In the 70s, the cognitive psychologists Amos Tversky and Daniel Kahneman decided to put to the test the axioms of the Expected Utility hypothesis. They performed a set of experiments in which they demonstrated that people usually violate the Expected Utility hypothesis and the laws of logic and probability in decision scenarios under uncertainty [5–9]. This means that, when people need to make a decision under scenarios with high levels of uncertainty, ambiguity and risk, they tend to violate the laws of probability theory, leading to decision paradoxes [3, 4].

One of these paradoxes was demonstrated in the article of Tversky and Shafir [10] and corresponds to the violation of Savage's Sure Thing Principle, also known as disjunction effects, under the Prisoner's Dilemma Game. This principle is fundamental in classical probability theory and states that, if one prefers action A over B under the state of the world X, and if one also prefers A over B under the complementary state of the world X, then one should always prefer action A over B even when the state of the world is unspecified [2]. Violations of the Sure Thing Principle imply violations of the classical law of total probability [11].

Quantum cognition has emerged as a research field that aims to build cognitive models using the mathematical principles of quantum mechanics. Given that classical probability theory is very rigid in the sense that it poses many constraints and assumptions (single trajectory principle, obeys set theory, etc.), it becomes too limited (or even impossible) to provide simple models that can capture human judgments and decisions since people are constantly violating the laws of logic and probability theory [12–14].

In this sense, psychological (and cognitive) models benefit from the usage of quantum probability principles because they have many advantages over classical counterparts [15]. They can represent events in vector spaces through a superposition state, which comprises the occurrence of all events at the same time. In quantum mechanics, the superposition principle refers to the property that particles must be in an indefinite state. That is, a particle can be in different states at the same time. Under a psychological point of view, a quantum superposition can be related to the feeling of confusion, uncertainty or ambiguity [16]. This vector space representation does not obey the distributive axiom of Boolean logic and to the law of total probability. This enables the construction of more general models that can mathematically explain cognitive phenomena such as violations of the Sure Thing Principle [17, 18], which is the focus of this study. Quantum probability principles have also been successfully applied in many different fields of the literature, namely in biology [19, 20], economics [21, 22], perception [23, 24], jury duty [25], etc.

One of the pioneering contributions to the Quantum Cognition field comes from the works of Aerts and Aerts [26]. The authors designed a quantum machine, which consists in a particle that can move across the surface of a sphere. An elastic, representing some experiment is introduced in this sphere. The particle then moves orthogonally to the elastic and the elastic breaks uniformly into two parts. With this geometric representation, one can easily compute the probabilities of the particle falling into each side of the elastic. The model was extended with an ϵ parameter that represents the evolution from a quantum structure to a classical one. This parameter varies between [0, 1], where 0 corresponds to maximum lack of knowledge (quantum structure) and 1 to zero lack of knowledge (classical knowledge). Between this interval, there is the possibility of exploring other types of structures that are neither classical nor quantum. The authors also made several experiments to test the variation of probabilities when posing yes∕no questions. According to their experiment, most participants formed their answer at the moment the question was posed. This behavior goes against classical theories, because in classical probability, it would be expected that the participants have a predefined answer to the question and not form it at the moment of the question. A further discussion about this study can be found in the works of Aerts [27–29], Gabora and Aerts [30], and Aerts et al. [31].

In other subsequent works, namely in Aerts [32], the author uses the formalisms of quantum mechanics in order to accommodate disjunction effects. The author, represents concepts as vectors and membership weights as quantum weights, in a complex Hilbert Space. By using quantum interference effects and quantum superpositions, the author was able to model accurately the disjunction of concepts present in experimental data.

