Quantum Theoretical Approach to the Integrate-and-Fire Model of Human Decision Making

Transcription

1 nternational Journal of Psychological Stuies; Vol. 6, No. 4; 014 SSN E-SSN X Publishe by Canaian Center of Science an Eucation Quantum Theoretical Approach to the ntegrate-an-fire Moel of Human Decision Making Till D. Frank 1 & Preecha P. Yupapin 1 CESPA, Department of Psychology, University of Connecticut, Storrs, USA Avance Stuies Centre, Department of Physics, King Mongkut's nstitute of Technology Lakrabang, Bangkok, Thailan Corresponence: Till D. Frank, CESPA, Department of Psychology, University of Connecticut, 406 Babbige Roa, Storrs, CT 0669, USA. Tel: till.frank@uconn.eu Receive: September 1, 014 Accepte: November 3, 014 Online Publishe: November 10, 014 oi: /ijps.v6n4p95 URL: Abstract We evelop a quantum mechanical moel for ecision making an iscuss preictions mae by the moel. The moel is base on the corpuscularity principle of quantum mechanics accoring to which all natural processes have a particle-like character. t is shown that the quantum mechanical approach allows us to etermine the factors that affect the variability of ecision making for iniviual ecision makers. Two main effects on mean ecision making times are preicte: iniviual bias an influence of society. The possibility of an interaction effect between iniviual bias an society impact is iscusse from a moel-base perspective. Moreover, it is argue that the moel can be regare as a counterpart to the classical integrate-an-fire moel of ecision making. Keywors: ecision making, iniviual bias, social impact, quantum mechanics, integrate-an-fire moel 1. ntrouction Recent evelopments in psychology have pointe out the necessity to apply quantum mechanical concepts to human perception an cognition. For example, probability jugement errors an interference effects have been aresse by means of quantum mechanical methos (Aerts, 009; Busemeyer et al., 011). Emotional states with positive an negative valence (happiness an saness) have been consiere as two sies of the same phenomenon an have been aresse similar to the Einstein-Poolsky-Rosen paraox as single paraox quantum states (Phunthawanunt & Yupapin, 014; in press). Bistable perception an cognition has been moelle by means of quantum mechanical moels (Atmanspacher, Filk, & Römer, 004) as alternative to classical ynamical systems moels of bistable perception an action (Frank et al., 009; Haken, 1991). Benchmark examples for avocating the introuction of quantum mechanical concepts into psychology are the so-calle pet-fish problem (Aerts & Gabora; 005) an the so-calle isjunction effect in ecision-making uner uncertainty. Accoring to Blutner, Pothos an Bruza (013) a seminal experiment by Tversky an Shafir (199) can be interprete to make the nee for quantum psychology evient. n the experiment, stuents were tol to imagine that they have just written an exam. Subsequently, they were aske by the researchers whether they woul buy a non-refunable holiay trip to Hawai for an extreme low price that woul take place a few ays before the make-up exam. They were aske uner three conitions. n the baseline conition, stuents were tol that they o not know the result of the exam. That is, they ha to make ecisions uner the uncertainty of the exam outcome. n the two other conitions, stuents were tol they have faile the exam or that they have passe the exam. The baseline conition is consiere to represent an unconitione ecision. n contrast, in the two other conitions stuents were aske to make conitional ecisions. That is, they were aske whether they woul buy the trip uner the conition that they know that they have passe or faile the exam. Blutner et al. (013) extracte from the experiment by Tversky an Shafir (199) the probabilities that stuents ecie to buy the trip. n oing so, they obtaine three probabilities. The probability for the baseline conition is consiere to be an unconitione probability p(buy). The two other probabilities are consiere to be conitional probabilities p(buy, knowing having faile) an p(buy, knowing having passe). The precise values were p(buy)=0.3, b(buy, knowing having faile)=0.57 an p(buy, knowing having passe)=0.54. t can be shown that when applying the orinary rules of probability theory these three probabilities are inconsistent with each other. However, the 95

