1 Introduction and Related Literature
Modeling the behavior of multi-agent social systems and social networks has recently attracted a substantial amount of activities of researchers, as can be inferred from the review articles [1, 2, 3] and numerous original papers, of which we can cite just a few recent [4, 5, 6, 7, 8]. High interest to such a modeling is due to two reasons. First, being able of describing the behavior of societies is of great importance by itself. Second, modeling the collective interactions of autonomous multientity 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, inhabiting a shared environment, can decide their actions through mutual negotiations. There can be two types of agents, cooperative and self-interested. Cooperative agents cooperate with each other to reach a common goal . Self-interested agents try to maximize their own payoff without concern 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 decision and take action in a distributed manner . (iii) Studies of how the dependence among multi-agents can lead to emerging social structures, such as groups and agent clusters .
The main goal of a decision process in a multi-agent system is to find the optimal policy that maximizes expected utility or expected reward for either a single agent or for the society as a whole. Various models of multi-agent systems can be found in the books [14, 15, 16, 17, 18, 19].
The maximization of expected utility, expected reward, or other functionals is based on the assumption that agents are perfectly rational and no restrictions on computational power and available resources are imposed. However, since Simon , it is well known that only bounded rationality can exist, so that any real decision maker, in addition to having limited computational power and finite time for deliberations, is subject to such behavioral effects as irrational emotions, subconscious feelings, and subjective biases [20, 21, 22, 23].
The behavioral effects become especially important when decisions are made under uncertainty. There are two sides of uncertainty that can be caused either by objective lack of complete knowledge or by subjective preferences and biases. Even when the agents in the society are assumed to possess complete knowledge, they cannot become absolutely rational decision makers due to the inherent dual property of human behavior that combines conscious evaluation of utility with subconscious feelings and emotions .
Thus, real decision making is a complex procedure of dual nature, simultaneously integrating the rational, conscious, objective evaluation of utility with behavioral effects, such as irrational emotions, subconscious feelings, and subjective biases. This feature of realistic decision making can be called rational-irrational duality, or conscious-subconscious duality, or objective-subjective duality. Keeping in mind these points, we can call this feature the behavioral duality 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 account of behavioral effects [25, 26, 27]
. To take the latter into account, several modifications of utility theory 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, 29, 30, 31]. However, such so-called nonexpected utility models listed in reviews [29, 30, 31] have been refuted as being merely descriptive and having no predictive power [32, 33, 34, 35, 36]. A more detailed discussion can be found in Refs. [37, 38, 39].
As has been shown by Safra and Segal , none of non-expected utility theories can resolve all problems and paradoxes typical of decision making of humans. The best that could be achieved is a kind of fitting for interpreting just one or, in the best case, a few problems, while the other remained unexplained. In addition, spoiling the structure of expected utility theory results in the appearance of several complications and inconsistences. As has been concluded in the detailed analysis of Al-Najjar and Weinstein [35, 36], any variation of the classical expected utility theory “ends up creating more paradoxes and inconsistences than it resolves”.
Stochastic decision theories are usually based on deterministic decision theories complemented by random variables with given distributions[40, 41]. Therefore, such stochastic theories inherit the same problems as deterministic theories embedded into them. Moreover, stochastic theories are descriptive, containing fitting parameters that need to be defined from empirical data. In addition, different stochastic specifications of the same deterministic core theory may generate very different, and sometimes contradictory, conclusions .
One more difficulty in modeling decision making of real humans is that they often vary their decisions, under the same invariant expected utility, after information exchange between decision makers, as has been observed in many empirical studies [43, 44, 45, 46, 47, 48, 49, 50]. This implies that agent interactions through information exchange can influence decision makers emotions, without touching their evaluation of utility.
In all previous works, behavioral effects, when being considered, have been treated as stationary. The principal difference of the present paper from all previous publications is that we develop a model that takes into account the dual nature of decision making, and allows for the description of dynamical behavioral effects caused by agent interactions through information exchange.
