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A formulation of computational trust based on quantum decision

theory

Mehrdad Ashtiani

1

&Mohammad Abdollahi Azgomi

1

Published online: 30 April 2015

#Springer Science+Business Media New York 2015

Abstract In this paper, we propose a new formulation of

computational trust based on quantum decision theory

(QDT). By using this new formulation, we can divide the

assigned trustworthiness values to objective and subjective

parts. First, we create a mapping between the QDT definitions

and the trustworthiness constructions. Then, we demonstrate

that it is possible for the quantum interference terms to appear

in the trust decision making process. By using the interference

terms, we can quantify the emotions and subjective prefer-

ences of the trustor in various contexts with different amounts

of uncertainty and risk. The non-commutative nature of quan-

tum probabilities is a valuable mathematical tool to model the

relative nature of trust. In relative trust models, the evaluation

of a trustee candidate is not only dependent on the trustee

itself, but on the other existing competitors. In other words,

the first evaluation is performed in an isolated context whereas

the rest of the evaluations are performed in a comparative one.

It is shown that a QDT-based model of trust can account for

these order effects in the trust decision making process. Final-

ly, based on the principles of risk and uncertainty aversion,

interference alternation theorem and interference quarter law,

quantitative values are assigned to interference terms. By

performing empirical evaluations, we have demonstrated that

various scenarios can be better explained by a quantum model

of trust rather than the commonly used classical models.

Keywords Trust model .Quantum decision theory (QDT) .

Quantum interference .Attraction factors .Superposition

axiom .Delegation .Order effect

1 Introduction

The concept of trust is becoming more and more important in

computational domains. Whether for selecting a service in

service-oriented environments or communications and con-

nectivity in social networks or even in holistic security ap-

proaches, the social and psychological concept of trust is

playing a key role. This fact is the main motivation behind

the rapid growth of computational trust modeling research.

Many trust models are introduced so far with their unique

advantages and downsides. Researchers are still expanding

the breath of trust formulations in order to find the most suit-

able model for computational domains. Regardless of the pro-

posed approach, presenting a model that provides the most

accurate representation of the social phenomenon of trust in

computational environments has been the fundamental goal of

the researchers in this domain.

Trust, is a subjective concept, which depends on the emo-

tions and attitudes of the trustor in various contexts with dif-

ferent amounts of risk and uncertainty. One of the primary

goals of computational trust modeling is to quantify these

seemingly non-quantifiable concepts. Trust is a concept asso-

ciated with a context, which depends on the experience,

knowledge and recommendation. Although Bayesian net-

works as an example of one of the most commonly used

approaches in the context of trust inference can help make

context-aware trust evaluation and aggregation in a rational

framework with sound and theoretical basis, they miss the

irrationalities and biases of human decision making. In

*Mohammad Abdollahi Azgomi

azgomi@iust.ac.ir

Mehrdad Ashtiani

m_ashtiani@comp.iust.ac.ir

1

Trustworthy Computing Laboratory, School of Computer

Engineering, Iran University of Science and Technology, Hengam

St., Resalat Sq., Tehran, Iran Postal Code: 16846-13114

Inf Syst Front (2016) 18:735–764

DOI 10.1007/s10796-015-9555-4

addition, it is worth-mentioning that most of the current com-

monly used classical models of trust are based on two

assumptions:

1. The trustor is a rational entity and therefore, the trust de-

cision making process is assumed to be rational.

2. The decision making context is isolated. In other words, it

is assumed that the evaluations of the trustee candidates

are performed independently from each other and an eval-

uation does not have any effect on other evaluations (i.e.,

non-comparative context).

In other words, two types of trust models can be recognized

in the literature:

1. Independent trust models, in which the trustor is consid-

ered rational and therefore is thought of having no bias to

calculate trust. Trust is based on experience and there is a

certain decay of trust. In these models, the trustee candi-

dates are evaluated in isolation. In other words, the eval-

uation of one trustee candidate will not have any effect on

subsequent evaluations. Unfortunately, most of the trust

models introduced so far, evaluate the trustee candidates

based on isolated direct experiences and independently

computed reputation values.

2. Relative trust models, in which trustees are considered

competitors and trust in a trustee candidate depends on

the experience with that trustee relative to other existing

trustees.

Recently, a new field of research has been developed

around the concept of quantum decision making and cogni-

tion. Researchers in this field have demonstrated that the

mathematical foundation of quantum theory can provide an

accurate representation of the decision making process of hu-

man beings and better describe the common irrationalities and

subjective biases that exist in the human decision making

process. In order to distinguish themselves from assuming a

quantum brain and emphasizing that they are just using the

mathematics of quantum theory as a tool to better describe and

explain thecognitive and decision making processes of human

beings, researchers refer to these approaches as quantum-like

or generalized quantum. These new findings were the primary

motivation for us to propose a quantum-like model of trust. In

a recent work, we have introduced the benefits of the quantum

Bayesian inference rule for modeling trust inference compared

to its classical counterpart (Ashtiani and Azgomi 2014).

In (Ashtiani and Azgomi 2014), we demonstrated the flex-

ibility of quantum mathematics for modeling the context and

bias in computational trust models. We discussed that by pre-

senting the trust state as a vector in various bases, a quantum-

like model of trust can provide a powerful mechanism to rep-

resent and change the context. But, the primary focus of our

previous work was to model the role of bias (either positive or

negative) in trust decision making. Quantum Bayesian infer-

ence as a generalized form of classical Bayesian inference was

used to incorporate this bias in the trust inference process. We

showed that by using this type of formulation, various phe-

nomena in computational trust modeling can be represented.

Phenomena such as: (1) the problem of exploration versus

exploitation in the domain of trust modeling, (2) the pseudo-

transitivity property of trust, (3) defending against good-

mouthing and bad-mouthing in trust models, and (4) recency

effects. In this paper, we focus our attention on the trust deci-

sion making itself instead of trust inference. Rather than ap-

plying the quantum Bayesian inference scheme, we use the

principles of quantum decision theory (QDT) in order to dis-

cuss and argue in a more fundamental manner about the ben-

efits of quantum mathematics in trust modeling. The quantum

decision theory is introduced and subsequently developed by

Yukalov et al. in (Yukalov and Sornette 2011; Yukalov and

Sornette 2010a; Yukalov and Sornette 2010b; Yukalov and

Sornette 2009a; Yukalov and Sornette 2009b; Yukalov and

Sornette 2012; Yukalov and Sornette 2008a; Yukalov and

Sornette 2008b). We create a mapping between the core con-

cepts of QDTand the building blocks of trust decision making

process. By creating this mapping, one can observe that not

only the appearance of quantum phenomena such as quantum

interference or quantum entanglement is inevitable in trust

decision making, but they can even play a fundamental role

in representing the subjective part of this process and enable

the model to correctly describe the trust decision making sce-

narios, which are more challenging to describe and represent

in the commonly used classical approaches. In addition, we

have formulated and shown in this paper that this kind of

formulation can provide the required mathematical infrastruc-

ture to introduce a relative model of trust capable of taking

into account the order effects that may occur in the trust deci-

sion making process when considering a comparative context.

Therefore, in order to be able to divide the concept of trust

to objective and subjective parts, quantify the emotions and

attitudes of the trustor in different contexts, and provide a

relative model of trust, we propose a formulation based on

the mathematics of QDT. The main advantage of formulating

trust based on QDT is that, due to the non-commutative nature

of the underlying mathematics of quantum theory, the effect of

evaluating different trustee candidates in various orders can be

modeled (i.e., order effects in trust evaluation). It is worth-

mentioning that many of the works performed in the

quantum-like modeling domain have investigated the ef-

fects of quantum phenomena such as quantum interfer-

ence and quantum entanglement in the human judg-

ments and decision making. A distinguishing property

of QDT is that it provides a more general and mathe-

matically rigorous description of these phenomena as

well as when or how they occur.

736 Inf Syst Front (2016) 18:735–764

In our formulation, we prove that the trust decision making

process has all the necessary conditions for the appearance of

quantum interference terms. The interference terms (also

called by Yukalov et al., the attraction factors) are the main

sources of explaining the biases and irrationalities of human

decision making and they constitute the subjective part of the

trust probability equation. By using the interference alterna-

tion theorem, interference quarter law and the principle of risk

and uncertainty aversion, we can quantify these terms and

provide a mean value for an average trustor entity. The assign-

ment of these initial values to the interference terms is invalu-

able for the bootstrapping process of trust. We use the attrac-

tion factors to determine what trustee candidate the trustor will

select and provide a better selection process in situations that

most of the classical (i.e., non-quantum theoretic) trust models

will consider equal. In order to take into account the relative

nature of trust, we define two new terms called local and

global attraction/repulsion factors to model the order effects

in evaluating different trustee candidates. By considering the

trust prospects through time, we can use the value discounting

functions to consider the effect of time and the forgetting

factor of different trustors. Also, based on this formulation

and by using the attraction factors, we can introduce a mech-

anism to increase the cost of malicious behavior plus a for-

giveness function to model the healing property of time.

The proposed QDT-based trust model is defined in a

Hilbert-space, which is a vector space with inner-product.

