## 1. Basic Ideas and Historical Retrospective

This section serves as an introduction to the problem, describing the historical retrospective, related studies, and basic ideas of the quantum approach to decision making. First of all, in order that the reader would not be lost in details, we need to stress the main goal of the approach.

**Principal Goal:** The principal goal of the quantum approach to decision making is to develop a unified theory that, from one side, could formalize the process of taking decisions by human decision makers in terms of quantum language and, from another side, would suggest a scheme of thinking quantum systems that could be employed for creating artificial intelligence.

Generally, decision theory is concerned with identifying what are the optimal decisions and how to reach them. Traditionally, it is a part of discrete mathematics. Most of decision theory is normative and prescriptive, and assumes that people are fully-informed and rational. These assumptions have been questioned early on with the evidence provided by the Allais paradox [

1] and many other behavioral paradoxes [

2], showing that humans often seem to deviate from the prescription of rational decision theory due to cognitive and emotion biases. The theories of bounded rationality [

3], of behavioral economics and of behavioral finance have attempted to account for these deviations. As reviewed by Machina [

4], alternative models of preferences over objectively or subjectively uncertain prospects have attempted to accommodate these systematic departures from the expected utility model, while retaining as much of its analytical power as possible. In particular, non-additive nonlinear probability models have been developed to account for the deviations from objective to subjective probabilities observed in human agents [

5,

6,

7,

8,

9,

10]. However, many paradoxes remain unexplained or are sometimes rationalized on an ad hoc basis, which does not provide much predictive power.

Another approach to decision theory can be proposed, being part of the mathematical theory of Hilbert spaces [

11] and employing the mathematical techniques that are used in quantum theory. (see, e.g., the special issue [

12] and references therein). However, no self-consistent quantum theory of decision making has been developed, which would have predictive power.

Recently, we introduced a general framework, called the

Quantum Decision Theory (QDT), in which decisions involve composite intended actions which, as we explain below, provides a unifying explanation of many paradoxes of classical decision theory in a quantitative predictive manner [

13]. Such an approach can be thought of as the mathematically simplest and most natural extension of objective probabilities into nonlinear subjective probabilities. The proposed formalism allows one to explain

quantitatively different anomalous phenomena, e.g., the disjunction and conjunction effects. The disjunction effect is the failure of humans to obey the sure-thing principle of classical probability theory. The conjunction effect is a logical fallacy that occurs when people assume that specific conditions are more probable than a single general one. The QDT unearths a deep relationship between the conjunction fallacy and the disjunction effect, the former being sufficient for the latter to exist.

QDT uses the same underlying mathematical structure as the one developed to establish a rigorous formulation of quantum mechanics [

14]. Based on the mathematical theory of separable Hilbert spaces on the continuous field of complex numbers, quantum mechanics showed how to reconcile and combine the continuous wave description with the fact that waves are organized in discrete energy packets, called quanta, which behave in a manner similar to particles. Analogously, in the QDT framework, the qualifier quantum emphasizes the fact that a decision is a discrete selection from a large set of entangled options. The key idea of QDT is to provide the simplest generalization of the classical probability theory underlying decision theory, so as to account for the complex dynamics of the many nonlocal hidden variables that may be involved in the cognitive and decision making processes of the brain. The mathematical theory of complex separable Hilbert spaces provides the simplest direct way to avoid dealing with the unknown hidden variables, and at the same time reflecting the complexity of nature [

15]. In decision making, unknown states of nature, emotions, and subconscious processes play the role of hidden variables.

Before presenting the QDT approach, it is useful to briefly summarize previous studies of decision making and of the associated cognitive processes of the brain which, superficially, could be considered as related to the QDT approach. This exposition will allow us to underline the originality and uniqueness of the approach. We do not touch here purely physiological aspects of the problem, which are studied in medicine and physiological cognitive sciences. Concerning the functional aspects of the brain, we focus our efforts towards its formal mathematical modeling.

One class of approaches is based on the theory of neural networks and of dynamical systems (see, e.g., [

16,

17,

18,

19]). These bottom-up approaches suffer from the obvious difficulties of modeling the emergence of upper mental faculties from a microscopic constructive neuron-based description.

Two main classes of theories invoke the qualifier “quantum”. In the first class, one finds investigations which attempt to represent the brain as a quantum or quantum-like object [

20,

21,

22], for which several mechanisms have been suggested [

23,

24,

25,

26,

27,

28,

29]. The existence of genuine quantum effects and the operation of any of these mechanisms in the brain remain however controversial and have been criticized by Tegmark as being unrealistic [

30]. Another approach in this first class appeals to the mind-matter duality, treating mind and matter as complementary aspects and considering consciousness as a separate fundamental entity [

31,

32,

33,

34]. This allows one, without insisting on the quantum nature of the brain processes, if any, to ascribe quantum properties solely to the consciousness itself, as has been advocated by Stapp [

35,

36]. Actually, the basic idea that mental processes are similar to quantum-mechanical phenomena goes back to the founder of the old quantum mechanics, Niels Bohr. One of the first publications on this analogy is his paper [

37]. Later on, he returned many times to the similarity between quantum mechanics and the function of the brain, for instance in [

38,

39,

40]. This analogy proposes that mental processes could be modeled by a quantum-mechanical wave function, whose evolution would be characterized by a dynamical equation, like the Schrödinger equation.

The second class of theories does not necessarily assume quantum properties of the brain or that consciousness is a separate entity with quantum characteristics. Rather, these approaches use quantum techniques, as a convenient language to generalize classical probability theory. An example is provided by the quantum games [

41,

42,

43,

44,

45,

46,

47,

48,

49,

50]. With the development of quantum game theory, it has been shown that many quantum games can be reformulated as classical games by allowing for a more complex game structure [

51,

52,

53,

54]. But, in the majority of cases, it is more efficient to play quantum game versions, as less information needs to be exchanged. Another example is the Shor algorithm [

55], which is purely quantum-mechanical but is solving the classical factoring problem. This shows that there is no contradiction in using quantum techniques to describe classical problems. Here “classical” is contrasted with “quantum”, in the sense consecrated by decades of discussions on the interpretation of quantum mechanics. In fact, some people go as far as stating that quantum mechanics is nothing but an effective theory describing very complicated classical systems [

56,

57,

58]. Interpretations of this type have been made, e.g., by de Broglie and Bohm. An extensive literature in this direction can be found in de Broglie [

59] and Bohm [

60]. In any case, whether we deal really with a genuinely quantum system or with an extremely complex classical system, the language of quantum theory can be a convenient effective tool for describing such systems [

15]. In the case of decision making performed by real people, the subconscious activity and the underlying emotions, which are difficult to quantify, play the role of the hidden variables appearing in quantum theory.

The QDT belongs to this second class of theories, i.e., we use the construction of complex separable Hilbert spaces as a mathematical language that is convenient for characterizing the complicated processes in the mind, which are associated with decision making. This approach encompasses in a natural way several delicate features of decision making, such as its probabilistic nature, the existence of entangled decisions, the possible non-commutativity of decisions, and the interference between several different decisions. These terms and associated concepts are made operationally clear in the sequel.

As a bonus, the QDT provides natural algorithms which could be used in the future for quantum information processing, the operation of quantum computers, and in creating artificial intelligence.

The classical approaches to decision making are based on utility theory [

61,

62]. Decision making in the presence of uncertainty about the states of nature is formalized in the statistical decision theory [

63,

64,

65,

66,

67,

68,

69,

70,

71,

72,

73]. Some problems, occurring in the interpretation of the classical utility theory and its application to real human decision processes have been discussed in numerous literature (e.g., [

4,

68,

74,

75]).

