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Information Processing by Networks of Quantum

Decision Makers

V.I. Yukalov1,2,∗, E.P. Yukalova1,3, and D. Sornette1,4

1Department of Management, Technology and Economics, ETH Zurich,Swiss Federal Institute of Technology, Zurich CH-8032, Switzerland

2Bogolubov Laboratory of Theoretical Physics,Joint Institute for Nuclear Research, Dubna 141980, Russia

3Laboratory of Information Technologies,Joint Institute for Nuclear Research, Dubna 141980, Russia

4Swiss Finance Institute, c/o University of Geneva,40 blvd. Du Pont d’Arve, CH 1211 Geneva 4, Switzerland

Abstract

We suggest a model of a multi-agent society of decision makers taking decisions beingbased on two criteria, one is the utility of the prospects and the other is the attractivenessof the considered prospects. The model is the generalization of quantum decision theory,developed earlier for single decision makers realizing one-step decisions, in two principalaspects. First, several decision makers are considered simultaneously, who interact witheach other through information exchange. Second, a multistep procedure is treated, whenthe agents exchange information many times. Several decision makers exchanging infor-mation and forming their judgement, using quantum rules, form a kind of a quantuminformation network, where collective decisions develop in time as a result of informationexchange. In addition to characterizing collective decisions that arise in human societies,such networks can describe dynamical processes occurring in artificial quantum intelli-gence composed of several parts or in a cluster of quantum computers. The practicalusage of the theory is illustrated on the dynamic disjunction effect for which three quan-titative predictions are made: (i) the probabilistic behavior of decision makers at theinitial stage of the process is described; (ii) the decrease of the difference between theinitial prospect probabilities and the related utility factors is proved; (iii) the existence ofa common consensus after multiple exchange of information is predicted. The predictednumerical values are in very good agreement with empirical data.

Keywords: Multi-agent society, artificial quantum intelligence, quantum decision theory, mul-tistep decision procedure, dynamics of collective opinion, quantum information networks, in-telligence networks, dynamic disjunction effect

∗Corresponding author: V.I. Yukalov

E-mail: yukalov@theor.jinr.ru

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1 Introduction and Related Literature

Modeling the behavior of multi-agent social systems and social networks has recently attracteda substantial amount of activities of researchers, as can be inferred from the review articles [1–3]and numerous original papers, of which we can cite just a few recent [4–8]. High interest tosuch a modeling is due to two reasons. First, being able of describing the behavior of societiesis of great importance by itself. Second, modeling the collective interactions of autonomousmultientity systems has already been widely envisioned to be a powerful paradigm for multi-agent computing.

Nowadays, there exists an extensive literature on decision making in multi-agent systems,which can be categorized into three main directions: (i) Studies of how the agents, inhabitinga shared environment, can decide their actions through mutual negotiations. There can be twotypes of agents, cooperative and self-interested. Cooperative agents cooperate with each otherto reach a common goal [9]. Self-interested agents try to maximize their own payoff withoutconcern to the global good, choosing the best negotiation strategy for themselves [10, 11]. (ii)Studies of how a network of agents with initially different opinions can reach a collective decisionand take action in a distributed manner [12]. (iii) Studies of how the dependence among multi-agents can lead to emerging social structures, such as groups and agent clusters [13].

The main goal of a decision process in a multi-agent system is to find the optimal policythat maximizes expected utility or expected reward for either a single agent or for the societyas a whole. Various models of multi-agent systems can be found in the books [14–19].

The maximization of expected utility, expected reward, or other functionals is based onthe assumption that agents are perfectly rational and no restrictions on computational powerand available resources are imposed. However, since Simon [20], it is well known that onlybounded rationality can exist, so that any real decision maker, in addition to having limitedcomputational power and finite time for deliberations, is subject to such behavioral effects asirrational emotions, subconscious feelings, and subjective biases [20–23].

The behavioral effects become especially important when decisions are made under un-certainty. There are two sides of uncertainty that can be caused either by objective lack ofcomplete knowledge or by subjective preferences and biases. Even when the agents in thesociety are assumed to possess complete knowledge, they cannot become absolutely rationaldecision makers due to the inherent dual property of human behavior that combines consciousevaluation of utility with subconscious feelings and emotions [24].

Thus, real decision making is a complex procedure of dual nature, simultaneously inte-grating the rational, conscious, objective evaluation of utility with behavioral effects, suchas irrational emotions, subconscious feelings, and subjective biases. This feature of realisticdecision making can be called rational-irrational duality, or conscious-subconscious duality, orobjective-subjective duality. Keeping in mind these points, we can call this feature the behavioralduality of decision making.

As a result of this behavioral duality, a correct description of decision making in a real multi-agent system has to deal with two sides – maximization of expected reward, and taking accountof behavioral effects [25–27]. To take the latter into account, several modifications of utilitytheory have been suggested, such as prospect theory, weighted-utility theory, regret theory,optimism-pessimism theory, dual-utility theory, ordinal-independence theory, and quadratic-probability theory, whose description can be found in the review articles [28–31]. However,

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such so-called nonexpected utility models listed in reviews [29–31] have been refuted as beingmerely descriptive and having no predictive power [32–36]. A more detailed discussion can befound in Refs. [37–39].

As has been shown by Safra and Segal [34], none of non-expected utility theories can resolveall problems and paradoxes typical of decision making of humans. The best that could beachieved is a kind of fitting for interpreting just one or, in the best case, a few problems, whilethe other remained unexplained. In addition, spoiling the structure of expected utility theoryresults in the appearance of several complications and inconsistences. As has been concluded inthe detailed analysis of Al-Najjar and Weinstein [35,36], any variation of the classical expectedutility theory “ends up creating more paradoxes and inconsistences than it resolves”.

Stochastic decision theories are usually based on deterministic decision theories comple-mented by random variables with given distributions [40,41]. Therefore, such stochastic theoriesinherit the same problems as deterministic theories embedded into them. Moreover, stochastictheories are descriptive, containing fitting parameters that need to be defined from empiricaldata. In addition, different stochastic specifications of the same deterministic core theory maygenerate very different, and sometimes contradictory, conclusions [42].

One more difficulty in modeling decision making of real humans is that they often varytheir decisions, under the same invariant expected utility, after information exchange betweendecision makers, as has been observed in many empirical studies [43–50]. This implies that agentinteractions through information exchange can influence decision makers emotions, withouttouching their evaluation of utility.

In all previous works, behavioral effects, when being considered, have been treated as sta-tionary. The principal difference of the present paper from all previous publications is that wedevelop a model that takes into account the dual nature of decision making, and allows for thedescription of dynamical behavioral effects caused by agent interactions through informationexchange.

2 Main Features of Quantum Approach

To take into account the dual nature of decision making, in the previous papers [37–39], wehave formulated Quantum Decision Theory (QDT) as a mathematical approach for describingdecision making under uncertainty. This approach generalizes classical utility theory [51] tothe cases of decisions under strong uncertainty, when utility theory fails. The necessity ofdeveloping a new approach has been justified by numerous empirical observations proving thefailure of classical utility making in realistic situations.

