter papers an attempt to use quantum probabilsitic description was performed.

In this paper we present the rigorous proofs that

a) the data from the Shafir and Tversky [?], [?] experiments can not be embeded in a Kolmogorov model^{3};

b) these data are neither quantum – they cannot be described by probabilities given by Dirac-von Neumann’s formalism.

Finally, authors proceed towards the description of data from Shafir and Tversky [?], [?] experiments by a quantum Markov model, [?], [?], [?].

2 Statistical tests of rational behavior

2.1 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 classic 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.

3In particular, this explains the reasons of the failure of the attempts to generate this data by using Markov models.

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 to do. 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 is to betray as you then walk free instead of receiving the minor sentence. If you knew the other prisoner would betray, your best move is still to betray, as you receive a lesser sentence than by silence. Betraying is a 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, self-interested play results in each prisoner being worse off than if they had stayed silent, see e.g. Wikipedia – “Prisoner’s dilemma.”

This is the principle of rational behavior which is basic for rational choice theory which is 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 [?] found that players frequently behave irrationally.

2.2 Gambling experiment

Shafir and Tversky also performed a simpler version of PD’s gamble experiment, see [?]. They proposed to test rational behavior for the following gambling experiment.

In this experiment, you are presented with two possible plays of a gamble that is equally likely to win 200 USD or lose 100USD. You are instructed that the first play has been completed, and now you are 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 non classical 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 n_{A} possible behaviors:

a_{1}, …, a_{n_{A}}

– B can choose among n_{B} possible behaviors:

b_{1}, …, b_{n_{B}}

Events

[A = a_{α}] ≡ A chooses the behavior a_{α}
[B = b_{β}] ≡ B chooses the behavior b_{β}

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 = a_{α}|B = b_{β}) := probability that A chooses the behavior a_{α} knowing that B has chosen the behavior b_{β}

P (B = b_{β}|A = a_{α}) := probability that B chooses the behavior b_{β} knowing that A has chosen the behavior a_{α}

P (A = a_{α}|C) := probability that A chooses the behavior a_{α} given that A knows nothing about the choice of B and conversely.

Analogue interpretation can be used for P (B = b_{β}|C).

3.2 General statement of the problem

Given the experimental data

P (B|A) ≡ (P (B = b_{β}|A = a_{α}))  (1)
P (A|B) ≡ (P (A = a_{α}|B = b_{β}))  (2)
P_{A} ≡ (P (A = a_{α}|C))  (3)
P_{B} ≡ (P (B = b_{β}|C))  (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 [?] 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 [?], and the Bell inequality are of this form as well as interference conditions considered in [?]–[?].

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 (??), (??) and the transition probabilities (??), (??) satisfy the following compatiblity condition (which essentially amounts to Bayes’ theorem):

∑_{β} P (B = b_{β}|C)P (A = a_{α}|B = b_{β}) = P (A = a_{α}|C)  (5)

∑_{α} P (A = a_{α}|C)P (B = b_{β}|A = a_{α}) = P (B = b_{β}|C)  (6)

Replacing the value of P (A = a_{α}|C) in (??) by the left hand side of (??), one finds the necessary condition:

∑_{β,α} P (B = b_{β}|C)P (A = a_{α}|B = b_{β})P (B = b\overline{β}|A = a_{α}) = P (B = b\overline{β}|C)

and the one obtained by exchanging the roles of (A, B), (α, β). These are equivalent to the fixed point equations:

P_{B}P (A|B)P (B|A) = P_{B}

and the equation obtained exchanging the roles of (A, B) i.e.:

P_{A}P (B|A)P (A|B) = P_{A}

3.4 Simple necessary conditions: II

Necessary conditions even simpler than those discussed in section (??) are deduced from the following remark:

P (A_{α} ∩ B_{β}) := P (B_{β}|C)P (A_{α}|B_{β})
P (A_{α} ∩ B_{β}) = P (A_{α}|C)P (B_{β}|A_{α})

which implies the so–called generalized detailed balance condition:

P (B_{β}|C)P (A_{α}|B_{β}) = P (A_{α}|C)P (B_{β}|A_{α})

which is a necessary condition for the existence of the K–model even if the fixed point conditions (??) or (??) 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 subsection ??, see Tversky and Shafir [?], then we will consider the PD experiment (discussed in subsection ??, see Shafir and Tversky [?]).

In this experiment B is a computer and its choices are determined by a random generator with probability 1/2, i.e.

