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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 preferences of the trustor in various contexts with different amounts of uncertainty and risk. The non-commutative nature of quantum 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. Finally, 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.
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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) 18:735764
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) 18:735764
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) 18:735764 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,|k1.
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
:þαk1k1
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
k1
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
αk1
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 observables 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) 18:735764
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
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
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 |ψcollapses to the |0state is
cos
2
θand the probability that |ψcollapses to the |1state is
sin
2
θ. Hence, measurement of |ψin 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 |ψcan 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 UU¼1ð5Þ
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 systems Hamiltonian H, which is the
observable corresponding to the total energy of the system.
This dynamics is determined by the Schrödingersequation
defined as below:
idψ
ji
dt ¼Hψ
ji ð6Þ
where, is the reduced Plancks 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) 18:735764 739
waves of light took in order to get to the detectorssurface.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
Aj
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) 18:735764
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) 18:735764 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 prisoners 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) 18:735764
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) 18:735764 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
þbnCompetence
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 trustors
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 |truststate. 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 |ψon 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) 18:735764
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.
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) 18:735764 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
}:
MiAiμ
 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
|Āwhere,
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 tthe prospect basis is defined as {|A,|Ā}andforthe
intention state |Delegating a task to Bat time tthe 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
:
Men
jifg
¼iMið20Þ
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
A
and
B
respectively, we will have:
Men
jfg
¼iMi¼MAMBð21Þ
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) 18:735764
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 {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

ψstðÞπj
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 Aand I want to delegate my task to Bare
considered. Each one of these intentions have two intention
representations, ItrustA(denoted as A), IdistrustA(de-
noted as Ā)andItrustB(denoted as B)andIdistrustB
(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) 18:735764 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() = 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
trustees 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) 18:735764
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
ðÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pABðÞpAB

qð35Þ
where, φ(π
A
) is called the uncertainty factor and is formulated
as follows:
φπ
A
ðÞcosΔπ
A
ðÞ ð36Þ
where, Δ(π
A
) is the uncertainty angle. The uncertainty angle
can be written as:
Δπ
A
ðÞarg 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
ðÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pA
B

pBðÞpA
B

p B

r
qπA

¼2φπ
A
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
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
ðÞ¼qπB
ðÞ>0ð45Þ
The prospect probabilities can be expressed in a more com-
pact form. For example, the probability of a prospect π
j
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
ðÞpπBeα
ðÞ½>qπB
ðÞqπA
ðÞ ð48Þ
Inf Syst Front (2016) 18:735764 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 trustors 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:
pABðÞpBAðÞ¼qABðÞ 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 models
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 trustors
behavior in QDT Trustors behavior Corresponding condition
Prefers trusting Arather than distrusting it.
α
pπAeα
ðÞ½pπAeα

>qπA

qπA
ðÞ
Prefers trusting Brather than distrusting it.
α
pπBeα
ðÞ½pπBeα

>qπB

qπB
ðÞ
Prefers trusting Aover BScenario1:
α
pπAeα
ðÞ½pπAeα

>qπA

qπA
ðÞ
α
pπBeα
ðÞ½pπBeα

<qπB

qπB
ðÞ
Scenario2:
α
pπAeα
ðÞ½pπAeα

>qπA

qπA
ðÞ
α
pπBeα
ðÞ½pπBeα

>qπB

qπB
ðÞ
α
pπAeα
ðÞ½pπBeα
ðÞ>qπB
ðÞqπA
ðÞ
Performs nothing. The trustor is more willing
to search for better trustee candidates.
α
pπAeα
ðÞ½pπAeα

<qπA

qπA
ðÞ
α
pπBeα
ðÞ½pπBeα

<qπB

qπB
ðÞ
750 Inf Syst Front (2016) 18:735764
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, lets 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. Lets 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

¼qADirect

>0ð54Þ
With the same kind of reasoning, we will have:
qArep

¼qA
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) 18:735764 751
Therefore, the Eq. 51 canbere-writtenasbelow:
pA
Direct

¼0:6þϑ
pADirect

¼0:4ϑ
pA
Reputation

¼0:4γ
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:
γ¼0:25μð58Þ
where, the parameter μ, can be viewed as the risk-taking factor
and can be used to define the trustors 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.
Lets 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) 18:735764
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

¼qA
Rep

>0
qBRep

¼qB
Rep

>0ð63Þ
Based on the above equations, we can re-write the Eq. 60,
as below:
pA
Reputation

¼0:5ϑ
pAReputation

¼0:5þϑ
pB
Reputation

¼0:5γ
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) 18:735764 753