The mathematical basis of the QDT is the generalization of the von Neumann [52] theoryof quantum measurements to the case of inconclusive quantum measurements [53] involvingcomposite events with intermediate operationally untestable steps [54,55]. In decision making,the intermediate inconclusive events characterize subconscious feelings and deliberations of adecision maker. It is this dual feature of decision making, including conscious logical reasoningand subconscious intuitive feelings, which explains the successful application of quantum tech-niques to describing human decision making, without the necessity of assuming any quantumnature of decision makers. The mathematics of quantum theory turns out to be well suited fordescribing the intrinsic conscious-subconscious duality of human cognition [56, 57].

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It is worth mentioning that, after Bohr [58] put forward the idea that the dual natureof consciousness requires the use of quantum description, a number of attempts have beenmade to apply quantum rules to cognition, as can be inferred from the review works [59–66].The previous attempts, however, were limited by models fitted to particular cases and havingno general mathematical structure valid for arbitrary events. Moreover, as has been recentlyshown [67, 68], these models contradict empirical facts.

Contrary to this, our QDT [37–39] is general, being formulated for arbitrary compositeevents. Its mathematics is based on the theory of quantum measurements [53–55], whichallows it to be used also for the problem of creating artificial quantum intelligence [39, 69, 70].As has been recently demonstrated [37–39], QDT is the sole theory that provides the possibilityof making quantitative predictions, without fitting parameters, even in such difficult situationswhen classical decision making is not even applicable qualitatively.

In our previous papers [37–39], quantum decision theory has been formulated for the caseof a single decision maker taking a one-step decision. But in a society of decision makers, theagents exchange information between each other and can make multistep decisions, with theirdecisions varying with time due to the information exchange. A simplified situation, when alldecision makers receive the same information from outside, without mutual exchange, has beenconsidered in [37, 71]. The necessity of developing, in the frame of QDT, a model describingthe multistep decision-making procedure, taking into account mutual exchange of informationbetween social agents, has been discussed in Ref. [39].

In the present paper, we suggest a principally important extension of QDT allowing for thedescription of dynamical collective effects. The main features, distinguishing this paper fromthe previous publications, are as follows.

• The considered system is a society of several decision makers, and not just a single decisionmaker.

• Each agent of the society is a decision maker transforming the information, received fromother agents, according to quantum decision theory.

• At each decision step, every agent generates an outcome that is a probability distributionover a given set of prospects.

• Agent interactions are not parameterized by a fixed interaction matrix, but are char-acterized by an information functional over the probability distribution quantifying theamount of information gained from other agents.

• The system is not static, but dynamical, the information functional and generated prob-ability distributions are functions of time.

• To describe a realistic situation, the information exchange is not simultaneous, but de-layed. This means that, if the agents at time t generate probability distributions {pj(t)},then the probability distribution that follows, which is caused by the information ex-change, is generated at a delayed time t+ τ .

• The usage of the theory is illustrated by a concrete example involving the dynamic dis-junction effect. The predicted numerical results are in very good agreement with empiricaldata.

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The proposed system is not a set of simple quantum devices representing players, as thoseconsidered in quantum games [72–76], but rather a collection of complex subjects exchang-ing information, each representing a quantum intelligence. Therefore the society of decisionmakers, acting according to the rules of QDT, is a kind of collective quantum intelligence, orsuperintelligence.

Such a collective quantum intelligence can describe any ensemble of agents generating prob-ability distributions according to the rules of QDT. This can be a human society makingdecisions on complex problems under uncertainty. This can also be a set of quantum comput-ers, or a complex artificial intelligence composed of parts, each of which being itself an artificialquantum intelligence, that is, a kind of superbrain.

The behavioral model we develop is principally new. The interaction structure of a multi-agent system, with agents interacting through the exchange of information occurring by quan-tum rules, has never been considered before to the best of our knowledge.

The organization of this article is as follows. Section 3 explains the basic ideas characterizingthe interaction structure of agents in a multi-agent society, where the interactions are dueto the information exchange. In Sec. 4, we very briefly summarize the main ingredientsdetermining the probability measure of a single given agent, which is based on quantum decisiontheory. Section 5 presents an extension of the theory of quantum decision making to the caseof interacting agents forming a society. In these first sections, we introduce the notationsand definitions that are necessary for the following considerations. Section 6 treats the caseof agents with long-term memory. Section 7 considers the case of agents with reconstructivememory, while Sec. 8 studies the situation of agents with short-term memory. In Section 9, weshow that the society of decision makers, acting by the rules of QDT, can be interpreted as anovel type of networks – a quantum information network or quantum intelligence network. Toillustrate the practical usage of the approach, in Sec. 10, we consider a concrete example ofdynamic decision making. We analyze the dynamic disjunction effect, predicting the behaviorof decision makers both, at the initial stage as well as in the long run, when there decisionsconverge to a common consensus. Our numerical predictions are in very good agreement withempirical data. Section 11 concludes.

3 Interaction Structure for Agents Exchanging Informa-

tion

Here we explain the main ideas of how the interaction structure between agents exchanginginformation is organized.

Let us consider decision making in a society of N agents choosing between NL prospects(lotteries) πn, with n = 1, 2, . . . , NL. At time t, a j-th agent generates a probability measurePj(t) ≡ {pj(πn, t)} giving the probabilities with which the prospects are to be chosen. As aresult of the human decision making duality, the probability measure consists of two parts, arational part quantifying the utility of the considered prospects and forming a classical proba-bility measure Fj(t) ≡ {fj(πn, t)} and a quantum part Qj(t) ≡ {qj(πn, t) : n = 1, 2, . . . , NL}characterizing irrational emotions and subconscious feelings, making a prospect subjectivelyattractive or not for an agent.

Explicit definitions required for the practical usage will be given in the following sections.

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Figure 1: Information exchange between two agents.

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Figure 2: Simplified scheme of interaction through information exchange between two agents.

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Figure 3: Scheme of interaction through information exchange between three agents.

Meanwhile, it is sufficient to remember that the probability measure, Pj(t) generated by eachagent, is the union of Fj(t) and Qj(t).

The exchange of information between the agents implies that each agent receives informationon the probabilities generated by other agents. Thus the information exchange between twoagents is represented by a directed graph in Fig. 1, showing that the first agent receivesinformation from the second agent, and the latter, from the first one. For simplicity, thisgraph can be represented as in Fig. 2. For three agents, their interactions through informationexchange can be shown as in Fig. 3. Similarly, the interaction scheme can be extended to manyagents.

After receiving information from other members of the society, each agent generates thenew probability measure that becomes available to other agents, and so on. Schematically, thedynamics of decision making with information exchange between two agents is shown in Fig. 4.The temporal variation of the probability measure Pj(t) includes the evolution of two parts, theclassical probability measure Fj(t), describing the utility or reward, whose dynamics is governedby the utility maximization, and the dynamics of the quantum part Qj(t) characterizing thevariation of emotions. Generally, the two types of dynamics do not need to be necessarilyconnected. Recall that a number of experimental studies [43–50] demonstrated the evolutionof behavioral effects, caused by information exchange, under a fixed utility.