This property creates a powerful mechanism to represent the

context in which the trust decision making is taking place. By

using the unitary rotation operators one can change the context

and measure the trustworthiness value in the newly

established context. Also, the measurement of the quantum

system is consistent with the trust evaluation process. When

a trustor, measures her/his trust state, the vague, uncertain and

ambivalent superposition state will collapse to a determinist

state. This mechanism can be interpreted as making the final

decision by the trustor. Finally, considering the mind state of

the trustor as a superposition state with the basic states of trust

and distrust has the highest consistency with the ambivalent

and uncertain nature of trust.

The rest of the paper is organized as follows. In Section 2,

the required basic notions of trust are given. In Section 3,a

brief introduction to relevant concepts of quantum theory is

presented. In Section 4, related work of this research is

discussed. In Section 5, reasons for adopting the QDT-based

formulation in trust decision making are explained. In Sec-

tion 6, our proposed QDT-based trust formulation is intro-

duced. In Section 7, multiple scenarios to distinguish the pro-

posed formulation from the classical models are given. In

addition, empirical evaluations are provided to demonstrate

and verify the behavior of the proposed formulation. Finally,

concluding remarks and future work are mentioned in

Section 8.

2 Related basic notions of trust

Many definitions exist in the literature for the concept of trust.

Frankel et al., state that trust enters where more exact knowl-

edge is not available (Frankel 2005). Lewis and Weigert on the

other hand mention that trust is a functional alternative to

rational prediction for the reduction of complexity (Lewis

and Weigert 2012). Indeed, trust succeeds where rational pre-

diction alone would fail, because to trust, is to live as if certain

rationally possible future will not occur. Thus, trust reduces

complexity far more quickly compared to prediction.

From these definitions, we can argue that trust is a

mixture of feeling and rational thinking. In other words,

if we want to model the process of making decision to

trust someone (or an entity in general), we have to

consider the main patterns of rational decision making

plus the emotions towards risk and uncertainty in the

context of trust. This is the additional element that Lew-

is and Weigert have mentioned in their work. Therefore,

we have to expect some irrationality in the context of

trust decision making.

Mayer et al., define trust as an intention to delegate a

task, which makes the trustor vulnerable (Mayer and

Davis 1999; Mayer et al. 1995). Based on this defini-

tion,trustinganentitywillinitiatewithanintentionof

delegating a task.

A wide range of parameters are involved in the context of

trust modeling. Some of the more fundamental parameters that

trust models should take into account are as follows:

1. The intention to delegate a task to an external enti-

ty. In Delegation, the delegating entity needs or

likes an action of the delegated agent and includes

it in her/his own plan. Delegating entity, plans to

achieve her/his goal through the delegated agent

(Castelfranchi and Falcone 2010). After the intention

of delegating a task to an external entity is created

in the mind of the trustor, the behavior (i.e., the

intentional act of trusting) will take place. In other

words, if the trustor has not decided to depend on

an external entity, she/he would have no reason to

care about deciding to trust the entity or considering

theriskofselectingsuchanentity.

2. The trustor, which necessarily is an intentional entity.In-

tentionality means that the trustor is considering multiple

alternatives in order to perform her/his task toward a spe-

cific goal.

3. A trustee, capable of causing some effects as the result of

her/his behavior. There is no meaning in trusting an entity,

which is not capable of causing any effect.

4. The trust relation, which is primarily a belief or an act (i.e.,

intention presentation in QDT-based formulation) in the

mind of the trustor. Trust, is a mental attitude resulting

Inf Syst Front (2016) 18:735–764 737

from a complex comparison and matching process

amongst the preventive evaluations of the various poten-

tial trustees. It is about the risks and the costs as well as

about the applicability of these evaluations to the actual

environments and contexts (Castelfranchi and Falcone

2010).

5. The risk: without uncertainty and risk, there is no trust. In

other words, if we are absolutely certain about how the

trustee will behave, talking about trust does not make any

sense. Any intentional action and any decision exposes,

makes us vulnerable to some risk (Castelfranchi and

Falcone 1998). Additionally, any act of trust or relying

on actions of others, exposes us to risk. There exist two

types of risks:

–Objective risk, which comes from the point of view

of an ideal external observer.

–Subjective perceived risk, which is the subjectively

perceived risk value from the viewpoint of each in-

dividual trustor. This is the dominant kind of risk in

the context of trust. By using the QDT-based formu-

lation, we can quantify this kind of risk in the trust

decision making process.

6. The context: Trust is a context sensitive notion. The de-

gree of trust assigned to an entity in various contexts is

different. This difference comes from the diverse degrees

of risk and uncertainty involved in trusting a trustee in a

specific context. For example, trusting a teacher in the

educational context has much lower risk and uncertainty

compared to trusting the same teacher in the context of

fixing cars.

7. The time: Time plays a great role in the context of trust.

Time, affects the validity of evidences as well as the in-

tentions and decisions. In our formulation, we use value

discounting for trust degrees corresponding to different

trustee candidates through time.

3 A brief introduction to quantum theory

In this section, we present a brief introduction to quantum

theory concepts and its three fundamental axioms (Nielsen

and Chuang 2010). Quantum theory is basically probabilistic.

With the difference that classical probabilistic theories are

based on Boolean algebra whereas the mathematics of quan-

tum theory is based on vector spaces (also called Hilbert

spaces). In quantum theory, the events and propositions are

considered as vector subspaces. The algebra of subspaces is

non-Boolean and geometrical. In other words, a Hilbert space,

which is at the core of quantum theory, is a vector space with

defined inner product. In quantum mechanics, a two-state

system is a system, which has two possible quantum states.

An example of a two-state system is the spin of a spin −1

2

particle such as an electron, whose spin can have values þħ

2or

−ħ

2,whereħis the reduced Planck constant. Three fundamental

axioms namely the superposition, measurement and time evolu-

tion axioms are defined at the core of quantum theory. These

axioms as well as the quantum interference and quantum entan-

glement phenomena are explained in the rest of this section.

3.1 The superposition axiom

Ak-level quantum system is a system that has kdistin-

guishable or classical states. Possible classical states for

this system is represented with a Bra-Ket notion as |0〉,

|1〉,…|k−1〉.

The superposition principle states that if a system can be in

one of the kstates, it can also be in any linear superposition of

the kstates. In other words, the system in general is in a state

such as the one shown below:

ψ

ji

¼α00

ji

þα11

ji

þα22

ji

…:þαk−1k−1

ji ð1Þ

where, α

i

is a complex number and is called the probability

amplitude. In quantum theory, these probability amplitudes do

not need to sum up to 1 unlike the classical probability, in

which the law of total probability holds. But on the other hand,

for a normalized superposition state, the square of these prob-

ability amplitudes should sum up to 1. Therefore, we have:

X

k−1

i¼0

αi

jj

2¼1ð2Þ

Two level systems are called qubits (i.e., k=2). For exam-

ple, ψ

ji

¼1ﬃﬃ2

p0

ji

þ1

2þi

2

1

jiis a qubit that represents a

two-level system. The geometrical presentation of a k-level

quantum system is given as

α0

α1

…

αk−1

2

6

6

43

7

7

5,whichisaunitvector.

3.2 The measurement axiom

Any physical quantity, attached to the quantum system, which

can be measured such as position, velocity, energy and so on is

represented by a self-adjoint operator Âon the Hilbert-space

H. These quantities are called observables of the system.

The value of the measurement of an observable is one

of the observable’s eigenvalues. The probability of

obtaining one specific eigenvalue can be calculated as

the modulus square of the inner product of the state vector

of the system with the corresponding eigenvector. On the

other hand, the state of the system immediately after

738 Inf Syst Front (2016) 18:735–764

performing the measurement is the normalized projection

of the state prior to the measurement onto the eigenvector

subspace. If we assume that Âhas eigenvalues a

k

and

eigenvectors |k〉:Â|k〉=a

k

|k〉, then if the system is in the

state |ψ〉, the probability of obtaining a

k

as the outcome

of the measurement of Âin this system can be calculated

as (Axioms of Quantum Mechanics et al. 2015):

pαk

ðÞ¼k

hψE

2ð3Þ

The above equation can be also written in terms of the k

th

eigenvector projection P

k

=|k〉〈k|asp(α

k

)=〈ψ|P

k

|ψ〉.Wecan

write the normalized output state of the system after performing

the measurement in terms of the projection operator as below:

ψ0

ji

¼Pkψ

ji

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ψ

hPk

jj

ψi

pð4Þ

It is worth-mentioning that if we repeat the measurement,

without the occurrence of any event on the system, we will get

the same outcome with probability equal to 1. Therefore, mea-

surement of a quantum system disturbs the system.

1

In Fig. 1,

the un-normalized state ψ

ji

¼1ﬃﬃ2

p0

ji

−ﬃﬃ3

p

21

ji

is represented in a

unit circle. The probability that |ψ〉collapses to the |0〉state is

cos

2

θand the probability that |ψ〉collapses to the |1〉state is

sin

2

θ. Hence, measurement of |ψ〉in an orthonormal basis {b

1

,

b

2

,…,b

n

} is a projection onto basis vectors. If we assume that

the outcome is b

i

, then the probability is equal to |〈ψ|b

i

〉|

2

in

which 〈ψ|b

i

〉is the inner product of the row vector

2

〈ψ| and the

column vector |b

i

〉. The average or the expectation value

of an observable for a system in state |ψ〉can be calcu-

lated as 〈Â〉=〈ψ|Â|ψ〉.

