## Abstract

Recent experiments report violations of the classical law of total probability and incompatibility of certain mental representations when humans process and react to information. Evidence shows promise of a more general quantum theory providing a better explanation of the dynamics and structure of real decision-making processes than classical probability theory. Inspired by this, we show how the behavioral choice-probabilities can arise as the unique stationary distribution of quantum stochastic walkers on the classical network defined from Luce’s response probabilities. This work is relevant because (i) we provide a very general framework integrating the positive characteristics of both quantum and classical approaches previously in confrontation, and (ii) we define a cognitive network which can be used to bring other connectivist approaches to decision-making into the quantum stochastic realm. We model the decision-maker as an open system in contact with her surrounding environment, and the time-length of the decision-making process reveals to be also a measure of the process’ degree of interplay between the unitary and irreversible dynamics. Implementing quantum coherence on classical networks may be a door to better integrate human-like reasoning biases in stochastic models for decision-making.

## Introduction

Systematic violations of assumptions in the standard model for rationality were reported
by Tversky and Kahneman^{1} with great impact^{2,3}. Numerous
empirical findings in cognitive psychology and behavioral sciences exhibit anomalies
with respect to the classical benchmark defined by the Kolmogorovian probability axioms
and the rules of Boolean logic, suggesting a whole new scope of research: the
development of models for decision-making based on the quantum formulation of
probability theory^{4,5,6,7,8}. The similarity between the
epistemological content of the quantum theory and the situation in philosophy and
psychology, where there is a vague separation between the subject and the object under
study, was already at the heart of the seminal debate on the notion of
complementarity^{9,10,11}. Of course, we are not proposing any quantum
physical implication for brain physiology, which remains as a controversial debate^{12,13}.

We are interested in the potential of using the quantum formulation of probability theory
into modelling decision-making schemes capable of integrating those behavioral aspects
which the classical and normative paradigm of rationality needs to describe as
mistakes^{14}. The use of certain mathematical tools of quantum
probability theory for the description of macroscopic systems usually thought as
‘just classical’ is a growing field of research^{15}. In
particular, and concerning the context of decision-making, beim Graben and
Atmanspacher^{16} show that quantum statistics can arise from neural
systems, even though the original model is not quantum but classical.

When applied to modelling cognition, the quantum-based model makes a universal,
non-parametric prediction for the presence of order effects in attitude judgments which
have been observed, for example, in large-scale American representative surveys^{17}. Recently, Yearsley and Pothos^{18} defined a test for
violations of the Leggett-Garg inequalities (temporal-Bell) in order to falsify the
so-called ‘cognitive realism’ hypothesis. As the authors explain,
“the observation of such a violation would indicate a failure of the top-down
approach to cognition, in a classical, realist way”. Their standpoint is closely
related to the proposal by Atmanspacher and Filk^{19} for the study of
bistable perception. Kvam *et al.*^{20} demonstrate how deliberation
modelled as a constructive process for evidence accumulation is better described with a
quantum model than with the popular classical random walk.

Within the domain of economics and game theory, the link between Bell’s notion of
nonlocality and Harsanyi’s theory of games with incomplete information has been
object of novel attention^{21}, as well as the comparison between the
effects of classical and quantum signalling in games^{22,23,24}.
Concerning the crucial Aumann’s agreement theorem^{25}, Khrennikov
and Basieva^{26} show how agents using a quantum probability system for
decision-making can indeed agree to disagree even if they have common priors, and their
posteriors for a given event are common knowledge. In addition, Lambert-Mogiliansky
*et al.*^{27} show how violations of transitivity of preferences in
observed choices emerge naturally when dealing with non-classical agents, in line with
the works by Makowski *et al.*^{28,29} who analyze how an agent
achieves the optimal outcome through a sequence of intransitive choices in a
quantum-like context.

As a consequence of the nature of the cognitive processes being better explained from the
quantum probabilistic (or logic) viewpoint, Busemeyer *et al.*^{30,31}
propose a quantum dynamic model of decision-making, as opposed to the Markovian settings
previously established. Asano *et al.*^{32,33} elaborate deeper on the
representability of these effects by understanding the decision-maker as a quantum open
system, with the dynamics of the global system driven by the quantum analogy of the
master equation.