In this article, we provide an overview and discussion of the most important state-of-the-art quantum cognitive models that are able to explain the paradoxical findings of experiments that violate the Sure Thing Principle (ex: the Prisoner's Dilemma game [33]). We conduct a deep comparison of and discussion on several quantum models: the Quantum-Like Approach [34], the Quantum Dynamical Model [35], the Quantum Prospect Decision Theory [36] and Quantum Bayesian Networks [37–40]. We discuss these models in terms of three metrics: (1) incorporation of quantum interference effects, (2) how to find values for quantum parameters, and (3) scalability of the model for more complex decision problems.

The first metric checks if the model uses quantum interference effects to predict actions chosen under uncertainty. Following the work of Yukalov and Sornette [36], toward uncertainty, human beings tend to have aversion preferences. They prefer to choose an action that brings them a certain but lower propensity/utility instead of an action that is uncertain but can yield a higher propensity/utility [41]. This can be simulated through quantum interference effects, in which one outcome is enhanced (or diminished) toward the opposite outcome.

The second metric takes into account the problem of finding values for quantum parameters. In quantum mechanics, a quantum state is modeled by probability amplitudes [42]. These amplitudes are a component of the wave function and this wave function represents a quantum state. Associated with each probability amplitude is a quantum parameter representing the phase of the wave. The interpretation of this parameter under the psychology literature is still not clear, although various works have presented interpretations [17]. Moreover, when applying quantum principles to cognition (or to any other subject), one will need to set these quantum parameters in such a manner that they will lead to accurate predictions. In this metric, we will check how easy it is for the analyzed models to set these parameters.

The third and last metric consists of determining if the model can be extended to more complex scenarios. Although there are many experiments that report violations of the Sure Thing Principle [17, 35, 43, 44], these experiments consist of very small scenarios that are modeled by, at most, two random variables. Therefore, many of the proposed models in the literature are only effective under such small scenarios and become intractable (or even cannot be applied) under more complex situations. These metrics will be analyzed with more detail in Section 8 of the present work.

It is important to note that the goal of this work is the following: we have collected a set of models from the literature that attempt to tackle violations of the Sure Thing Principle in a quantum fashion, and then we compare the collected models. For this comparison, we just show, through a mathematical description of each model, their advantages and disadvantages. That is, we compare these models with the three metrics proposed: number of parameters involved in the model, the scalability of the quantum interference effects and their usage. We will also show that classical models also suffer from the same parameter growth problem as quantum approaches. However, because these models must obey set theory and the laws of classical probability, it is not possible to use them to make predictions in situations where the Sure Thing Principle is being violated.

We will start this article with a motivational problem, in which the Sure Thing Principle is found to be violated under the Prisoner's Dilemma Game (Section 2). In Section 3, we will show that a classical approach cannot accommodate violations of the Sure Thing Principle because these approaches obey set theory and consequently the laws of probability theory. We will make a full step-by-step description of the most influential models of the literature. We will show how one could apply them to predict the results concerned with violations of the Sure Thing Principle in the Prisoner's Dilemma Game. In Section 4, we will cover the Quantum-Like Approach [34]. In Section 5, we will analyze the Quantum Dynamical Model [17]. In Section 6, we will describe the Quantum Prospect Decision Theory [36]. In Section 7, we will provide an overview of Quantum-Like Bayesian Networks [37–40]. We then engage in a deeper discussion of these approaches and give thought to the advantages/disadvantages of each model in Section 8. We finish this article by presenting the main conclusions of this work by providing some insights regarding various trends in quantum probabilistic models (Section 9).

In the Växjö Model, the context relates to the circumstances that form the setting for an event in terms of which it can be fully understood, clarifying the meaning of the event. For instance, in domains outside of physics, such as cognitive science, one can have mental contexts. In social sciences, we can have a social context. The same idea is applied to many other domains, such as economics, politics, game theory, and biology.

Associated with a context, there is a set of observables. In quantum mechanics, an observable corresponds to a self-adjoint operator on a complex Hilbert Space. Under the Växjö Model, these observables correspond to the set of possible events with their respective values.

