QuLBIT: Quantum-Like Bayesian Inference Technologies forCognition and Decision

Catarina Moreira1 (catarina.pintomoreira@qut.edu.au)

Matheus Hammes1 (matheus.hammes@connect.qut.edu.au)

Rasim Serdar Kurdoglu2 (r.s.kurdoglu@bilkent.edu.tr)

Peter Bruza1 (p.bruza@qut.edu.au)

1School of Information Systems, Queensland University of Technology, Brisbane, Australia2Faculty of Business Administration, Bilkent University, Ankara, Turkey

Abstract

This paper provides the foundations of a uni-fied cognitive decision-making framework (QulBIT)which is derived from quantum theory. The mainadvantage of this framework is that it can caterfor paradoxical and irrational human decision mak-ing. Although quantum approaches for cognitionhave demonstrated advantages over classical prob-abilistic approaches and bounded rationality mod-els, they still lack explanatory power. To addressthis, we introduce a novel explanatory analysis ofthe decision-maker’s belief space. This is achievedby exploiting quantum interference effects as a wayof both quantifying and explaining the decision-maker’s uncertainty. We detail the main modulesof the unified framework, the explanatory analy-sis method, and illustrate their application in situ-ations violating the Sure Thing Principle.

Keywords: QuLBIT; quantum cognition; quantum-like Bayesian networks; quantum-like influence di-agrams; bounded rationality; explanatory analysis

IntroductionThe primary motivation behind QuLBIT1 is thechallenge to formally account for seemingly para-doxical human decision making. It is widely knownin the literature of cognitive science and economicsthat when it comes to decision-making under un-certainty, humans usually make decisions that areinconsistent with the axioms of expected utilitytheory, leading to decisions that are either sub-optimal, paradoxical or even irrational (Kahneman& Tversky, 1979; Aerts et al., 2017). These paradox-ical decisions can result in cognitive biases (Tversky& Kahneman, 1974), violations of major economicprinciples (like the Sure Thing Principle) (Savage,1954; Allais, 1953; Ellsberg, 1961) or violations tothe laws of probability theory and logic (Busemeyeret al., 2006; Pothos & Busemeyer, 2009). In short,decades of research has found a whole range of hu-man judgements that deviates substantially from

1https://github.com/catarina-moreira/QuLBiT

what would be considered normatively correct ac-cording to logic or probability theory.

The field of Quantum Cognition emerged to re-spond to this challenge, a major feature being theuse of quantum probability theory to model hu-man cognition, including decision making (Buse-meyer & Bruza, 2012). Quantum probability canbe viewed as generalisation of Bayesian probabilitytheory. In quantum-like cognitive models, eventsare modelled as sub-spaces of a Hilbert spaces,a vector space of complex numbers (amplitudes)which enables the calculation of probabilities byprojection: performing the squared magnitude ofan amplitude. This representation allows eventsto interfere with each other, which influences theirassociated probabilities. These interference effectsgenerate a set of new parameters that can be usedto either accommodate violations in Bayesian the-ory (Busemeyer & Bruza, 2012) or paradoxical hu-man decisions Kahneman et al. (1982). Interferenceis a core concept in the QuLBIT framework whichenables an alternative quantification of uncertainty,as well as the representation of conflicting, ambigu-ous beliefs. The vector representation of superposi-tion obeys neither the distributive axiom of Booleanlogic nor the law of total probability (Moreira &Wichert, 2016b) (See Figure 1).

Quantum cognitive models make use of addi-tional parameters, which allows fitting to empir-ical data but which “do not necessarily explainthem” (Blutner & beim Graben, 2016). QuLBITaddresses this lack of explanatory power by em-ploying a novel analysis method which allows in-terpretation of the decision-maker’s belief space,e.g., when a decision-maker prefers one choice overanother. This paper illustrates QuLBIT’s novel ex-planatory analysis method in regard to violations ofthe Sure Thing Principle in the Prisoner’s DilemmaGame (Shafir & Tversky, 1992).

2520©2020 The Author(s). This work is licensed under a CreativeCommons Attribution 4.0 International License (CC BY).