P (B_{1}|C) = P (B_{2}|C) = \frac{1}{2}  (7)

A and C are human and – the reaction of A to the choice of B, i.e. P_{A|B},

– the reaction of A to the choice of C, i.e. P_{A|C}, are found in experiments. They are given by:

P_{A|B} := ( \begin{matrix}P (A_{1}|B_{1}) & P (A_{1}|B_{2}) \\ P (A_{2}|B_{1}) & P (A_{2}|B_{2}) \end{matrix} ) = ( \begin{matrix}0, 16 & 0, 03 \\ 0, 84 & 0, 97 \end{matrix} )

P_{A} = P (A = ·|C) = ( \begin{matrix}0, 63 \\ 0, 37 \end{matrix} )

Notice that, in this experiment, the knowledge of P_{B|A} is not required.

A necessary condition for the existence of a Kolmogorov model is the validity of the Bayes’ rule:

P (A_{1}|B_{1})P (B_{1}|C) + P (A_{1}|B_{2})P (B_{2}|C) = P (A_{1}|C)

which, given (??), becomes

\frac{1}{2} [P (A_{1}|B_{1}) + P (A_{1}|B_{2})] = P (A_{1}|C)

However in our case:

\frac{1}{2} [P (A_{1}|B_{1}) + P (A_{1}|B_{2})] = (0, 5)[0, 16 + 0, 03] =

= (0, 5)(0, 19) = 0, 95 ≠ 0, 63 = P (A_{1}|C)

3.6 Shafir and Tversky PD experiment

n_{A} = n_{B} = 2

P_{B|A} := ( \begin{matrix}P (B_{1}|A_{1}) & P (B_{1}|A_{2}) \\ P (A_{2}|A_{1}) & P (B_{2}|A_{2}) \end{matrix} ) = ( \begin{matrix}0, 16 & 0, 03 \\ 0, 84 & 0, 97 \end{matrix} )

Postulate : A and B have similar psychology:

P_{A} = P_{B} = P (·|C) = ( \begin{matrix}0, 63 \\ 0, 37 \end{matrix} )

P_{A|B} := ( \begin{matrix}P (A_{1}|B_{1}) & P (A_{1}|B_{2}) \\ P (A_{2}|B_{1}) & P (A_{2}|B_{2}) \end{matrix} ) =

= ( \begin{matrix}P (B_{1}|A_{1}) & P (B_{1}|A_{2}) \\ P (B_{2}|A_{1}) & P (B_{2}|A_{2}) \end{matrix} ) = P_{B|A}

We have seen that a necessary condition for the existence of the K– model is the generalized detailed balance condition:

P (B_{1}|C)P (A_{2}|B_{1}) = P (A_{2}|C)P (B_{1}|A_{2})

However

(0, 63) · (0, 84) = 0, 5292 ≠ (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 [?], [?] cannot be embeded in a Kolmogorov probability space. The next natural question is whether it can be embeded in Dirac-von Neumann’s (pure) quantum model. One can easily see that neither this is the case because the matrices of the transition probabilities are not doubly stochastic! Thus one should look for some generalization of Diracvon Neumann’s pure model. Notice that Dirac-von Neumann’s non pure 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 [?], [?], [?] which goes beyond the Dirac-von Neumann’s 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 [?] and is a more subtle notion because it is an operator generalization of the basic intuition that quantum theory deals with amplitudes rather than with directly probabilities. Since, intuitively, amplitudes are square roots of probabilities, given a classical Markov transition matrix (p_{ij}) with state space {1, …, d} (d ∈ ℕ) a natural way to associate to it a conditional density amplitude is to define

K = ∑_{i,j} \sqrt{p_{ij}}e_{ii} ⊗ e_{jj} ∈ M_{d} ⊗ M_{d}

where M_{d} denotes the d×d complex matrices, (e_{i}) an orthonormal basis of ℂ^{d}, (e_{ij}) the associated system of matrix units (e_{ij}e_{hk} = δ_{j,h}e_{ik}), and \sqrt{p_{ij}} are arbitrary (usually complex) square roots of p_{ij}. Any conditional density amplitude determines a linear lifting (see [?]) E^{∗} : S(M_{d}) → S(M_{d} ⊗ M_{d}) from density matrices on M_{d} to density matrices on M_{d} ⊗ M_{d} or, dually, a transition expectation, i.e. a completely positive, identity preserving linear map