The convergence property of the decision making dynamics depends on the specificationof the agents’ memory. When the process converges, then a j-th decision maker comes to a

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Figure 4: Dynamics of multistep decision making governed by information exchange betweentwo agents. The last step represents the asymptotic convergence to fixed values of the proba-bilities that contribute to the stationary values of the probabilities for a specific agent to choosea given prospect.

stationary probability measure P∗j composed of the set of probabilities p∗j (πn), each of which

implies the probability of choosing a prospect πn. Comparing these probabilities defines theoptimal, for the j-th agent, prospect π∗

j for which the probability is maximized: p∗j (π∗j ) =

maxn{p∗j(πn)}. As will be shown below, for short-term memory, the decision making process

may become not convergent.

4 Probability Measure Generated by a Single Agent

The theory of quantum probability distributions, generated by separate agents, has been de-veloped in the previous papers [37, 39, 77–81]. The mathematical techniques of the theory areequally applicable for describing quantum measurements as well as quantum decision mak-ing [54,55,77]. As far as all mathematical details have been thoroughly exposed in our previouspublications, here we only briefly recall the main points of the theory and introduce notationsthat will be necessary for the generalization in the next sections to the case of multi-agentsystems.

The information processing by a single agent proceeds as follows. An agent receives infor-mation characterized by different types of events An and Bα, labelled by the indices n and α,and represented by quantum states,

An → |n〉 , Bα → |α〉 (1)

in the corresponding Hilbert spaces

HA = span{|n〉} , HB = span{|α〉} . (2)

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The difference between the events is in their degree of testability. The events An are opera-tionally testable, allowing for their unambiguous observation or measurement. In contrast, theevents Bα are not operationally testable, being characterized by random amplitudes bα. Theset of the random events B = {Bα} is an inconclusive event [53–55] represented by a state inthe Hilbert space HB,

B = {Bα} → |B〉 =∑

α

bα|α〉 . (3)

Generally, information is provided through composite events

πn = Am

⊗

B → |πn〉 = |nB〉 (4)

represented by states in the Hilbert space

H = HA

⊗

HB . (5)

These composite events are termed prospects. Note that the states |πn〉 are not necessarilyorthonormalized.

The prospects induce the prospect operators

P (πn) ≡ |πn〉〈πn| , (6)

which, in general, are not necessarily projectors, because the related prospect states may benot orthonormalized. The set of prospect operators forms a positive operator-valued measure[82,83]. The family of all prospect operators is equivalent to the algebra of local observables inquantum theory.

An agent is associated with a strategic state ρ that is non-negative and normalized, suchthat Trρ = 1, with the trace operation over space (5). The agent generates the prospectprobabilities

p(πn) = Trρ P (πn) (7)

forming a probability measure, so that

∑

n

p(πn) = 1 , 0 ≤ p(πn) ≤ 1 . (8)

A prospect probability consists of two parts,

p(πn) = f(πn) + q(πn) , (9)

where the first termf(πn) =

∑

α

| bα |2〈nα| ρ |nα〉 (10)

is positive-definite, while the second term

q(πn) =∑

α6=β

b∗αbβ〈nα| ρ |nβ〉 (11)

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is not positive defined. The appearance in the probability of a not positive-defined term is dueto the quantum definition of the probability and the interference of inconclusive events. Sucha quantum term would be absent in classical probability.

Quantum information processing has to include, as a particular case, classical processing.Therefore the quantum-classical correspondence principle [84] has to be valid. In our case,this requires that the quantum probability should reduce to the classical probability when thequantum term tends to zero:

p(πn) → f(πn) , q(πn) → 0 . (12)

Thus, the positive-definite terms play the role of classical probabilities, with the standardproperties

∑

n

f(πn) = 1 , 0 ≤ f(πn) ≤ 1 . (13)

Then the quantum terms satisfy the conditions

∑

n

q(πn) = 0 , −1 ≤ q(πn) ≤ 1 . (14)

Keeping in mind conditions (8) and (13), we see that the quantum term can be either negative,such that

−f(πn) ≤ q(πn) ≤ 0 ,

or positive, when0 ≤ q(πn) ≤ 1− f(πn) .

In QDT, the term f(πn) describes the utility of the prospect πn, which justifies to call it theutility factor, while q(πn) characterizes the attractiveness of the prospect and is called theattraction factor. Two terms, appearing in decision making, reflect the duality of the latter,including the logical conscious evaluation of the prospect utility and subconscious intuitiveestimation of its attractiveness. The properties of these terms and their practical determinationhave been described in detail in the previous papers [37–39].

In this way, the information processing by a single agent consists in generating the proba-bility measure over the given information characterized by the set of prospects. In the case ofsocial networks, the classical probability f(πn) can be defined by using a kind of Luce choiceaxiom [41,85]. More generally, it can be defined as a minimizer of an information functional orby conditional entropy maximization [37, 39]. Note that unconditional entropy maximizationmay occur to be not sufficient for correctly defining probabilities, in which case one has torespect additional constraints making the ensemble representative [37, 39, 86].

The quantum term q(πn) depends on the amount of information M received by the agent.In the absence of information, the initial value q0(πn) can be random, although satisfyingconditions (14). After getting the amount of information M , the quantum term can be written[37, 71] as

q(πn) = q0(πn) exp(−M) . (15)

In quantum decision theory, the quantum term characterizes the attractiveness of the consideredprospects, and it is thus called the attraction factor.

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Upon the receipt of additional information, the non-utility part of the probability measuregenerated by the agent is reduced. In the previous papers [37, 71], we have considered thesimple case where the amount of information M is the same for all agents, being given by anexternal source or by a control algorithm. Now we shall study a more realistic situation, whenthe agents receive information by exchanging it with other members of the society.