2 Main Features of Quantum Approach
To take into account the dual nature of decision making, in the previous papers [37, 38, 39], we have formulated Quantum Decision Theory (QDT) as a mathematical approach for describing decision making under uncertainty. This approach generalizes classical utility theory  to the cases of decisions under strong uncertainty, when utility theory fails. The necessity of developing a new approach has been justified by numerous empirical observations proving the failure of classical utility making in realistic situations.
The mathematical basis of the QDT is the generalization of the von Neumann  theory of quantum measurements to the case of inconclusive quantum measurements  involving composite events with intermediate operationally untestable steps [54, 55]. In decision making, the intermediate inconclusive events characterize subconscious feelings and deliberations of a decision maker. It is this dual feature of decision making, including conscious logical reasoning and subconscious intuitive feelings, which explains the successful application of quantum techniques to describing human decision making, without the necessity of assuming any quantum nature of decision makers. The mathematics of quantum theory turns out to be well suited for describing the intrinsic conscious-subconscious duality of human cognition [56, 57].
It is worth mentioning that, after Bohr  put forward the idea that the dual nature of consciousness requires the use of quantum description, a number of attempts have been made to apply quantum rules to cognition, as can be inferred from the review works [59, 60, 61, 62, 63, 64, 65, 66]. The previous attempts, however, were limited by models fitted to particular cases and having no general mathematical structure valid for arbitrary events. Moreover, as has been recently shown [67, 68], these models contradict empirical facts.
Contrary to this, our QDT [37, 38, 39] is general, being formulated for arbitrary composite events. Its mathematics is based on the theory of quantum measurements [53, 54, 55], which allows it to be used also for the problem of creating artificial quantum intelligence [39, 69, 70]. As has been recently demonstrated [37, 38, 39], QDT is the sole theory that provides the possibility of making quantitative predictions, without fitting parameters, even in such difficult situations when classical decision making is not even applicable qualitatively.
In our previous papers [37, 38, 39], quantum decision theory has been formulated for the case of a single decision maker taking a one-step decision. But in a society of decision makers, the agents exchange information between each other and can make multistep decisions, with their decisions varying with time due to the information exchange. A simplified situation, when all decision makers receive the same information from outside, without mutual exchange, has been considered in [37, 71]. The necessity of developing, in the frame of QDT, a model describing the multistep decision-making procedure, taking into account mutual exchange of information between social agents, has been discussed in Ref. .
In the present paper, we suggest a principally important extension of QDT allowing for the description of dynamical collective effects. The main features, distinguishing this paper from the previous publications, are as follows.
The considered system is a society of several decision makers, and not just a single decision maker.
Each agent of the society is a decision maker transforming the information, received from other agents, according to quantum decision theory.
At each decision step, every agent generates an outcome that is a probability distribution over a given set of prospects.
Agent interactions are not parameterized by a fixed interaction matrix, but are characterized by an information functional over the probability distribution quantifying the amount of information gained from other agents.
The system is not static, but dynamical, the information functional and generated probability distributions are functions of time.
To describe a realistic situation, the information exchange is not simultaneous, but delayed. This means that, if the agents at time generate probability distributions , then the probability distribution that follows, which is caused by the information exchange, is generated at a delayed time .
The usage of the theory is illustrated by a concrete example involving the dynamic disjunction effect. The predicted numerical results are in very good agreement with empirical data.
The proposed system is not a set of simple quantum devices representing players, as those considered in quantum games [72, 73, 74, 75, 76], but rather a collection of complex subjects exchanging information, each representing a quantum intelligence. Therefore the society of decision makers, acting according to the rules of QDT, is a kind of collective quantum intelligence, or superintelligence.