3.3 Time evolution of quantum systems

For the time evolution of a quantum system, we can define a

time evolution operator Uas below (Axioms of Quantum

Mechanics et al. 2015):

ψ0

ji

¼Uψ

ji

where U†U¼1ð5Þ

where, the dagger operator denotes the Hermitian conjugate of

the operator U. Since the state has all the required information

about the system at time step t, the state of the system at time

step t+Δtonly depends on the current state at time tplus the

evolution operator U(t,t+Δt). The dynamics of the system are

generated by the system’s Hamiltonian H, which is the

observable corresponding to the total energy of the system.

This dynamics is determined by the Schrödinger’sequation

defined as below:

idψ

ji

dt ¼Hψ

ji ð6Þ

where, is the reduced Planck’s constant. It is worth-mention-

ing that the above equation is only true for time-independent

Hamiltonians.

3.4 Quantum interference effect

The interference effect and the way it appears was first dem-

onstrated by the well-known double-slit experiment (Carnal

and Mlynek 1991). This experiment demonstrated that light

has a wave-particle duality property. This experiment has two

different setups. In the first setup, an emitter is assumed with a

plane with two slits in front of it as shown in Fig. 2.Infrontof

the plane with two slits, there is a detector that senses the

collision of entities and reports their distance value (x)corre-

sponding to the place that the entities hit the detector. It is

assumed that the emitter is shooting particles. The location x

in which the particles were sensed by the detector is plotted as

the curve shown in Fig. 2. This outcome is consistent with

what is expected in a classical manner. The two maximum

points of the curve correspond to the locations of the two slits.

If we denote the number of particles that hit the slit 1 and slit 2

with N

1

(x)andN

2

(x) then, for the output curve corresponding

to each point we will have N

12

(x)=N

1

(x)+N

2

(x).

For the second setup, it is assumed that the emitter emits

waves rather than particles. This setup and the corresponding

outcome are shown as Fig. 3. As it is shown in the figure, the

outcome is an interference pattern. For some points, this inter-

ference has a constructive nature and for others it has a de-

structive one. The interesting observation is that, the interfer-

ence pattern appears when we do not know which path the

1

This property is so fundamental in quantum systems that it is the basis of

quantum cryptography. Any third party observation along the wire will

destroy the initial state of the system. This is critical for detection of

unauthorized observation of quantum encrypted state.

Fig. 1 Presenting ψ

ji

¼1

ﬃﬃ2

p0

ji

−ﬃﬃ3

p

21

ji

on unit circle

2

〈ψ| is the conjugate transpose of the column vector |ψ〉.

Inf Syst Front (2016) 18:735–764 739

waves of light took in order to get to the detector’ssurface.If

we observe which path the light waves are coming from (i.e.,

we measure the system) the interference term disappears and a

classical outcome as demonstrated in Fig. 2will be produced.

3.5 Quantum entanglement

Quantum entanglement is one of the unique characteristics of

the realm of quantum theory. Quantum entanglement can re-

sult in deeper correlations that cannot be found in classical

systems. The most fascinating part is that such correlation in

entangled quantum systems can continue its existence even

when the two systems are spatially separated and fat apart.

This is the main reason that Einstein Podolsky and Rosen

(commonly known as EPR) in their well-known paper and

in order to show that a quantum description of the physical

world cannot be considered as complete, referred to this phe-

nomenon as spooky action at the distance (Einstein et al.

1935). Later on, Bell, in an experiment demonstrated and

described the occurrence of quantum entanglement and

ruled-out the local realism theory favored by Einstein (Bell

1966).

If we consider two non-interacting systems denoted by A

and B, corresponding to Hilbert spaces H

A

and H

B

, the Hilbert

space of the composite system is the tensor product of the

Hilbert spaces. This tensor product can be written as H

A

⊗

H

B

. If the system Ais in the state |ψ

A

〉and the system Bis in

the state |ψ

B

〉, then the state of the composite system is |ψ

A

〉

⊗|ψ

B

〉. Such states that can be represented as the tensor prod-

uct of two states are called separable states or product states.

If we assume the basis {|i

A

}correspondingtoH

A

and the basis

{|j〉

B

}correspondingtoH

B

, then a general state in H

A

⊗H

B

can

be written in the following form:

ψ

ji

AB ¼X

i;j

cij i

ji

A⊗j

ji

Bð7Þ

The above state is separable if there exist c

i

A

,c

j

B

in a

way that c

ij

=c

i

A

c

j

B

. This will result in separated states

ψ

ji

A¼∑

icA

ii

ji

Aand ψ

ji

B¼∑

icB

jj

ji

B. The above state is

Fig. 3 Double-slit experiment-

second setup

Fig. 2 Double-slit experiment-

first setup

740 Inf Syst Front (2016) 18:735–764

inseparable if for all c

i

A

,c

j

B

,wehavec

ij

≠c

i

A

c

j

B

.Ifastate

is inseparable, it is called an entangled state. For exam-

ple, states such as ψ

ji

¼1

ﬃﬃ2

p00

ji

þ1ﬃﬃ2

p11

ji

(which are

called the Bell states) are well-known examples of max-

imally entangled quantum states. When two qubits are

in the form of an entangled state as described, it does

not matter how far they are apart. The measurement of

one qubit will result to certain measurement of the basic

states of the other qubit. Today, quantum entanglement

plays a fundamental role in areas such as quantum tele-

portation and quantum cryptography.

4 Related work

A point that comes to mind by looking at the works performed

in the context of computational trust modeling is that, in the

existing literature, most of the proposed models concerning

human-based trust decision making assume some level of ra-

tionality and paradoxes of human decision making and the

corresponding biases are not considered. This is in contrast

to the vast amount of evidence provided against these assump-

tions (Eddy 1982; Gilovich et al. 2002; Shah and Oppenhei-

mer 2008). Hoogendoorn et al. in (Hoogendoorn et al. 2008a),

interestingly argue that in order for intelligent agents acting as

the personal assistants of human beings to provide good rec-

ommendations or support them in task allocation, a suitable

model of what humans prefer is essential. They continue by

proposing that the measurement of trust should include the

notion of relativeness since we are dealing with multiple al-

ternatives. In the proposed work, they provide such a model in

which a number of parameters can be set for the better repre-

sentation of the characteristics of human beings.

Hoogendoorn et al. in (Hoogendoorn et al. 2010)extend

their work and discuss that for the case of multiple trustees,

there might exist dependencies between the trust levels in

different trustees. Two types of models are introduced in this

work. The first one is the development of a new trust model,

which incorporates this relativeness explicitly and the second

one is an extension of the existing trust models capable of

expressing the relative nature of trust by using a translation

mechanism from objective experiences to subjective ones. A

number of simulations are performed in order to demonstrate

the better consistency of the proposed relative trust model with

the human trust.

As a further development, Hoogendoorn et al. presented a

validation and verification of a relative model of trust in com-

parison with a basic non-relative model (Hoogendoorn et al.

2008b). In this work, custom made military-like software was

used in order to gather empirical data. In this software, based

on the different properties of geographical locations, players

could command to attack, ask for help or perform nothing. In

other words, human players were asked to perform a classifi-

cation task based on the information received from different

sources. It is shown in this paper that relative models of trust

provide a better explanation of real world data. In addition, in

(Hoogendoorn et al. 2011), a modeling and validation of bi-

ased human trust is performed. In this paper, a number of

possibilities to model biased human trust in a computational

manner are discussed and evaluated. In order to evaluate these

biased trust models, which mostly are the same work per-

formed in (Hoogendoorn et al. 2008a) and (Hoogendoorn et

al. 2008b), and to demonstrate that they achieve better results

in predicting human trust than non-biased trust models, the

models have been validated on empirical data obtained from

the same software mentioned in (Hoogendoorn et al. 2008b).

Klüwer et al., on the other hand, presented a theory about trust

in cases where there is more than one trustee, assuming that a

relation, according to a dimension given by a rule or regularity

between trustee candidates is given (Klüwer and Waaler

2006a). These relationships are modeled through a lattice-

based model. With the help of the lattice structure, authors

aimed at making the implicit relative properties into explicit

ones.

Although these approaches provide significant insight into

the concept of relative trust modeling, they lack the essential

mechanisms required for a trust model such as a way to de-

scribe context and a well-defined structure for trust decision

making. Most of these approaches work by adding extra pa-

rameters to the existing commonly used models. Providing a

holistic model of trust that has inherent capabilities for taking

into account this relative nature is one of the advantages of our

proposed model.

One of the newly developed research areas is the area of

quantum cognition and decision making also more commonly

known as quantum interaction. Researchers in this domain use

the mathematics of quantum theory in order to model the

cognitive and decision making processes and propose models

that are consistent with actual biases and paradoxes of human

decision making.

Pothos et al., introduced a quantum probabilistic explana-

tion of conjunction effect and inverse fallacy (Pothos and

Busemeyer 2009). The conjunction effect explanation for ex-

ample is performed with the help of a well-known question

called feminist Linda in which participants needed to decide

whether Linda is a feminist activist or a bank teller based on

the provided descriptions. In (Busemeyer and Trueblood

2011), Busemeyer et al. propose and argue about the reasons

why the quantum probability theory can be applied to cogni-

tive domains and human decision making.

In (Anderson and Hubert 1963; Hogarth and Einhorn 1992;

Trueblood and Busemeyer 2011), it is shown that quantum

probabilities can explain and correctly model order effects

due to their geometric foundation. In other words, the way

that probabilities are calculated in quantum mathematics is

Inf Syst Front (2016) 18:735–764 741

by projecting the state vector in the corresponding subspace.