Quantum approach to decision making, suggested in Reference [

13], is principally different from the classical utility theory. In this approach, the action probability is defined as is done in quantum mechanics, using the mathematical theory of complex separable Hilbert spaces. This proposition can be justified by invoking the following analogy. The probabilistic features of quantum theory can be interpreted as being due to the existence of the so-called nonlocal hidden variables. The dynamical laws of these nonlocal hidden variables could be not merely extremely cumbersome, but even not known at all, similarly to the unspecified states of nature in decision theory. The formalism of quantum theory is then formulated in such a way as to avoid dealing with unknown hidden variables, but at the same time, to reflect the complexity of nature [

15]. In decision making, the role of hidden variables is played by unknown states of nature, by emotions, and by subconscious processes, for which quantitative measures are not readily available.

In the following sections, we develop the detailed description of the suggested program, explicitly constructing the action probability in quantum-mechanical terms. The probability of an action is intrinsically subjective, as it must characterize intended actions by human beings. For brevity, an intended action can be called an intention or just an action. And, in compliance with the terminology used in the theories of decision-making, a composite set of intended actions, consisting of several sub-actions, is called a prospect. An important feature of the quantum approach is that, in general, it deals not with separate intended actions, but with composite prospects, including many incorporated intentions. Only then it becomes possible, within the frame of one general theory, to describe a variety of interesting unusual phenomena that have been reported to characterize the decision making properties of real human beings.

The pivotal point of the approach, formalized in QDT, is that mathematically it is based on the von Neumann theory of quantum measurements [

14]. The formal relation of the von Neumann measurement theory to quantum information processing has been considered by [

76]. QDT generalizes the quantum measurement theory to be applicable not merely to simple actions, but also to composite prospects, which is of paramount importance for the appearance of decision interference. The principal difference of QDT from the measurement theory is the existence of a specific

strategic state characterizing each particular decision maker.

A brief account of the axiomatics of QDT has been published in the recent letter [

13]. The principal scheme of functioning of a thinking quantum system, imitating the process of decision making, has been advanced [

77]. The applicability of the suggested quantum approach for analyzing the phenomena of dynamic inconsistency has been illustrated [

78]. The aim of the present survey is to provide a detailed explanation of the theory and to demonstrate that it can be successfully applied to real-life problems of decision making. We also show that the method of conditional entropy maximization, which is equivalent to the minimization of an information functional, yields an explicit relation between the quantum decision theory and the classical decision theory based on the standard notion of expected utility.

## 2. Mathematical Foundation of Quantum Decision Theory

In order to formulate in precise mathematical terms the scheme of information processing and decision making in quantum decision theory, it is necessary to introduce several definitions. To better understand these definitions, we shall give some very simple examples, although much more complicated cases can be invented. The entity concerned with the decision making task can be a single human, a group of humans, a society, a computer, or any other system that is able or enables to make decisions. Throughout the paper, for the operations with intended actions, we shall use the notations that are accepted in the literature on decision theory [

62,

63,

64,

65,

66,

67,

68,

69,

70,

71,

72,

73] and for the physical states, we shall employ the Dirac notations widely used in quantum theory [

79].

Definition 1. Action ring

The process of taking decisions implies that one is deliberating between several admissible actions with different outcomes, in order to decide which of the intended actions to choose. Therefore, the first element arising in decision theory is an intended action A.

An intended action which, for brevity, can be called an intention or an action, is a particular thought about doing something. Examples of intentions could be as follows: “I would like to marry” or “I would like to be rich” or “I would like to establish a firm”. There can be a variety of intentions, which we assume to be enumerated by an index $i=1,2,3,\dots ,N$, where the total number N of actions can be finite or infinite.

The whole family of all these actions forms the

action set
The elements of this set are assumed to be endowed with two binary operations, addition and multiplication, so that, if

A and

B pertain to

$\mathcal{A}$, then

$AB$ and

$A+B$ also pertain to

$\mathcal{A}$. The addition is associative, such that

$A+(B+C)=(A+B)+C$, and reversible, in the sense that

$A+B=C$ implies

$A=C-B$. The multiplication is distributive,

$A(B+C)=AB+BC$. The multiplication is not necessarily commutative, so that, generally,

$AB$ is not the same as

$BA$.

Among the elements of the action set (

1), there is an identity action 1, for which

$A1=1A=A$. The identity action 1 is not to “do nothing”, since inaction is actually an action. This is well recognized for instance in the field of risk management. Consider for instance the famous quotes: “The man who achieves makes many mistakes, but he never makes the biggest mistake of all—doing nothing” (Benjamin Franklin), or “Life is inherently risky. There is only one big risk you should avoid at all costs, and that is the risk of doing nothing” (Denis Waitley). This also resonates with the standard recommendations in risk management: “If you do not actively attack risks, they will attack you” or “Risk prevention is cheaper than reconstruction.” Thus, “not acting” is not the identity action. We interpret the identity action 1 as the action of keeping running the present action an individual is involved in. For instance, if action

A is “to marry someone”, the action

$1A$ is to marry someone and to confirm this action. The action

$A1$ can be interpreted as first “being open to decide an action” and then to “decide to marry someone”.

And there exists an impossible action 0, for which

$A0=0A=0$. Two actions are called disjoint, when their joint action is impossible, giving

$AB=BA=0$. The action set (

1), with the described structure, is termed the

action ring.

We recall that, in mathematics, a ring is an algebraic structure consisting of a set together with two binary operations (usually called addition and multiplication), where each operation combines two elements to form a third element (closure property). This closure property here embodies the fact that choosing between alternative actions or combining several actions still correspond to actions.

In the algebra of the elements of the action ring, the meaning of the operations of addition and multiplication is the same as is routinely used in the literature [

62,

63,

64,

65,

66,

67,

68,

69,

70,

71,

72,

73]. The sum

$A+B$ means that either the action

A or action

B is intended to be realized. And the product of actions

$AB$ implies that both these actions are to be accomplished together. Instead of writing the sum

${A}_{1}+{A}_{2}+\cdots $, it is often convenient to use the shorter summation symbol

${\bigcup}_{i}{A}_{i}\equiv {A}_{1}+{A}_{2}+\cdots $, which is also the standard abbreviation. Similarly, for a long product

${A}_{1}{A}_{2}\cdots $, it is convenient to use the shorter notation

${\bigcap}_{i}{A}_{i}\equiv {A}_{1}{A}_{2}\cdots $. The use of these standard notations can lead to no confusion.

Definition 2. Action modes

An action is simple, when it cannot be decomposed into the sum of other actions. An action is composite, when it can be represented as a sum of several other actions. If an action is represented as a sum

whose terms are mutually incompatible, then these terms are named the

action modes. Here,

${M}_{i}$ denotes the number of modes in action

${A}_{i}$. The modes correspond to different possible ways of realizing an action. According to the meaning emphasized above, the summation symbol in Equation (

2) implies that one of the actions is intended to be realized.

Action representations, or action modes, are concrete implementations of an intended action. For instance, the intention “to marry” can have as representations the following variants: “to marry A” or “to marry B", and so on. The intention “to be rich” can have as representations “to be rich by working hard” or “to be rich by becoming a bandit”. The intention “to establish a firm" can have as representations “to establish a firm producing cars” or “to establish a firm publishing books” and so on. We number all representations of an i-intended action by the index $\mu =1,2,3,\dots $. Note that intention representations may include not only positive intention variants “to do something” but also negative variants such as “not to do something”. For example, Hamlet’s hesitation “to be or not to be” is the intended action consisting of two representations, one positive and the other negative.