Such a collective quantum intelligence can describe any ensemble of agents generating probability distributions according to the rules of QDT. This can be a human society making decisions on complex problems under uncertainty. This can also be a set of quantum computers, or a complex artificial intelligence composed of parts, each of which being itself an artificial quantum 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 quantum 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 characterizing the interaction structure of agents in a multi-agent society, where the interactions are due to the information exchange. In Sec. 4, we very briefly summarize the main ingredients determining the probability measure of a single given agent, which is based on quantum decision theory. Section 5 presents an extension of the theory of quantum decision making to the case of interacting agents forming a society. In these first sections, we introduce the notations and definitions that are necessary for the following considerations. Section 6 treats the case of agents with long-term memory. Section 7 considers the case of agents with reconstructive memory, while Sec. 8 studies the situation of agents with short-term memory. In Section 9, we show that the society of decision makers, acting by the rules of QDT, can be interpreted as a novel type of networks – a quantum information network or quantum intelligence network. To illustrate the practical usage of the approach, in Sec. 10, we consider a concrete example of dynamic decision making. We analyze the dynamic disjunction effect, predicting the behavior of decision makers both, at the initial stage as well as in the long run, when there decisions converge to a common consensus. Our numerical predictions are in very good agreement with empirical data. Section 11 concludes.
3 Interaction Structure for Agents Exchanging Information
Here we explain the main ideas of how the interaction structure between agents exchanging information is organized.
Let us consider decision making in a society of agents choosing between prospects (lotteries) , with . At time , a -th agent generates a probability measure giving the probabilities with which the prospects are to be chosen. As a result of the human decision making duality, the probability measure consists of two parts, a rational part quantifying the utility of the considered prospects and forming a classical probability measure and a quantum part characterizing irrational emotions and subconscious feelings, making a prospect subjectively attractive or not for an agent.
Explicit definitions required for the practical usage will be given in the following sections. Meanwhile, it is sufficient to remember that the probability measure, generated by each agent, is the union of and .
The exchange of information between the agents implies that each agent receives information on the probabilities generated by other agents. Thus the information exchange between two agents is represented by a directed graph in Fig. 1, showing that the first agent receives information from the second agent, and the latter, from the first one. For simplicity, this graph can be represented as in Fig. 2. For three agents, their interactions through information exchange can be shown as in Fig. 3. Similarly, the interaction scheme can be extended to many agents.
After receiving information from other members of the society, each agent generates the new probability measure that becomes available to other agents, and so on. Schematically, the dynamics of decision making with information exchange between two agents is shown in Fig. 4. The temporal variation of the probability measure includes the evolution of two parts, the classical probability measure , describing the utility or reward, whose dynamics is governed by the utility maximization, and the dynamics of the quantum part characterizing the variation of emotions. Generally, the two types of dynamics do not need to be necessarily connected. Recall that a number of experimental studies [43, 44, 45, 46, 47, 48, 49, 50] demonstrated the evolution of behavioral effects, caused by information exchange, under a fixed utility.
The convergence property of the decision making dynamics depends on the specification of the agents’ memory. When the process converges, then a -th decision maker comes to a stationary probability measure composed of the set of probabilities , each of which implies the probability of choosing a prospect . Comparing these probabilities defines the optimal, for the -th agent, prospect for which the probability is maximized: . 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 developed in the previous papers [37, 39, 77, 78, 79, 80, 81]. The mathematical techniques of the theory are equally applicable for describing quantum measurements as well as quantum decision making [54, 55, 77]. As far as all mathematical details have been thoroughly exposed in our previous publications, here we only briefly recall the main points of the theory and introduce notations that will be necessary for the generalization in the next sections to the case of multi-agent systems.
The information processing by a single agent proceeds as follows. An agent receives information characterized by different types of events and , labelled by the indices and , and represented by quantum states,
in the corresponding Hilbert spaces
The difference between the events is in their degree of testability. The events are operationally testable, allowing for their unambiguous observation or measurement. In contrast, the events are not operationally testable, being characterized by random amplitudes . The set of the random events is an inconclusive event [53, 54, 55] represented by a state in the Hilbert space ,
Generally, information is provided through composite events
represented by states in the Hilbert space
These composite events are termed prospects. Note that the states are not necessarily orthonormalized.
The prospects induce the prospect operators
which, in general, are not necessarily projectors, because the related prospect states may be not 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 in quantum theory.