5 Generalization to Multiple Agents Exchanging Infor-

mation

5.1 Arbitrary number of prospects

Let us now generalize the procedure of information processing by a single agent to the case ofmany agents forming a society. Let the agents be enumerated by j = 1, 2, . . . , N . At the initialtime, the information is presented through a set of prospects πn enumerated by n = 1, 2, . . . , NL.The process of getting additional information requires time, so that the probability measuregenerated by the j-th agent is a function of time:

pj(πn, t) ≡ pjn(t) . (16)

Respectively, the utility and attraction factors are also functions of time:

fj(πn, t) ≡ fjn(t) , qj(πn, t) ≡ qjn(t) . (17)

There always exist the normalization conditions for the probabilities,

NL∑

n=1

pjn(t) = 1 , 0 ≤ pjn(t) ≤ 1 , (18)

for the utility factors,NL∑

n=1

fjn(t) = 1 , 0 ≤ fjn(t) ≤ 1 , (19)

and for the attraction factors,

NL∑

n=1

qjn(t) = 0 , −1 ≤ qjn(t) ≤ 1 . (20)

In realistic situations, the probability measure is not immediately generated by the j-thagent but after a delay time τ . Taking this into account and measuring time in units of τ , wecan write

pjn(t + 1) = fjn(t) + qjn(t) . (21)

The first term is a utility factor, whose value is prescribed by the objective utility of theproblem, hence assumed to be defined by prescribed rules [37–39]. The second term has beenshown [37, 71] to be a function of the information measure Mj(t) quantifying the amount ofinformation received by the j-th agent until time t,

qjn(t) = qjn(0) exp{−Mj(t)} . (22)

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The initial value qjn(0) is a random quantity satisfying the above normalization conditions.It is worth stressing the importance of taking into account the delay in receiving the infor-

mation, which makes the consideration realistic, since in real life acquiring information alwaysrequires finite time. On the other side, the occurrence of time delay can essentially change thedynamics of multi-agent systems.

The total information received by the j-th agent until time t can be represented as a sum

Mj(t) =t

∑

k=1

ϕj(t, k)µj(k) , (23)

where µj(k) is the information gained by the j-th agent at the k-th time step and ϕj(t, k) is amemory operator to be specified below, which defines how much of the information at time k

is retained at the later time t > k.Information measures can be chosen in different ways, for instance as transfer entropy [87–90]

or in the standard form of the Kullback-Leibler relative information [91–93]. We prefer the latterapproach. Then, the information gain for the j-th agent concerning the n-th prospect can bewritten as

µj(k) =

NL∑

n=1

pjn(k) lnpjn(k)

hjn(k), (24)

where

hjn(k) =1

N − 1

∑

i(6=j)

pin(k) (25)

is the average probability for the n-th prospect over all agents of the society, except the j-thagent.

5.2 Case of two prospects

The problem simplifies when there are only two prospects (NL = 2). This is actually a situationthat is very often met, corresponding to choosing between just two alternatives, when decidingon “yes” or “no”. Then, keeping in mind the normalization conditions (18) to (20), it issufficient to consider only one of the prospects, say π1, thus simplifying the notation for thefirst-prospect probability,

pj(t) ≡ pj1(t) = pj(π1, t) , (26)

since the second-prospect probability is

pj(π2, t) ≡ pj2(t) = 1− pj(t) . (27)

Similar simplified notations can be used for the utility factors,

fj(t) ≡ fj1(t) = fj(π1, t) ,

fj(π2, t) ≡ fj2(t) = 1− fj(t) , (28)

and the attraction factors,qj(t) ≡ qj1(t) = qj(π1, t) ,

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qj(π2, t) ≡ qj2(t) = −qj(t) , (29)

each associated with the j-th agent.The utility factors, reflecting the basic utility of prospects, can be treated as time-independent,

which we shall use in what follows:

fj(t) = fj(0) = fj . (30)

To proceed further, we need to specify the memory operator characterizing the type ofmemory typical of the considered agents.

5.3 Types of memory

Different types of memory have been classified in psychology and neurobiology [94–96]. Forthe purpose of the present consideration, we need to distinguish the types of memory definingin different ways the temporal behavior of the information measure Mj(t). It is possible todistinguish three qualitatively different kinds of temporal memory:

(i) Long-term memory, when there is no memory attenuation and all information, gainedin the past, is perfectly retained. This implies that the memory operator is identical to unity,ϕj(t, k) = 1.

(ii) Reconstructive memory, when only the closest events are kept in mind, while all previoustemporal blanks are filled in by reconstructing the past by analogy with the present time [97–99].This kind of memory can be represented by the action of the memory operator ϕj(t, k)µj(k) =µj(t).

(iii) Short-term memory, when the memory of the past quickly attenuates. In the very short-term variant, only the memory from the last step is retained, while nothing is remembered fromthe previous temporal steps. This assumes the local memory operator ϕj(t, k) = δkt, whichdefines the Markov-type memory in decision process.

Below, we analyze in turn the behavior of the agents possessing these three principallydifferent types of memory.

6 Agents with Long-Term Memory

Let us consider the limiting case of long-term memory corresponding to a non-decaying memorycharacterized by the memory operator ϕj(t, k) = 1. As a consequence, the total accumulatedinformation (23) is the sum of the information gains at each temporal step,

Mj(t) =t

∑

k=1

µj(k) . (31)

For simplicity, we again consider the case of two prospects (NL = 2). Moreover, we assumethat agents can be divided into two groups, so that all agents within each group have thesame initiation conditions. This amounts to consider a situation with two “group-agents”, or“superagents” representing two agent groups, exchanging information with each other. We thushave the equations for the probabilities:

pj(t + 1) = fj + qj(t) , (32)

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with j = 1, 2. The utility factors fj are assumed to be constants characterizing the intrinsicutility of the prospects for the agents. Generally, the agents of a society are heterogeneous, andthus the values of fj are different for different agents. The attraction factors vary with time as

qj(t) = qj(0) exp{−Mj(t)} , (33)

where qj(0) are the chosen initial conditions. The temporal variation is influenced by theaccumulated information defined in Eq. (31). The information gain (24) of the j-th agent, ata k-th step, reads as

µj(k) = pj(k) lnpj(k)

pi(k)+ [1− pj(k)] ln

1− pj(k)

1− pi(k), (34)

with j 6= i. The initial conditions for the probabilities are

pj(0) = fj + qj(0) . (35)

We also assume that there is no additional information at the beginning, when t = 0, sothat Mj(0) = 0. Time varies in discrete steps as t = 0, 1, 2, . . .. Note that, for any k ≥ 0, theinformation gains µj(k) defined in (34) are always positive. It follows that the total informationgain Mj(t) > 0 is positive for j = 1, 2 and t ≥ 0.

Analyzing the behavior of the probabilities for varying initial conditions, we find that thereare two qualitatively different types of solutions, depending on whether there exists an initialconflict between the utility factors and probabilities, or not. The existence or absence of aninitial conflict is understood in the following sense.

(i) There is no initial conflict when either

f1 > f2 , p1(0) > p2(0) (no conflict) , (36)

orf1 < f2 , p1(0) < p2(0) (no conflict) . (37)

This implies that, at the initial time, the agents already prefer the prospect with a larger utility.(ii) There exists an initial conflict when either

f1 < f2 , p1(0) > p2(0) (conflict) , (38)

orf1 > f2 , p1(0) < p2(0) (conflict) . (39)

This occurs when, at the initial time, the agents prefer the less useful prospects. Let us recallthat such a conflicting choice very often occurs in decision making under uncertainty [37–39].

If there is no initial conflict, the probabilities pj(t) tend to the corresponding fj , whent → ∞, as is shown in Fig. 5. But in the presence of an initial conflict, the numerical analysisof equations (32) to (35) shows that both probabilities tend to the common limit p∗, defined as

p∗ =f1q2(0)− f2q1(0)

q2(0)− q1(0), (40)

within an accuracy of 10−3. This is illustrated in Fig. 6.