Definition 3. Elementary prospects

Generally, decision taking is not necessarily associated with a choice of just one action among several simple given actions, but it involves a choice between several complex actions. The simplest such complex action is defined as follows. Let the multi-index

$n=\{{\nu}_{1},{\nu}_{2},...,{\nu}_{N}\}$ be a set of indices enumerating several chosen modes, under the condition that each action is represented only by one of its modes. The

elementary prospect is the conjunction

of the chosen modes, one for each of the actions from the action ring (

1). The total set of all elementary prospects will be denoted as

$\left\{{e}_{n}\right\}$.

Definition 4. Composite prospects

A prospect is composite, when it cannot be represented as an elementary prospect (

3). Generally, a composite prospect is a conjunction

of several composite actions of form (

2), where each of the factors

${A}_{{j}_{n}}$ pertains to the action ring (

1). While expression (

4) has the similar form as (

3), the difference is that the actions

${A}_{{j}_{n}}$ in (

4) are composite while the actions

${A}_{i{\nu}_{i}}$ in (

3) are elementary action modes.

A prospect is a set of several intended actions or several intention representations. In reality, a decision maker is always motivated by a variety of intentions, which are mutually interconnected. Even the realization of a single intention always involves taking into account many other related intended actions.

Definition 5. Prospect lattice

All possible prospects, among which one needs to make a choice, form a set

The set is assumed to be equipped with the binary relations

$>,<,=,\ge ,\le $, so that each two prospects

${\pi}_{i}$ and

${\pi}_{j}$ in

$\mathcal{L}$ are related as either

${\pi}_{i}>{\pi}_{j}$, or

${\pi}_{i}={\pi}_{j}$, or

${\pi}_{i}\ge {\pi}_{j}$, or

${\pi}_{i}<{\pi}_{j}$, or

${\pi}_{i}\le {\pi}_{j}$. For a while, it is sufficient to assume that such an ordering exists. Then, the ordered set (

5) is called a

lattice. The explicit ordering procedure associated with decision making will be given below.

Definition 6. Mode space

To each action mode

${A}_{i\mu}$, there corresponds the

mode state $|{A}_{i\mu}\rangle $, which is a complex function

$\mathcal{A}\to \mathcal{C}$, and its Hermitian conjugate

$\langle {A}_{i\mu}|$. Here we employ the Dirac notation [

79]. We assume that a scalar product is defined, such that the mode states, pertaining to the same action, are orthonormalized:

The

mode space is the closed linear envelope

spanning all mode states. By this definition, the mode space, corresponding to an

i-action

${A}_{i}$, is a Hilbert space of dimensionality

${M}_{i}$. The elements of the mode space will be called the

intention states.

Definition 7. Mind space

To each elementary prospect

${e}_{n}$, there corresponds the

basic state $|{e}_{n}\rangle $, which is a complex function

${\mathcal{A}}^{N}\to \mathcal{C}$, and its Hermitian conjugate

$\langle {e}_{n}|$. The structure of a basic state is

The scalar product is assumed to be defined, such that the basic states are orthonormalized:

The

mind space is the closed linear envelope

spanning all basic states (

8). Hence, the mind space is a Hilbert space of dimensionality

The vectors of the mind space represent all possible actions and prospects considered by a decision maker.

The family of the basic states forms the mind basis $\left\{\right|{e}_{n}>\}$ in the mind space. Different states belonging to the mind basis are assumed to be disjoint, in the sense of being orthogonal. Since the modulus of each state has no special meaning, these states are also normalized to one. This is formalized as the orthonormality of the basis.

Definition 8. Prospect states

To each prospect

${\pi}_{j}$, there corresponds a state

$|{\pi}_{j}\rangle \in \mathcal{M}$ that is a member of the mind space (

10). Hence, the prospect state can be represented as an expansion over the basic states

The expansion coefficients in Equation (

11) are assumed to be defined by the decision maker, so that

$|{a}_{jn}{|}^{2}$ gives the weight of the state

$|{e}_{n}>$ into the general prospect.

The prospects are enumerated with the index $j=1,2,\dots $. The total set $\left\{\right|{\pi}_{j}>\}$ of all prospect states $|{\pi}_{j}>$, corresponding to all admissible prospects, forms a subset of the space of mind. The set $\left\{\right|{\pi}_{j}>\}\subset \mathcal{M}$ can be called the prospect-state set.

The prospect states are not required to be mutually orthogonal and normalized to one, so that the scalar product

is not necessarily a Kronecker delta. The normalization condition will be formulated for the prospect probabilities to be defined below.

The fact that different prospect states are not necessarily orthogonal assumes that the related prospects are not necessarily incompatible. The incompatibility is supposed only for the elementary prospects (

3), whose states form the basis in the mind space (

10) and are orthogonal to each other, according to Equation (

9). But an arbitrarily defined composite prospect, generally, is not required to be orthogonal to all other considered prospects. In particular, this can be so, but, in general, we do not need this property.

The prospect states are not normalized to one, since, imposing such a condition would over define these states. The normalization condition will be imposed below on the prospect probabilities. Generally, imposing two normalization conditions could make them inconsistent with each other. So, we need just one normalization condition for the prospect probabilities, which is necessary for the correct definition of the related probability measure.

Being, generally, not orthonormalized, the prospect states do not form a basis in the mind space.

Definition 9. Strategic state

Among the states of the mind space, there exists a special fixed state $|s\rangle \in \mathcal{M}$, playing the role of a reference state, which is termed the strategic state. The strategic state of mind is a fixed vector characterizing a particular decision maker, with his/her beliefs, habits, principals, etc., that is, describing each decision maker as a unique subject. Hence, each space of mind possesses a unique strategic state. Different decision makers possess different strategic states.

Being in the mind space (

10), this state can be represented as the decomposition

Being a unique state, characterizing each decision maker like its fingerprints, it can be normalized to one:

From Equations (

12) and (

13), it follows that

The existence of the strategic state, uniquely defining each particular decision maker, is the principal point distinguishing an

active thinking quantum system from a

passive quantum system subject to measurements from an external observer. For a passive quantum system, predictions of the outcome of measurements are performed by summing (averaging) over all possible statistically equivalent states, which can be referred to as a kind of “annealed” situation. In contrast, decisions and observations associated with a thinking quantum system occur in the presence of this unique strategic state, which can be thought of as a kind of fixed “quenched” state. As a consequence, the outcomes of the applications of the quantum-mechanical formalism will thus be different for thinking versus passive quantum systems.

Definition 10. Prospect operators

Each prospect state

$|{\pi}_{j}\rangle $, together with its Hermitian conjugate

$\langle {\pi}_{j}|$, defines the

prospect operator
By this definition, the prospect operator is self-adjoint. The family of all prospect operators forms the involutive bijective algebra that is analogous to the algebra of local observables in quantum theory. Since the prospect states, in general, are neither mutually orthogonal nor normalized, the squared operator

contains the scalar product

which does not equal to one. This tells us that the prospect operators, generally, are not idempotent, thus, they are not projection operators. It is only when the prospect is elementary that the related prospect operator

becomes idempotent and is a projection operator. But, in general, this is not so.

The properties of the prospect operators follow immediately from those of the prospect states and definition (

14). Recall that the prospect operators are analogous to the operators of local observables in quantum theory. The latter operators are not required to be idempotent. So, the prospect operators are also not required to be such. The intuition of why the prospect operators are not idempotent could be justified by understanding that, in general, a prospect realized twice results in the consequences that are not necessarily the same as a sole prospect realization. For instance, to marry twice is not the same as to marry once.

Definition 11. Prospect probabilities

In quantum theory, the averages over the system state, for the operators from the algebra of local observables, define the observable quantities. In the same way, the averages, over the strategic state, for the prospect operators define the observable quantities, the

prospect probabilities
These are assumed to be normalized to one:

where the summation is over all prospects from the prospect lattice (

5). By their definition, the quantities (

15) are non-negative, since Equation (

15) reduces to the modulus of the transition amplitude squared

The normalization in Equation (

16) is necessary for the set

$\left\{p\left({\pi}_{j}\right)\right\}$ be the scalar probability measure. In plane words, the fact that all prospects probabilities are summed to one implies that one of them is to be certainly realized.