An agent is associated with a strategic state that is non-negative and normalized, such that , with the trace operation over space (5). The agent generates the prospect probabilities
forming a probability measure, so that
A prospect probability consists of two parts,
where the first term
is positive-definite, while the second term
is not positive defined. The appearance in the probability of a not positive-defined term is due to the quantum definition of the probability and the interference of inconclusive events. Such a 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  has to be valid. In our case, this requires that the quantum probability should reduce to the classical probability when the quantum term tends to zero:
Thus, the positive-definite terms play the role of classical probabilities, with the standard properties
Then the quantum terms satisfy the conditions
or positive, when
In QDT, the term describes the utility of the prospect , which justifies to call it the utility factor, while
characterizes the attractiveness of the prospect and is called the attraction factor. Two terms, appearing in decision making, reflect the duality of the latter, including the logical conscious evaluation of the prospect utility and subconscious intuitive estimation of its attractiveness. The properties of these terms and their practical determination have been described in detail in the previous papers[37, 38, 39].
In this way, the information processing by a single agent consists in generating the probability measure over the given information characterized by the set of prospects. In the case of social networks, the classical probability can be defined by using a kind of Luce choice axiom [85, 41]. More generally, it can be defined as a minimizer of an information functional or by conditional entropy maximization [37, 39]. Note that unconditional entropy maximization may occur to be not sufficient for correctly defining probabilities, in which case one has to respect additional constraints making the ensemble representative [37, 39, 86].
The quantum term depends on the amount of information received by the agent. In the absence of information, the initial value can be random, although satisfying conditions (14). After getting the amount of information , the quantum term can be written [37, 71] as
In quantum decision theory, the quantum term characterizes the attractiveness of the considered prospects, and it is thus called the attraction factor.
Upon the receipt of additional information, the non-utility part of the probability measure generated by the agent is reduced. In the previous papers [37, 71], we have considered the simple case where the amount of information is the same for all agents, being given by an external source or by a control algorithm. Now we shall study a more realistic situation, when the agents receive information by exchanging it with other members of the society.
5 Generalization to Multiple Agents Exchanging Information
5.1 Arbitrary number of prospects
Let us now generalize the procedure of information processing by a single agent to the case of many agents forming a society. Let the agents be enumerated by . At the initial time, the information is presented through a set of prospects enumerated by . The process of getting additional information requires time, so that the probability measure generated by the -th agent is a function of time:
Respectively, the utility and attraction factors are also functions of time:
There always exist the normalization conditions for the probabilities,
for the utility factors,
and for the attraction factors,
In realistic situations, the probability measure is not immediately generated by the -th agent but after a delay time . Taking this into account and measuring time in units of , we can write
The first term is a utility factor, whose value is prescribed by the objective utility of the problem, hence assumed to be defined by prescribed rules [37, 38, 39]. The second term has been shown [37, 71] to be a function of the information measure quantifying the amount of information received by the -th agent until time ,
The initial value is a random quantity satisfying the above normalization conditions.
It is worth stressing the importance of taking into account the delay in receiving the information, which makes the consideration realistic, since in real life acquiring information always requires finite time. On the other side, the occurrence of time delay can essentially change the dynamics of multi-agent systems.
The total information received by the -th agent until time can be represented as a sum
where is the information gained by the -th agent at the -th time step and is a memory operator to be specified below, which defines how much of the information at time is retained at the later time .
Information measures can be chosen in different ways, for instance as transfer entropy [87, 88, 89, 90] or in the standard form of the Kullback-Leibler relative information [91, 92, 93]. We prefer the latter approach. Then, the information gain for the -th agent concerning the -th prospect can be written as
is the average probability for the -th prospect over all agents of the society, except the -th agent.
5.2 Case of two prospects
The problem simplifies when there are only two prospects (). This is actually a situation that is very often met, corresponding to choosing between just two alternatives, when deciding on “yes” or “no”. Then, keeping in mind the normalization conditions (18) to (20), it is sufficient to consider only one of the prospects, say , thus simplifying the notation for the first-prospect probability,
since the second-prospect probability is
Similar simplified notations can be used for the utility factors,
and the attraction factors,
each associated with the -th agent.
