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Quantum-Like Bayesian Inference Technologies for Cognition and Decision

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Quantum-Like Bayesian Inference Technologies for Cognition and Decision QuLBIT: Quantum-Like Bayesian Inference Technologies for Cognition and Decision Catarina Moreira1 ([email protected]) Matheus Hammes1 ([email protected]) Rasim Serdar Kurdoglu2 ([email protected]) Peter Bruza1 ([email protected]) 1School of Information Systems, Queensland University of Technology, Brisbane, Australia 2Faculty of Business Administration, Bilkent University, Ankara, Turkey Abstract what would be considered normatively correct ac- cording to logic or probability theory. This paper provides the foundations of a uni- fied cognitive decision-making framework (QulBIT) The field of Quantum Cognition emerged to re- which is derived from quantum theory. The main spond to this challenge, a major feature being the advantage of this framework is that it can cater use of quantum probability theory to model hu- for paradoxical and irrational human decision mak- ing. Although quantum approaches for cognition man cognition, including decision making (Buse- have demonstrated advantages over classical prob- meyer & Bruza, 2012). Quantum probability can abilistic approaches and bounded rationality mod- be viewed as generalisation of Bayesian probability els, they still lack explanatory power. To address this, we introduce a novel explanatory analysis of theory. In quantum-like cognitive models, events the decision-maker’s belief space. This is achieved are modelled as sub-spaces of a Hilbert spaces, by exploiting quantum interference effects as a way a vector space of complex numbers (amplitudes) of both quantifying and explaining the decision- maker’s uncertainty. We detail the main modules which enables the calculation of probabilities by of the unified framework, the explanatory analy- projection: performing the squared magnitude of sis method, and illustrate their application in situ- an amplitude. This representation allows events ations violating the Sure Thing Principle. to interfere with each other, which influences their Keywords: QuLBIT; quantum cognition; quantum- like Bayesian networks; quantum-like influence di- associated probabilities. These interference effects agrams; bounded rationality; explanatory analysis generate a set of new parameters that can be used to either accommodate violations in Bayesian the- Introduction ory (Busemeyer & Bruza, 2012) or paradoxical hu- The primary motivation behind QuLBIT1 is the man decisions Kahneman et al. (1982). Interference challenge to formally account for seemingly para- is a core concept in the QuLBIT framework which doxical human decision making. It is widely known enables an alternative quantification of uncertainty, in the literature of cognitive science and economics as well as the representation of conflicting, ambigu- that when it comes to decision-making under un- ous beliefs. The vector representation of superposi- certainty, humans usually make decisions that are tion obeys neither the distributive axiom of Boolean inconsistent with the axioms of expected utility logic nor the law of total probability (Moreira & theory, leading to decisions that are either sub- Wichert, 2016b) (See Figure 1). optimal, paradoxical or even irrational (Kahneman Quantum cognitive models make use of addi- & Tversky, 1979; Aerts et al., 2017). These paradox- tional parameters, which allows fitting to empir- ical decisions can result in cognitive biases (Tversky ical data but which “do not necessarily explain & Kahneman, 1974), violations of major economic them” (Blutner & beim Graben, 2016). QuLBIT principles (like the Sure Thing Principle) (Savage, addresses this lack of explanatory power by em- 1954; Allais, 1953; Ellsberg, 1961) or violations to ploying a novel analysis method which allows in- the laws of probability theory and logic (Busemeyer terpretation of the decision-maker’s belief space, et al., 2006; Pothos & Busemeyer, 2009). In short, e.g., when a decision-maker prefers one choice over decades of research has found a whole range of hu- another. This paper illustrates QuLBIT’s novel ex- man judgements that deviates substantially from planatory analysis method in regard to violations of the Sure Thing Principle in the Prisoner’s Dilemma 1https://github.com/catarina-moreira/QuLBiT Game (Shafir & Tversky, 1992). 