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It is worth stressing that the consensual limit (40), under conflicting initial conditions (38)or (39), satisfies the inequalities

0 < p∗ < 1 ,

which is proved as follows.Suppose that conditions (38) are valid. Then, using (35), we get

q2(0)− q1(0) < f1 − f2 < 0 .

In the case of conditions (39), we find

0 < f1 − f2 < q2(0)− q1(0) .

In both these cases,

0 <f1 − f2

q2(0)− q1(0)< 1 .

Since 0 ≤ pj(0) ≤ 1 and 0 ≤ fj ≤ 1, with −1 ≤ qj(0) ≤ 1, for any C ∈ (0, 1), we have

0 < fj + Cqj(0) < 1 (0 < C < 1) .

Substituting here

C =f1 − f2

q2(0)− q1(0),

we come to the result 0 < p∗ < 1.

7 Agents with Reconstructive Memory

Let us consider the situation corresponding to agents with reconstructive memory, when theyput large weight to the last information gain at time t, and use it to fill up the blanks over allprevious times. This implies the action of the memory operator ϕj(t, k)µj(k) = µj(t). Thenthe total information received by the j-th agent at time t is

Mj(t) =

t∑

k=1

µj(t) = µj(t) t . (41)

Except for the total information (41), all other formulas are the same as in the previous section.The analysis shows that, for all initial conditions with different fj and any qj , the probabil-

ities pj(t) tend to their utility factors fj,

limt→∞

pj(t) = fj . (42)

However, this tendency can be of three types. If the initial conditions are not conflicting, in thesense of inequalities (36) or (37), the tendency can be either monotonic, as in Fig. 7 or withoscillations, as in Figs 8 and 9. But when the initial conditions are conflicting, in the sense ofEqs. (38) or (39), then there always appear oscillations at intermediate stages, as is shown inFigs. 10 and 11. In the marginal case, when f1 and f2 coincide, the oscillations last forever,

14

0 50 100 150 200 2500

0.2

0.4

0.6

0.8

1

t

pj(t) (a)

p1(t)

p2(t)

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

t

pj(t)

p1(t)

p2(t)

(b)

Figure 5: Long-term memory under no conflict. Dynamics of the probabilities p1(t) for su-peragent 1 (solid line) and p2(t) for superagent 2 (dashed line), when there is no initial con-flict, for two cases of close and rather different initial probabilities: (a) initial conditions aref1 = 0.8 > f2 = 0.49, with q1(0) = −0.1 and q2(0) = 0.2, so that the initial probabilitiesp1(0) = 0.7 > p2(0) = 0.69 are close to each other; (b) initial conditions are f1 = 0.5 > f2 = 0.3,with q1(0) = 0.4 and q2(0) = −0.2, so that the initial probabilities p1(0) = 0.9 > p2(0) = 0.1are far from each other. In both cases, there is no conflict and the probabilities pj(t) tend totheir respective limits fj .

0 20 40 60 80 1000.6

0.7

0.8

0.9

1

t

pj(t)

p2(t)

p1(t)

(a)

0 100 200 300 400 5000.45

0.46

0.47

0.48

0.49

0.5

t

pj(t)

p1(t)

p2(t)

(b)

Figure 6: Long-term memory under conflict. Dynamics of the probabilities p1(t) (solid line)and p2(t) (dashed line) in the presence of an initial conflict: (a) initial conditions are f1 =0.8 > f2 = 0.7, with q1(0) = −0.1 and q2(0) = 0.2, so that p1(0) = 0.7 < p2(0) = 0.9, theconsensual limit is p∗ = 0.77; (b) initial conditions are f1 = 0.2 < f2 = 0.65, with q1(0) = 0.3and q2(0) = −0.2, so that p1(0) = 0.5 > p2(0) = 0.45, the consensual limit is p∗ = 0.47.

as illustrated in Fig. 9. However, this regime is not stable and disappears under any smalldifference between the fj ’s, leading to damped oscillations.

15

0 4 8 12 16 200

0.2

0.4

0.6

0.8

1

t

pj(t) (a)p

1(t)

p2(t)

0 4 8 12 16 20−8

−7

−6

−5

−4

−3

−2

−1

0

1

t

λj (b)

λ2

λ1

Figure 7: Reconstructive memory under no conflict. Dynamics of the probabilities pj(t) andlocal Lyapunov exponents for the initial conditions f1 = 0.8 > f2 = 0.4, with q1(0) = −0.1 andq2(0) = 0.2, so that p1(0) = 0.7 > p2(0) = 0.6: (a) probabilities; (b) local Lyapunov exponents.

In the case where there is no conflict, the oscillations appear more regular while, in the casewith an initial conflict, they look more chaotic. In order to understand better the oscillationcharacteristics, we calculate the local Lyapunov exponents, as is explained in Ref. [100]. Forthis purpose, we define the multiplier matrix m(t) ≡ [mij(t)] with the elements

mij(t) =δpi(t+ 1)

δpj(t)=

δqi(t)

δpj(t), (43)

whose eigenvalues are

e1,2 =1

2

[

Tr m±√

(Tr m)2 − 4det m]

, (44)

whereTr m ≡ m11 +m22 , det m ≡ m11m22 −m12m21 .

Then for the local Lyapunov exponents we have

λ1,2(t) ≡1

tln |e1,2| . (45)

At the intermediate stage characterized by strong oscillations of the probabilities, at leastone of the local Lyapunov exponents becomes transiently positive, thus, demonstrating localinstability. But in the long run, the dynamics is always stable since, at large t, the Lyapunovexponents are negative, as is seen from Figs. 7 to 11.

8 Agents with Short-Term Memory

The other limiting case is the society of agents with very short-term memory, remembering onlythe information from the last temporal step and keeping no track of any previous information

16

0 20 40 60 80 1000.1

0.2

0.3

0.4

0.5

0.6

0.7

t

pj(t) (a)

p2(t)

p1(t)

0 20 40 60 80 100−2

−1.5

−1

−0.5

0

0.5

t

λj

(b)

λ1

λ2

Figure 8: Reconstructive memory under no conflict. Weak oscillations of the probabilities (a)and smooth local Lyapunov exponents (b) for the initial conditions f1 = 0.3 < f2 = 0.4, withq1(0) = −0.1 and q2(0) = 0.2, so that p1(0) = 0.2 < p2(0) = 0.6.