The utility factors, reflecting the basic utility of prospects, can be treated as time-independent, which we shall use in what follows:
To proceed further, we need to specify the memory operator characterizing the type of memory typical of the considered agents.
5.3 Types of memory
Different types of memory have been classified in psychology and neurobiology[94, 95, 96]. For the purpose of the present consideration, we need to distinguish the types of memory defining in different ways the temporal behavior of the information measure . It is possible to distinguish three qualitatively different kinds of temporal memory:
(i) Long-term memory, when there is no memory attenuation and all information, gained in the past, is perfectly retained. This implies that the memory operator is identical to unity, .
(ii) Reconstructive memory, when only the closest events are kept in mind, while all previous temporal blanks are filled in by reconstructing the past by analogy with the present time [97, 98, 99]. This kind of memory can be represented by the action of the memory operator .
(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 from the previous temporal steps. This assumes the local memory operator , which defines the Markov-type memory in decision process.
Below, we analyze in turn the behavior of the agents possessing these three principally different types of memory.
6 Agents with Long-Term Memory
Let us consider the limiting case of long-term memory corresponding to a non-decaying memory characterized by the memory operator . As a consequence, the total accumulated information (23) is the sum of the information gains at each temporal step,
For simplicity, we again consider the case of two prospects (). Moreover, we assume that agents can be divided into two groups, so that all agents within each group have the same initiation conditions. This amounts to consider a situation with two “group-agents”, or “superagents” representing two agent groups, exchanging information with each other. We thus have the equations for the probabilities:
with . The utility factors are assumed to be constants characterizing the intrinsic utility of the prospects for the agents. Generally, the agents of a society are heterogeneous, and thus the values of are different for different agents. The attraction factors vary with time as
with . The initial conditions for the probabilities are
We also assume that there is no additional information at the beginning, when , so that . Time varies in discrete steps as . Note that, for any , the information gains defined in (34) are always positive. It follows that the total information gain is positive for and .
Analyzing the behavior of the probabilities for varying initial conditions, we find that there are two qualitatively different types of solutions, depending on whether there exists an initial conflict between the utility factors and probabilities, or not. The existence or absence of an initial conflict is understood in the following sense.
(i) There is no initial conflict when either
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
If there is no initial conflict, the probabilities tend to the corresponding , when , as is shown in Fig. 5. But in the presence of an initial conflict, the numerical analysis of equations (32) to (35) shows that both probabilities tend to the common limit , defined as
within an accuracy of . This is illustrated in Fig. 6.
which is proved as follows.
7 Agents with Reconstructive Memory
Let us consider the situation corresponding to agents with reconstructive memory, when they put large weight to the last information gain at time , and use it to fill up the blanks over all previous times. This implies the action of the memory operator . Then the total information received by the -th agent at time is
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 and any , the probabilities tend to their utility factors ,
However, this tendency can be of three types. If the initial conditions are not conflicting, in the sense of inequalities (36) or (37), the tendency can be either monotonic, as in Fig. 7 or with oscillations, as in Figs 8 and 9. But when the initial conditions are conflicting, in the sense of Eqs. (38) or (39), then there always appear oscillations at intermediate stages, as is shown in Figs. 10 and 11. In the marginal case, when and coincide, the oscillations last forever, as illustrated in Fig. 9. However, this regime is not stable and disappears under any small difference between the ’s, leading to damped oscillations.