For instance, for a context C ∈ C and for an observable a ∈ O having values α, the probability of the value of one observable is expressed in terms of the conditional (contextual) probability involving the values of an observable. That is, the probability distribution π is given by:

If we move into the quantum mechanics realm, Equation (9) can be interpreted as the selection with respect to the result a = α of a measurement performed in a.

For the contextual probability model, the Växjö framework corresponds to a model M described by M = (C, O, π(O, C)). Again, C is a set of contexts, O is the set of observables, and π(O, C) corresponds to a probability distribution of some observables belonging to a specific context.

In addition, assume for a context C ∈ C that there are two dichotomous observables a, b ∈ O and that each of these observables can take some values α ∈ a and β ∈ b, respectively.

The Växjö Model can be built from the general structure of the quantum law of total probability. That is, the formula is a combination of the classical probability theory with a supplementary term called the interference term (Equation 10). This term does not exist in classical probability and enables the representation of interferences between quantum states.

Under this representation, we can replace Classical_Probability by the classical total probability and also replace the quantum Interference_Term by a supplementary measure, represented by δ(β | a, C). Under the Växjö Model, the term δ(β | a, C) corresponds to:

Equation (11) can be written in a similar way to the classical probability in the following manner:

If we perform the normalization of the probability measure of supplementary δ(β | a, C) by the square root of the product of all probabilities, we obtain:

From Equation (13), the general probability formula of the Växjö Model can be derived. For two variables, it is given by:

If we look closely at Equation (14), we can see that the first summation of the formula corresponds to the classical law of total probability. The second term of the formula (the one that contains the λ_θ parameter) does not exist in the classical model and is called the interference term.

Although the Quantum-Like Approach provides great possibilities compared with the classical one, it appears that it cannot completely cover data from psychology and that a quantum formalism was not enough to explain some paradoxical findings (see [50]), so hyperbolic spaces were proposed [51–53].

From Equation (14), if Pr(b=β)-∑α∈aPr(a=α|C)Pr(b=β|a=α,C) is different from zero, then various interference effects occur. To determine which type of interference occurred, one tests the Växjö Model for quantum probabilities. This can be determined by normalizing the supplementary measure in a quantum fashion, just as presented in Equation (13).

If we are under a quantum context, then the quantum interference term will be:

In a quantum context because the supplementary term δ(β | a, C) is being normalized in a quantum fashion, then we automatically know that the indicator term λ_θ will always have to be smaller than 1 to obtain quantum probabilities, λ_θ ≤ 1. Thus, under trigonometric contexts, the Växjö Model for quantum probabilities becomes:

If, however, the probability Pr(b = β) was not computed in a trigonometric space (that is, it is not quantum), then, it is straightforward that the quantum normalization applied in Equation (13) will yield a value larger than 1. Because we are not in the context of quantum probabilities, the quantum normalization factor will fail to normalize the interference term and will produce a number larger than the normalization factor. Under these circumstances, the Växjö Model incorporates the generalization of hyperbolic probabilities, arguing that the context in which these probabilities were computed was Hyperbolic [49, 53, 54].

Under Hyperbolic contexts, the Växjö Model contextual probability formula becomes:

In summary, according to the values computed by the indicator function λ_θ, the Växjö Model enables the computation of probabilities in the following contexts:

If | λ_θ | = 0, then there is no interference, and the Växjö Model collapses to classical probability theory.

The probabilities that emerge from the Växjö model for trigonometric spaces (i.e., quantum probabilities), do not provide a complete description of a quantum system because it can violate the positivity axiom of probability theory [49].

In this sense, an algorithm was proposed in the literature that extends the Växjö model and is able to accommodate the positivity axiom. The algorithm proposed is the Quantum-Like Representation Algorithm (QLRA), and it was proposed by Khrennikov [55–59].

As already mentioned, quantum complex amplitudes can be obtained from classical probability by using Born's rule [60, 61]. In the QLRA, for any trigonometric context C, one can simplify Born's rule for two dichotomous variables using (Equation 19) [49].

