1 The Prisoner’s Dilemma Game and Violations to the Sure Thing Principle
Several experiments in the literature have shown that people violate this principle in decisions under uncertainty, leading to paradoxical results and violations of the law of total probability (Tversky Kahneman, 1974; Aerts ., 2004; Birnbaum, 2008; Li Taplin, 2002; Hristova Grinberg, 2008). The prisoner’s dilemma is an example where, under uncertainty, people violate the sure thing principle, by being more cooperative.
To test the veracity of the Sure Thing Principle under the Prisoner’s Dilemma game, experiments were made in where three conditions were tested:

Participants were informed the other participant chose to Defect (Condition 1: Known to defect);

Participants were informed the other participant chose to Cooperate (Condition 2: Known to cooperate);

Participants had no information about the other participant’s decision (Condition 3 : Unknown).
Condition  Pr( P2 = Defect) 

Condition 1 (P1 Known to Defect):  0.97 
Condition 2 (P1 Known to Cooperate) :  0.84 
Condition 3: (P1 Unknown)  0.63 
Table 1 summarizes the results of these experiments for the three conditions. The column classical prediction shows the classical probability of a player choosing to , given that the decision of Player 1 is unknown (Condition 3). The payoff matrix used in Shafir Tversky (1992) Prisoner’s Dilemma experiment can be found in Table 2.
P2 = Def  P2 = Coop  

P1 = Def  30  25 
P1 = Coop  85  36 
2 QuLBIT: QuantumLike Bayesian Inference Technologies
The QuLBIT framework provides a unifying decision model for cognitive decision making, which is susceptible to cognitive biases. In addition, the framework caters for datadriven computational decisions, which are based on optimization algorithms. The main advantage of this framework is that it can cater for paradoxical and irrational human decisions during the inference process. This not only enhances the understanding of cognitive decisionmaking, but is also relevant for providing effective decision support systems. The nature of the quantumlike approach allows the system to capture optimal, suboptimal (bounded rational), or even irrational decisions which play an important role in a “humanistic system”, which are systems strongly influenced by human judgment, and behaviour.
2.1 Views on Rationality
The QuLBIT framework caters for a spectrum of views in relation to rational decision making, depending on the degrees of rationality that the decisionmaker employs. The views presented under the proposed quantumlike approach are similar to the ones put forward by Gigerenzer Goldstein (1996) and Lieder Griffiths (2020). They include decisions bounded in terms of time, processing power, information, etc. We extend this view to incorporate the notion of the irrational mind
, a point in the decisionmaker’s belief space where heuristics are no longer sufficient to produce satisficing outcomes. Figure
2 illustrates the different views on rationality that are represented in the QuLBIT framework depicted in terms of a nonlinear quantum interference wave. We define these views in the following way:
Belief space:
Corresponds to the set of possible beliefs that are held by a decisionmaker from the moment that (s)he is faced with a decision until (s)he actually makes the decision. Beliefs as inputs of thought, and desires as motivational sources of reasoning, guide decisionmaking
(Cushman, 2019). 
Unbounded Rationality: Corresponds to the ideal scenario where the decisionmaker has unlimited cognitive resources: time, processing power and information in order to transact the decision. Note that the ideal scenario is often not within reach for a human decisionmaker.

Bounded Rationality: Corresponds to the scenario where the decisionmaker makes decisions bounded in terms of cognitive resources with limited information, time, and processing power. Consequently, fast and frugal heuristics are applied sometimes resulting in suboptimal decisions (Kahneman ., 1982), but also yielding goodenough adaptive decisions (Gigerenzer Gaissmaier, 2019; Spiliopoulos Hertwig, 2020), i.e. favourable in terms of meeting desires satisfactorily without creating illogical or improbable conclusions. Fast and frugal heuristics, by their definition, are not produced by strictly logical or probabilistic calculations, but equally they do not violate logical or probabilistic principles.

Resource Rationality: Corresponds to bounded rationality in the decisionmaker’s belief space that lead to the maximum performance for a given level of uncertainty Lieder Griffiths (2020).