In the case where there is no conflict, the oscillations appear more regular while, in the case with an initial conflict, they look more chaotic. In order to understand better the oscillation characteristics, we calculate the local Lyapunov exponents, as is explained in Ref. . For this purpose, we define the multiplier matrix with the elements
whose eigenvalues are
Then for the local Lyapunov exponents we have
At the intermediate stage characterized by strong oscillations of the probabilities, at least one of the local Lyapunov exponents becomes transiently positive, thus, demonstrating local instability. But in the long run, the dynamics is always stable since, at large , the Lyapunov exponents 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 only the information from the last temporal step and keeping no track of any previous information gains. This is described by the local memory operator . As a result, the total information coincides with the information gain from the last step,
All other equations are the same as above, with the same initial conditions. In particular, at the initial time there is no yet any information, as has been assumed above, so that in the present case we have
Numerical analysis shows that there exist three types of dynamics. One is a smooth tendency to limiting states from below or from above, as is illustrated in Fig. 12. The second type is the 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 or not intersecting with each other, as is shown in Fig. 14. The existence of these three types of dynamics happens for positive as well as for negative attraction factors. The limits of the trajectories at infinite time, when they exist, depend on initial conditions and do not equal the related utility factors. Slightly varying the initial conditions leads to small finite changes in the limiting trajectory values, so that the motion is Lyapunov stable. But it is not asymptotically stable, 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-term memory, but does not occur for agents with other types of memory.
9 Societies of Decision Makers as Intelligence Networks
A society of agents generating, in the course of decision making, probability distributions over a given set of prospects, and interacting with each other through information exchange, is analogous to a network, which can be termed a social information network. Recall that the mathematics of taking decisions according to the rules of QDT is equivalent to the activity of an artificial intelligence, since an artificial intelligence, mimicking human cognition, has to take account of the conscious-subconscious duality typical of human brains [37, 38, 39]. And this duality is well represented by QDT, where the prospect probabilities consist of two terms, utility factor and attraction factor.
Therefore the society of decision makers, acting in the frame of QDT, is equivalent to a quantum information network or quantum intelligence network. In other words, such a network can characterize a composite artificial intelligence, or superintelligence, formed by an assembly of 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-call networks, citation networks, linguistic networks, ecological and biological networks, cellular networks, protein networks, neural networks, etc.[101, 102, 103, 104, 105]. All above mentioned networks are characterized by classical models. There are also quantum networks that are usually modelled by 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 of nodes, or vertices, or agents, enumerated by an index and a set of edges, or lines, or links, or arcs, connecting the nodes. The pair is called a graph. The graph is termed ‘directed’ if there is a map . For a directed graph, an edge may be different from . A pair of edges and , connecting two nodes, is called a circuit. Generally, a network is the triple representing a directed graph. In standard quantum 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 by wave functions. A node operator transforms a given wave function into another function, thus realizing a gate [106, 107].
A society of decision makers, or an assembly of artificial intelligences, functioning by the rules 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 by other agents. Keeping in mind that the network dynamics generates probability distributions varying with time, it is straightforward to interpret this activity as time-dependent decision making . The delayed information exchange and link directionality reflect the causality of interactions . The network dynamics describes the information flow . This is why the intelligence 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 makes decisions, while the standard nodes, like spins, do not do this. The type of intelligence networks we propose represents networks of agents acting by the quantum rules of QDT, with outcomes that are not just simple two-bit signals, as yes or no, and which could be modelled by spin systems with spins up or down. In contrast, in the quantum network proposed here, each agent generates a quantum probability measure over a given set of prospects. Being based on common mathematics, the intelligence networks can characterize either an assembly of interacting human decision makers, or a cluster of several quantum computers, or the activity of a composite artificial quantum intelligence consisting of several parts, each of which is an intelligence itself. The network dynamics models the formation of a collective decision as a collective 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 decision making. There are three main problems in describing such a repeated decision making.
At the beginning, decisions under uncertainty often contradict utility theory, as is discussed in the Introduction. Decision makers often choose the prospect with smaller utility factor, in contradiction with the prescription of utility theory. Why this is so and how to correctly predict the behavioral choice of decision makers at the initial stage?
There exist numerous empirical works [43, 44, 45, 46, 47, 48, 49, 50, 111, 112] showing that the difference between the probability of choosing a prospect and the utility factor decrease with time. For a society of decision makers, we define the experimental probability of a given prospect as the fraction of agents choosing that prospect. How to prove mathematically that this difference between the prospect probability and utility factor does diminish with time?