2 nternational Journal of Psychological Stuies Vol. 6, No. 4; 014 observe probabilities can be explaine using the quantum mechanical concept of probability illustrating a nee for a quantum theoretical approach to psychology. n line with the aforementione stuies, we will consier the quantum mechanical concept of the corpuscular nature of natural processes an examine implications of that concept for human ecision making. That is, just as funamental processes like electromagnetic waves an vibrations are treate in quantum mechanics via particle-like escriptions (photons an phonons), we will propose a particle-like treatment of ecision making within the framework of quantum theory an will stuy the implications of such a treatment. As a by-prouct of this corpuscular quantum mechanical approach, we will be able to etermine from a theoretical perspective the structure of the noise term that accounts for the variability of human ecision making. Finally, as we will show below, the propose quantum mechanical moel can be consiere as a counterpart to the integrate-an-fire moel of human ecision making (Busemeyer, 1985; Busemeyer & Townsen, 1993) that has been stuie extensively in the psychological literature.. Derivation of the Quantum Theoretical Moel for Decision Making n previous stuies a moel for ecision making was propose that escribes the preference of a human agent to ecie in favour of an issue (e.g., buying a car) by means of a real-value scalar variable. The preference state evolves accoring to the following time-iscrete evolution equation (Busemeyer, 1985; Busemeyer & Townsen, 1993) P ( t 1) P( z( t 1) (1) Time is measure on iscrete events. The variable enotes a real number an escribes the bias towars the favourable ecision. The variable z is a normally istribute ranom number with zero mean an a stanar eviation that oes not change over time. n what follows we focus on the very funamental situation in which a human agent is confronte with a one-sie or single-boune ecision making problem. For example, a car owner consiers the possibility to buy a new car. A tenant renting an apartment consiers buying a home. A person in a permanent employment situation consiers applying for another position. That is, the ecision maker can make an affirmative ecision in a particular matter at han anytime in the future. As long as this ecision is not mae, the ecision maker implicitly ecies against the issue at han. n orer to escribe such a one-sie ecision making process, the moel is supplemente with a threshol value. f the preference state becomes equal to or larger than the ecision threshol, then the agent makes the ecision in favour of the issue at han. f the impact of the ranom variable can be neglecte, then the preference state increases linearly for >0 until the ecision threshol is reache an the ecision is mae. n other wors, the eterministic moel escribes the integration of a positive input until a critical value is reache. For this reason, it belongs to the class of integrate-an-fire moels use in neuroscience. f the impact of the ranom variable can not be neglecte, then the integration process is subjecte to fluctuations. The stochastic integrate-an-fire moel preicts that for ifferent agents the threshol is reache at ifferent time points because the ecision making process involves an erratic component. n what follows, we erive a quantum mechanical counterpart to the integrate-an-fire moel that accounts to a certain extent for the iscrete nature of natural phenomena an in aition allows us to etermine from theoretical consierations the structure of the ranom component involve in human ecision making. n general, quantum mechanics escribes natural phenomena in terms of Hamiltonian operators that act on wave functions of the particles uner consieration. The Hamiltonian operators, in turn, are relate to energy expressions of the particles. Similarly, our eparture point is a particle-base corpuscular perspective of ecision making. Accoringly, a preference in favor of an issue comes in iscrete units. n other wors, a ecision making process involves the prouction (or collection) of preference particles or the annihilation of collecte preference particles. The particles, in turn, come with certain, appropriately efine Hamiltonian operators. n orer to etermine these Hamiltonian operators, we assign to each preference particle the single particle energy L () This form is chosen because it correspons to the stanar form of particle energies in quantum mechanics (e.g., photon or phonon particle energies). Accoringly, the particle energy is the prouct of the Planck quantum an a positive constant L. The precise value of L oes not matter because as we will see below the parameter L will rop out of the escription entirely. Following the stanar proceure of quantum mechanics, the Hamiltonian operator of an isolate or free collection of preference particles is given by H free L a a (3) 96