Figure 1: The representation of the decision-maker’s beliefs under uncertainty where beliefs can enter intoa superposition and suffer non-linear quantum interference effects that can destroy or reinforce certain

beliefs in a single time step, leading to decisions that deviate from classical theory.

The Prisoner’s Dilemma Game andViolations to the Sure Thing Principle

Several experiments in the literature have shownthat people violate this principle in decisions un-der uncertainty, leading to paradoxical results andviolations of the law of total probability (Tver-sky & Kahneman, 1974; Aerts et al., 2004; Birn-baum, 2008; Li & Taplin, 2002; Hristova & Grin-berg, 2008). The prisoner’s dilemma is an examplewhere, under uncertainty, people violate the surething principle, by being more cooperative.

To test the veracity of the Sure Thing Principleunder the Prisoner’s Dilemma game, experimentswere made in where three conditions were tested:

• Participants were informed the other participantchose to Defect (Condition 1: Known to defect);

• Participants were informed the other participantchose to Cooperate (Condition 2: Known to coop-erate);

• Participants had no information about the otherparticipant’s decision (Condition 3 : Unknown).

Condition Pr( P2 = Defect)Condition 1 (P1 Known to Defect): 0.97Condition 2 (P1 Known to Cooperate) : 0.84Condition 3: (P1 Unknown) 0.63

Table 1: Experimental results from Shafir &Tversky (1992) PD experiment.

Table 1 summarizes the results of these experi-ments for the three conditions. The column clas-sical prediction shows the classical probability of aplayer choosing to Def ect, given that the decisionof Player 1 is unknown (Condition 3). The payoffmatrix used in Shafir & Tversky (1992) Prisoner’sDilemma experiment can be found in Table 2.

P2 = Def P2 = CoopP1 = Def 30 25

P1 = Coop 85 36

Table 2: Payoff matrix used in Shafir & Tversky(1992) Prisoner’s Dilemma experiment.

QuLBIT: Quantum-Like BayesianInference Technologies

The QuLBIT framework provides a unifying deci-sion model for cognitive decision making, whichis susceptible to cognitive biases. In addition, theframework caters for data-driven computationaldecisions, which are based on optimization algo-rithms. The main advantage of this framework isthat it can cater for paradoxical and irrational hu-man decisions during the inference process. Thisnot only enhances the understanding of cognitivedecision-making, but is also relevant for providingeffective decision support systems. The nature ofthe quantum-like approach allows the system tocapture optimal, sub-optimal (bounded rational),or even irrational decisions which play an impor-tant role in a “humanistic system”, which are sys-tems strongly influenced by human judgment, andbehaviour.

Views on RationalityThe QuLBIT framework caters for a spectrum ofviews in relation to rational decision making, de-pending on the degrees of rationality that thedecision-maker employs. The views presented un-der the proposed quantum-like approach are simi-lar to the ones put forward by Gigerenzer & Gold-stein (1996) and Lieder & Griffiths (2020). They in-clude decisions bounded in terms of time, process-ing power, information, etc. We extend this view toincorporate the notion of the irrational mind, a point

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in the decision-maker’s belief space where heuris-tics are no longer sufficient to produce satisficingoutcomes. Figure 2 illustrates the different viewson rationality that are represented in the QuLBITframework depicted in terms of a non-linear quan-tum interference wave. We define these views inthe following way:

• Belief space: Corresponds to the set of pos-sible beliefs that are held by a decision-makerfrom the moment that (s)he is faced with a deci-sion until (s)he actually makes the decision. Be-liefs as inputs of thought, and desires as mo-tivational sources of reasoning, guide decision-making (Cushman, 2019).

• Unbounded Rationality: Corresponds to theideal scenario where the decision-maker has un-limited cognitive resources: time, processingpower and information in order to transact thedecision. Note that the ideal scenario is often notwithin reach for a human decision-maker.

• Bounded Rationality: Corresponds to the sce-nario where the decision-maker makes decisionsbounded in terms of cognitive resources withlimited information, time, and processing power.Consequently, fast and frugal heuristics are ap-plied sometimes resulting in sub-optimal deci-sions (Kahneman et al., 1982), but also yieldinggood-enough adaptive decisions (Gigerenzer &Gaissmaier, 2019; Spiliopoulos & Hertwig, 2020),i.e. favourable in terms of meeting desires sat-isfactorily without creating illogical or improb-able conclusions. Fast and frugal heuristics, bytheir definition, are not produced by strictly log-ical or probabilistic calculations, but equally theydo not violate logical or probabilistic principles.