E : M_{d} ⊗ M_{d} → M_{d}

defined by

E(x) = \overline{T r_{2}}(K^{∗}xK)  ;  x ∈ M_{d} ⊗ M_{d}  (8)

where \overline{T r_{2}} denotes the partial trace over the second factor (\overline{T r_{2}}(a ⊗

b) := aT r(b)) and T r denotes the non normalized trace on M_{d}. More explicitly, if X, Y ∈ M_{d} one has:

E(X⊗Y) = \overline{T r_{2}}(K^{∗}(X⊗Y)K) = ∑_{i,j} ∑_{h,k} \sqrt{p_{ij}}^{∗}\sqrt{p_{hk}} \overline{T r_{2}}(e_{ij}Xe_{hk}⊗e_{ji}Y e_{kk})

= ∑_{i,j} ∑_{h,k} \sqrt{p_{ij}}^{∗}\sqrt{p_{hk}}e_{ii}Xe_{hk}δ_{jk}T r(Y e_{jj})

therefore, using the identities:

e_{jj}Y e_{jj} = ⟨e_{j}, Y e_{j}⟩e_{jj}
e_{ii}e_{αβ}e_{hh} = δ_{ix}δ_{βh}e_{ih}
e_{ii}Xe_{hh} = ⟨e_{i}, X_{e_{h}}⟩e_{ih}

one obtains:

E(X ⊗ Y) = ∑_{i,j,h} \sqrt{p_{ij}}^{∗}\sqrt{p_{hj}}⟨e_{i}, Xe_{h}⟩e_{ih}⟨e_{j}, Y e_{j}⟩  (9)

Notice that, if X, Y ∈ M_{d} are diagonal in the (e_{i})–basis, then this formula reduces to the usual formula for the conditional expectation in classical probability:

E(X ⊗ Y) = ∑_{i,j} p_{ij}X_{ii}e_{ii}Y_{jj} = ∑_{i} (X_{ii} ∑_{j} p_{ij}Y_{jj}) e_{ii} = XP (Y)

Given any density matrix in M_{d}

ρ_{0} = ∑_{α′,γ′} ρ_{α′,γ′}e_{α′,γ′}

the pair {ρ_{0}, E} uniquely determines the joint n–point correlations

φ (A_{0} ⊗ A_{1} ⊗ … A_{n}) := T r (ρ_{0}E(A_{0} ⊗ E(A_{1} ⊗ E(A_{2} ⊗ … E(A_{n} ⊗ 1) …))

(A_{0}, A_{1}, A_{2}, …, A_{n} ∈ M_{d}). We are interested in a time horizon with two instants: initial and final. The corresponding joint probabilities are, in this case:

T rρ_{0}E(B ⊗ E(A ⊗ 1))

Using (??) we obtain:

E(A ⊗ 1) = ∑_{i,j,h} \sqrt{p_{ij}}^{∗}\sqrt{p_{hj}}⟨e_{i}, Ae_{h}⟩e_{ih}

Therefore

E(B ⊗ E(A ⊗ 1)) = ∑_{i,j,h} \sqrt{p_{ij}}^{∗}\sqrt{p_{hj}}⟨e_{i}, Ae_{h}⟩E(B ⊗ e_{ih}) =

= ∑_{i,j,h} \sqrt{p_{ij}}^{∗}\sqrt{p_{hj}}⟨e_{i}, Ae_{h}⟩ ∑_{α,β,γ} \sqrt{p_{αβ}}^{∗}\sqrt{p_{γβ}}⟨e_{α}, Be_{γ}⟩e_{αγ}⟨e_{β}, e_{ih}e_{β}⟩ =

= ∑_{i,j,α,γ} \sqrt{p_{ij}}^{∗}\sqrt{p_{ij}}⟨e_{i}, Ae_{i}⟩⟨e_{α}, Be_{γ}⟩e_{αγ}\sqrt{p_{αi}}^{∗}\sqrt{p_{γi}}.