3 nternational Journal of Psychological Stuies Vol. 6, No. 4; 014 The most right staning symbols enote the particle annihilation operator an the particle creation operator. Roughly speaking, in quantum mechanics, the creation operator escribes the creation or generation of a particle, whereas the annihilation operator escribes the estruction of a particle. Taking a quantum optical perspective, the creation (prouction) of particles can be escribe by a pumping Hamiltonian operator. n our context, the operator reflects the bias of a human agent to ecie in favor of the issue uner consieration. The pumping or bias operator is efine by (Agrawal, 01; Gariner, 1997) H bias i a exp( il * aexp( il (4) The lower case letter i in the equation for the operator is the imaginary unit (i.e., the square root of minus one). The epsilon variable is a complex-value variable that shows up in the operator both in its orinary form an in its complex-conjugate form. The magnitue of the bias can be efine by * (5) The magnitue is a positive real-value variable. t correspons to the pumping strength in quantum optical evices. The total Hamiltonian of the ecision making process is given by H agent H free H bias (6) n orer to moel the impact of the society, we follow again establishe methos in quantum mechanics. Accoringly, the environment is consiere as a heat bath that has its own Hamiltonian operator. n our context, this Hamiltonian operator escribes the society excluing the agent uner consieration H society (7) Note that the explicit form of this operator is not of interest at this stage. Rather, at issue is that we have at han two Hamiltonian operators: one escribing the agent an another one escribing the society (environmen. n orer to arrive at a complete escription, we nee to escribe the coupling between the agent an the society. n quantum mechanics this correspons to the coupling of the system uner consieration an the heat bath. Again, this coupling is typically escribe by means of a Hamiltonian operator H coupling (8) As a result, a complete escription of the agent an the society incluing the interaction between agent an society is formally given in terms of a total Hamiltonian operator that is compose of three terms that reflect agent behaviour, society behaviour, an agent-society coupling H total H agent H coupling H society (9) The ecision making process of the agent an the on-going processes in the society are then formally escribe by means of the evolution of an appropriately efine wave function. The evolution equation of the wave function is the Schröinger equation an reas i H total (10) t From the wave function all observables of interest can be calculate in terms of expectation values involving the relevant operators. For our purposes, we are intereste in calculating the mean number of preference particles. This mean number is known to correspon to the expectation value of the prouct of the annihilation an creation operator like n a a (11) Note that the mean number of preference particles can be regare as the counterpart to the preference state P efine by the integrate-an-fire moel iscusse above. While the preference state variable P is a phenomenologically introuce real-value state variable, the mean number of preference particles is a real-value observable that is base on a quantum mechanical particle-base treatment of the ecision making process. n what follows, we will use stanar techniques of quantum mechanics to obtain a close escription of the system uner consieration that oes not require us to escribe the behaviour of the environment in etail but nevertheless accounts for the impact of the environment on the system uner consieration. That is, we will erive a close escription for the ecision making process that only involves the annihilation an creation operators of the preference particles but nevertheless takes the society ynamics an the interaction between the society an the agent into account. To this en, we first note that the Schröinger equation (10) can be transforme into a Liouvielle equation for the so-calle ensity operator of the total system (see e.g., Puri, 001) 97