0 50 100 150 2000

0.2

0.4

0.6

0.8

1

t

pj(t) (a)

p2(t)

p1(t)

0 50 100 150 200−1.5

−1

−0.5

0

0.5

t

λj

(b)λ2

λ1

Figure 9: Reconstructive memory under no conflict. Strong oscillations of the probabilities (a)and oscillating local Lyapunov exponents (b) for the initial conditions f1 = 0.6 > f2 = 0.55,with q1(0) = 0.3 and q2(0) = −0.2, so that p1(0) = 0.9 > p2(0) = 0.35.

gains. This is described by the local memory operator ϕj(t, k) = δkt. As a result, the totalinformation coincides with the information gain from the last step,

Mj(t) = µj(t) . (46)

All other equations are the same as above, with the same initial conditions. In particular, atthe initial time there is no yet any information, as has been assumed above, so that in thepresent case we have

Mj(0) = µj(0) = 0 . (47)

17

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

t

pj(t) (a)

p1(t)

p2(t)

0 10 20 30 40 50−5

−4

−3

−2

−1

0

1

t

λj

(b)

λ2λ

1

Figure 10: Reconstructive memory under conflict. Dynamics of the probabilities (a) and localLyapunov exponents (b) for the initial conditions f1 = 0.2 < f2 = 0.55, with q1(0) = 0.3 andq2(0) = −0.2, so that p1(0) = 0.5 > p2(0) = 0.35.

0 50 100 150 2000.5

0.6

0.7

0.8

0.9

1

t

pj(t) (a)

p1(t)

p2(t)

0 50 100 150 200−2

−1.5

−1

−0.5

0

0.5

t

λj

(b)

λ1

λ2

Figure 11: Reconstructive memory under conflict. Dynamics of the probabilities (a) and localLyapunov exponents (b) for the initial conditions f1 = 0.8 > f2 = 0.7, with q1(0) = −0.1 andq2(0) = 0.2, so that p1(0) = 0.7 < p2(0) = 0.9. At the intermediate stage, the probabilitiesexhibit chaotic oscillations.

Numerical analysis shows that there exist three types of dynamics. One is a smooth tendencyto limiting states from below or from above, as is illustrated in Fig. 12. The second type isthe tendency to limiting states through several or a number of oscillations, as in Fig. 13.And the third kind of dynamics is the occurrence of everlasting oscillations, intersecting ornot intersecting with each other, as is shown in Fig. 14. The existence of these three typesof dynamics happens for positive as well as for negative attraction factors. The limits of thetrajectories at infinite time, when they exist, depend on initial conditions and do not equal the

18

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

t

pj(t)

p2(t)

p1(t)

(a)

0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

t

pj(t)

p1(t)

p2(t)

(b)

Figure 12: Short-term memory. Dynamics of the probabilities for the initial conditions: (a)f1 = 0.99, f2 = 0.1, with q1(0) = −0.4 and q2(0) = 0.2, so that p1(0) = 0.59 > p2(0) = 0.3, and(b) f1 = 0.2, f2 = 0.6, with q1(0) = −0.15 and q2(0) = −0.4, so that p1(0) = 0.05 < p2(0) = 0.2.The probabilities exhibit monotonic behavior irrespectively of conflict or no-conflict initialconditions.

0 5 10 15 200

0.2

0.4

0.6

0.8

1

t

pj(t)

p2(t)p

1(t)

(a)

0 10 20 30 40 50 60 700

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

t

pj(t)

p2(t)

p1(t)

(b)

Figure 13: Short-term memory. Dynamics of the probabilities for the initial conditions (a)f1 = 0.5, f2 = 0.6, with q1(0) = −0.4 and q2(0) = 0.15, so that p1(0) = 0.1 < p2(0) = 0.75, and(b) f1 = 0.6, f2 = 0.9, with q1(0) = −0.1 and q2(0) = −0.89, so that p1(0) = 0.5 > p2(0) = 0.01.The probabilities exhibit decaying oscillations irrespectively of conflict or no-conflict initialconditions.

related utility factors. Slightly varying the initial conditions leads to small finite changes in thelimiting trajectory values, so that the motion is Lyapunov stable. But it is not asymptoticallystable, contrary to the cases of societies of agents with long-term or reconstructive memories.The existence of stable everlasting oscillations is also typical only for the agents with short-termmemory, but does not occur for agents with other types of memory.

19

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

t

pj(t) p

2(t) (a)

p1(t)

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

t

pj(t)

p1(t)

(b)p2(t)

Figure 14: Short-term memory. Dynamics of the probabilities for the initial conditions: (a)f1 = 0.8, f2 = 0.6, with q1(0) = −0.7 and q2(0) = 0.2, and (b) f1 = 0.99, f2 = 0.5, with q1(0) =−0.8 and q2(0) = 0.4. The probabilities exhibit everlasting oscillations without intersection (a)or with intersecting probabilities (b). The everlasting oscillations exist irrespectively of conflictor no-conflict initial conditions.

9 Societies of Decision Makers as Intelligence Networks

A society of agents generating, in the course of decision making, probability distributions overa given set of prospects, and interacting with each other through information exchange, isanalogous to a network, which can be termed a social information network. Recall that themathematics of taking decisions according to the rules of QDT is equivalent to the activity ofan artificial intelligence, since an artificial intelligence, mimicking human cognition, has to takeaccount of the conscious-subconscious duality typical of human brains [37–39]. And this dualityis well represented by QDT, where the prospect probabilities consist of two terms, utility factorand attraction factor.

Therefore the society of decision makers, acting in the frame of QDT, is equivalent to aquantum information network or quantum intelligence network. In other words, such a networkcan characterize a composite artificial intelligence, or superintelligence, formed by an assemblyof intelligences. In that sense, it is essentially more complicated than other known networks.

There exists a large variety of networks: internet and web networks, computer networks,social networks, business networks, radio and television networks, electric networks, phone-callnetworks, citation networks, linguistic networks, ecological and biological networks, cellularnetworks, protein networks, neural networks, etc. [101–105]. All above mentioned networks arecharacterized by classical models. There are also quantum networks that are usually modelledby quantum spin systems or exciton systems that can be reduced to spin models [106, 107].

In mathematical terms, a network is defined as follows. There is a set A = {aj} of nodes,or vertices, or agents, enumerated by an index j = 1, 2, . . . , N and a set L = {lij} of edges, orlines, or links, or arcs, connecting the nodes. The pair {A,L} is called a graph. The graph istermed ‘directed’ if there is a map m : L −→ R+ ≡ [0,∞). For a directed graph, an edge lijmay be different from lji. A pair of edges lij and lji, connecting two nodes, is called a circuit.Generally, a network is the triple {A,L, m} representing a directed graph. In standard quantum

20

networks, nodes are represented by quantum operators, usually by spin or quasi-spin operators,and their interactions by parameters of an interaction matrix. Signals are characterized bywave functions. A node operator transforms a given wave function into another function, thusrealizing a gate [106, 107].

A society of decision makers, or an assembly of artificial intelligences, functioning by therules of QDT, can be classified as a network. The set of nodes is represented by decision makers,whose links are described by the exchanges of information. The corresponding graph is directed,since the information received by an agent, generally speaking, can differ from that received byother agents. Keeping in mind that the network dynamics generates probability distributionsvarying with time, it is straightforward to interpret this activity as time-dependent decisionmaking [108]. The delayed information exchange and link directionality reflect the causality ofinteractions [109]. The network dynamics describes the information flow [110]. This is why theintelligence networks can also be termed the information networks.