Equation (18) can be simplified in the following manner:

Equation (19) corresponds to the representation of the quantum law of total probability through the Växjö model. In this equation, the angle θ_{β|α, C} corresponds to the phase of a random variable and incorporates the phase of both A = α₁ and A = α₂ in the following manner: θ_{β|α, C} = θ_β|α1 − θ_β|α2.

One should note that the Quantum-Like Approach can be extended to more complex decision scenarios, that is, with more than two random variables. However, this will lead to the very difficult task of tuning an exponential number of quantum θ parameters. Peter Nyman noticed this problem when he generalized the Quantum-Like Approach for 3 dichotomous variables [52, 62–64].

If we want to compute the average probabilities reported in Table 1 for the Prisoner's Dilemma game, then we would need to make the following substitutions to Equation (18):

The main problem of the Växjö model and the Quantum-Like Approach is that it can only address very small decision scenarios and the fitting of the θ parameter has to be done fitted to data. To compute the probability of a player choosing to defect, Pr(P₂ = Defect), one would proceed as follows:

To achieve the observed result, θ must be equal to 1.7779 to achieve the final probability Pr(P₂ = Defect) = 0.64. However, this method does not provide any other means to find this θ parameter except by extrapolating the observed data.

To accommodate the violations of the Sure Thing Principle, the uncertainty factor must be set in such a way that it will enable accurate predictions. Two methods were proposed by Yukalov and Sornette [36] to estimate the uncertainty factor: the Interference Alternation method and the Interference Quarter Law.

In this section, we apply the Quantum Prospect Decision Theory to try to predict the average results for the Prisoner's Dilemma Game reported in Table 1.

The probability of a player defecting (and cooperating), given that one does not know what the action of the other player was, is given by Equation (38). For simplicity, we will assume the following notation: Defect (D) and Cooperate (C).

The interference terms are given by:

The uncertainty factors are given by:

According to the Interference Quarter Law and to the Alternation Law, the probabilities for acting under uncertainty are given by:

For the Prisoner's Dilemma Game,

Then, the final predicted probabilities are given by:

The average probability to defect for the Prisoner's Dilemma Game in Table 1 when the first player's action is unknown is 0.64. That means that, with the Quarter Interference Law together with the Interference Alternation property, the Prospect Quantum Decision Theory obtained an error of 13%.

A classical Bayesian Network can be defined by a directed acyclic graph structure in which each node represents a different random variable from a specific domain and each edge represents a direct influence from the source node to the target node. The graph represents independence relationships between variables, and each node is associated with a conditional probability table that specifies a distribution over the values of a node given each possible joint assignment of values of its parents [73].

The full joint distribution [74] of a Bayesian Network, where X is the list of variables, is given by:

The formula for computing classical exact inferences on Bayesian Networks is based on the full joint distribution (Equation 43). Let e be the list of observed variables and let Y be the remaining unobserved variables in the network. For some query X, the inference is given by:

The summation is over all possible y, i.e., all possible combinations of values of the unobserved variables y. The α parameter corresponds to the normalization factor for the distribution Pr(X|e) [74]. This normalization factor comes from some assumptions that are made in Bayes rule.

We represent the Prisoner's Dilemma Game under a Bayesian Network structure in which we assume neutral priors: there is a 50% of a player choosing the actions Defect or Cooperate (Figure 5). The decision of the first participant is then followed by the decision of the second participant. The probability distribution of the second player is obtained (or learned) from the experimental data for the averaged results in Table 1 when the actions of the first player are observed. Using this data, the goal is to try to determine the probability of the second player choosing to defect given that it is not known what action the first player chose.

To compute the probability Pr(P₂ = Defect), two operations are required: the computation of the full joint probability distribution (Equation 43) and the computation of the marginal probability.

The full joint probability distribution can be easily computed by multiplying all possible assignments of the network with each other. Table 2 shows the computation of these probabilities.