4 nternational Journal of Psychological Stuies Vol. 6, No. 4; 014 t (1) n Equation (1) the square bracket enotes the commutator of two operators A,B as efine by [A,B]=AB-BA. By means of an appropriate projection, we obtain from the ensity operator of the total system the operator of the agent alone. Moreover, we can iscuss the evolution of that operator in the so-calle interaction picture of quantum mechanics. That is, we take the following steps agentsociety agent agent (13) n orer to arrive at a close escription for the agent alone, we nee to make the weak coupling assumption use in quantum mechanics (Gariner, 1997; Walls & Milburn, 008). n the case of a weak coupling between agent an society the Liouvielle equation for the total system can be simplifie to arrive at an evolution equation for the agent ensity operator. The evolution equation for the agent ensity operator (in our notation) reas (Gariner, 1997; Walls & Milburn, 008) t with agentsociety agent [ H [ H pump tot,, agent agentsociety ] G( aza a az za a N( a, z G ( z), ] agent a (15) Note that in the first term on the right han sie the free Hamiltonian of the agent with the unknown parameter L oes not occur. The reason for this is that in the interaction picture this operator oes not show up any more. Moreover, note that the secon term on the right han sie accounts for the evolution of the society an the interaction between society an the agent. The secon term involves two parameters. The parameter gamma can be regare as a coupling parameter measuring the strength of the coupling (interaction) between agent an society. f the coupling parameter is put equal to zero, then we consier the isolate agent. The secon parameter is N(. Roughly speaking, in quantum mechanics this parameter reflects the mean number of particles in the environment that have the ability to affect an perturb the system embee in the environment provie that there is a coupling between the system an environment. We may interpret this parameter as the volatility of the society, that is, the magnitue of erraticness or the magnitue of "chaos" in the society that affects a human agent provie that the agent interacts with the society. n orer to obtain a quantum theoretical moel for ecision making that can be consiere as counterpart to the integrate-an-fire moel introuce above, we consier the so-calle Glauber-Suarshan representation of the agent ensity operator an the particle creation an annihilation operators (Gariner, 1997; Puri, 001; Walls & Milburn, 008). n this representation, the creation an annihilation operators are replace by a complex-value variable an its complex-conjugate value. Moreover, the ensity operator is replace by a probability ensity efine on the omain of the complex-value variable. That is, the following associations are mae (Gariner, 1997; Puri, 001; Walls & Milburn, 008) a a S * P(, *, n particular, the mean number of preference particles can be compute in the Glauber-Suarshan representation like a a n( * * P(, *, This relationship allows us to efine trajectories for the number of preference particles: ) (14) (16) (17) a a * (18) n * The trajectory j escribes the preference level of an iniviual agent j as a function of time. That is, the trajectory escribes an iniviual ecision making process. n contrast, the mean number reflects the behaviour of an 98

5 nternational Journal of Psychological Stuies Vol. 6, No. 4; 014 ensemble (large group) of agents that all are confronte with the same situation. As mentione above, accoring to the integrate-an-fire moel, an iniviual agent j makes a ecision when the preference level reaches or excees a certain threshol. This means that the agent j makes a ecision in favor of the issue at han when ( ) n j (19) n orer to unerstan how the trajectories can be calculate, we note that from the Liouvielle equation we can erive the partial ifferential equation for the probability ensity (Drummon, McNeil, &Walls, 1981) This partial ifferential equation takes the form of a Fokker-Planck equation (Risken, 1989; Frank, 005). Trajectories can be compute from the corresponing complex-value Langevin equations that correspon to the Fokker-Planck equation (0). The Langevin equations rea These ifferential equations involve a complex-value fluctuating force that can be ecompose into its real an imaginary parts like x i y () The real an imaginary parts represent real-value statistically inepenent Langevin forces, which are a special kin of fluctuating forces (Risken, 1989). From the Fokker-Planck equation (0) it follows that the Langevin forces are normalize with respect to the Dirac elta function like k k (3) for k=x,y. The secon complex-value stochastic ifferential equation in (1) is just the first equation conjugate. Therefore, in orer to compute trajectories we nee to focus only on the first stochastic ifferential equation. 3. Decision Making from a Quantum Theoretical Perspective 3.1 Decision Making of the solate Decision Maker Let us consier an agent (ecision maker) who is not influence in the ecision making process by the society. We may refer to such an agent as being isolate. f the coupling parameter between agent an society is put equal to zero, the ifferential equation for the Glauber-Suarshan representation of the creation operator reas This is a eterministic evolution equation. The solution is given by or P ( ) ( * *) P N( P t * * N( ( * * * N( * 1 ( s) ( t s) t n t t Consequently, the time to reach the ecision threshol is T ( n ) This equation preicts that the ecision time becomes longer when the threshol is higher an it becomes shorter when the bias is stronger in magnitue. Both preictions are intuitively plausible. (0) (1) (4) (5) (6) (7) 99