The principal difference of a quantum intelligence network from the usual quantum networks,with the nodes given by spins or atoms, is that each agent in an intelligence network makesdecisions, while the standard nodes, like spins, do not do this. The type of intelligence networkswe propose represents networks of agents acting by the quantum rules of QDT, with outcomesthat are not just simple two-bit signals, as yes or no, and which could be modelled by spinsystems with spins up or down. In contrast, in the quantum network proposed here, eachagent generates a quantum probability measure over a given set of prospects. Being basedon common mathematics, the intelligence networks can characterize either an assembly ofinteracting human decision makers, or a cluster of several quantum computers, or the activityof a composite artificial quantum intelligence consisting of several parts, each of which is anintelligence itself. The network dynamics models the formation of a collective decision as acollective outcome developing in the process of information exchange.

10 Example of Dynamic Decision Making

10.1 Position of the problem

To illustrate the usage of the theory, we treat below a particular case of dynamic decisionmaking. There are three main problems in describing such a repeated decision making.

(i) At the beginning, decisions under uncertainty often contradict utility theory, as is dis-cussed in the Introduction. Decision makers often choose the prospect with smaller utilityfactor, in contradiction with the prescription of utility theory. Why this is so and how tocorrectly predict the behavioral choice of decision makers at the initial stage?

(ii) There exist numerous empirical works [43–50,111,112] showing that the difference betweenthe probability of choosing a prospect and the utility factor decrease with time. For asociety of decision makers, we define the experimental probability of a given prospect asthe fraction of agents choosing that prospect. How to prove mathematically that thisdifference between the prospect probability and utility factor does diminish with time?

(iii) Empirical works [43–50,111,112] also show that, even if at the initial time the behavioralprobabilities strongly deviate from the prescription of utility theory, in the long run they

21

converge to the related utility factors. The convergence of initially dispersed opinions toa common consensus is the basis of the so-called Delphi method that has been devised inorder to obtain the most reliable opinion consensus of a group of experts by subjectingthem to a series of questionnaires interspersed with discussions providing feedback [113,114]. The empirical studies raise the following question: How to describe the fact that,for real decision makers exchanging information, the initially dispersed opinions convergeto a common consensus and, furthermore, the limiting prospect probabilities converge toan objective utility factor?

We illustrate the above questions by a concrete example involving the dynamic disjunctioneffect, and compare our theoretical predictions with empirical data. Since the disjunction effectis specified as a violation of the sure-thing principle [115], we first briefly recall the meaningof this principle. Then we give a real-life example of the effect, concentrating on the case ofa composite game studied by Tversky and Shafir [116]. After this, we analyze the disjunctioneffect dynamics by using the Tversky and Shafir data to specify the numerical values neededin the numerical solution of our equations to obtain the limiting values of the correspondingprospect probabilities.

10.2 Sure-thing principle

Let us consider a two-stage composite prospect. In the first stage, the events B1 and B2 canoccur. One can either know for sure that one of these concrete events has occurred or one canonly be aware that one of them has happened, not knowing which of them actually occurred.The latter case can be denoted as B = {B1, B2}. At the second stage, either an event A1 or A2

occurs. We are thus confronted with the composite prospects πn = An

⊗

B, with n = 1, 2. Thesure-thing principle [115] states: If the alternative A1 is preferred to the alternative A2, whenan event B1 occurs, and it is also preferred to A2, when an event B2 occurs, then A1 should bepreferred to A2, when it is not known which of the events, either B1 or B2, has occurred.

This principle is easily illustrated in classical probability theory, where the probability ofthe prospect πn is

f(πn) = p(AnB1) + p(AnB2) . (48)

Then, if p(A1Bα) > p(A2Bα) for α = 1, 2, it follows that f(π1) > f(π2), which explains thesure-thing principle.

10.3 Disjunction effect

However, empirical studies have discovered numerous violations of the sure-thing principle,which was called disjunction effect [116–119]. Such violations are typical for two-step compositegames of the following structure. First, a group of agents takes part in a game, where eachagent can either win (event B1) or loose (event B2), with equal probability p(Bα) = 0.5. Theyare then invited to participate in a second game, having the right either to accept the secondgame (event A1) or to refuse it (event A2). The second stage is realized in different variants:One can either accept or decline the second game under the condition of knowing the result ofthe first game. Or one can either accept or decline the second game without knowing the result

22

of the first game. We define the probabilities, as usual, in the frequentist sense as the fractionsof individuals taking the corresponding decision.

In their experimental studies, Tversky and Shafir [116] find that the fraction of peopleaccepting the second game, under the condition that the first was won, is p(A1|B1) = 0.69 andthe fraction of those accepting the second game, under the condition that the first was lost isp(A1|B2) = 0.59. From the definition of conditional probabilities, one has the normalization

p(A1|Bα) + p(A2|Bα) = 1 (α = 1, 2) ,

which yields p(A2|B1) = 0.31 and p(A2|B1) = 0.41. Therefore the related joint probabilitiesp(AnBα) = p(An|Bα)p(Bα) are

p(A1B1) = 0.345 , p(A1B2) = 0.295 ,

p(A2B1) = 0.155 , p(A2B2) = 0.205 .

According to equation (48), one gets

f(π1) = 0.64 , f(π2) = 0.36 .

This implies that, by classical theory, the probability of accepting the second game, not knowingthe results of the first one, is larger than that of refusing the second game, not knowing theresult of the first game.

Surprisingly, in their experiments, Tversky and Shafir [116] observe that human decisionmakers behave opposite to the prescription of the sure-thing principle, with the majority refus-ing the second game, if the result of the first one is not known,

pexp(π1) = 0.36 , pexp(π2) = 0.64 .

But in QDT, such a paradox does not appear. Recall that, in QDT, the probability of aprospect is the sum of f(πn) and q(πn). The sign of the attraction factor is prescribed by theuncertainty aversion [37–39,59], while its noninformative prior value is found [37–39] to satisfythe quarter law, with the average absolute value of the attraction factor |q| = 0.25. Thus, inQDT, we have

p(π1) = f(π1)− 0.25 = 0.39 ,

p(π2) = f(π2) + 0.25 = 0.61 .

These theoretical results for p(πn) are in very good agreement with the experimental datapexp(πn) by Tversky and Shafir, actually being indistinguishable within the accuracy of exper-imental statistics.

Similar results, demonstrating the disjunction effect, have been obtained in a variety of otherempirical studies having the same two-stage games structure [116–119]. All such paradoxes finda simple explanation in QDT, similarly to the case treated above.

23

10.4 Disjunction-effect dynamics

We now take the data from the previous subsection as initial conditions of our dynamicalequations of Sec. 5 and study the time dependence of the decision makers’ opinions.

There are two prospects. One is the prospect π1 = A1

⊗

B of accepting the second gamenot knowing the result of the first game. And the second is the prospect π2 = A2

⊗

B ofrefusing from the second game, when the result of the first game is not known. As is explainedin Sec. 5, using the normalization conditions for the probabilities, it is sufficient to considerone of the prospects, say π1.