Irrationality: Corresponds to the situation where bounded rationality can no longer apply fast and frugal heuristics that satisfy the desires due to extreme levels of uncertainty or due to the inability of the decisionmaker to make adequate decisions. These decisions generally have poor utility and performances. The fast and frugal heuristics start to fail, and the decisionmaker experiences cognitive dissonance effects and starts to violate rules of logic and probability, which in turn leads to unfavourable heuristics and their paradoxical and irrational outcomes. These decisions are hard (or even impossible) to be captured by current computational decision systems.
2.2 QuantumLike Bayesian Networks
The fundamental core of the QuLBIT framework is the notion of graphical probabilistic inference using the formalism of quantum theory. The Quantumlike Bayesian network, originally proposed in Moreira Wichert (2014, 20161) is the fundamental building block of the QuLBIT framework and is also an example of a model that has been extensively studies in the literature for predicting and accommodating paradoxical human decisions across different decision scenarios, ranging from psychological experiments (Moreira Wichert, 20161, 2017; Wichert ., 2020) to realworld credit application scenarios (Moreira ., 2018). The difference between the traditional Bayesian network (BN) and a quantumlike Bayesian Network (QLBN) is the way one specifies the values of the conditional probability tables. While in the traditional BNs, one uses real numbers to express probabilities, in quantum mechanics these probabilities are expressed as probability amplitudes, which are represented by complex numbers. Figure 3 shows a representation of a QuantumLike Bayesian Network.
Exact inference in QLBNs is given by three steps:

Definition of the superposition state. In quantum theory, all individual quantum states contained in a Hilbert Space are defined by a superposition state which is represented by a quantum state vector
comprising the occurrence of all events of the system. This can be analogous to the classical full joint probability distribution, with the difference that instead of using real numbers to express probabilities, one uses complex probability amplitudes. For example, for
Shafir Tversky (1992)PD experiment, the full joint distribution and the corresponding superposition state can be represented by the following vectors:

A density matrix which describes the quantum system. The density operator, , aims to describe a system where we can compute the probabilities of finding each state in the network. One way to achieve this is by computing a density operator through the product between the superposition state and the corresponding conjugate transpose , that is .
The density operator also contains quantum interference terms in the offdiagonal elements, which are the core of this model. It is precisely through these offdiagonal elements that, during the inference process, one is able to obtain quantum interference effects, and consequently, deviations from normative probabilistic inferences.

QuantumLike Marginal Distribution. The quantumlike marginal probability can be defined by two selection operators and , which are vectors that select the entries of the classical joint distribution that match the query.
Computing the quantumlike probability of Player 2 defecting, given that (s)he is uncertain about Player 1’s strategy, , corresponds to summing out all entries of the joint related to Player 1’s strategy. This probability can be computed by applying the selection operator to the density matrix, , and normalising the results with a normalisation factor , and where is the transpose of operator ,
(1) This is the same as having the classical probability, , together with a quantum interference term, , which corresponds to the emergence of destructive / constructive interference effects, associated with the uncertainty that the player is experiencing,
(2) In the same way, we can compute the probability of Player 2 cooperating, ,
(3) This suggests that the proposed model provides a hierarchy of mental representations ranging from quantumlike effects to pure classical ones.
Note that if one sets or to , then and . This means that the interference term is canceled and the quantumlike Bayesian network collapses to its classical counterpart. Setting , will reproduce the disjuction effect observed in Shafir Tversky (1992), . In Moreira Wichert (20161); Andreas Wichert Bruza (2020), the authors proposed a similarity heuristic and later a law of balance that are able to automatically find the values of and without manually fitting the data.
2.3 QuantumLike Influence Diagrams
QuantumLike Influence diagrams (Moreira Wichert, 2018) are a directed acyclic graph structure that represents a full probabilistic description of a decision problem by using probabilistic inferences performed in quantumlike Bayesian networks together with an utility function.
Given a set of possible decision rules, , a classical Influence Diagram computes the decision rule that leads to the Maximum Expected Utility in relation to decision . In a classical setting, this formula makes use of a full joint probability distribution, , over all possible outcomes, , given different actions belonging to the decision rules where the goal is to choose some action that maximises the expected utility with respect to some decision rule, .
(4) 
The quantumlike approach of the influence diagrams consists in replacing the classical probability, , by its quantum counterpart, . The general idea is to take advantage of the quantum interference terms produced in the quantumlike Bayesian network to influence the probabilities used to compute the expected utility.
Mathematically, one can define utility operators, and , that represent the payoff that Player 2 receives if (s)he chooses to and to , respectively. And the quantumlike influence diagram simply consists in replacing the classical probability in Equation 4, by the probability computed by the quantumlike Bayesian network . Details of these formalisms can be found in the publicly available notebook^{2}^{2}2https://git.io/JfKKB and Moreira Wichert (2018).
The expected utility of Player 2 defecting becomes
(5) 
(6) 
(7) 
In the same way, the expected utility of Player 2 cooperating becomes
(8) 
From this formalism, the region of the belief space where the decisionmaker will always perceive that (s)he will have a higher utility for cooperating, , is given by , and .
3 A Novel Explanatory Analysis in QuantumLike Decision Models
In the QuLBIT framework, for decisions under uncertainty, the decisionmaker’s beliefs are represented as waves during the reasoning process. Only when the decisionmaker makes a decision, these beliefs collapse to the chosen decision with a certain probability and utility. Before reaching a decision, the decisionmaker can experience uncertainty regarding Player 1’s actions. This corresponds to beliefs of and competing with each other causing constructive/destructive interferences (quantum interference parameters and ).
Figure 5 (right) shows the combined graphical representations of the utilities that a player can obtain when reasoning about considering a strategy (Equations 7) or a strategy (Equation 8) according to the uncertainty that (s)he feels about Player 1’s actions.
It follows from Figure 5 (right) that this model allows different levels of representations of decisions under uncertainty ranging from (1) fully rational and optimal decisions (fully classical), (2) suboptimal decisions, to (3) irrational decisions (Shafir Tversky, 1992) (fully quantum). Figures 5 (left) and Figures 5 (center) show the evolution of the decisionmaker’s beliefs through the belief space enabling a novel analysis and interpretation of the quantum interference waves in terms of the different levels of rationality.