In addition, Franco et al. in (Franco and Busemeyer 2008),

have shown that quantum probability can naturally account

for the inverse fallacy and a couple of other human decision

making biases. Also, Yukalov et al., discuss that people violate

the sure thing principle of decision theory in two stage gam-

bling game and prisoner’s dilemma (Yukalov and Sornette

2009a; Yukalov and Sornette 2012). In their work, a quantum

theoretic model is proposed in order to correctly and naturally

explains these violations. Therefore, a quantum model of trust

can be considered as a potential candidate for considering the

biases and paradoxes of human trust decision making.

As for the potential of quantum theoretic models to explain

order effects and decision making in comparative contexts,

Trueblood et al., presented a quantum probabilistic approach

to define order effects in human judgments (Trueblood and

Busemeyer 2011). The authors correctly argue that the order

of information plays a fundamental role in the process of

updating beliefs across time. The presence of order effects

creates difficulties for applying a classical probabilistic ap-

proach to such situations. The proposed quantum model is

fitted to data collected in a medical diagnostic task as well as

a jury decision making task. By comparing their model with a

couple of classical models, they reach the conclusion that the

quantum model provides a more coherent account for order

effects that was not possible before.

On the other hand, although many of the proposed compu-

tational trust models provide mechanisms to take the context

into account (Yao et al. 2013; Busacca and Castaldo 2011;Ali

et al. 2010; Nguyen et al. 2010), context is very much just a

simple label. Although considering the context as a label that

indicates the goals and priorities of the trustor is a simple and

easy to use approach, it is not capable of answering more

fundamental questions such as “what would happen if we

change the context?”,“how similar the two contexts are?”or

“if we trust a trustee candidate to a degree in one context, how

much we will trust her/him if the context is changed?”.By

considering the context as a basis and defining the trust states

relative to different bases, we can approach the problem of

context modeling in our QDT-based trust model in a much

more intuitive and flexible manner.

Another important facet in computational trust models is

how they define the trust state. A wide range of trust models

assume that trust and distrust are two separate and indepen-

dent constructions (Falcone and Castelfranchi 2012; Verbiest

et al. 2012; DuBois et al. 2011;KlüwerandWaaler2006b;

Lesani and Bagheri 2006). Another line of thought is to con-

sider distrust as the non-existence or lack of trust. But, these

assumptions are not consistent with the definitions of trust

provided by well-known trust researchers. In the proposed

QDT-based trust model, by considering trust and distrust as

the basic states of a superposition state, we can define a trust

state that considers the co-existence and ambivalence of trust

and distrust within a relationship or transaction. In such a state

the trustor can only talk about the tendency or potentiality to

trust or distrust a trustee candidate.

Besides, many trust models that use a purely probabi-

listic or statistical approach to computational trust model-

ing (ElSalamouny et al. 2010; Huang and Wang 2008;

Feng and Huizhong 2008;H

angetal.2008;Shietal.

2005) always compute the trust or distrust disposition

based on a pre-calculated probability. Thus, every time a

trustor asks about the trustworthiness of a trustee candi-

date, the result is calculated probabilistically. This ap-

proach creates a problem when subsequent trust queries

are sent without the reception of any new evidences. In

such a situation the trust disposition should be the same as

the previous one in a deterministic manner. This problem

can be tackled in the proposed model by using the con-

cept of measurement. When we measure a quantum state,

the superposition state indicating the ambiguous and un-

certain trust state will collapse to a basic state denoting

trust or distrust with a probability consistent with the

probability amplitude of the trust and distrust basic states.

Hence, if a trustor asks about the trustworthiness of the

same trustee candidate without the occurrence of any

event or the reception of new evidence, she/he will get a

deterministic outcome equal to the previous trust query.

5 Reasons for adopting a QDT-based formulation

of trust

The main aim of this paper is to propose a model of trust that

can take into account both the objective and subjective parts of

the trust decision making process. The proposed model should

be capable of quantifying the risk and uncertainty associated

with different contexts and provides a suitable framework for

modeling the relativeness property of trust evaluation and its

corresponding order effects.

By using the foundations of quantum decision theory, a

sound, powerful, natural, and even general model of trust with

the above properties can be introduced. The reasons we state

that a quantum decision theoretic formulation is beneficial for

introducing a more suitable computational trust model are as

follows:

1. A superposition state can be used to represent the mind

state of the trustor regarding trusting or distrusting a trust-

ee candidate. The classical notion of uncertainty corre-

sponds to the lack of knowledge about an entity or a state.

But, in quantum mathematics, representing a state in a

superposition form induces a deeper notion of uncertainty.

A superposition state is not consistent with any single

possible outcome (i.e., trust or distrust in the trust decision

making process). Instead, there is a potentiality for each of

742 Inf Syst Front (2016) 18:735–764

the outcomes and if the state evolves in time, these poten-

tialities will also change. Therefore, if there is weight for

all the possibilities, the person is in a superposition of all

these possibilities and it is impossible to assume that the

person is in a specific state. It can be argued that such a

characteristic is best suited for presenting emotional am-

bivalence, which is the ability to represent both the posi-

tive and negative tendencies simultaneously. This is very

much consistent with the definitions provided by well-

known trust researchers. As an example, Marsh and

Dibben in their well-known research (Marsh and Dibben

2005), argue that neither full trust nor distrust is actually

present in practical situations. A trustor never fully trust or

distrust a trustee candidate. The mind state of the trustor

regarding the trustworthiness of the trustee candidate is

always a combination of trust and distrust with different

degrees. In a similar argument, Lewicki et al. discuss that

trust and distrust usually coexist within a relationship

(Lewicki et al. 1998). The proposed QDT-based trust

model tries to model these definitions with the concept

of superposition.

2. The act of measuring observables in quantum sys-

tems is very consistent with the evaluation process

in the context of trust decision making compared to

classical models of trust. In most of the probabilistic

trust models, every time we ask about the trustwor-

thiness of an entity, we get a probabilistic outcome

based on the probability assigned to that candidate.

But, in quantum systems, upon measuring the sys-

tem (i.e., evaluating the trustee candidate) the super-

position corresponding to a trust state will collapse

to a basic state and until no further event has oc-

curred, the trust degree will be a deterministic value

(Ashtiani and Azgomi 2014).

3. Non-commutativity property of the Hilbert space-based

mathematical foundation of quantum theory will be a

great tool to model the relativity and order effects in the

context of trust. In simpler words, the probability of eval-

uating Afirst then Bis not equal to evaluating Bfirst then

A, which is due to the comparative context of trust deci-

sion making.

4. Many situations exist in which the human judgment does

not obey the laws of classical logic and probability

(Busemeyer et al. 2009a; Aerts et al. 2011;Franco

2009). Specifically, situations that demonstrate order and

context effects usually deviate from the common laws of

classical logic. Trust decision making contains both the

order and context sensitivity characteristics. Hence, clas-

sical probabilistic-based approaches cannot be viewed as

the best mechanisms to represent the trust constructions.

As we have discussed in the previous section, quantum

mathematics have shown promising results corresponding

to these types of situations.

5. The quantum interference terms (i.e., attraction factors in

QDT) can occur in trust decision making and are founda-

tional in quantifying the risk and uncertainty associated

with different contexts as well as the attitude of the trustor

towards them.

6 A trust model based on quantum decision theory

In this section, we introduce a model of trust based on QDT.

We start the discussion by first introducing the fundamental

building blocks of our quantum-like trust model such as the

quantum trust state, quantum measurement and expressing

context in the quantum model. In the preliminaries section,

we only provide a general overview of the core concepts of

the proposed framework for computational trust modeling.

For more detailed information please refer to our previous

work (Ashtiani and Azgomi 2014). Then, we use the QDT

developed by Yukalov et al. as our basis, in order to propose

a new model of trust that can meet the introduced motivations

of this work.

6.1 Preliminaries

In this section, the basic mappings between the trust building

blocks and the axioms of quantum theory are defined.

Quantum trust state The concept of superposition differs

from the concept of a mixed state in classical probability. In

other words, classical approaches must assume that a system

is always in a particular and determined state even if the

knowledge of itis uncertain. Superposition, on the other hand,

only talks about the potentiality or tendency of being in each

of the existing basic states. Hence, superposition can be con-

sidered as an intuitive mechanism to characterize the fuzzi-

ness, uncertainty, ambiguity and ambivalence of human judg-

ment (Busemeyer et al. 2009b). In classical trust models, the

trustor is assumed to either trust or distrust a trustee candidate

at each moment in time. But, because we do not know with

certainty which one, we assign a probability to each of these

two possibilities. On the other hand, in the quantum probabil-

ity theory, when a state vectorisexpressedas|ψ〉=α

0

|distrust〉+

α

1

|trust〉, the trustor is neither distrusting nor trusting the trustee

candidate. The trustor is considering and entertaining both of

these possibilities concurrently. But, until the final decision

is made (i.e., the trust state is measured or evaluated) she/he

is committed to none of them. In a trust superposition state,

the only thing that we can talk about is the potential or

tendency that the trustor is going to decide whether she/he

trusts the trustee candidate or not. Representing trust as a

superposition state plays a fundamental role in our quan-

tum-like formulation and can be considered as a natural

Inf Syst Front (2016) 18:735–764 743

mapping between the social definitions of trust and the re-

quirements of a quantitative computational trust model.