Taking into account the heterogeneity of agents, we assume that there are two main groupsof agents differing in their initial opinions. The corresponding prospect probabilities are denotedas pj(t) ≡ pj(π1, t). Respectively, we use the abbreviated notation for the utility factors fj(t) ≡fj(π1, t) and for the attraction factors qj(t) ≡ qj(π1, t). The utility factor, being an objectivequantity, is assumed to be constant and defined as in the previous subsection: fj(t) = 0.64. Asa result of the inequalities 0 ≤ pj(t) ≤ 1 for the probabilities, the attraction factor can vary inthe range

−fj(t) ≤ qj(t) ≤ 1− fj(t) ,

which in the present case implies that −0.64 < qj(t) < 0.36. The initial values of the attractionfactors must fall within this interval.

We consider the more realistic situation where the decision makers take repeated decisionsand remember their previous choices. We solve the evolution equations of Sec. 5 for this typeof memory, taking as initial conditions for the attraction factors uniformly distributed valuestaken in the interval (−0.64, 0.36). The typical obtained behavior of the prospect probabilitiesfor different initial conditions are shown in Fig. 15.

Two important conclusions follow from the temporal behavior of the solutions. First, for anyinitial conditions, the difference between the probabilities pj(t) and the utility factor decreaseswith time. Second, all solutions, for arbitrary initial conditions, tend to the same limit 0.64.This demonstrate that, despite a large variation in the initial opinions, the agents come to acommon consensus by exchanging information in a multi-step procedure.

11 Conclusion

We have suggested a model of a society of decision makers taking their decisions according tothe quantum decision theory. This model generalizes the theory, formulated earlier for one-stepdecisions of single decision makers to multistep decision making of interacting agents in a society.Such a society of decision makers, acting according to the rules of QDT, represents a new kindof networks whose nodes are the QDT agents, interacting through the exchange of information.Each agent generates a probability measure over a set of prospects. The generated probabilitiesare defined according to quantum rules, which results in their representation as sums of twoterms, positive definite and sign indefinite. The quantum-classical correspondence principlemakes it possible to interpret the positive-definite term as a classical probability or utilityfactor. The sign-indefinite term, having purely quantum origin, represents the attractivenessof the given prospects, and is referred to as the attraction factor. The utility factor can beconsidered as given and time-invariant. In contrast, the attraction factor is random at theinitial time and varies with time due to the information exchange between the agents.

24

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

t

pj(t) (a)

3

1

2

2

3

1

500 2500 4500 6500 8500 100000.62

0.63

0.64

0.65

0.66

t

pj(t) (b)

32

1

21

3

Figure 15: Dynamic disjunction effect. The prospect probabilities p1(t) (solid lines) andp2(t) (dashed-doted lines) as functions of time t for different initial attraction factors: (1)q1(0) = −0.35 and q2(0) = 0.05; (2) q1(0) = −0.25 and q2(0) = 0.15; (3) q1(0) = −0.2 andq2(0) = 0.2. Figures (a) and (b) show the behaviour of the probabilities in the initial andlong-term time intervals t ∈ [0, 12] and t ∈ [500, 10000], respectively. When t → ∞, thenp1(t) → 0.64 from below, and p2(t) → 0.64 from above.

It is worth stressing that our approach is principally different form stochastic decision theory.The variants of such theories are usually based on deterministic decision theories complementedby random variables with given distributions [40,41]. Therefore such stochastic theories inheritthe same problems as deterministic theories embedded into them. Moreover, stochastic theoriesare descriptive, containing fitting parameters that need to be defined from empirical data. Inaddition, different stochastic specifications of the same deterministic core theory may generatevery different, and sometimes contradictory, conclusions [42]. In stochastic decision theory,random terms are usually added to expected utility, while in our case the quantum prospectprobability acquires an additional term, named attraction factor. What is the most important,our approach is not descriptive and does not contain fitting parameters.

Our model of a society of agents interacting through information exchange is principallynew and has never been considered before to the best of our knowledge. We have suggestedthe first model describing the opinion dynamics of real decision makers subject to behavioraleffects.

The probability dynamics depends on the form of the total information accumulated byeach agent. We have analyzed three qualitatively different limiting cases, long-term memory,reconstructive memory filling the past gaps on the basis of the most recent information gain,and short-term memory. In the case of long-term memory, the probability dynamics is smooth.When the initial conditions are not conflicting, the probabilities tend to their related utilityfactors (pj → fj). But when the initial conditions are conflicting, the probabilities tend to aconsensual common limit (pj → p∗). In all the cases, the motion is asymptotically Laypunovstable.

The probabilities for the agents with reconstructive memory always tend to their relatedutility factors (pj → fj), never exhibiting consensus. At intermediate stages of their dynamics,

25

the probabilities can experience oscillations, with positive local Lyapunov exponents, whichimplies local instability. But in the long run, the dynamics becomes asymptotically Lyapunovstable.

The society of agents with short-term memory behaves rather differently from the behaviorof the agents in the previous cases. Two types of dynamics can occur. The probability trajec-tories can tend to fixed points (pj → p∗j ) strongly depending on initial conditions, so that themotion is not asymptotically Lyapunov stable, although Lyapunov stable. These fixed pointsare different from the utility factors. The other type of motion is characterized by everlastingoscillations, which seems to be natural for agents with short-term memory, where there is noinformation accumulation, because of which the agents cannot make precise stable decisions.Also, for such agents, a consensus is impossible.

The dynamics of decisions is due to the time dependence of the attraction factors, sincethe utility factors have been taken as invariants representing the objective utility of the relatedprospects. The decrease of the attraction factors with time, due to the exchange of informationbetween decision makers, amounts to a reduction of the inconsistencies with utility theory.This decay of the deviations from utility theory, caused by the information exchange betweendecision makers, has been observed in many empirical studies [43–50, 113, 114].

From another side, as stressed above, a society of decision makers, acting in the frame ofQDT, is equivalent to an intelligence network, where the agents take decisions respecting theconscious-subconscious duality. Since this duality is typical of human decision makers, thenetworks, attempting to mimic the activity of human brains, are to be based on the rulesof QDT. This should be taken into account in creating artificial intelligence and networks ofartificial intelligences.

The usage of the approach is illustrated by a concrete realistic example involving the dy-namic disjunction effect. Treating this effect in the frame of our theory yields three importantpredictions. (i) The initial values of the prospect probabilities are predicted in very goodagreement with empirical data. (ii) The difference between the initial probabilities and thecorresponding utility factors monotonically decrease with time. (iii) In a society of hetero-geneous agents with long-term memory and randomly chosen initial conditions, all prospectprobabilities tend to a common limit coinciding with the utility factor, thus demonstrating theexistence of a consensus as a result of the repeated information exchange. These predictionsare in agreement with empirical data.

Acknowledgement

We acknowledge financial support from the ETH Zurich Risk Center.

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