Fully classical decisions:
the majority of the decisionmakers are stable in the regions of the belief space where the MEU of defect is maximised. This notion is in accordance with predictions from expected utility theory and concepts from Game Theory, where in strictly dominant strategies, the decisionmaker stays stable in the Nash equilibrium state, in this case, engaging in a defect strategy. In this region, it seems that the decisionmaker is not experiencing much uncertainty, and consequently quantum interference effects are minimum.

SubOptimal decisions: the lighter regions of the figure indicate decisions where the MEU of defecting is close to the MEU of cooperate, , but still the decisionmaker prefers to defect, because there are not enough heuristic cues to convince him/her to switch from defect to cooperate. Quantum interference effects occur and uncertainty is high, however the quantum interference effects are not strong enough to make the decisionmaker change his mind and for that reason (s)he continues to choose according to expected utility.

Irrational decisions: correspond to the central, blue regions of the Figure 5. In these regions, uncertainty is maximised and quantum interference effects are significant enough to make the decisionmaker change his mind. The decisionmaker irrationally engages in wishful thinking, or beliefs that are far stretched from available data on hand. It is in this region where the decisionmaker perceives that , and consequently makes a decision that deviates from the classical notions of the Expected Utility theory. Notions of game theory actually accept the fact that the decisionmaker might not always obey to the formalisms of expected utility theory. This can occur when players did not understand the rules of the game, or simply because they played randomly. What game theory notions tell us is that the decisionmaker will not stay stable in these irrational states (in this case the state), which is in accordance with Figure 5.
4 Conclusions
It is the purpose of this paper to provide a set of contributions of quantumbased models applied to cognition and decision as an alternative mathematical approach for decisionmaking in order to better understand the structure of human behaviour.
The QuLBIT framework is opensource and allows the combination of both Bayesian and nonBayesian influences in cognition, where classical representations provide a better account of data as individuals gain familiarity, and quantumlike representations can provide novel predictions and novel insights about the relevant psychological principles involved during the decision process.
Our contributions so far show that QuLBIT is a unified framework for cognition and decisionmaking that is able (1) to accommodate and predict paradoxical human decisions (namely disjunction errors), (2) to analyse the belief space of the decisionmaker through quantum interference, (3) to quantify uncertainty and (4) to provide a nonlinear view on the different levels of rationality, ranging from fully optimal decisions (classical) to irrational decisions (quantumlike).
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