The marginalization formula is used when we want to perform queries to the network. For instance, in the Prisoner's Dilemma Game, we want to know what the probability is of the second player choosing to defect given that we do not know what the other player has chosen, Pr(P₂ = Defect). This is obtained by summing the entries of the full joint probability (Table 2) that have P₂ = Defect. That is, we sum up the first and third rows of this table. Equation (45) shows this operation. For simplicity, we have used the following notation: D = Defect and C = Cooperate.

In Equation (45), one can see that the classical Bayesian Network was not able to predict the observed results in Table 1 using classical inference. One might think that, if we parameterize the Bayesian Network to take into account the player's actions and dissonance effects, there could be a possibility of obtaining the required results. This line of thought is legitimate, but one must take into account that, in the end, the probabilistic inferences computed through the Bayesian Network must obey set theory and the law of total probability. This means that, even if we parameterize the network, we cannot find any closed form optimization that could lead to the desired results. This happened with the previous example of the Markov Model in Section 3. Although we parameterized the player's actions and dissonance effects, we could not arrive at the desired results because they go against the laws of probability theory, and Markov Models (as well as Bayesian Networks) must obey these laws.

There are two main works in the literature that have contributed to the development and understanding of Quantum Bayesian Networks. One belongs to Tucci [37] and the other to Leifer and Poulin [38].

In the work of Tucci [37], it is argued that any classical Bayesian Network can be extended to a quantum one by replacing real probabilities with quantum complex amplitudes. This means that the factorization should be performed in the same manner as in a classical Bayesian Network. Thus, the Bayesian Network of Figure 5 could be represented by a Quantum Bayesian Network with the following matrices tables (the ordering of the probability amplitudes in the matrices are the same as the ones in Figure 5):

One significant problem with Tucci's work is related to the non-existence of any methods to set the phase parameters e^iθ. The author states that one could have infinite Quantum Bayesian Networks representing the same classical Bayesian Network depending on the values that one chooses to set the parameter. This requires that one knows a priori which parameters would lead to the desired solution for each node queried in the network (which we never know). Thus, for these experiments, Tucci's model cannot predict the results observed because one does not have any information about the quantum parameters.

In the work of Leifer and Poulin [38], the authors argue that, to develop a Quantum Bayesian Network, quantum versions of probability distributions, quantum marginal probabilities and quantum conditional probabilities are required (Table 3). The authors performed a preliminary study of these concepts. Generally speaking, a quantum probability distribution corresponds to a density matrix contained in a Hilbert space, with the constraint that the trace of this matrix must sum to 1. In quantum probability theory, a full joint distribution is given by a density matrix, ρ. This matrix provides the probability distribution of all states that a Bayesian Network can have. The marginalization operation corresponds to a quantum partial trace [75, 76].

In the end, the models of Tucci [37] and Leifer and Poulin [38] fail to provide any advantage relative to the classical models because they cannot take into account interference effects between random variables. Thus, they provide no advantages in modeling decision-making problems that try to predict decisions that violate the laws of total probability.

A more recent work from Moreira and Wichert [65] suggested defining the Quantum-Like Bayesian Network in the same manner as in the work of Tucci [37], replacing real probability numbers by quantum probability amplitudes.

In this sense, the quantum counterpart of the full joint probability distribution corresponds to the application of Born's rule to Equation (43):

The general idea of a Quantum-Like Bayesian network is that, when performing probabilistic inference, the probability amplitude of each assignment of the network is propagated and influences the probabilities of the remaining nodes. In other words, every assignment of every node of the network is propagated until the node representing the query variable is reached. Note that, by taking multiple assignments and paths at the same time, these trails influence each other in producing interference effects.