6 nternational Journal of Psychological Stuies Vol. 6, No. 4; Entirely Society-Driven Decision Making For an agent who exhibits no bias towars making an affirmative ecision in the matter of issue, the behavior is completely etermine by the interaction between the agent an the society. n this case, the evolution equation for the Glauber-Suarshan representation of the creation operator reas N( ( (8) The complex-value variable can be ecompose into its real an imaginary parts like x i y (9) We then obtain two separate stochastic ifferential equations for the real-value variables: x x y y N( N( Moreover, we have x y (31) Consequently, the trajectory of the preference level of an iniviual agent j can be compute from (3) n [ x ] [ y ] Since the stochastic processes for the two real-value variables are equivalent an since the mean values of both processes are zero, we can compute the mean number of the preference particles from n( x x (33) where σ x is the variance of x(. The evolution equation (30) for the real part x( is an Ornstein-Uhlenbeck process (Risken, 1989). The stationary variance of the Ornstein-Uhlenbeck process is N( x, st 4 Moreover, the transient evolution of the variance is given by (Risken, 1989) x x, st 1 exp which implies for the mean number of preference particles that N( n 1 exp( t ) x y (36) Let us stuy the impact of the two moel parameters on the evolution of the mean number of preference particles. The top panel of Figure 1 illustrates the impact of the coupling parameter. When the coupling parameter is increase, then the approach to the stationary value happens faster. The bottom panel of Figure 1 illustrates the impact of the volatility of the society. f the society is more volatile, then the stationary value is shifte to higher values. (30) (34) (35) 100

7 nternational Journal of Psychological Stuies Vol. 6, No. 4; 014 Figure 1. mpact of the coupling parameter gamma (top panel) an volatility (bottom) Top panel parameters: N(=.0, gamma was varie from 0.1 to 1 (bottom to top). Bottom panel parameters: gamma was equal to 1; N( was varie from to 11 (bottom to top). All graphs were compute from Equation (36). The analytically obtaine graphs can alternatively be compute from numerical simulations. This is illustrate in Figure. The ashe line in Figure shows the analytical function. n contrast the soli line show the graph obtaine from numerical simulation of a set of 100 trajectories. We foun that the numerical result approximates to some extent the analytical function. Since the graph obtaine by numerical simulations was calculate from a finite number (100) of agent trajectories, the numerically obtaine result fluctuates an oes not correspon to a smooth function. However, if the number of agents is increase then the numerically obtaine graph becomes closer an closer to the analytically obtain function (ata not shown). Figure. Stochastic simulation (soli line) versus analytical result of the eterministic moel (ashe line) The latter was compute from Equation (36) just as in Figure 1. Parameters: N(=, gamma was equal to 1. The stochastic simulation of Equation (30) was conucte by means of an Euler-forwar scheme with single time step (Frank, 005; Risken, 1989). The graph (soli line) shows the average of a total of 100 repetitions. Note that so far we i not take the ecision making step into consieration. That is, the consierations presente above just aress the increase of the number of preference particles an o not account for the fact that if for an iniviual agent the number reaches a critical threshol then a ecision is mae an the ecision making process is terminate. Let us aress next this ecision making event. To this en, we consier the mean ecision time of an ensemble of agents. This mean ecision time is given by the so-calle mean first passage time (MFPT) to reach the 101