Definition 12. Utility factor

The diagonal form

plays the role of the expected utility in classical decision making, justifying its name as the

utility factor. In order to be generally defined and to be independent of the chosen units of measurement, the utility factor (

17) can be normalized as

The fact that the utility factor (

17) is really equivalent to the classical expected utility follows from noticing that

hence Equation (

17) acquires the form

where

$<{e}_{n}\left|\widehat{P}\left({\pi}_{j}\right)\right|{e}_{n}>$ plays the role of a utility function, weighted with the probability

$|{c}_{n}{|}^{2}$.

Definition 13. Attraction factor

The nondiagonal term

corresponds to the quantum interference effect. Its appearance is typical of quantum mechanics. Such nondiagonal terms do not occur in classical decision theory. This term can be called the

interference factor. Interpreting its meaning in decision making, we can associate its appearance as resulting from the system deliberation between several alternatives, when deciding which of the latter is more attractive. Thence, the name “attraction factor”. Using expansion (

12) in Equation (

19) yields

which shows that the interference occurs between different elementary prospects in the process of considering a composite prospect

${\pi}_{j}$. It is worth stressing that the interference factor is nonzero only when the prospect

${\pi}_{j}$ is composite. If it were elementary, say

${\pi}_{j}={e}_{k}$ then, since

we would have

and no interference would arise.

Between two prospects, the one which enjoys the larger attraction factor is more attractive.

Definition 14. Prospect ordering

In defining the prospect lattice (

5), we have assumed that the prospects could be ordered. Now, after introducing the scalar probability measure, we are in a position to give an explicit prescription for the prospect ordering. We say that the prospect

${\pi}_{1}$ is

preferable to

${\pi}_{2}$ if and only if

Two prospects are called indifferent if and only if

And the prospect

${\pi}_{1}$ is preferable or indifferent to

${\pi}_{2}$ if and only if

These binary relations provide us with an explicit prospect ordering making the prospect set (

5) a lattice.

Definition 15. Optimal prospect

Since all prospects in the lattice are ordered, it is straightforward to find among them that one enjoying the largest probability. This defines the

optimal prospect ${\pi}_{*}$ for which

Finding the optimal prospect is the final goal of the decision-making process. Since the prospect probabilities are non-negative, it is possible to find the minimal prospect in the lattice (

5) with the smallest probability. And the largest probability defines the optimal prospect

${\pi}_{*}$. Therefore the prospect set (

5) is a complete lattice.

**Remark**. Generally speaking, all states of the mind space can depend on time t. We do not write explicitly the time dependence, when this makes no difference for the considerations developed below. When this is important, we shall denote the time dependence explicitly.

## 5. Interference of Intended Actions

Interference is the effect that is typical of all those phenomena which are described by wave equations. Following the Bohr’s idea [

37,

38,

39,

40] of describing mental processes in terms of quantum mechanics, one is immediately confronted with the interference effect, since the physical states in quantum mechanics

are characterized by wave functions. The possible occurrence of interference in the problems of decision making has been discussed before on different grounds (see, e.g., [

93]). However, no general theory has been suggested, which would explain why and when such a kind of effect would appear, how to predict it, and how to give a quantitative analysis of it that can be compared with empirical observations. In our approach, interference in decision making arises only when one takes a decision involving composite intentions. The corresponding mathematical treatment of these interferences within QDT is presented in the following subsections.

#### 5.1. Illustration of Interference in Decision Making

As an illustration, let us consider the following situation of two intended actions, “to get a friend” and “to become rich”. Let the former intention have two mode representations “to get a friend

A" and “to get a friend

B”. And let the second intention also have two representations, “to become rich by working hard" and “to become rich by being a gangster”. The corresponding strategic mind state is given by Equation (

12), with the evident notation for the basic states

$|{e}_{n}>$ and the coefficients

${c}_{n}$ represented by the identities

Suppose that one does not wish to choose between these two friends in an exclusive manner, but one hesitates of being a friend to A as well as B, with the appropriate weights. This means that one considers the intended actions A and B, while the way of life, either to work hard or to become a gangster, has not yet been decided.

The corresponding composite prospects

are characterized by the prospect states

Let us stress that the weights correspond to the intended actions, among which the choice is yet to be made. And one should not confuse the intended actions with the actions that have already been realized. One can perfectly deliberate between keeping this or that friend, in the same way, as one would think about marrying

A or

B in another example above. This means that the choice has not yet been made. And before it is made, there exist deliberations involving stronger or weaker intentions to both possibilities. Of course, one cannot marry both (at least in most Christian communities). But before marriage, there can exist the dilemma between choosing this or that individual.

Calculating the scalar products

we find the prospect probabilities.

Recall that the prospects are characterized by vectors pertaining to the space of mind

$\mathcal{M}$, which are not necessarily normalized to one or orthogonal to each other. The main constraint is that the total set of prospect states

$\left\{\right|{\pi}_{j}>\}$ be such that the related probabilities

be normalized to one, according to the normalization condition (

16).

The probabilities (

35) can be rewritten in another form by introducing the partial probabilities

and the interference terms

Then the probabilities (

35) become

Let us define the

uncertainty angles
and the

uncertainty factors
Using these, the interference terms (

37) take the form

The interference terms characterize the existence of deliberations between the decisions of choosing a friend and, at the same time, a type of work.

This example illustrates the observation that the phenomenon of decision interference appears when one considers a composite prospect with several intention representations assumed to be realized simultaneously. Treating a composite prospect as a combination of several sub-prospects, we could consider the global decision as a collection of sub-decisions. Then the arising interference would occur between these sub-decisions. From the mathematical point of view, it appears more convenient to combine several sub-decisions into one global decision and to analyze the interference of different intentions. Thus, we can state that interference in decision making appears only when one decides about a composite prospect.

For the above example of decision making in the case of two intentions, “to get a friend” and “to be rich”, the appearance of the interference can be understood as follows. In real life, it is too problematic, and practically impossible, to become a very close friend to several persons simultaneously, since conflict of interests often arises between the friends. For instance, doing a friendly action to one friend may upset or even harm another friend. Any decision making, involving mutual correlations between two persons, necessarily requires taking into account their, sometimes conflicting, interests. This is, actually, one of the origins of the interference in decision making. Another powerful origin of intention interference is the existence of emotions, as will be discussed in the following sections.

#### 5.2. Conditions for Interference Appearance

The situations for which intention interferences is impossible can be classified into two cases, which are examined below. From this classification, we conclude that the necessary conditions for the appearance of intention interferences are that the dimensionality of mind should be not lower than two and that there should be some uncertainty in the considered prospect.

**One-dimensional mind**

Suppose there are several intended actions

$\left\{{A}_{i}\right\}$, enumerated by the index

$i=1,2,\dots $, whose number can be arbitrary. But each intention possesses only a single representation

$|{A}_{i}>$. Hence, the dimension of “mind” as defined in Definition 7, is

$dim\mathcal{M}=1$. Only a single basic vector exists, which forms the strategic state

In this one-dimensional mind, all prospect states are disentangled, being of the type

Therefore, only one probability exists:

Thus, despite the possible large number of arbitrary intentions, they do not interfere, since each of them has just one representation. There can be no intention interference in one-dimensional mind.