The quantum counterpart of the Bayesian exact inference formula corresponds to the application of Born's rule to Equation (44), leading to:

Expanding Equation (47), it will lead to the quantum interference formula:

In the Quantum Dynamical Model, because it uses unitary operators, the double symmetric property of these operators does not require the normalization of the computed values. However, in this approach, because we do not have the constraints of double stochasticity operators, we need to normalize the final scores that are computed to achieve a probability value. In classical Bayesian inference, normalization of the inference scores is also necessary due to assumptions made in Bayes rule. The normalization factor corresponds to α in Equation (48).

Note that, in Equation (48), if one sets (θ_i − θ_j) to π∕2, then cos(θ_i − θ_j) = 0, which means that the quantum Bayesian Network collapses to its classical counterpart. That is, they can behave in a classical way if one sets the interference term to zero. Moreover, in Equation (48), if the Bayesian Network has N binary random variables, we will end up with 2^N free quantum θ parameters. We represent each set of quantum parameters as a single parameter of the full joint probability distribution just like it is presented in Table 4. Approaches to tune those parameters under a Quantum-Like Bayesian Network approach are still an open research question.

In the model of Moreira and Wichert [65], if there are many unobserved nodes in the network, then the levels of uncertainty are very high and the interference effects produce changes in the final likelihoods of the outcomes. However, in the opposite scenario, when there are very few unobserved nodes, then the proposed quantum model tends to collapse into its classical counterpart because the uncertainty levels are very low. This work only provides a study on the impact of the quantum parameters in complex decision scenarios. On later works, the same authors have proposed the usage of heuristics to automatically assign values to quantum parameters [39, 77].

In this section, we will demonstrate how the proposed Bayesian Network can be applied to the average results presented in Table 1 for the Prisoner's Dilemma game, just as was proposed in the work of Moreira and Wichert [65].

We begin applying the Quantum-Like formalism by creating a Bayesian Network out of the decision problem, in which real classical probabilities are replaced by quantum amplitudes (Figure 6). In the Prisoner's Dilemma Game, if nothing is told to the participants, then there is a 50% chance of the first participant choosing to defect or cooperate. The decision of the first participant is then followed by the decision of the second participant.

To compute the probability Pr(P₂ = Defect), two operations are required: the computation of the quantum version of the full joint probability distribution (Equation 46) and the computation of the quantum version of the marginal probability (Equation 48).

The full joint probability distribution can be easily computed by multiplying all possible assignments of the network with each other. For instance, the quantum full joint probability amplitude ψ_{(P1 = Defect, P2 = Defect)} is given by multiplying the prior probability amplitude ψ_{(P1 = Defect)} with the conditional probability amplitude ψ_{(P2 = Defect|P1 = Defect)}. Table 4 shows the computation of these quantum probability amplitudes.

From the quantum version of the full joint probability distribution, one can compute the quantum version of the marginal probability distribution by summing all the entries of Table 4 that contain the assignment P2 = Defect (Equation 49). For simplification purposes, we will consider the following abbreviations: Defect = D and Cooperate = C.

To compute the normalization factor α, we also need to compute Pr(P2 = C):

The normalization factor α is given by Equation (54).

Equation (54) contains two quantum parameters θ. Setting these parameters is still an open research question in the literature, although in some works, various heuristics have been proposed to address this problem [39, 40, 77].

In this section, we analyze the presented works in the literature regarding three different metrics: interference effects, parameter tuning, and scalability.

Table 5 presents a summary of the evaluation of the models presented in this work with respect to the three metrics described above. The parameter growth column is based on the number of parameters that each model generates when we increase the number of unknown random variables in the decision model

Starting the discussion with the classical models presented in Sections 3 and 7.1, the probabilistic inferences computed through Bayesian/Markov Networks must obey set theory and the law of total probability. This means that, even if we parameterize the networks, we cannot find any closed form optimization that could lead to the desired results. These networks can be modeled with no parameters (just as was presented in Sections 3 and 7.1), or they can be parameterized. This parameterization can end up with the same size as the full joint probability distribution of the networks. Although these models do not make use of any quantum interference effects and consequently cannot accommodate violations of the Sure Thing Principle, it is worth noting that one can always classically explain behavioral results through appropriate conditionalizations and extensions of classical probabilistic models [16].