8 nternational Journal of Psychological Stuies Vol. 6, No. 4; 014 threshol. That is, the mean ecision time is the time on average it takes for the preference level to reach the threshol. Mathematically speaking, we have T MFPT 1 lim R R R j1 T with n ( t T ) (38) The mean ecision time can be compute numerically using a large but finite set of agent trajectories. That is, in the numerical simulations R is finite. Before we present the results of such simulations, let us arrive at a hypothesis that can be teste in such computer experiments. As illustrate in Figure 1, an increase of the coupling parameter gamma yiels a faster increase of the mean number of preference particles. Therefore, we speculate that the mean ecision time as measure in terms of the MFPT becomes shorter when the coupling strength is increase. n orer to test this hypothesis the mean ecision time was compute for three values of coupling parameters gamma. The results are reporte in Table 1. As expecte, we foun that the mean ecision time ecaye monotonically with increasing coupling strength. Hypothesis testing (ANOVA) showe that there was a statistical significant effect of the coupling strength on the mean ecision time, F(,7)=9.5, p< (37) Table 1. Mean ecision time obtaine from computer simulations of Equation (30) as functions of coupling strength gamma Society coupling Mean ecision parameter [a.u.] time [a.u.] (0.5).5 (0.) (0.) Other parameters: N(=. Simulations were conucte by means of Euler-forwar metho (single time step 0.01). Mean ecision times were compute from a total of 100 repetitions. 3.3 Decision Making Affecte by niviual Bias an the Social Environment n orer to stuy the impact of the iniviual bias an the social environment, we consier the full stochastic moel. n view of the moel structure, it is plausible to assume that for a relative low bias the time will be relative large, whereas for a relative strong bias the time will be relative short. n aition, we speculate that increasing the coupling parameter shortens mean ecision time. Let us test the two aforementione hypotheses. To this en, we consier again the stochastic ifferential equations of the real an imaginary parts rather than the stochastic ifferential equation of the complex-value variable. These equations rea x x x y y y N( N( x y with x i y (40) The stochastic ifferential equations put us in the position to etermine the mean ecision time numerically. The mean ecision time is efine by the mean first passage time, just as in the previous section. To investigate the two impacts we consiere computer simulation using two inepenent variables (V) each with two levels. The first V was the magnitue of the bias with two levels: zero bias an non-zero bias. The secon V was the coupling between agent an society. We focuse on two levels: low an high coupling. For the zero bias (39) 10

9 nternational Journal of Psychological Stuies Vol. 6, No. 4; 014 conition we took the ata from the simulation reporte in Table 1. For the non-zero bias conitions we conucte two aitional sets of computer experiments. The results are summarize in Table an illustrate in Figure 3 Table. Mean ecision times (with stanar eviations in brackets) for the four ifferent conitions as obtaine by solving numerically Equation (39) Zero iniviual bias (0 units) Non-zero iniviual bias (1 uni Society coupling Low (1 uni Society coupling High (3 units) Society coupling low (1 uni Society coupling high (3 units) 4.6 (0.5) 1.8 (0.) 1.7 (0.1) 1.5 (0.1) Other parameters: N(=. For further etails of the simulation proceure see Table 1. Figure 3. Graphical illustrations of the mean ecision times observe in the four test conitions We foun a main effect of iniviual bias. The mean ecision time ecrease ue to the iniviual bias. We also foun a main effect of coupling between agent an society. The mean ecision time ecrease with increasing coupling between agent an society. Both main effects were expecte. nterestingly, we also foun an interaction effect. The impact of the society is meiate by the impact of the iniviual bias. The impact of the society was weaker in the presence of an iniviual bias. That is, while for the unbiase agent an increase of coupling by two units prouce a rop of the mean ecision time by more than time units, for the biase agent the same increase of coupling strength prouce only a rop of the mean ecision time by 0. time units. Hypothesis testing (two way between subjects ANOVA) showe the (main) effect of bias (epsilon) was significant, F(1,36)=401, p< Likewise, the (main) effect of coupling (gamma) was significant, F(1,36)=37, p< Finally, the interaction effect was significant as well, F(1,36)=56, p< Discussion Decision making is an activity that abouns in the lifetime of a human being. t may apply to business-relate activities (see e.g., Esa, Alias, & Sama, 014) or everyay activities such as when to eat. t is clear that in general ecision making is influence by many factors. For example, foo intake may be affecte by working shifts (Nagashima, Masutani, & Wakamura, 014). Here, we consiere a very funamental situation in which a ecision maker is making a (one sie) ecision such as buying a new car or buying a home. The eparture point of our work was to acknowlege that a key concept in quantum mechanics is the notion of the corpuscular nature of natural processes. For example, an electromagnetic wave (e.g., a light beam) exhibits particle-like properties. 103