**Absence of uncertainty**

Another important condition for the appearance of intention interference is the existence of uncertainty. To understand this statement, let us consider a given mind with a large dimensionality

$dim\mathcal{M}>1$, characterized by a strategic state

$|s>$. Let us analyze a certain prospect with the state

with an arbitrary strategic state

$|s>$. Then again, the corresponding prospect probability is the same as in Equation (

44), and no interference can arise.

Thus, the necessary conditions for the appearance of interference are the existence of uncertainty and the dimensionality of mind not lower than 2.

#### 5.3. Interference Alternation

The interference terms, forming the attraction factor (

19), enjoy a very important property that can be called the

theorem of interference alternation.

**Theorem 1:** The process of decision making, associated with the prospect probabilities (15) and occurring under the normalization conditions (16) and (18), is characterized by the alternating interference terms, such that the sum of all attraction factors vanishes:

Proof.

The proof follows directly from Definitions 13, 15, and 19, taking into account the normalization conditions (

16) and (

18).

In order to illustrate in more detail the meaning of the above theorem, let us consider a particular case of two intentions, one composing a set

$\left\{{A}_{i}\right\}$ of

${M}_{1}$ representation modes, and another one forming a set

$\left\{{X}_{j}\right\}$ of

${M}_{2}$ modes. The total family of intended actions is therefore

The basis in the mind space is the set

$\left\{\right|{A}_{i}{X}_{j}>\}$. The strategic state of mind can be written as an expansion over this basis,

with the coefficients satisfying the standard normalization

Let us assume that we are mainly interested in the representation set

$\left\{{A}_{i}\right\}$, while the representations from the set

$\left\{{X}_{j}\right\}$ are treated as secondary. A prospect that is formed of a fixed intention representation

${A}_{i}$, and which can be realized under the occurrence of any of the representations

${X}_{j}$, corresponds to the prospect state

where

$X={\bigcup}_{j}{X}_{j}$. The probability of realizing the considered prospect is

according to Definition 11.

Following the above formalism of describing the intention interferences, we use the notation

for the joint probability of

${A}_{i}$ and

${X}_{j}$; and we denote the partial interference terms as

Then, the probability of

${A}_{i}X$, given by Equation (

48), becomes

The interference terms appear due to the existence of uncertainty. Therefore, we may define the

uncertainty factor
where the uncertainty angle is

Then, the interference term (

51) takes the form

The attraction factor (

19) here is nothing but the sum of the interference terms:

This allows us to rewrite probability (

52) as

The joint and conditional probabilities are related in the standard way

We assume that the family of intended actions is such that at least one of the representations from the set

$\left\{{A}_{i}\right\}$ has to be certainly realized, which means that

and that at least one of the representations from the set

$\left\{{X}_{j}\right\}$ also necessarily happens, that is,

Along with these conditions, we keep in mind that at least one of the representations from the set

$\left\{{A}_{i}\right\}$ must be realized for each given

${X}_{j}$, which implies that

Then we immediately come to the equality

which is just a particular case of the general condition (

46).

This equality shows that, if at least one of the terms is non-zero, some of the interference terms are necessarily negative and some are necessarily positive. Therefore, some of the probabilities are depressed, while others are enhanced. This alternation of the interference terms will be shown below to be a pivotal feature providing a clear explanation of the disjunction effect. It is worth emphasizing that the violation of the sure-thing principle, resulting in the disjunction effect, will be shown not to be due simply to the existence of interferences as such, but, more precisely, to the interference alternation.

For instance, the depression of some probabilities can be associated with uncertainty aversion, which makes less probable an action under uncertain conditions. In contrast, the probability of other intentions, containing less or no uncertainty, will be enhanced by positive interference terms. This interference alternation is of crucial importance for the correct description of decision making, without which the known paradoxes cannot be explained.

#### 5.4. Less is More

The title of this subsection is taken from a poem of the nineteenth century English poet Robert Browning [

94].

In the present context, this expression means that sometimes excessive information is not merely difficult to get, but can even be harmful, resulting in wrong decisions. It often happens that decisions, based on smaller amount of information, are better than those based on larger amount of information. This may happen because, with increasing the amount of information, the choice between alternatives can become more complicated as a result of which uncertainty grows. Increasing complexity often increases uncertainty.

To describe the “less is more” phenomenon in mathematical language, let us consider a prospect

${\pi}_{k}^{*}$ that is optimal under a fixed information set

${X}_{k}$, with the probability

Suppose, we increase the amount of information by going to the information set

${X}_{k+1}$, such that

${X}_{k}\in {X}_{k+1}$, and obtain the related optimal prospect

${\pi}_{k+1}^{*}$, with the probability

Assume that the utilities of these two prospects are the same,

while the uncertainty in the decision making process increases, so that the attraction factor decreases,

Then, the relation between the prospect probabilities

tells us that the decision process leading to choosing prospect

${\pi}_{k}^{*}$ is clearer than for prospect

${\pi}_{k+1}^{*}$, because the larger value of the corresponding probability makes the signal stronger for the decision maker, resulting in a larger frequency of choices

${\pi}_{k}^{*}$. As the information set is increased, in the presence of many alternatives, the preferred prospect becomes less clearly defined as the top choice. As a consequence, a lack of efficiency, a growing indeterminacy and ultimately the freezing of the decision process can ensue.

When dealing with complex nonlinear problems, excessive information can lead to incorrect conclusions because of the extreme sensitivity of nonlinear problems to minor details. As simple examples, when excessive information can be harmful, we may mention the following typical cases from physics.

**Example 1**. How to describe the state of air in a room? The unreasonable decision would be to analyze the motion of all molecules in the room, specifying all their interactions, positions and velocities. Such a decision would lead to not merely extremely overcomplicated calculations, but even can result in incorrect conclusions. The reasonable decision is to characterize the state of the air by defining the room temperature, volume, and atmospheric pressure.

**Example 2**. How to characterize the water flow in a river? A silly decision would be to consider the motion of all water molecules in the river describing their locations, velocities, interactions, and so on. Contrary to this, a clever decision is to use the hydrodynamic equations.

**Example 3**. How to describe a large social system? Again, the unreasonable decision would be to collect all possible information on each member of the society. Then, being overloaded by senseless information, one would be lost in secondary details, being unable to make any clever conclusion. Instead of this, it is often (though may be not always) sufficient to consider the society composed of typical (or “representative”) agents.

## 6. Disjunction Effect

The disjunction effect was first specified by Savage [

62] as a violation of the “sure-thing principle”, which can be formulated as follows:

If the alternative A is preferred to the alternative B, when an event ${X}_{1}$ occurs, and it is also preferred to B, when an event ${X}_{2}$ occurs, then A should be preferred to B, when it is not known which of the events, either ${X}_{1}$ or ${X}_{2}$, has occurred.

#### 6.1. Sure-Thing Principle

Let us now show how the sure-thing principle arises in classical probability theory.

Let us consider a field of events

$\{A,B,{X}_{j}|j=1,2,\dots \}$ equipped with the classical probability measures [

95]. We denote the classical probability of an event

A by the capital letter

$P\left(A\right)$ in order to distinguish it from the probability

$p\left(A\right)$ defined in the previous sections by means of quantum rules. We shall denote, as usual, the conditional probability of

A under the knowledge of

X by

$P\left(A\right|X)$ and the joint probability of

A and

X, by

$P\left(AX\right)$. We assume that at least one of the events

${X}_{j}$ from the set

$\left\{{X}_{j}\right\}$ certainly happens and that the

${X}_{i}$ are mutually exclusive and exhaustive, which implies that

The probability of

A, when

${X}_{j}$ is not specified, that is, when at least one of

${X}_{j}$ happens, is denoted by

$P\left(AX\right)$, where

$X={\bigcup}_{j}{X}_{j}$. The same notations are applied to

B. Following the common wisdom, we understand the statement “

A is preferred to

B" as meaning that

$P\left(AX\right)>P\left(BX\right)$. Then the following theorem is valid.