• Resource Rationality: Corresponds to boundedrationality in the decision-maker’s belief spacethat lead to the maximum performance for agiven level of uncertainty Lieder & Griffiths(2020).

• Irrationality: Corresponds to the situationwhere bounded rationality can no longer applyfast and frugal heuristics that satisfy the desiresdue to extreme levels of uncertainty or due to theinability of the decision-maker to make adequatedecisions. These decisions generally have poorutility and performances. The fast and frugalheuristics start to fail, and the decision-maker ex-periences cognitive dissonance effects and startsto violate rules of logic and probability, which inturn leads to unfavourable heuristics and their

paradoxical and irrational outcomes. These deci-sions are hard (or even impossible) to be capturedby current computational decision systems.

Figure 2: Wave-like interpretation of the QuLBITframework in terms of the notions of rationality.

Quantum-Like Bayesian NetworksThe fundamental core of the QuLBIT frameworkis the notion of graphical probabilistic inferenceusing the formalism of quantum theory. TheQuantum-like Bayesian network, originally pro-posed in Moreira & Wichert (2014, 2016a) is thefundamental building block of the QuLBIT frame-work and is also an example of a model thathas been extensively studies in the literature forpredicting and accommodating paradoxical hu-man decisions across different decision scenarios,ranging from psychological experiments (Moreira& Wichert, 2016a, 2017; Wichert et al., 2020) toreal-world credit application scenarios (Moreiraet al., 2018). The difference between the tradi-tional Bayesian network (BN) and a quantum-likeBayesian Network (QLBN) is the way one speci-fies the values of the conditional probability ta-bles. While in the traditional BNs, one uses realnumbers to express probabilities, in quantum me-chanics these probabilities are expressed as proba-bility amplitudes, which are represented by com-plex numbers. Figure 3 shows a representation of aQuantum-Like Bayesian Network.

Exact inference in QLBNs is given by three steps:

• Definition of the superposition state. In quan-tum theory, all individual quantum states con-tained in a Hilbert Space are defined by a super-position state which is represented by a quantumstate vector |S〉 comprising the occurrence of allevents of the system. This can be analogous to the

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Figure 3: Representation of Shafir & Tversky(1992) PD experiment using a quantum-like BN

(left) and a classical BN.

classical full joint probability distribution, withthe difference that instead of using real numbersto express probabilities, one uses complex proba-bility amplitudes. For example, for Shafir & Tver-sky (1992) PD experiment, the full joint distri-bution and the corresponding superposition statecan be represented by the following vectors:

Joint =

0.4850.0150.4200.080

S =

√

0.485 eiθ1√0.015 eiθ2√0.420 eiθ3√0.080 eiθ4

• A density matrix which describes the quantum

system. The density operator, ρ, aims to describea system where we can compute the probabilitiesof finding each state in the network. One way toachieve this is by computing a density operatorthrough the product between the superpositionstate S and the corresponding conjugate trans-pose S†, that is ρ = SS†.

Figure 4: Density matrix, ρ = SS†

The density operator also contains quantum in-terference terms in the off-diagonal elements,

which are the core of this model. It is preciselythrough these off-diagonal elements that, duringthe inference process, one is able to obtain quan-tum interference effects, and consequently, devi-ations from normative probabilistic inferences.

• Quantum-Like Marginal Distribution. Thequantum-like marginal probability can be de-fined by two selection operators Def and Coop,which are vectors that select the entries of theclassical joint distribution that match the query.