Empirical works [43, 44, 45, 46, 47, 48, 49, 50, 111, 112] also show that, even if at the initial time the behavioral probabilities strongly deviate from the prescription of utility theory, in the long run they converge to the related utility factors. The convergence of initially dispersed opinions to a common consensus is the basis of the so-called Delphi method that has been devised in order to obtain the most reliable opinion consensus of a group of experts by subjecting them 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 converge to a common consensus and, furthermore, the limiting prospect probabilities converge to an objective utility factor?
We illustrate the above questions by a concrete example involving the dynamic disjunction effect, and compare our theoretical predictions with empirical data. Since the disjunction effect is specified as a violation of the sure-thing principle , we first briefly recall the meaning of this principle. Then we give a real-life example of the effect, concentrating on the case of a composite game studied by Tversky and Shafir . After this, we analyze the disjunction effect dynamics by using the Tversky and Shafir data to specify the numerical values needed in the numerical solution of our equations to obtain the limiting values of the corresponding prospect probabilities.
10.2 Sure-thing principle
Let us consider a two-stage composite prospect. In the first stage, the events and can occur. One can either know for sure that one of these concrete events has occurred or one can only be aware that one of them has happened, not knowing which of them actually occurred. The latter case can be denoted as . At the second stage, either an event or occurs. We are thus confronted with the composite prospects , with . The sure-thing principle  states: If the alternative is preferred to the alternative , when an event occurs, and it is also preferred to , when an event occurs, then should be preferred to , when it is not known which of the events, either or , has occurred.
This principle is easily illustrated in classical probability theory, where the probability of the prospect is
Then, if for , it follows that , which explains the sure-thing principle.
10.3 Disjunction effect
However, empirical studies have discovered numerous violations of the sure-thing principle, which was called disjunction effect [116, 117, 118, 119]. Such violations are typical for two-step composite games of the following structure. First, a group of agents takes part in a game, where each agent can either win (event ) or loose (event ), with equal probability . They are then invited to participate in a second game, having the right either to accept the second game (event ) or to refuse it (event ). The second stage is realized in different variants: One can either accept or decline the second game under the condition of knowing the result of the first game. Or one can either accept or decline the second game without knowing the result of the first game. We define the probabilities, as usual, in the frequentist sense as the fractions of individuals taking the corresponding decision.
In their experimental studies, Tversky and Shafir  find that the fraction of people accepting the second game, under the condition that the first was won, is and the fraction of those accepting the second game, under the condition that the first was lost is . From the definition of conditional probabilities, one has the normalization
which yields and . Therefore the related joint probabilities are
According to equation (48), one gets
This implies that, by classical theory, the probability of accepting the second game, not knowing the results of the first one, is larger than that of refusing the second game, not knowing the result of the first game.
Surprisingly, in their experiments, Tversky and Shafir  observe that human decision makers behave opposite to the prescription of the sure-thing principle, with the majority refusing the second game, if the result of the first one is not known,
But in QDT, such a paradox does not appear. Recall that, in QDT, the probability of a prospect is the sum of and . The sign of the attraction factor is prescribed by the uncertainty aversion [37, 38, 39, 59], while its noninformative prior value is found [37, 38, 39] to satisfy the quarter law, with the average absolute value of the attraction factor . Thus, in QDT, we have
These theoretical results for are in very good agreement with the experimental data by Tversky and Shafir, actually being indistinguishable within the accuracy of experimental statistics.
10.4 Disjunction-effect dynamics
We now take the data from the previous subsection as initial conditions of our dynamical equations of Sec. 5 and study the time dependence of the decision makers’ opinions.
There are two prospects. One is the prospect of accepting the second game not knowing the result of the first game. And the second is the prospect of refusing from the second game, when the result of the first game is not known. As is explained in Sec. 5, using the normalization conditions for the probabilities, it is sufficient to consider one of the prospects, say .
Taking into account the heterogeneity of agents, we assume that there are two main groups of agents differing in their initial opinions. The corresponding prospect probabilities are denoted as . Respectively, we use the abbreviated notation for the utility factors and for the attraction factors . The utility factor, being an objective quantity, is assumed to be constant and defined as in the previous subsection: . As a result of the inequalities for the probabilities, the attraction factor can vary in the range
which in the present case implies that . The initial values of the attraction factors must fall within this interval.