2520 ©2020 The Author(s). This work is licensed under a Creative Commons Attribution 4.0 International License (CC BY). Figure 1: The representation of the decision-maker’s beliefs under uncertainty where beliefs can enter into a 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 and P2 = Def P2 = Coop Violations to the Sure Thing Principle P1 = Def 30 25 P1 = Coop 85 36 Several experiments in the literature have shown that people violate this principle in decisions un- Table 2: Payoff matrix used in Shafir & Tversky der uncertainty, leading to paradoxical results and (1992) Prisoner’s Dilemma experiment. violations of the law of total probability (Tver- sky & Kahneman, 1974; Aerts et al., 2004; Birn- baum, 2008; Li & Taplin, 2002; Hristova & Grin- QuLBIT: Quantum-Like Bayesian berg, 2008). The prisoner’s dilemma is an example Inference Technologies where, under uncertainty, people violate the sure thing principle, by being more cooperative. The QuLBIT framework provides a unifying deci- To test the veracity of the Sure Thing Principle sion model for cognitive decision making, which under the Prisoner’s Dilemma game, experiments is susceptible to cognitive biases. In addition, the were made in where three conditions were tested: framework caters for data-driven computational decisions, which are based on optimization algo- • Participants were informed the other participant rithms. The main advantage of this framework is chose to Defect (Condition 1: Known to defect); that it can cater for paradoxical and irrational hu- man decisions during the inference process. This • Participants were informed the other participant not only enhances the understanding of cognitive chose to Cooperate (Condition 2: Known to coop- decision-making, but is also relevant for providing erate); effective decision support systems. The nature of • Participants had no information about the other the quantum-like approach allows the system to participant’s decision (Condition 3 : Unknown). capture optimal, sub-optimal (bounded rational), or even irrational decisions which play an impor- tant role in a “humanistic system”, which are sys- Condition Pr( P2 = Defect) tems strongly influenced by human judgment, and Condition 1 (P1 Known to Defect): 0.97 Condition 2 (P1 Known to Cooperate) : 0.84 behaviour. Condition 3: (P1 Unknown) 0.63 Views on Rationality Table 1: Experimental results from Shafir & The QuLBIT framework caters for a spectrum of Tversky (1992) PD experiment. views in relation to rational decision making, de- pending on the degrees of rationality that the Table 1 summarizes the results of these experi- decision-maker employs. The views presented un- ments for the three conditions. The column clas- der the proposed quantum-like approach are simi- sical prediction shows the classical probability of a lar to the ones put forward by Gigerenzer & Gold- player choosing to Def ect, given that the decision stein (1996) and Lieder & Griffiths (2020). They in- of Player 1 is unknown (Condition 3). The payoff clude decisions bounded in terms of time, process- matrix used in Shafir & Tversky (1992) Prisoner’s ing power, information, etc. We extend this view to Dilemma experiment can be found in Table 2. incorporate the notion of the irrational mind, a point 2 2521 in the decision-maker’s belief space where heuris- paradoxical and irrational outcomes. These deci- tics are no longer sufficient to produce satisficing sions are hard (or even impossible) to be captured outcomes. Figure 2 illustrates the different views by current computational decision systems. on rationality that are represented in the QuLBIT framework depicted in terms of a non-linear quan- tum interference wave. We define these views in the following way: • Belief space: Corresponds to the set of pos- sible beliefs that are held by a decision-maker from 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 the ideal scenario where the decision-maker has un- limited cognitive resources: time, processing power and information in order to transact the Figure 2: Wave-like interpretation of the QuLBIT decision. Note that the ideal scenario is often not framework in terms of the notions of rationality. within reach for a human decision-maker. • Bounded Rationality: Corresponds to the sce- Quantum-Like Bayesian Networks nario where the decision-maker makes decisions The fundamental core of the QuLBIT framework bounded in terms of cognitive resources with is the notion of graphical probabilistic inference limited information, time, and processing power. using the formalism of quantum theory. The Consequently, fast and frugal heuristics are ap- Quantum-like Bayesian network, originally pro- plied sometimes resulting in sub-optimal deci- posed in Moreira & Wichert (2014, 2016a) is the sions (Kahneman et al., 1982), but also yielding fundamental building block of the QuLBIT frame- good-enough adaptive decisions (Gigerenzer & work and is also an example of a
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