Def =

1010

Coop =

0101

Computing the quantum-like probability ofPlayer 2 defecting, given that (s)he is uncertainabout Player 1’s strategy, P rq(P 2 = Def ), corre-sponds to summing out all entries of the joint re-lated to Player 1’s strategy. This probability canbe computed by applying the selection operatorto the density matrix, ρ, and normalising the re-sults with a normalisation factor γ = 1/(P rq(P 2 =Def ) + P rq(P 2 = Coop), and where Def T is thetranspose of operator Def ,

P rq(P 2 =Def ) = γ |Def ρDef T | == γ |0.905 + 2× 0.451331cos(θDef )

(1)

This is the same as having the classical probabil-ity, P r(P 2 =Def ), together with a quantum inter-ference term, InterfDef , which corresponds to theemergence of destructive / constructive interfer-ence effects, associated with the uncertainty thatthe player is experiencing,

P rq(P 2 =Def ) = γ |P r(P 2 =Def ) + 2InterfDef | =γ |0.905 + 2× 0.451331cos(θDef )|

(2)

In the same way, we can compute the probabilityof Player 2 cooperating, P rq(P 2 = Coop),

P rq(P 2 = Coop) = γ |CoopρCoopT | == γ |0.095 + 2× 0.034641cos(θCoop)

(3)

This suggests that the proposed model providesa hierarchy of mental representations rangingfrom quantum-like effects to pure classical ones.

Note that if one sets θDef or θCoop to π/2, thencos(θDef ) = 0 and cos(θCoop) = 0. This means that

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the interference term is canceled and the quantum-like Bayesian network collapses to its classicalcounterpart. Setting θDef = θCoop = 2.8057, willreproduce the disjuction effect observed in Shafir& Tversky (1992), P rq(P 2 = Def ) = 0.63. In Mor-eira & Wichert (2016a); Andreas Wichert & Bruza(2020), the authors proposed a similarity heuristicand later a law of balance that are able to auto-matically find the values of θDef and θCoop withoutmanually fitting the data.

Quantum-Like Influence Diagrams

Quantum-Like Influence diagrams (Moreira &Wichert, 2018) are a directed acyclic graph struc-ture that represents a full probabilistic descriptionof a decision problem by using probabilistic in-ferences performed in quantum-like Bayesian net-works together with an utility function.

Given a set of possible decision rules, δA, a clas-sical Influence Diagram computes the decision rulethat leads to the Maximum Expected Utility in re-lation to decision D. In a classical setting, this for-mula makes use of a full joint probability distribu-tion, P rδA(x|a), over all possible outcomes, x, givendifferent actions a belonging to the decision rulesδA where the goal is to choose some action a thatmaximises the expected utility with respect to somedecision rule, δA.

EU [D [δA]] =∑x,a

P rδA (x,a)U (x,a) . (4)

The quantum-like approach of the influence di-agrams consists in replacing the classical proba-bility, P rδA, by its quantum counterpart, P rqδA.The general idea is to take advantage of the quan-tum interference terms produced in the quantum-like Bayesian network to influence the probabilitiesused to compute the expected utility.

Mathematically, one can define utility opera-tors, UDef and UCoop, that represent the payoffthat Player 2 receives if (s)he chooses to Def ectand to Cooperate, respectively. And the quantum-like influence diagram simply consists in replac-ing the classical probability P rδA(x,a) in Equation4, by the probability computed by the quantum-like Bayesian network P rqδD (x|a). Details of theseformalisms can be found in the publicly availablenotebook2 and Moreira & Wichert (2018).

2https://git.io/JfKKB

UDef =

30 0 0 00 0 0 00 0 85 00 0 0 0

UCoop =

0 0 0 00 25 0 00 0 0 00 0 0 36

The expected utility of Player 2 defecting be-

comes

EUq[D[δDef

]]=

∑x,def

P rδDef (x,def )U (x,def ) (5)

EUq [P 2 =Def ] = T race[P rq(P 2 =Def )UDef

](6)

EUq [P 2 =Def ] =115cos(θDef ) + 115.02

0.131cos(θCoop + cos(θDef ) + 1.132(7)

In the same way, the expected utility of Player 2 co-operating becomes

EUq [P 2 = Coop] = T race[P rq(P 2 = Coop)UCoop

]=

100cos(θCoop) + 137.121

cos(θCoop) + 13.0288cos(θDef ) + 14.4338(8)

From this formalism, the region of the beliefspace where the decision-maker will always per-ceive that (s)he will have a higher utility for cooper-ating, EUq [P 2 = Coop] = X > X = EUq [P 2 =Def ],is given by θDef = π, and 0 ≥ θCoop ≤ 2π.