The Quantum-Like Approach [72] is based on the direct mapping of classical probabilities to quantum probability amplitudes through Born's rule. This means that one can perform inferences for more complex decision-making scenarios by using the quantum counterpart of the classical marginal probability formula. Thus, the model generates quantum interference effects. The main problem of the Quantum-Like approach concerns the quantum parameters. The current works of the literature do not provide any means to assign values to these quantum parameters. They have to be fitted to explain the observed outcome. Thus, the Quantum-Like approach, although it can be (mathematically) extended to more complex decision scenarios, does not provide any means to assign quantum parameters. Note that, in the Quantum-Like approach, just like in many other models, it is required a mathematical fitting of a set of parameters to make an optimal prediction of the probabilities. So, the Quantum-Like Approach is considered to be a predictive model.

The Quantum Dynamical Model proposed by Pothos and Busemeyer [17] and Busemeyer et al. [35] incorporates quantum interference effects not from the quantum law of probability but by the usage of unitary operators and Hamiltonians. One of the main disadvantages of this model concerns the definition of the Hamiltonian matrices. Creating a Hamiltonian is a very hard problem. It is required that all possible interactions of the decision problem are known, and this specification must be made in such a way that the matrix is doubly stochastic. The unitary matrix also grows exponentially with the complexity of the decision problem, and the computation of a unitary operator from such matrices is a very complex process. Most of the time, approximations are used because of the complexity of the calculations involved in the matrix exponentiation operation. Just as in the Quantum-Like Approach, one needs to fit the quantum parameters so that the final model can give the observed outcome. It is important to note that, in the Quantum Dynamical Model, the parameters used are based on a psychological setting. The incorporation of parameters to model dissonance effects and the payoffs of the players provide an approximation to the psychology of the problem that is not observed in other quantum cognitive models in the literature.

Finally, the Quantum Prospect Theory proposed by Yukalov and Sornette [36] also incorporates quantum interference effects from the quantum law of total probability. This model is very similar, from a mathematical point of view, to the Quantum-Like Approach, with the difference that it proposes laws to compute the quantum interference parameters: the alternation and the quantum quarter laws. Although the model is very precise for very small decision problems (such as the Prisoner's Dilemma), it is not clear how the quantum quarter law and the alternation law would work for more complex problems. For this reason, the Quantum Prospect Theory is a model that enables the usage of quantum interference terms to make predictions under paradoxical scenarios and also provides an automatic mechanism to set the quantum parameters under very small scenarios with a static interference term (q = ±0.25). That is, the interference term is always the same, even for different contextual problems. For this reason, the model is not able to generalize well for more complex decision scenarios.

Regarding Bayesian Networks, it is hard to apply the model proposed in the work of Tucci [37] in paradoxical findings that violate the Sure Thing Principle because the author makes no mention of how to set these parameters. He even argues that a classical Bayesian Network can be represented by an infinite number of quantum Bayesian Networks depending on how one tunes the quantum parameters. Because the model is a Bayesian Network, one is able to perform inferences for any scenario by using the quantum counterpart of the classical marginal probability formula. Thus, in the end, the quantum Bayesian Network proposed by Tucci [37] is scalable and takes into account quantum interference effects; however, it does not give any insights into how to set the quantum parameters that result from the interference.

In the work of Leifer and Poulin [38], the authors create a direct mapping from classical probability to quantum theory. Because they made a quantum Bayesian Network, this model enabled probabilistic inference, and consequently, it can be generalized for any number of random variables through the use of the quantum part of the marginal probability formula. By making the direct mapping from classical to quantum probabilities, the full joint probability distribution is mapped into a density matrix. This means that the interference terms are canceled. The authors also take into account the order in which the operations are performed. Because, the commutativity axiom is not valid in quantum mechanics, we obtain different outcomes if the calculations are performed in a different order. Thus, the quantum Bayesian Network proposed by Leifer and Poulin [38] is scalable and takes into account quantum interference effects; however, by making a direct mapping from classical to quantum, these interference effects will cancel because the network will collapse into its classical counterpart. Thus, in the end, this model does not take advantage of quantum interferences to explain paradoxical decision scenarios.