Definition 1 Absolute trust and distrust

The extreme cases of absolute trust and distrust are repre-

sented in the model as below (Ashtiani and Azgomi 2014):

ψ

ji

¼trust

ji

absolute trust ð8Þ

ψ

ji

¼distrust

ji

absolute distrust ð9Þ

Definition 2 Internal and external trust

In (Falcone and Castelfranchi 2001), Falcone et al. de-

fined internal trust by the trustworthiness criteria such as

competence,motivation,willingness andsoonthatare

related to the internal characteristics of the trustee. While

the external trustworthiness criteria such as danger and

opportunity correspond to the external factors that can

affect the performance of the trustee. In a quantum-like

model of trust, we can use sub-states in order to define the

trust state from these points of view. The definition of

trust in terms of internal/external trustworthiness criteria

can be formulated as below:

Trust

ji

¼α0Internal Criteria

ji

þα1External Criteria

ji

ð10Þ

Internal Criteria

ji

¼b0Motivation

ji

þb1Willingness

jiþ

þ…bnCompetence

ji ð11Þ

External Criteria

ji

¼c0Opportunity

ji

þc1Danger

ji

ð12Þ

Trust evaluation through the act of quantum

measurement Trust is not a prediction but it is an evaluation

that changes the state of the mind of trustor (Castelfranchi and

Falcone 2000). Quantum theory is very much consistent with

this definition. The evaluation and its change in the trustor’s

mind can be modeled by the act of measurement. The mea-

surement of the quantum system will change the state of the

quantum system and will collapse the superposition to a def-

inite state. Imagine you evaluate a trustee and decide that she/

he is trustworthy. This evaluation in the form of a measure-

ment will change your ambiguity defined by the superposition

state to the |trust〉state. If you (as the trustor) are asked imme-

diately after without any other evaluation or occurrence of an

external event regarding your belief about the trustee, you will

deterministically say that you trust her/him. This is a concept

that psychologists are very well aware of. For example, they

use proxy events (i.e., questions) inorder to change the state of

mind of their subjects in order to get different outcomes. This

is in contrast to classical probability. In classical probability

after any evaluation you will get a different outcome accord-

ing to an objective or subjective probability (Ashtiani and

Azgomi 2014).

Corresponding to each observable (i.e., properties that we

measure in order to characterize the quantum state of a sys-

tem) there exist a projection operator. The projection operator

corresponding to a sub-state is defined as an outer product as

below:

Pπ¼π

jiπ

hj ð13Þ

where ∑

n

j¼1

Pj¼Iin which Iistheidentitymatrixandnis the

total number of sub-states.

For calculating the probability of finding the quantum sys-

tem in a specific state, we multiply the projector with the

quantum state as:

pπðÞ¼Pπψjikk

2¼πjiπhjψikk

2ð14Þ

Representing the trust context in the quantum

model Trust is context sensitive. Quantum probability pro-

vides a powerful mechanism for changing the context. This

is performed by modifying the basis vectors. By changing the

basis vector, which itself can be performed by using the uni-

tary rotation matrices, we can measure trust in a trustee in

multiple contexts. The only issue that we should be concerned

about is the amount of similarity that exists between these two

contexts. The more similar the two contexts are, the smaller

the angle between the two basis vectors is. This is shown in

Fig. 4. The Fig. 4a represents two contexts that are more

similar than Fig. 4b.

The projection of the state vector |ψ〉on each of these bases,

determines the probability to trust the trustee in the respective

context. The initial state described with respect to the Context

1

basis is related to the initial state described with respect to the

Context

2

basis by the linear transformation Context

2

=U.

Context

1

or Context

1

=U

†

.Context

2

. Because the matrix is

unitary, the preservation of the length of the vector is guaran-

teed in the transformation.

6.2 Fundamentals and definitions

In this section, the fundamental concepts and definitions of

QDT are explained. We have tried to create a mapping be-

tween the concepts of QDT and the building blocks of trust

decision making. For each fundamental QDT definition, its

corresponding definition for the context of trust decision mak-

ing is introduced.

Definition 1 Intended actions

An intended action (or an intention) is a particular thought

about doing something (Yukalov and Sornette 2011). In the

context of trust, the whole process of establishing a trust-based

relationship will start by a simple intention of the trustor. This

initial intention is the intention of relying (i.e., depending) on

an external entity to perform a task τtowards a goal gfor the

744 Inf Syst Front (2016) 18:735–764

trustor. This intention can be interpreted as the intention to

delegate a task. The relationship between delegation and trust

is complicated. Delegation necessarily requires trust. In other

words, when the trustor comes to this conclusion that she/he

cannot (or do not want to) perform a task τ, a delegation

intention is created in the mind of the trustor. After this inten-

tion, the trustor decides to trust an external entity amongst

many alternatives to perform her/his task. This is why

Castelfranchi et al. in (Castelfranchi 2008), mention that the

trustor entity should be intentional. Therefore, the intention

that is created in the mind of the trustor is something like:

“I would like to delegate my task τto an external entity

Mto reach my goal gin a specific context C”.

Proposition 1 Minimum number of intended actions in the

context of trust decision making is two.

Proof An important point that we want to discuss here is that,

in the context of trust and upon deciding to delegate a task,

there should always be more than two intentions. This is be-

cause the number of intentions to delegate a task to an external

entity is proportional to the number of existing alternatives. In

the context of trust, the minimum number of existing alterna-

tives should be at least two. In other words, if we are stuck

with one choice or cannot freely decide what alternative to

choose, then talking about trust does not make much sense.

This is the fundamental difference between the concepts of

trust and confidence (Cofta 2007). Yukalov et al. in (Yukalov

and Sornette 2010b), define the set of intended actions

equipped with binary operators such as product and addition

as an action ring. The only difference between the action ring

defined in the context of trust and a general action ring is the

number of intended actions.

A¼ Ai:i¼2;3;…fg ð15Þ

where, each A

i

is defined as follows:

A

i

I would like to delegate my task to an external entity M

i

to

peform the task τtowards a goal g in a specific content C.

Definition 2 Internal and external trust

Intention representations are the realizations or as mentioned

in (Yukalov and Sornette 2011; Yukalov and Sornette 2010b;

Yuk a l ov a nd So r n ett e 2009a), the concrete implementations of

the intentions. For example, for the intention of “delegating a

task to A”, the intention representation is “ItrustA”.Thisisdue

to the reason that the most basic representation of the delegation

intention towards Ais the fact that we need to decide whether

we trust Aor not.

This relationship is also mentioned by Hoogendoorn et al. in

(Hoogendoorn et al. 2008b) where it is stated that, if a trustor

has not decided to depend on an entity, talking about trusting

that entity is pointless.

The intention representations may include not only positive

but negative presentations. Therefore, in a very basic scenario

where a trustor wants to delegate her/his task to one of the two

existing alternatives, the intentions and their corresponding in-

tention representations shown in Table 1can be assumed.

Definition 3 Mode states

The mode state or representation state of an intention pre-

sentation A

iμ

is denoted as the vector |A

iμ

〉where irepresents

the i

th

intention and μis the μ

th

intention representation. For

example, the mode state |A

11

〉represents the state where the

trustor wants to delegate her/his task to Aand also trusts A,

while |A

22

〉represents the state where the trustor wants to

delegate her/his task to Band also distrusts B.

Tab le 1 The intentions and their corresponding intention

representations in a basic trust scenario

Intention Intention presentation

I want to delegate my task to AItrustA

I distrust A

I want to delegate my task to BItrustB

I distrust B

Fig. 4 Representing different

contexts with different basis: a

Two contexts that are more

similar and bTwo contexts that

are less similar

Inf Syst Front (2016) 18:735–764 745

Definition 4 Mode space

The mode space consists of all the possible intention states.

The mode space is formed as the following closed envelope,

which spans the mode basis {|A

iμ

〉}:

Mi≡ℒAiμ

ð16Þ

Therefore, the mode space is a Hilbert space with the prop-

erties explained in the previous section.

Definition 5 Intention states

The intention state corresponding to the i

th

intention at time

tis defined as below:

ψitðÞ

ji

¼X

μ

ciμtðÞAiμ

ð17Þ

In the context of trust, this is the same superposition de-

fined in Eq. 1. The re-writing of the Eq. 17, in terms of an

intention state can be performed as follows:

Delegating a task to Aat time t

ji

¼c11 tðÞTrust Aat time t

ji

þc12 tðÞDistrust Aat time t

ji

ð18Þ

For the rest of the paper, we denote Aand Āas trust and

distrust towards an external entity Arespectively. Therefore,

we represent the above equation as |ψ

A

〉=c

11

|A〉+c

12

|Ā〉where,

for presentation simplicity, we have dropped the time param-

eter while keeping in mind that the complex probability am-

plitudes c

ij

are time-dependent.

Definition 6 Action prospects

A prospect r

j

is a conjunction of several intention represen-

tations. Especially in the context of trust, an individual is al-

ways motivated by various intentions. This is due to the exis-

tence of different trustee candidates that the trustor wants to

select from. Therefore, the realization of an intention to dele-

gate a task to a trustee candidate involves taking into account

the intentions of delegating to other available trustees. Thus, in

the context of trust, a prospect is an object of the composite

type and each action can be composed of at least two modes.