We consider the more realistic situation where the decision makers take repeated decisions and remember their previous choices. We solve the evolution equations of Sec. 5 for this type of memory, taking as initial conditions for the attraction factors uniformly distributed values taken in the interval. The typical obtained behavior of the prospect probabilities for different initial conditions are shown in Fig. 15.
Two important conclusions follow from the temporal behavior of the solutions. First, for any initial conditions, the difference between the probabilities and the utility factor decreases with time. Second, all solutions, for arbitrary initial conditions, tend to the same limit . This demonstrate that, despite a large variation in the initial opinions, the agents come to a common consensus by exchanging information in a multi-step procedure.
We have suggested a model of a society of decision makers taking their decisions according to the quantum decision theory. This model generalizes the theory, formulated earlier for one-step decisions 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 kind of 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 probabilities are defined according to quantum rules, which results in their representation as sums of two terms, positive definite and sign indefinite. The quantum-classical correspondence principle makes it possible to interpret the positive-definite term as a classical probability or utility factor. The sign-indefinite term, having purely quantum origin, represents the attractiveness of the given prospects, and is referred to as the attraction factor. The utility factor can be considered as given and time-invariant. In contrast, the attraction factor is random at the initial time and varies with time due to the information exchange between the agents.
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 complemented by random variables with given distributions [40, 41]. Therefore such stochastic theories inherit the same problems as deterministic theories embedded into them. Moreover, stochastic theories are descriptive, containing fitting parameters that need to be defined from empirical data. In addition, different stochastic specifications of the same deterministic core theory may generate very different, and sometimes contradictory, conclusions . In stochastic decision theory, random terms are usually added to expected utility, while in our case the quantum prospect probability 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 principally new and has never been considered before to the best of our knowledge. We have suggested the first model describing the opinion dynamics of real decision makers subject to behavioral effects.
The probability dynamics depends on the form of the total information accumulated by each 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 utility factors . But when the initial conditions are conflicting, the probabilities tend to a consensual common limit . In all the cases, the motion is asymptotically Laypunov stable.
The probabilities for the agents with reconstructive memory always tend to their related utility factors , never exhibiting consensus. At intermediate stages of their dynamics, the probabilities can experience oscillations, with positive local Lyapunov exponents, which implies local instability. But in the long run, the dynamics becomes asymptotically Lyapunov stable.
The society of agents with short-term memory behaves rather differently from the behavior of the agents in the previous cases. Two types of dynamics can occur. The probability trajectories can tend to fixed points strongly depending on initial conditions, so that the motion is not asymptotically Lyapunov stable, although Lyapunov stable. These fixed points are different from the utility factors. The other type of motion is characterized by everlasting oscillations, which seems to be natural for agents with short-term memory, where there is no information 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, since the utility factors have been taken as invariants representing the objective utility of the related prospects. The decrease of the attraction factors with time, due to the exchange of information between 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 between decision makers, has been observed in many empirical studies [43, 44, 45, 46, 47, 48, 49, 50, 113, 114].
From another side, as stressed above, a society of decision makers, acting in the frame of QDT, is equivalent to an intelligence network, where the agents take decisions respecting the conscious-subconscious duality. Since this duality is typical of human decision makers, the networks, attempting to mimic the activity of human brains, are to be based on the rules of QDT. This should be taken into account in creating artificial intelligence and networks of artificial intelligences.
The usage of the approach is illustrated by a concrete realistic example involving the dynamic disjunction effect. Treating this effect in the frame of our theory yields three important predictions. (i) The initial values of the prospect probabilities are predicted in very good agreement with empirical data. (ii) The difference between the initial probabilities and the corresponding utility factors monotonically decrease with time. (iii) In a society of heterogeneous agents with long-term memory and randomly chosen initial conditions, all prospect probabilities tend to a common limit coinciding with the utility factor, thus demonstrating the existence of a consensus as a result of the repeated information exchange. These predictions are in agreement with empirical data.
We acknowledge financial support from the ETH Zürich Risk Center.
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