**Theorem 2**:

If for all $j=1,2,\dots $, one hasthen Proof.

It is straightforward that, under

$X={\bigcup}_{j}{X}_{j}$, one has

and

From Equations (

69) and (

70), under assumption (

67), inequality (

68) follows immediately.

The above proposition is a theorem of classical probability theory. Savage [

62] proposed to use it as a normative statement on how human beings make consistent decisions under uncertainty. As such, it is no more a theorem but a testable assumption about human behavior. In other words, empirical tests showing that humans fail to obey the sure-thing principle must be interpreted as a failure of humans to abide to all the rules of classical probability theory.

#### 6.2. Disjunction-Effect Examples

Thus, according to standard classical probability theory which is held by most statisticians as the only rigorous mathematical description of risks, and therefore as the normative guideline describing rational human decision making, the sure-thing principle should be always verified in empirical tests involving real human beings. However, numerous violations of this principle have been investigated empirically [

62,

96,

97,

98,

99]. In order to be more specific, let us briefly outline some examples of the violation of the sure-thing principle, referred to as the disjunction effect.

(i) To gamble or not to gamble?

A typical setup for illustrating the disjunction effect is a two-step gamble [

96]. Suppose that a group of people accepted a gamble, in which the player can either win (

${X}_{1}$) or lose (

${X}_{2}$). After one gamble, they are invited to gamble a second time, being free to either accept the second gamble (

A) or to refuse it (

B). Experiments by Tversky and Shafir [

96] showed that the majority of people accept the second gamble when they know the result of the first one, in any case, whether they won or lost in the previous gamble. In the language of conditional probability theory, this translates into the fact that people act as if

$P\left(A\right|{X}_{1})$ is larger than

$P\left(B\right|{X}_{1})$ and

$P\left(A\right|{X}_{2})$ is larger than

$P\left(B\right|{X}_{2})$ as in Equation (

67). At the same time, it turns out that the majority refuses to gamble the second time when the outcome of the first gamble is not known. The second empirical fact implies that people act as if

$P\left(BX\right)$ overweighs

$P\left(AX\right)$, in blatant contradiction with inequality (

68), which should hold according to the theorem resulting from (

67). Thus, a majority accepted the second gamble after having won or lost in the first gamble, but only a minority accepted the second gamble when the outcome of the first gamble was unknown to them. This provides an unambiguous violation of the Savage sure-thing principle.

(ii) To buy or not to buy?

Another example, studied by Tversky and Shafir [

96], had to do with a group of students who reported their preferences about buying a non-refundable vacation, following a tough university test. They could pass the exam (

${X}_{1}$) or fail (

${X}_{2}$). The students had to decide whether they would go on vacation (

A) or abstain (

B). It turned out that the majority of students purchased the vacation when they passed the exam as well as when they had failed, so that condition (

67) was valid. However, only a minority of participants purchased the vacation when they did not know the results of the examination. Hence, inequality (

68) was violated, demonstrating again the disjunction effect.

(iii) To sell or not to sell?

The stock market example, analyzed by Shafir and Tversky [

100], is a particularly telling one, involving a deliberation taking into account a future event, and not a past one as in the two previous cases. Suppose we consider the USA presidential election, when either a Republican wins (

${X}_{1}$) or a Democrat wins (

${X}_{2}$). On the eve of the election, market players can either sell certain stocks from their portfolio (

A) or hold them (

B). It is known that a majority of people would be inclined to sell their stocks, if they would know who wins, regardless of whether the Republican or Democrat candidate wins the upcoming election. This is because people expect the market to fall after the elections. Hence, condition (

67) is again valid. At the same time, a great many people do not sell their stocks before knowing who really won the election, thus contradicting the sure-thing principle and the inequality (

68). Thus, investors could have sold their stocks before the election at a higher price but, obeying the disjunction effect, they were waiting until after the election, thereby selling at a lower price after stocks have fallen. Many market analysts believe that this is precisely what happened after the 1988 presidential election, when George Bush defeated Michael Dukakis.

There are plenty of other more or less complicated examples of the disjunction effect [

62,

96,

97,

98,

100,

101,

102]. The common necessary conditions for the disjunction effect to arise are as follows. First, there should be several events, each characterized by several alternatives, as in the two-step gambles. Second, there should necessarily exist some uncertainty, whether with respect to the past, as in the examples (i) and (ii), or with respect to the future, as in the example (iii).

Several ways of interpreting the disjunction effect have been analyzed. Here, we do not discuss the interpretations based on the existence of some biases, such as the gender bias, or the bias invoking the notion of decision complexity, which have already been convincingly ruled out [

97,

103]. We describe the reason-based explanation which appears to enjoy a wide-spread following and discuss its limits before turning to the view point offered by QDT.

#### 6.3. Reason-Based Analysis

The dominant approach for explaining the disjunction effect is the reason-based analysis of decision making [

96,

97,

100,

102,

104]. This approach explains choice in terms of the balance between reasoning for and against the various alternatives. The basic intuition is that when outcomes are known, a decision maker may easily come up with a definitive reason for choosing an option. However, in the case of uncertainty, when the outcomes are not known, people may lack a clear reason for choosing an option and consequently they abstain and make an irrational choice.

From our perspective, the weakness of the reason-based analysis is that the notion of “reason” is too vague and subjective. Reasons are not only impossible to quantify, but it is difficult, if possible at all, to give a qualitative definition of what they are. Consider example (i) “to gamble or not to gamble?” Suppose you have already won at the first step. Then, you can rationalize that gambling a second time is not very risky: if you now loose, this loss will be balanced by the first win on which you were not counting anyway, so that you may actually treat it differently from the rest of your wealth, according to the so-called “mental accounting” effect; and if you win again, your profit will be doubled. Thus, you have a “reason” to justify the attractiveness of the second gamble. But, it seems equally justified to consider the alternative “reason”: if you have won once, winning the second time may seem less probable (the so-called gambler’s fallacy), and if you loose, you will keep nothing of your previous gain. This line of reasoning justifies to keep what you already got and to forgo the second gamble.

Suppose now you have lost in the first gamble and know it. A first reasoning would be that the second gamble offers a possibility of getting out of the loss, which provides a reason for accepting the second gamble. However, you may also think that the win is not guaranteed, and your situation could actually worsen, if you loose again. Therefore, this makes it more reasonable not to risk so much and to refrain from the new gamble.

Consider now the situation where you are kept ignorant of whether you have won or lost in the first gamble. Then, you may think that there is no reason and therefore no motivation for accepting the second gamble, which is the standard reason-based explanation. But, one could argue that it would be even more logical if you would think as follows: Okay, I do not know what has happened in the first gamble. So, why should I care about it? Why don’t I try again my luck? Certainly, there is a clear reason for gambling that could propagate the drive to gamble a second time.

This discussion is not pretending to demonstrate anything other than that the reason-based explanation is purely ad-hoc, with no real explanatory power; it can be considered in a sense as a reformulation of the disjunction fallacy. It is possible to multiply the number of examples demonstrating the existence of quite “reasonable” justifications for doing something as well as a reason for just doing the opposite. It seems to us that the notion of “reason" is not well defined and one can always invent in this way a justification for anything. Thus, we propose that the disjunction effect has no direct relation to reasoning. In the following section, we suggest another explanation of this effect based on QDT, specifically the interference between the two uncertain outcomes resulting from an aversion to uncertainty (uncertainty-aversion principle), which provides a quantitative testable prediction.

#### 6.4. Quantitative Analysis Within Quantum Decision Theory

The disjunction effect, described above, finds a natural explanation in the frame of the Quantum Decision Theory, as is shown below.