By using the fact that ∑_{j} p_{ij} = 1, for any i, we obtain

T rρ_{0}E(B⊗E(A⊗1)) = ∑_{i,α,γ,α′,γ′}⟨e_{i}, Ae_{i}⟩⟨e_{α}, Be_{γ}⟩\sqrt{p_{αi}}^{∗}\sqrt{p_{γi}}ρ_{α′γ′}T re_{α′γ′}e_{αγ}

and, using the identity:

T re_{α′γ′}e_{αγ} = δ_{γ′α}δ_{α′γ}

and the notations

⟨e_{i}, Ae_{i}⟩ =: A_{ii} ; ⟨e_{α}, Be_{γ}⟩ =: B_{αγ}

one finally obtains

T rρ_{0}E(B ⊗ E(A ⊗ 1)) = ∑_{i,α,γ}⟨e_{i}, Ae_{i}⟩⟨e_{α}, Be_{γ}⟩\sqrt{p_{αi}}^{∗}\sqrt{p_{γi}}ρ_{γα} =

= ∑_{i,αγ} A_{ii}B_{αγ}\sqrt{p_{αi}}^{∗}\sqrt{p_{γi}}ρ_{γα} = ∑_{α,γ} ρ_{γα}B_{αγ}(∑_{i} A_{ii}\sqrt{p_{αi}}^{∗}\sqrt{p_{αi}}) =

=: ∑_{α,γ} ρ_{αγ}B_{αγ}L(A)_{αγ}

If the observables A, B have the form B = ∑ b_{j}|b_{j}⟩⟨b_{j}|

A = ∑ a_{j}|a_{j}⟩⟨a_{j}|

the marginal probabilities for B are obtaine by putting A = 1, i.e.

T r (ρ_{0}E(|b_{1}⟩⟨b_{1}| ⊗ 1)) = ∑_{i,j,h} \sqrt{p_{ij}}^{∗}\sqrt{p_{hj}}⟨e_{i}, b_{1}⟩⟨b_{1}, e_{h}⟩ρ_{hi} =

= ∑_{h,i} ρ_{hi}⟨e_{i}, b_{1}⟩⟨b_{1}, e_{h}⟩ ∑_{j} \sqrt{p_{ij}}^{∗}\sqrt{p_{hj}}

where we used the identity

T rρe_{ih} = ∑_{α,γ} ρ_{α,γ}T re_{αγ}e_{ih} = ∑_{α,γ} ρ_{αγ}δ_{αi}δ_{αh} = ρ_{hi}

Now we specialize to the case d = 2 so that ρ_{0} has the form

ρ_{11} = p ; ρ_{22} = 1 − p ; ρ_{12} = \overline{ρ_{21}} = 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 diagonalized in some basis: So, we work with this basis. This gives the equation

p|⟨e_{1}, b_{1}⟩|^{2} + (1 − p)|⟨e_{2}, b_{1}⟩|^{2} = P (B = b_{1}|C).

A similar calculation for the marginal probabilities for A gives the equation

T r_{ρ_{0}}E(1 ⊗ E(|a_{1}⟩⟨a_{1}|)) = ∑_{α} ρ_{αα} ∑_{i} |⟨a_{1}, e_{i}⟩|^{2}p_{αi} = P (A = a_{1}|C).

Expanding the eigenvectors of A and B in the e–basis, we obtain, for some p_{A}, p_{B} ∈ [0, 1]:

∑_{i} pp_{1i}|⟨a_{1}, e_{i}⟩|^{2} + ∑_{i}(1 − p)p_{2i}|⟨a_{1}, e_{i}⟩|^{2} = P (A = a_{1}|C)

|a_{1}⟩ =: \sqrt{p_{A}}e_{1} + \sqrt{1 − p_{A}}e_{2}
|b_{1}⟩ =: \sqrt{p_{B}}e_{1} + \sqrt{1 − p_{B}}e_{2}

and therefore the above equations become:

pp_{B} + (1 − p)(1 − p_{B}) = P (B = b_{1}|C)

pp_{11}p_{A}+pp_{12}(1−p_{A})+(1−p)p_{21}p_{A}+(1−p)p_{22}(1−p_{A}) = P (A = a_{1}|C) or

pp_{B} − (1 − p)p_{B} + (1 − p) = (2p − 1)p_{B} + (1 − p) = P (B = b_{1}|C)

p(p_{11} −p_{12})p_{A} +pp_{12} +(1−p)p_{A}(p_{21} −p_{22})+(1−p)p_{22} = P (A = a_{1}|C)

Recalling that the empirical values, found by Tversky and Shafir, are

P (B = b_{1}|C) = \frac{1}{2}

P (A = a_{1}|C) = 0, 63

the first equation becomes equivalent to:

(2p − 1)p_{B} + (1 − p) = \frac{1}{2}

which gives, independently of p:

p_{B} = (p − \frac{1}{2}) \frac{1}{2p − 1} = \frac{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 A-player(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 |b_{j}⟩, 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

|b_{1}⟩ = (e_{1} + e_{2})/\sqrt{2};

the second vector can be chosen as any normalized vector which is orthogonal to |b_{1}⟩, say

|b_{2}⟩ = (e_{1} − e_{2})/\sqrt{2}.