Definition 7 Elementary prospects

An elementary prospect e

n

is a simple prospect formed by a

conjunction of single action modes. In the context of trust with

two alternatives, the prospects AB;AB;AB and AB are the

elementary prospects.

Definition 8 Basic states

Basic states are the vectors defined as follows:

eni≡⊗iAivμ

≡Aiv1Aiv2Aiv3…

ji ð19Þ

In other words, corresponding to each elementary prospect,

a basic state is defined. In the context of trust with two alter-

natives, basic states are defined as AB

ji

;AB

;AB

;AB

Definition 9 Prospect basis

The prospect basis {|e

n

〉}, is the family of all basic states

corresponding to elementary prospects. Different states that

belong to the prospect basis are assumed to be disjoint due

to being orthogonal to each other.

For example, for the intention state |Delegating a task to A-

at time t〉the prospect basis is defined as {|A〉,|Ā〉}andforthe

intention state |Delegating a task to Bat time t〉the prospect

basis is given as A

ji

;B

.

Definition 10 Mind space

The mind space is defined as the closed linear envelope

over the prospect basis |e

n

〉:

M≡ℒen

jifg

¼⊗iMið20Þ

The closed linear envelope, covers all possible states that

can be expanded over the total basis |e

n

〉. For example, if we

consider the mode space consisting of the intention states cor-

responding to delegating a task to trustee candidates Aand B

with ℳ

A

and ℳ

B

respectively, we will have:

M≡ℒen

jfg

¼⊗iMi¼MA⊗MBð21Þ

Definition 11 Mind dimensionality

The dimensionality of the mind space is defined with the

following equation:

dim MðÞ≡∏

iMið22Þ

where, M

i

is the number of the i-intention modes (or

representations).

Proposition 2 The mind dimensionality in the context of trust

is at least four.

Proof The number of intentions in the mind of the trustor for

delegating a task as argued in Proposition 1 should be at least

two. This means that the trustor is at least presented with two

trustee candidates and she/he is free to delegate to any one of

them. If the trustor is only presented with one entity to dele-

gate, then talking about trust does not make much sense. This

is because she/he is forced to select the existing trustee candi-

date no matter how trustworthy it is considered. Also, the

intention presentation of a delegation intention consists of

two intention modes of trust and distrust, which is presented

as a superposition in the corresponding intention state. Thus,

based on the Eqs. (21,22), the mind dimensionality in the

context of trust should be at least four.

746 Inf Syst Front (2016) 18:735–764

Definition 13 Prospect states

Aprospectstate|π

j

〉is a member of the mind space. The

vectors |π

j

〉are not necessarily orthogonal to each other and in

general are not normalized.

Definition 14 Strategic state

The strategic state of mind at time tis defined as below:

ψstðÞ

ji

¼X

n

cntðÞen

ji ð23Þ

where, the complex probability amplitudes c

n

(t) are time-de-

pendent and their temporal evolution is based on a particular

individual and a specific context. The strategic state is a Hil-

bert space defined with an inner-product. This state is formu-

lated as follows:

ψst1

ðÞ

hψst2

ðÞ

ji

≡X

n

c*nt1

ðÞcnt2

ðÞ ð24Þ

The norm of the strategic state is defined by the following

equation:

ψstðÞ

jikk

≡ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ψstðÞ

hψstðÞ

ji

pð25Þ

Furthermore, for the normalization condition of the coeffi-

cients we have:

X

n

cntðÞ

jj

2¼1ð26Þ

Each strategic state of mind represents an individual trustor

with her/his beliefs and subjective desires. Thus, different

trustors possess different strategic states.

6.3 Trust decision making

The first step in the trust decision making process is evaluating

how useful and appealing it would be to choose different

trustee alternatives based on the viewpoint of the trustor. As

explained in Section 6.1, the quantum measurement operators

can be used for the evaluation process in the context of trust

decision making. If we assume that our prospect set,

consistingof all the prospects from which the trustor can make

a choice, is defined as ℒ≡{π

j

:j=1,2…}then,thecorrespond-

ing prospect evaluation operator will be defined as below:

b

Pπj

≡πj

πj

ð27Þ

For finding the probability of realizing a prospect, the ex-

pectation value of the prospect operator can be used as fol-

lows:

pπj

≡ψs;tðÞ

hb

Pπj

ψstðÞi¼ πj

ψstðÞi

2ð28Þ

The above prospect probability, quantifies the probability

that a given individual chooses the prospect π

j

given her/his

strategic state of mind at time t.

Because the strategic state of mind is time-dependent, the

trustor may choose different prospects at different times with

different probabilities. This can be interpreted in the context of

trust with the help of relative models of trust.

6.4 Prospect interference and trust decision making

In this section, we propose and demonstrate that quantum

interference terms are an inevitable part of trust decision mak-

ing and they can be used to quantify the emotions and subjec-

tive desires towards risk and uncertainty in this context. The

main difference between the classical and quantum decision

theories occurs here, where the former, only considers the

objective properties (i.e., expected utility) while the later, not

only considers the objective properties but the subjective de-

sires quantified by the interference terms.

As an illustration, consider the most basic trust decision

making scenario in which a trustor wants to delegate her/his

task to one of two existing trustee candidates denoted by Aand

B. In this situation, two main intentions, “Iwanttodelegate

my task to A”and “I want to delegate my task to B”are

considered. Each one of these intentions have two intention

representations, “ItrustA”(denoted as A), “IdistrustA”(de-

noted as Ā)and“ItrustB”(denoted as B)and“IdistrustB”

(denoted as B).

The corresponding strategic state of mind is defined by the

Eq. 29. To define the existing prospects, we need to consider

that the trustor is deliberate between delegating her/his task to

A(or B) or not. This can be defined with the following com-

posite prospects:

πA¼ABþB

πA¼ABþB

πB¼BAþA

πB¼BAþA

ð29Þ

The corresponding prospect states with the above compos-

ite prospects related to trustee candidates Aand Bare defined

as follows:

πA

ji

¼a1AB

ji

þa2AB

πA

¼a01AB

þa02AB

πB

ji

¼a1AB

ji

þa01AB

πB

¼a2AB

þa02AB

ð30Þ

where, for example, the prospect state |π

A

〉represents the state

where the trustor is deciding to trust Awhile the choice to trust

or distrust Bhas not been decided yet.

Inf Syst Front (2016) 18:735–764 747

Proposition 3 Decision making in the context of trust has all

the necessary conditions for the presence of interference

terms.

Proof In (Yukalov and Sornette 2011; Yukalov and Sornette

2010b; Yukalov and Sornette 2009a; Yukalov and Sornette

2008a), the necessary conditions for the presence of interfer-

ence terms are proposed. For proving this proposition, we will

mention these conditions and argue that they all exist in the

context of trust decision making. These conditions are as

follows:

1. The mind dimensionality should be greater than one.This

is because, if dim(ℳ) = 1 then, only a single basic

vector exists (i.e., |A

1

A

2

…〉), and all prospect states

are of the form |ψ〉=c|A

1

A

2

…〉. Therefore, only one

probability exists that is p=|〈A

1

A

2

…|ψ

s

〉|

2

=1. In oth-

er words, the phenomenon of decision interference

appears when one considers a composite entangled

prospect with several intention representations

assumed to be realized simultaneously. As it is

proved in Proposition 2, the mind dimensionality in

the context of trust decision making is at least

four. Also, as it is shown in this section, in the con-

text of trust decision making, we are dealing with

entangled composite prospects with at least two in-

tention representations assumed to be realized

simultaneously.

2. Uncertainty should be present. For example, even if we

have a large dimensional mind, if we have a certain

prospect with the state defined as |π

j

〉=c

j

|ψ

s

〉then, the

corresponding probability should be equal to 1. Un-

certainty in the prospects is a natural characteristic of

trust. If we are certain about the outcome of the

trustee’s performance, then talking about trust does

not make any sense. As it is stated in (DuBois et

al. 2011), trust is present when there is an amount

of uncertainty involved in the situation.

Therefore, because a quantum theoretic decision making

model can provide a mechanism for quantitative analysis of

these interference terms, it is capable of better explaining the

uncertain and subjective nature of trust by taking into account

the presence of these terms.

Interference alternation theorem In (Yukalov and

Sornette 2012), it is shown that if we have multiple

interference terms corresponding to different prospects

and if at least one of the terms are non-zero, then some

of the interference terms are necessarily negative and

some are necessarily positive. Therefore, some of the

probabilities are depressed while others are enhanced.