A Novel Explanatory Analysis inQuantum-Like Decision Models

In the QuLBIT framework, for decisions underuncertainty, the decision-maker’s beliefs are rep-resented as waves during the reasoning process.Only when the decision-maker makes a deci-sion, these beliefs collapse to the chosen deci-sion with a certain probability and utility. Be-fore reaching a decision, the decision-maker canexperience uncertainty regarding Player 1’s ac-tions. This corresponds to beliefs of cooperateand def ect competing with each other causing con-structive/destructive interferences (quantum inter-ference parameters θDef and θCoop).

Figure 5 (right) shows the combined graphi-cal representations of the utilities that a playercan obtain when reasoning about considering aDef ect strategy (Equations 7) or a Cooperate strat-egy (Equation 8) according to the uncertainty that(s)he feels about Player 1’s actions.

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Figure 5: Evolution of the quantum interference waves in the decision-maker’s belief space for Shafir &Tversky (1992) PD experiment.

It follows from Figure 5 (right) that this model al-lows different levels of representations of decisionsunder uncertainty ranging from (1) fully ratio-nal and optimal decisions (fully classical), (2) sub-optimal decisions, to (3) irrational decisions (Shafir& Tversky, 1992) (fully quantum). Figures 5 (left)and Figures 5 (center) show the evolution of thedecision-maker’s beliefs through the belief spaceenabling a novel analysis and interpretation of thequantum interference waves in terms of the differ-ent levels of rationality.

• Fully classical decisions: the majority of thedecision-makers are stable in the regions of thebelief space where the MEU of defect is max-imised. This notion is in accordance with predic-tions from expected utility theory and conceptsfrom Game Theory, where in strictly dominantstrategies, the decision-maker stays stable in theNash equilibrium state, in this case, engaging ina defect strategy. In this region, it seems that thedecision-maker is not experiencing much uncer-tainty, and consequently quantum interferenceeffects are minimum.

• Sub-Optimal decisions: the lighter regions ofthe figure indicate decisions where the MEUof defecting is close to the MEU of cooper-ate, MEU (Def ect) ≈ MEU (Cooperate), but stillthe decision-maker prefers to defect, becausethere are not enough heuristic cues to convincehim/her to switch from defect to cooperate.Quantum interference effects occur and uncer-tainty is high, however the quantum interfer-ence effects are not strong enough to make thedecision-maker change his mind and for that rea-son (s)he continues to choose according to ex-pected utility.

• Irrational decisions: correspond to the cen-

tral, blue regions of the Figure 5. In theseregions, uncertainty is maximised and quan-tum interference effects are significant enoughto make the decision-maker change his mind.The decision-maker irrationally engages in wish-ful thinking, or beliefs that are far stretchedfrom available data on hand. It is in this re-gion where the decision-maker perceives thatMEU (Cooperate) > MEU (Def ect), and conse-quently makes a decision that deviates from theclassical notions of the Expected Utility theory.Notions of game theory actually accept the factthat the decision-maker might not always obeyto the formalisms of expected utility theory. Thiscan occur when players did not understand therules of the game, or simply because they playedrandomly. What game theory notions tell us isthat the decision-maker will not stay stable inthese irrational states (in this case the Cooperatestate), which is in accordance with Figure 5.

ConclusionsIt is the purpose of this paper to provide a set ofcontributions of quantum-based models applied tocognition and decision as an alternative mathemat-ical approach for decision-making in order to betterunderstand the structure of human behaviour.

The QuLBIT framework is open-source and al-lows the combination of both Bayesian and non-Bayesian influences in cognition, where classicalrepresentations provide a better account of dataas individuals gain familiarity, and quantum-likerepresentations can provide novel predictions andnovel insights about the relevant psychologicalprinciples involved during the decision process.

Our contributions so far show that QuLBIT isa unified framework for cognition and decision-making that is able (1) to accommodate and pre-

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dict paradoxical human decisions (namely disjunc-tion errors), (2) to analyse the belief space of thedecision-maker through quantum interference, (3)to quantify uncertainty and (4) to provide a non-linear view on the different levels of rationality,ranging from fully optimal decisions (classical) toirrational decisions (quantum-like).

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