Of course, mathematically the situation is essentially more general. The vector representation of the state of B can involve complex phases:

|b_{1}⟩ = (e_{1} + e^{iθ_{B}}e_{2})/\sqrt{2}; |b_{2}⟩ = (e_{1} − e^{iθ_{B}}e_{2})/\sqrt{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 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:

p_{A}(p(p_{11} − p_{12}) + (1 − p)(p_{21} − p_{22})) + pp_{12} + (1 − p)p_{22} = 0, 63 ⇔

⇔ p_{A}(p(0, 16−0, 84)+(1−p)(0, 03−0, 97))+p0, 84+(1−p)0, 97 = 0, 63

⇔ −p_{A}p0, 68 − p_{A}(1 − p)0, 94 + p0, 84 + (1 − p)0, 97 = 0, 63 ⇔

⇔ p(0, 84 − p_{A}0, 68) + (1 − p)(0, 97 − p_{A}0, 94) = 0, 63

⇔ p = \frac{0, 63 − (0, 97 − p_{A}0, 94)}{0, 84 − p_{A}0, 68}

Thus any p_{A} ∈ [0, 1] such that

0 ≤ \frac{0, 63 − (0, 97 − p_{A}0, 94)}{0, 84 − p_{A}0, 68} ≤ 1  (10)

will give a unique p. The two inequalities (??) are equivalent to

0 ≤ 0, 63 − 0, 97 + p_{A}0, 94
0, 63 − 0, 97 + p_{A}0, 94 ≤ 0, 84 − p_{A}0, 68

The first is equivalent to

0, 34 ≤ p_{A}0, 94 ⇔ \frac{34}{94} ≤ p_{A}

The second is equivalent to

p_{A}(0, 94 + 0, 68) ≤ 0, 84 + 0, 34 ⇔ p_{A}1, 62 ≤ 1, 18 ⇔ p_{A} ≤ \frac{118}{162}

Since

\frac{34}{94} ≈ 0, 36 < \frac{118}{162} ≈ 0, 73

any p_{A} in the (non empty) interval [\frac{34}{94}, \frac{118}{162}] will give a solution.

If p_{A} = \frac{34}{94}, then p = 0. Thus in the density matrix ρ_{11} = 0 and ρ_{22} = 1, non-diagonal elements equal to zero. It is the pure state ρ = e_{2} ⊗ e_{2}. It is a “pure state of mind.” In the same way in the case p_{A} = \frac{118}{162} we get ρ = e_{1} ⊗ e_{1}.

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) ≡ P_{ρ}(A = +1) = T rρ(|a_{1}⟩ ⊗ ⟨a_{1}|) and P (B = +1|C) ≡ P_{ρ}(B = +1) = T rρ(|b_{1}⟩ ⊗ ⟨b_{1}|).

In the brain of the A-player there are created vectors |a_{j}⟩ and |b_{j}⟩. One can speculate that they provide a kind of “internal mental representation” of 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 p_{ij} 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 C_{a_{j}} = (A = a_{j}) and C_{b_{j}} = (B = b_{j}). 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 coming psychological interpretations will be different from our interpretation.

4 Discussion

As it was pointed out, the experimental data obtained by Shafir and Tversky contradict the postulate on rational behavior 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 to the postulate on rationality of agents of economics, e.g., in the form of Savage “Sure Thing Principle”, [?].

We would like to comment 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 P M, 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 behavior (based on mixture of economics and psychology) that their actions will be described by some P M. In this case, they behave rationally, but with respect to this model (created in their brains on the basis of context), i.e., P M-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 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 interesting open problems is study other databases induced by cognitive psychology and to check a possibility to describe them by the quantum Markov model. Our conjecture is that behaviour of cognitive systems can be completely described by this model (in the same way as behaviors of classical and quantum particles were described by Kolmogorov and Dirac-von Neumann models, respectively. Unfortunately, all known experiments are based on two steps 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 the 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 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 possibilities is to try to use the adaptive dynamics approach for a possible description of the internal processes in this black box, see [?].

This project was partially supported by Centro V. Volterra, Universita’ di Roma Torvergata (visit of A. Yu. Khrennikov to this center, June 2008), by International Center for Mathematical Modelling, University of Växjö (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).

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