The depression of probabilities can be associated with

748 Inf Syst Front (2016) 18:735–764

In order to find the prospect probabilities, we use the

Eq. 28. Therefore, we will have:

πA

hψs

ji

¼a*

1c11 þa*

2c12

πA

ψs

ji

¼a0*

1c21 þa0*

2c22

πB

hψs

ji

¼a*

1c11 þa0*

1c21

πB

ψs

ji¼a*

2c12 þa0*

2c22

ð31Þ

where, the corresponding prospect probabilities are as fol-

lows:

pp

A

ðÞ¼a*

1c11 þa*

2c12

2

pp

A

¼ja0*

1c21 þa0*

2c22j2

pp

B

ðÞ¼ja*

1c11 þa0*

1c21j2

pp

B

¼ja*

2c12 þa0*

2c22j2

ð32Þ

with the constraint that:

pπA

ðÞþpπA

þpπB

ðÞþpπB

¼1ð33Þ

The prospect probability p(π

A

) is defined as:

pABðÞþpA

B

¼a1c11

jj

2þa2c12

jj

2þqπA

ðÞ ð34Þ

where, q(π

A

), is called the interference term and is defined as:

qπA

ðÞ¼2φπ

A

ðÞﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

pABðÞpAB

qð35Þ

where, φ(π

A

) is called the uncertainty factor and is formulated

as follows:

φπ

A

ðÞ≡cosΔπ

A

ðÞ ð36Þ

where, Δ(π

A

) is the uncertainty angle. The uncertainty angle

can be written as:

Δπ

A

ðÞ≡arg a*

1c11 þa2c*

12

ð37Þ

The same equations can be provided for other existing

prospects. If we consider the standard equation of joint and

conditional probabilities given as:

pA

iBj

¼pA

iBj

pB

j

ð38Þ

Then, the interference terms in Eq. 35 can be written as

follows:

qπA

ðÞ¼2φπ

A

ðÞﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

pA

B

pBðÞpA

B

p B

r

qπA

¼2φπ

A

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

pAB

pBðÞpAB

p B

rð39Þ

exist between two prospects with different interference

terms:

1. Two p rosp ects π

A

and π

B

are equally repulsive (or equally

attractive) iff q(π

A

)=q(π

B

).

2. The prospect π

A

is more attractive than the prospect π

B

iff

q(π

A

)>q(π

B

).

3. Aprospectπ

A

is more attractive (or less repulsive) than π

B

when one of the followings is true:

a. π

A

leads to a less uncertain gain compared to π

B

.In

other words, π

A

leads to a more certain gain.

b. π

A

leads to less certain loss in comparison with π

B

.

The above statements imply that π

B

is perceived as more

uncertain or potentially more harmful than π

A

.Ifweconsider

the interference alternation theorem, then, for a two prospect

scenario we will have:

qπA

ðÞþqπB

ðÞ¼0ð43Þ

This means that if either the prospects are considered as

equally certain or uncertain, then we will have:

qπA

ðÞ¼qπB

ðÞ¼0ð44Þ

But, if one of the prospects, say π

A

,is considered as more

certain or less harmful, then we will have:

qπA

ðÞ¼−qπB

ðÞ>0ð45Þ

The prospect probabilities can be expressed in a more com-

pact form. For example, the probability of a prospect π

j

∈ℒ

can be written as (Yukalov and Sornette 2009a):

pπj

¼X

α

pπjeα

þqπj

ð46Þ

where, the summation ∑

α

pπjeα

over the elementary pros-

pects e

α

, can be considered as the objective part whereas the

interference term q(π

j

) denotes the subjective part of the pros-

pect probability. The probability p(π

j

e

α

)iscalledpartial

probability and is defined as (Yukalov and Sornette 2009a):

pπjeα

¼b

Pe

α

ðÞ

b

Pπj

b

Pe

α

ðÞ

DE ð47Þ

In other words, the probability of a conjunction prospect

π

j

e

α

expresses the partial probability of realizing an elemen-

tary prospect e

α

when deciding about the prospect π

j

.

In QDT, between two prospects π

A

and π

B

,prospectπ

A

is

preferred to π

B

if and only if:

X

α

pπAeα

ðÞ−pπBeα

ðÞ½>qπB

ðÞ−qπA

ðÞ ð48Þ

Inf Syst Front (2016) 18:735–764 749

attitudes towards risk and uncertainty. This can be for-

mulated as follows:

X

j

qπj

¼0ð40Þ

This is called the interference alternation theorem.

Interference quarter law The interference quarter law, intro-

duced and proved in (Yukalov and Sornette 2011; Yukalov

and Sornette 2010b), states that the average or mean absolute

value of interference terms for an average decision maker can

be assigned as below:

qπj

¼qeπj

¼1

4ð41Þ

For a binary prospect situation, this result is due to the

reason that without any prior information, the simplest prior

is to assume a uniform distribution of the uncertainty factors

defined in Eq. 35, so that their expected values are as follows:

φπ

j

¼φeπj

¼1

2ð42Þ

This quantification of the mean values for the interfer-

ence terms is very important for the bootstrapping process

of trust. We can assume a trustor with no prior personal

bias (as most of the average trust decision makers in com-

putational domains are), and use the mean values to as-

sign suitable initial probabilities to different trustee can-

didates in various contexts.

6.5 Selecting a trustee candidate

For selecting the most suitable trustee candidate, we need to

consider the prospect probabilities corresponding to the

existing trustee candidates. Prospect probabilities contain both

the objective and subjective terms from the viewpoint of the

trustor. The interference terms denoted as q(π

i

) play the role of

the subjective part of the prospect probability. The selection

process in the context of trust depends on the objective repu-

tation or direct evaluation feedbacks and the subjective pref-

erences of the trustor. In order to compare the subjective

values of the interference terms, the principle of risk and un-

certainty aversion will be used. This principle is explained

below.

Principle of risk and uncertainty aversion This principle

is defined in (Yukalov and Sornette 2011; Yukalov and

Sornette 2010b; Yukalov and Sornette 2009a; Yukalov

and Sornette 2012; Yukalov and Sornette 2008a), and de-

scribes how the interference terms will change in different

contexts with different amounts of risk and uncertainty

involved. Based on this principle, the following relations

This proposition is important because, it shows that the

preference of one prospect over another is not only dependent

on the objective probabilities but also on the quantum inter-

ference terms.

We use the Eq. 48 in the context of trust decision making in

order to predict the trustor’s behavior in selecting different

trustee candidates. Different outcomes for various scenarios

are shown in Table 2.

6.6 Non-commutativity and relative trust models

It is worth-mentioning that the action ring defined in Eq. 15 is

not commutative with respect to multiplication. Therefore, the

prospects AB and BA are different because p(AB) does not

coincide with p(AB)ingeneral.

In (Yukalov and Sornette 2009a), it is shown that the

following equation holds for the commutativity of two

prospects:

pABðÞ−pBAðÞ¼qABðÞ ð49Þ

where, Ais an arbitrary prospect and Bis a composite one.

Therefore,the order of realizing two prospects is indifferent

when q(AB)=q(BA)=0.

The non-commutativity property is a valuable mathemati-

cal tool in the context of trust modeling. As it was mentioned

in Section 1, relative trust models, take into account the com-

parative context that every evaluation will create for the sub-

sequent evaluations. In other words, evaluating a trustee can-

didate Afirst, and then evaluating the trustee candidate B,in

general will not produce the same result compared to, first

considering Band then A.Thisisincontrasttotheindepen-

dent trust models where every trustee candidate is evaluated in

an isolated context with total disregard to other evaluations

performed for other candidates.

If interference occurs between two prospects, then the order

of performing evaluations becomes important. In the context

of trust, some of the situations in which order effects can occur

are as follows:

1. In evaluating multiple trustee candidates, the evaluation

of the first trustee can influence the evaluation of others.

2. In evaluating a single trustee candidate based on multiple

evidences provided by equally-weighted recommenders,

order effects can occur based on the reception order of the

recommendation values.

7 Evaluation and comparison with the classical

models of trust

In this section, we will present a couple of illustrative exam-

ples plus empirical evaluations to demonstrate the benefits of

the proposed QDT-based computational trust model. We have

tried to show the differences of the proposed model with the

commonly used classical models and illustrate what the pro-

posed model can bring to the computational trust modeling

domain.

7.1 Scenario 1: Basic examples to demonstrate the model’s

behavior

In this scenario, we will present a series of basic examples to

demonstrate the effects of interference terms on trust decision

Tabl e 2 Formulated trustor’s

behavior in QDT Trustor’s behavior Corresponding condition

Prefers trusting Arather than distrusting it. ∑

α

pπAeα

ðÞ½−pπAeα

>qπA

−qπA

ðÞ

Prefers trusting Brather than distrusting it. ∑

α

pπBeα

ðÞ½−pπBeα

>qπB

−qπB

ðÞ

Prefers trusting Aover BScenario1:

∑

α

pπAeα

ðÞ½−pπAeα

>qπA

−qπA

ðÞ

∑

α

pπBeα

ðÞ½−pπBeα

<qπB

−qπB

ðÞ

Scenario2:

∑

α

pπAeα

ðÞ½−pπAeα

>qπA

−qπA

ðÞ

∑

α

pπBeα

ðÞ½−pπBeα

>qπB

−qπB

ðÞ

∑

α

pπAeα

ðÞ½−pπBeα

ðÞ>qπB

ðÞ−qπA

ðÞ

Performs nothing. The trustor is more willing

to search for better trustee candidates.

∑

α

pπAeα

ðÞ½−pπAeα

<qπA

−qπA

ðÞ

∑

α

pπBeα

ðÞ½−pπBeα

<qπB

−qπB

ðÞ

750 Inf Syst Front (2016) 18:735–764

making. In addition, the application of the principle of risk and

uncertainty aversion in assigning quantitative values to the

interference terms and the quarter interference law for

assigning initial crisp values to these terms are investigated.