**Application to Disjunction-Effect Examples**

The possibility of connecting the violation of the sure-thing principle with the occurrence of interference has been mentioned in several articles (see, e.g., [

93]). But all these attempts were ad hoc assumptions not based on a self-consistent theory. Our explanation of the disjunction effect differs from these attempts in several aspects. First, we consider the disjunction effect as just one of several possible effects in the frame of the

general theory. The explanation is based on the theorem of

interference alternation, which has never been mentioned, but without which no explanation can be complete and self-consistent. We stress the importance of the

uncertainty-aversion principle. Also, we offer a

quantitative estimate for the effect, which is principally new.

Let us discuss the two first examples illustrating the disjunction effect, in which the prospect consists of two intentions with two representations each. One intention “to decide about an action” has the representations “to act” (A) and “not to act” (B). The second intention “to know the results" (or “to have information”) has also two representations. One (${X}_{1}$) can be termed “to learn about the win” (gamble won, exam passed), the other (${X}_{2}$) can be called “to learn about the loss” (gamble lost, exam failed). With the numbers of these representations ${M}_{1}=2$ and ${M}_{2}=2$, the dimension of mind, given in Definition 7, is $dim\mathcal{M}={M}_{1}{M}_{2}=4$.

For the considered cases, the general set of Equations (

56) reduces to two equations

in which again

$X={\bigcup}_{j}{X}_{j}$ and the interference terms are the attraction factors

Of course, Equations (

71) and (

72) could be postulated, but then it would not be clear where they come from. In QDT, these equations appear naturally. Here

$\phi \left(AX\right)$ and

$\phi \left(BX\right)$ are the uncertainty factors defined in (

53). The normalizations (

58) and (

59) become

The normalization condition (

60) gives

The uncertainty factors can be rewritten as

with the interference terms being

The principal point is the condition of

interference alternation (

46), which now reads

Without this condition (

77), the system of equations for the probabilities would be incomplete, and the disjunction effect could not be explained.

In the goal of explaining the disjunction effect, it is not sufficient to merely state that some type of interference is present. It is necessary to determine (quantitatively if possible) why the probability of acting is suppressed, while that of remaining passive is enhanced. Our aim is to evaluate the expected size and signs of the interference terms $q\left(AX\right)$ (for acting under uncertainty) and $q\left(BX\right)$ (for remaining inactive under uncertainty). Obviously, it is an illusion to search for a universal value that everybody will use. Different experiments with different people have indeed demonstrated a significant heterogeneity among people, so that, in the language of QDT, this means that the values of the interference terms can fluctuate from individual to individual. A general statement should here refer to the behavior of a sufficiently large ensemble of people, allowing us to map the observed frequentist distribution of decisions to the predicted QDT probabilities.

**Attraction Factors as Interference Terms**

The interference terms (

72) can be rewritten as

The interference-alternation theorem (Theorem 1), which leads to Equation (

77), implies that

and

Hence, in the case where

$p\left(A\right|{X}_{j})>p\left(B\right|{X}_{j})$, which is characteristic of the examples illustrating the disjunction effect, one must have the uncertainty factors which exhibit the opposite property,

$\left|\phi \right(AX\left)\right|<\left|\phi \right(BX\left)\right|$, so as to compensate the former inequality to ensure the validity of the equality (

79) for the absolute values of the interference terms. The next step is to determine the sign of

$\phi \left(AX\right)$ (and thus of

$\phi \left(BX\right)$) from (

80) and their typical amplitudes

$\left|\phi \right(AX\left)\right|$ and

$\left|\phi \right(BX\left)\right|$.

**Signs of Uncertainty Factors**

A fundamental well-documented characteristic of human beings is their aversion to uncertainty,

i.e., the preference for known risks over unknown risks [

105]. As a consequence, the propensity/utility (and therefore the probability) to act under larger uncertainty is smaller than under smaller uncertainty. Mechanically, this implies that it is possible to specify the sign of the uncertainty factors, yielding

since

A (respectively

B) refers to acting (respectively, to remain inactive).

**Amplitudes of Uncertainty Factors**

As a consequence of Equation (

81) and also of their mathematical definition (

53), the uncertainty factors vary in the intervals

Without any other information, the simplest prior is to assume a uniform distribution of the uncertainty factors in each interval, so that their expected values are respectively

Choosing in that way the average values of the uncertainty factors is equivalent to using a representative agent, while the general approach is fully taking into account a pre-existing heterogeneity. That is, the values (

83) should be treated as estimates for the expected uncertainty factors, corresponding to these factors averaged with the uniform distribution over the large number of agents.

**Interference-Quarter Law**

To complete the calculation of

$q\left(AX\right)$ and of

$q\left(BX\right)$ given by Equations (

78), we also assume the non-informative uniform prior for all probabilities appearing below the square-roots, so that their expected values are all

$1/2$ since they vary between 0 and 1. Using these in Equation (

78) results in the interference-quarter law

valid for the four-dimensional mind composed of two intentions with two representations each.

As a consequence, the probabilities for acting or for remaining inactive under uncertainty, given by Equations (

71), can be evaluated as

The influence of intention interference, in the presence of uncertainty, on the decision making process at the basis of the disjunction effect can thus be estimated a priori. The sign of the effect is controlled by the aversion to uncertainty exhibited by people (uncertainty-aversion principle). The amplitude of the effect can be estimated, as shown above, from simple priors applied to the mathematical structure of the QDT formulation.

**Uncertainty-Aversion Principle**

The above calculation implies that the disjunction effect can be interpreted as essentially an emotional reaction associated with the aversion to uncertainty. An analogy can make the point: it is widely recognized that uncertainty frightens living beings, whether humans or animals. It is also well documented that fear paralyzes, as in the cartoon of the “rabbit syndrome,” when a rabbit stays immobile in front of an approaching boa instead of running away. There are many circumstantial evidences that uncertainty may frighten people as a boa frightens rabbits. Being afraid of uncertainty, a majority of human beings may be hindered to act. In the presence of uncertainty, they do not want to act, so that they refuse the second gamble, as in example (i), or forgo the purchase of a vacation, as in example (ii), or refrain from selling stocks, as in example (iii). Our analysis suggests that it is the aversion to uncertainty that paralyzes people and causes the disjunction effect.

It has been reported that, if people, when confronting uncertainty paralyzing them against acting, are presented with a detailed explanation of the possible outcomes, they then may change their mind and decide to act, thus reducing the disjunction effect [

96,

97]. Thus, by encouraging people to think by providing them additional explanations, it is possible to influence their minds. In such a case, reasoning plays the role of a kind of therapeutic treatment decreasing the aversion to uncertainty. This line of reasoning suggests that it should be possible to decrease the aversion to uncertainty by other means, perhaps by distracting them or by taking food, drink or drug injections. This provides the possibility to test for the dependence of the strength of the disjunction effect with respect to various parameters which may modulate the aversion response of individuals to uncertainty.

We should stress that our explanation departs fundamentally from the standard reason-based rationalization of the disjunction effect summarized above. Rather than using what we perceive is an hoc explanation, we anchor the disjunction effect on the very fundamental characteristic of living beings, that of the aversion to uncertainty. This allows us to construct a robust and parsimonious explanation. But this explanation arises only within QDT, because the latter allows us to account for the complex emotional, often subconscious, feelings as well as many unknown states of nature that underlie decision making. Such unknown states, analogous to hidden variables in quantum mechanics, are taken into account by the formalism of QDT through the interference alternation effect, capturing mental processes by means of quantum-theory techniques.