10 nternational Journal of Psychological Stuies Vol. 6, No. 4; 014 The light particles have been calle photons. Thermal vibrations of the atoms of a soli can be stuie by means of vibration particles (phonons). Earlier stuies have suggeste that there is a nee for consiering quantum mechanical aspects of psychological phenomena. n line with these stuies, we have examine implications of the quantum mechanical notion of the iscreteness of natural processes for human ecision making. We have erive a quantum theoretical moel for ecision making that can be regare as a quantum mechanical counterpart to the integrate-an-fire moel of ecision making that has been frequently use in the literature. Consistent with the integrate-an-fire moel, the quantum mechanical moel preicts a main effect of the bias of ecision makers on the mean ecision time as compute for an ensemble of ecision makers. The mean ecision time ecreases when the bias towars an affirmative ecision increases. We also foun a main effect of the coupling between ecision makers an society on the ecision making process. When the coupling increases then mean ecision time ecreases. We may think of such an increase coupling as interactions between a ecision maker an the society that become more intense or frequent. nteresting, we also foun an interaction effect between agent-society coupling an agent bias. We foun that the coupling parameter has less regulative power when the ecision makers have a strong bias. Let us speculate about the nature of the observe interaction effect. To this en, we woul like to recall first a general situation in which an interaction effect can arise from a flooring effect. Let us consier an experimental esign with two inepenent variables (Vs) an a positive efinite epenent variable (DV) such as ecision time. Let us assume that when V is at level A we observe that the DV is relatively high when V1 is at level A an then rops by an amount of X units when V1 is change from level A to B. Let us further assume that when V is at level B we observe that the DV is relatively low when V1 is at level A. Moreover, for that low value a rop by X units woul yiel a negative number which is physically impossible since the DV is a positive efinite observable. Therefore, when V1 is change from A to B (at level B of V) we will observe a rop of the epenent variable that is less than X. We will observe an interaction effect that is ue to a flooring effect. This scenario can be applie to the interaction effect reporte in Section 3. The observation that the impact of the society was meiate by the bias of the agent may reflects that for an agent with a strong bias the ecision time is relatively short in any case irrespective of the interactions between the agent an the society. Future stuies may well on this issue an eluciate the role of the flooring effect for the observe interaction. References Aerts, D. (009). Quantum structure in cognition. Journal of Mathematical Psychology, 53, Aerts, D., & Gabora, L. (005). A theory of concepts an their combinations. A Hilbert space representation. Kybernetes, 34, Agrawal, G. S. (01). Quantum optics. Cambrige: Cambrige University Press. Atmanspacher, H., Filk, T., & Römer, H. (004). Quantum Zeno features of bistable perception. Biological Cybernetics, 90, Busemeyer, J. R. (1985). Decision making uner uncertainty: A comparison of simple scalability, fixe-sample, an sequential sample moels. Journal of Experimental Psychology: Learning, Memory, an Cognition, 11, Busemeyer, J. R., Pothos, E. M., Franco, R., & Truebloo, J. S. (011). A quantum theoretical explanation of probability jugement errors. Psychological Review, 118, Busemeyer, J. R., & Townsen, J. T. (1993). Decision fiel theory: A ynamic-cognitive approach to ecision making in an uncertain environment. Psychological Review, 100, Blutner, R., Pothos, E. M., & Bruza, P. (013). A quantum probability perspective on borerline vagueness. Topics in Cognitive Sciences, 5, Drummon, K. J., McNeil, K. J., & Walls, D. F. (1981). Nonequilibrium transitions in subsecon harmonic generation. Optica Acta: nternational Journal of Optics, 8, 11-5 Esa, M., Alias, A., & Sama, Z. A. (014). Project managers cognitive style in ecision making: A perspective from construction inustry. nternational Journal of Psychological Stuies, 6, Frank, T. D. (005). Nonlinear Fokker-Planck equations: Funamental an applications. Berlin: Springer. 104

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