Example 1 Trust decision making based on the direct expe-

rience and recommendation values

As for the first example, let’s consider the scenario where a

trustor wants to decide whether she/he should trust a trustee

candidate denoted by A. As it is commonly assumed in com-

putational trust models, the trustor is relying on two sources of

information: (1) Direct evaluation in which the trustor assigns

trustworthiness values based on her/his own evaluations in the

performed transactions with the trustee candidates and (2)

Recommendation values received from the recommenders or

other evaluators that had previous experiences with the trustee

candidate. Let’s further assume that the gathered data from

these two information sources are as shown in Table 3.The

zeroes and ones represent distrust and trust evaluation or

recommendation outcomes respectively. Thus, based on the

assumed values, the outcome of the direct evaluation for the

first transaction with the considered trustee candidate is trust

whereas the outcome of the recommendation value for the

same transaction with the trustee candidate is distrust.The

values corresponding to 10 transactions with the trustee can-

didate are given in Table 3.

For this scenario, we have four prospects defined as:

πDirect

AπDirect

AπReputation

AπReputation

Að50Þ

where, the π

A

Direct

prospect represents the situation where the

trustor only evaluates the trustee candidate based on direct

experience with complete disregard to the reputation value

for that candidate. The same argument applies to π

A

Reputation

,

which represents the situation where the trustor only considers

the reputation value.

The corresponding probabilities for these prospects are de-

fined as follows:

pA

Direct

¼pA

DirectArep

þpA

Direct Arep

þqA

Direct

¼fA

Direct

þqA

Direct

pADirect

¼pADirectArep

þpADirectArep

þqADirect

¼fADirect

þqADirect

pA

Reputation

¼pA

repADirect

þpA

repADirect

þqA

rep

ðÞ¼fA

rep

ðÞþqA

rep

ðÞ

pAReputation

¼pArepADirect

þpArepADirect

þqArep

¼fArep

þqArep

ð51Þ

Based on the values given in Table 3, the above probabil-

ities can be computed as below:

pA

Direct

¼0:6þqA

Direct

pADirect

¼0:4þqADirect

pA

Reputation

¼0:4þqA

rep

ðÞ

pAReputation

¼0:6þqArep

ð52Þ

If we look at the above equations, we can infer that the

objective utility part of the equations for p(A

Direct

)and

p(Ā

Reputation

) is equal. This is also true for p(Ā

Direct

)and

p(A

Reputation

). Therefore, quantum interference terms that are

the subjective part of the equations play a key role in deter-

mining the final probability of trusting or distrusting the trust-

ee candidate A. In order to calculate the values of quantum

interference terms and subsequently decide about whether we

should trust or distrust the target trustee, we will use: (1) the

interference alternation theorem, (2) the principle of risk and

uncertainty aversion and (3) the interference quarter law.

From the interference alternation theorem we will have:

qA

Direct

þqADirect

þqA

rep

ðÞþq A rep

¼0ð53Þ

Based on the principle of risk and uncertainty aversion, we

have to decide which situation is more uncertain (or potential-

ly more harmful) compared to other ones. If we consider the

first two prospects, then the uncertainty regarding the situation

of distrusting the trustee candidate without knowing the rep-

utation values (i.e., by only considering the direct experience)

is higher than trusting the target entity in this situation (based

on the principle of risk and uncertainty aversion, a prospect is

more attractive if it leads to a more certain gain or a less certain

loss). Therefore, we have:

qA

Direct

¼−qADirect

>0ð54Þ

With the same kind of reasoning, we will have:

qArep

¼−qA

rep

ðÞ

jj

>0ð55Þ

Tab l e 3 Direct evaluations and recommendations outcomes for

scenario 1

Transaction number →12345678910

Direct 1110101001

Recommendations 0001011010

Inf Syst Front (2016) 18:735–764 751

Therefore, the Eq. 51 canbere-writtenasbelow:

pA

Direct

¼0:6þϑ

pADirect

¼0:4−ϑ

pA

Reputation

¼0:4−γ

pAReputation

¼0:6þγ

ð56Þ

As can be seen from the above equation, the probabilities

of trusting based on direct experience and distrusting based on

reputation value are enhanced whereas the probabilities for

distrusting based on direct experience and trusting based on

reputation value are depressed.

It should be noted that there is an objective probability of

0.6 in favor of trusting Abased on direct experience whereas

the same exact objective probability exist in favor of

distrusting Abased on reputation value. In order to resolve

this problem, we will use the principle of risk and uncertainty

aversion once more.

If we consider two prospects π

A

Direct

and π

Ā

Reputation

,thenitis

evident that the certainty about the direct experience is higher

than the certainty regarding the received recommendations

from unknown recommender entities. In other words, it is

commonly perceived that humans usually trust their own di-

rect evaluation more than what others recommend. Hence, we

will have:

ϑ>γ→

pA

Direct

¼0:6þϑ>pAReputation

¼0:6þγ→

pA

Direct

>pAReputation

ð57Þ

Thus, the probability of trusting based on direct experience

is more enhanced compared to the probability of distrusting

basedonreputationvalue.Therefore,thetrustorwilldecideto

trust the trustee candidate. In order to assign crisp values to the

probabilities, we will use the interference quarter law. Based

on this law, the mean value of 0.25 can be selected for ϑ. Thus,

the value assigned for γwill be:

γ¼0:25−μð58Þ

where, the parameter μ, can be viewed as the risk-taking factor

and can be used to define the trustor’s attitude towards risk and

uncertainty.Example 2 Effect of interference terms on seem-

ingly equal reputation values

As a second demonstration of how attraction factors can

influence the trust decision making process, consider the case

where a trustor wants to select between two trustee candidates

based on the recommendation values in seemingly similar but

different situations.

Let’s assume that the trustor wants to select from the trustee

candidates Aor Bbased on the recommendation values re-

ceived from different evaluators. The total number of recom-

mendations and their values are shown in Table 4.

For this scenario, the prospects considered by the trustor

are as follows:

πReputation

AπReputation

AπReputation

BπReputation

Bð59Þ

The corresponding probabilities for these prospects are:

pA

Reputation

¼pA

RepBrep

þpA

RepBrep

þqA

Rep

¼fA

Rep

þqA

Rep

pAReputation

¼pARepBrep

þpARepBrep

þqARep

¼fARep

þqARep

pB

Reputation

¼pB

repArep

ðÞþpB

repArep

þqB

rep

ðÞ¼fB

rep

ðÞþqB

rep

ðÞ

pBReputation

¼pBrepArep

þpBrepArep

þqBrep

¼fBrep

þqBrep

ð60Þ

Based on the data in Table 4, the above equations can be

written as below:

pA

Reputation

¼0:5þqA

Rep

pAReputation

¼0:5þqARep

pB

Reputation

¼0:5þqB

rep

ðÞ

pBReputation

¼0:5þqBrep

ð61Þ

From the above probabilities, it is evident that the

objective part of all the equations is equal. In order

to determine which trustee candidate the trustor will

select, we use the three mentioned principles of risk

and uncertainty aversion, interference alternation

theorem and interference quarter law. By using the in-

terference alternation theorem, we will have the

following equation:

Tabl e 4 Recommendation values for the existing trustee candidates

Total number of received

recommendations

Number of trust

recommendations

Number of distrust

recommendations

A211

B100 50 50

752 Inf Syst Front (2016) 18:735–764

qA

Rep

þqARep

þqB

rep

ðÞþq B rep

¼0ð62Þ

For this scenario and for both cases, there is a lot of uncer-

tainty involved (i.e., a 50 % probability for being trustworthy

is an extremely uncertain situation). Based on the principle of

risk and uncertainty aversion, people will tend to remain in-

active in the presence of uncertainty. Therefore, distrusting the

target entity for both Aand Bseems more attractive than

trusting them.

Based on this argument we will have:

qARep

¼−qA

Rep

>0

qBRep

¼−qB

Rep

>0ð63Þ

Based on the above equations, we can re-write the Eq. 60,

as below:

pA

Reputation

¼0:5−ϑ

pAReputation

¼0:5þϑ

pB

Reputation

¼0:5−γ

pBReputation

¼0:5þγ

ð64Þ

As can be seen from the above equation, the proba-

bilities to trust the target trustee candidates are de-

pressed while the probabilities to distrust the trustee

candidates are enhanced. Although in this scenario there

is more repulsion than attraction to trust the existing

trustee candidates, if the trustor has no other choices

but to select one of these entities, then we need an

approach for determining which one. The problem in

this scenario is that the objective probabilities to trust

the trustee candidates are both equal to 0.5. Thus, we

need to rely on the subjective attraction factors to de-

termine the winner. This is one of the important differ-

ences between the QDT-based trust formulation and the

classical formulations of trust. Most of the classical

models of trust will not be able to distinguish between

trustee candidates in this scenario as they only consider

the objective reputation value with well-known equa-

tions like below:

Trustreputation ¼trust recommendations

total number of recommendations ð65Þ

The uncertainty involved in the reputation value of A,is

higher than the uncertainty involved in the reputation value

of B. This is due to the extreme lack of evidence for A(i.e.,

only two recommendations are provided where one of them

is in favor of trust and the other one is in favor of distrust).

Therefore, based on the principle of risk and uncertainty

aversion, the attraction factor for q(B

Rep

) should be higher

than q(A

Rep

). Additionally, it should be noted that both the

Fig. 5 The global and local

attraction/repulsion effects

Tabl e 5 First order of evaluation of trustee candidates

Order1 0.6 0.18 0.73 0.43 0.3 0.27 0.75 0.25 0.45 0.76

Relative trustworthiness probabilities 0.6 0.012 0.84 0.31 0.188 0.172 0.846 0.05 0.43 0.822

Inf Syst Front (2016) 18:735–764 753