It is appropriate here to remind once more that it was Bohr who advocated throughout all his life the idea that mental processes do bear close analogies with quantum processes (see, e.g., [

37,

38,

39,

40]). Since interference is one of the most striking characteristic features of quantum processes, the analogy suggests that it should also arise in mental processes as well. The existence of interference in decision making disturbs the classical additivity of probabilities. Indeed, we take as an evidence of this the nonadditivity of probabilities in psychology which has been repeatedly observed [

106,

107,

108], although it has not been connected with interference.

**Numerical Analysis of Disjunction Effect**

In the frame of QDT, it is possible, not merely to connect the existence of the disjunction effect with interference, but to give quantitative predictions. Below, this is illustrated by the numerical explanation of the examples described above.

(i) To gamble or not to gamble?

Let us turn to the example of gambling. The statistics reported by Tversky and Shafir [

96] are

Then Equations (

73) and (

74) give

Recall that the disjunction effect here is the violation of the sure-thing principle, so that, although

$p\left(A\right|{X}_{j})>p\left(B\right|{X}_{j})$ for

$j=1,2$, one observes nevertheless that

$p\left(AX\right)<p\left(BX\right)$. In the experiment reported by Tversky and Shafir [

96], the probabilities for winning or for losing were identical:

$p\left({X}_{1}\right)=p\left({X}_{2}\right)=0.5$. Then, using relation (

57), we obtain

For the interference terms, we find

The uncertainty factors (

75) are therefore

They are of opposite sign, in agreement with condition (

80). The probability

$p\left(AX\right)$ of gambling under uncertainty is suppressed by the negative interference term

$q\left(AX\right)<0$. Reciprocally, the probability

$p\left(BX\right)$ of not gambling under uncertainty is enhanced by the positive interference term

$q\left(BX\right)>0$. This results in the disjunction effect, when

$p\left(AX\right)<p\left(BX\right)$.

It is important to stress that the observed amplitudes in (

86) are close to the interference-quarter law (

84). Actually, within the experimental accuracy with a statistical error about

$20\%$, the found interference terms cannot be distinguished from the value 0.25. Thus, even not knowing the results of the considered experiment, we are able to

quantitatively predict the strength of the disjunction effect.

(ii) To buy or not to buy?

For the second example of the disjunction effect, the data, taken from [

96], read

Following the same procedure as above, we get

Given again that the two alternative outcomes are equiprobable,

$p\left({X}_{1}\right)=p\left({X}_{2}\right)=0.5$, we find

For the interference terms, we obtain

The uncertainty factors are

Again, the values obtained in (

87) are close to our predicted interference-quarter law (

84). More precisely, these values are actually undistinguished from 0.25 within the statistical error

$20\%$, typical of the discussed experiments.

Because of the uncertainty aversion, the probability $p\left(AX\right)$ of purchasing a vacation is suppressed by the negative interference term $q\left(AX\right)<0$. At the same time, the probability $p\left(BX\right)$ of not buying a vacation under uncertainty is enhanced by the positive interference term $q\left(BX\right)>0$. This alternation of interferences causes the disjunction effect resulting in $p\left(AX\right)<p\left(BX\right)$. It is necessary to stress it again that without this interference alternation no explanation of the disjunction effect is possible in principle.

In the same way, our approach can be applied to any other situation related to the disjunction effect associated with the violation of the sure-thing principle. We now turn to another deviation from rational decision making, known under the name of the conjunction fallacy.

## 10. Conclusions

We have presented a quantum theory of decision making. By its nature, it can, of course, be realized by a quantum object, say, by a quantum computer or another quantum system. This theory provides a guide for creating

thinking quantum systems [

77]. It can be used as a scheme for quantum information processing and for creating artificial intelligence based on quantum laws. This, however, is not compulsory. And the developed theory can also be applied to non-quantum objects with an equal success. It just turns out that the language of quantum theory is a very convenient tool for describing the process of decision making performed by any decision maker, whether quantum or not. In this language, it is straightforward to characterize entangled decisions, non-commutativity of subsequent decisions, and intention interference. These features, although being quantum in their description, at the same time, have natural and transparent interpretations in the simple everyday language and are applicable to the events of real life. To stress the applicability of the approach to the decision making of human beings, we have provided a number of simple illustrative examples.

We have demonstrated the applicability of the approach to the cases when the Savage sure-thing principle is violated, resulting in the disjunction effect. Interference of intentions, arising in decision making under uncertainty, possesses specific features caused by aversion to uncertainty. The theorem on interference alternation that we have derived connects the aversion to uncertainty to the appearance of negative interference terms suppressing the probability of actions. At the same time, the probability of the decision maker not to act is enhanced by positive interference terms. This alternating nature of the intention interference under uncertainty explains the occurrence of the disjunction effect.

The theory has led naturally to a calculational method of the interference terms, based on considerations using robust assessment of probabilities, which makes it possible to predict their influence in a quantitative way. The estimates are in good agreement with experimental data for the disjunction effect.

The conjunction fallacy is also explained by the presence of the interference terms. A series of experiments are analyzed and shown to be in excellent agreement with the a priori evaluation of interference effects. The conjunction fallacy is also shown to be a sufficient condition for the disjunction effect, and novel experiments testing the combined interplay between the two effects are suggested.

We have emphasized that the intention interference results in the non-commutativity of subsequent decisions, which follows from the theorem on non-commutativity of intended actions.

The approach of entropy maximization, or information-functional minimization, is employed for deriving a relation between the quantum and classical decision theories.

The specific features of the Quantum Decision Theory, distinguishing it from other approaches known in the literature on decision making and information processing, can be summarized as follows.

(1) QDT is a general mathematical approach that is applicable to arbitrary situations. We do not try to adjust the QDT to fit particular cases; the same theory is used throughout the paper to treat quite different effects.

(2) Each decision maker is characterized by its own strategic state. This strategic state of mind is, generally, not a trivial wave function, but rather a composite vector, incorporating a great number of intended competing actions.

(3) QDT allows us to characterize not a single unusual, quantum-like, property of the decision making process, but several of these characteristics, including entangled decisions, non-commutative decisions, and the interference between intentions.

(4) The literature emphasizes that aversion with respect to uncertainty is an important feeling regulating decision making. This general and ubiquitous feeling is formulated under the uncertainty-aversion principle, connecting it to the signs of the alternating interference terms.

(5) The theorem on interference alternation is proved, which shows that the interference between several intentions, arising under uncertainty, consists of several terms alternating in sign, some being positive and some being negative. These terms are the source of the different paradoxes and logical fallacies presented by humans making decisions in uncertain contexts.

(6) Uncertainty aversion and interference alternation, combined together, are the key factors that suppress the probability of acting and, at the same time, enhance the probability of remaining passive, in the case of uncertainty.

(7) The principal point is that it is not simply the interference between intentions as such, but specifically the interference alternation, together with the uncertainty aversion, which are responsible for the violation of the Savage’s sure-thing principle at the origin of the disjunction effect.

(8) The conjunction fallacy is another effect that is caused by the interference of intentions, together with the uncertainty-aversion principle. Without the latter, the conjunction effect cannot be explained.

(9) The conjunction fallacy is shown to be a sufficient condition for the disjunction effect to occur, exhibiting a deep link between the two effects.

(10) The general “interference-quarter law” is formulated, which provides a quantitative prediction for the amplitude of the interference terms, and thus of the quantitative level by which the sure-thing principle is violated.

(11) Detailed quantitative comparisons with experiments documenting the disjunction effect and the conjunction fallacy confirm the validity of the derived laws.

(12) Subsequent decisions are shown, in general, to be not commutative with each other, by proving a theorem on non-commutativity of decisions.

(13) The minimization of an information functional, which is equivalent to the conditional maximization of entropy, makes it possible to connect the quantum probability with expected utility.

(14) The relation between the quantum and classical decision theories is established, showing that the latter is the limit of the former under vanishing interference terms.