In the work of Moreira and Wichert [65], the authors also make a direct mapping from classical theory to quantum probability by replacing classical real probability values by complex quantum probability amplitudes using Born's rule. They also applied the same mechanism to derive a quantum-like full joint probability distribution formula and a quantum-like marginal probability distribution for exact inference. In the end, the model is very similar to the Quantum-Like Approach, and it can be modeled for more complex decision-making scenarios very easily due to its graphical structure. Because this model uses quantum probability amplitudes, quantum interference effects arise from the quantum-like exact inference formula. However, the number of parameters grows exponentially large when the levels of uncertainty are high, that is, when there are many unobserved nodes in the network. Although the authors have proposed some dynamic heuristics to address this problem in recent works [39, 40, 77], one needs to take into account that they are heuristics, which means that it can lead to the expected outcome, but it can also lead to completely inaccurate results.

Note that we are aware that the problems that we note in this discussion section about the quantum models are the same in many cognitive science models. However, we are not claiming that it is difficult to find the parameters for a game such as the Prisoner's Dilemma. What we are claiming is that the several models analyzed in this work (Quantum-Like Approach, Quantum Dynamical Model, Quantum-Like Bayesian Networks) contain a set of parameters that need to be fitted to match the desired predictions.

For instance, the Quantum Dynamical Model requires three parameters for such a small decision scenario, whereas the Quantum-Like Approach only needs one, and the Quantum Prospect Decision Theory does not need any parameters, because it has a static heuristic to replace the interference term. The purpose of this discussion section is simply to compare the existing quantum models in terms of the evaluation metrics specified in Table 5.

All models analyzed in this work present different growth rates in what concerns parameters. For instance, the Dynamical Model parameterizes the player's actions plus an additional parameter to model cognitive dissonance effects. Thus, the number of parameters would be static if we consider the N-Person Prisoner's Dilemma Game. That is, instead of having only 2 players, it is extended to N players. In the case of the Quantum-Like Approach, we would have 2^N parameters for the N-Person Prisoner's Dilemma Game. The number 2 comes from the fact that each player has two actions (either defect or cooperate). The same applies to the Classical Networks, the Quantum-Like Bayesian Networks and the Quantum Prospect Theory Model. However, because the authors of this last model presented the Quantum Quarter Law of Interference as a static heuristic, this model does not require any parameters.

At this point, the reader might be thinking that the Quantum Dynamical Model provides great advantages vs. the existing models because the number of parameters required corresponds to the player's actions with an additional cognitive dissonance parameter. Although this line of thought is correct, one should also take into account how the model unfolds. Although the numbers of parameters do not grow exponentially large as in the Quantum-Like Approach, the size of the Hamiltonian does. In fact, it grows exponentially large with the following size: NactionsNplayers×NactionsNplayers, where N_actions represents the number of actions of the players and N_players corresponds to the number of players.

We conclude this section by clarifying that most of the quantum cognitive models proposed in the literature have been directed toward small decision scenarios because of the scarcity of datasets representing complex decision scenarios and violations of the Sure Thing Principle. Consequently, the models proposed are simply overfitting simple decision scenarios. Moreover, we believe that the violations of the Sure Thing Principle tend to diminish with the complexity of the decision scenario. Imagine, for instance, a Three-Stage Gambling game. It will be very hard to find significant data that shows a player wishing to play the last gamble given that he has lost the two previous gambles. More experimental data and more studies are needed for more complex decision scenarios to test the viability of quantum models vs. their classical counterparts.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.