Inspired by these latter developments, we propose a way to reconcile the novel
application of quantum techniques with the classical origin of the problem of
understanding human decision-making. Once that we accept the need for a non-classical
extension of the standard models for decision-making, this paper addresses the question
of *“how we can model the deliberation process generating the behavioral
probabilities in a quantum manner”*.

We show how a quantum stochastic evolution of the relevant decision-making variables can be defined in terms of a linear superoperator deeply rooted in two fundamental elements of classical decision theory: (i) the ability of the decision-maker to discriminate between the available options, and (ii) the process of formation of beliefs in situations of uncertainty. Besides, the relaxation time reveals to be driven by the level of interpolation between the purely quantum and the purely classical random walk, in addition to a tradeoff with the relative weight the decision-maker assigns between the comparison of alternatives’ profitabilities and the formation of expectations on the possible states of the world.

The paper is organized as follows. We first define the evolution of the cognitive state as a case of quantum stochastic walks, and show how the dynamics of the walker can be represented by a network. We obtain the cognitive network as a natural extension of the well established decision-making trees, relying on the classical probabilistic choice model. Finally, we illustrate the class of quantum stochastic walks on networks for decision-making with the famous example of the Prisoner’s Dilemma game, a task which implies situations of strategic uncertainty. We provide a Methods section with rigorous discussion on the mathematical properties of the model.

## Results

### The cognitive state evolving as a quantum open system

We describe the *cognitive state* of the agent in a Hilbert space
, and we denote a state by
|*ψ*〉. Let the state be definite, then the
Schrödinger equation
*d*|*ψ*(*t*)〉/*dt* = −*iH*|*ψ*(*t*)〉
formalizes its time evolution if the system is isolated, where *H* is the
Hamiltonian (a Hermitian operator acting on ) and
*i*^{2} = −1. Nevertheless, in a
general case we do not know the state, so the system has to be described by a
mixed state or density matrix *ρ*. This *ρ* is a
statistical mixture of pure states and formally it is a Hermitian, non-negative
operator, whose trace is equal to one. The natural extension of the
Schrödinger equation to density matrices is the von Neumann equation

with [*H*, *ρ*] the commutator
*H**ρ* − *ρ**H*.

Furthermore, we consider that the best description for the ‘mind’
of an agent involved in a decision-making process is not as an isolated system,
but one subject to *some* interaction with the environment. Therefore, its
evolution is not given by the simple von Neumann equation.

Let our system of interest be a composite of two constituents: *M* and
*E*, mind and environment. Due to the whole system (mind *and*
environment) being isolated, *ρ*_{M+E} evolves
according to the dynamics given in Eq. (1) by definition.
From this, we can focus specifically on the state of *M* if we take partial
trace over the Hilbert space of *E* such that the subsystem of the mind is
*ρ*_{M} = Tr_{E}(*ρ*_{M+E})^{34}. Henceforth we drop the subindex *M* when referring to the
state of the mind.

In order to know the equation of motion of *ρ*, we should take
partial trace in Eq. (1), which is generally impossible.
However, under the assumption of Markovianity (the evolution can be factorized as
given a sequence of instants *t*_{0}, *t*_{1},
*t*_{2}) one can find the most general form of this time
evolution based on a time local master equation ,
with a differential superoperator (it acts over
operators) called Lindbladian, embedding the standard form in which any
Markovian master equation can be considered, and given by the
Lindblad-Kossakowski equation^{35} such that

Here, *H* is the Hamiltonian of the subsystem of interest (in our case, the
‘mind’ of the decision-maker), the matrix with elements
*γ*_{(m, n)} is positive semidefinite,
*L*_{(m, n)} is a set of linear operators, and
denotes the anticommutator . The meaning of the subindices (*m*, *n*)
shall become clear later.

The second part in the master equation Eq. (2) contains the
dissipative term responsible for the irreversibility in the decision-making
process, weighted by the coefficient *α* such that the parameter
*α* ∈ [0, 1] interpolates between the
von Neumann evolution (*α* = 0) and the completely
dissipative dynamics (*α* = 1). The section Methods
covers the basics required to reach this formulation. See also Fig. 1 for an axiomatic construction of the quantum stochastic
walks.

### From the tree to the network

A usual feature of many standard models for the analysis of decision-making
problems is their representation as a graph with characteristics of a directed
tree. We can understand such models as a root node 0 connected to each possible
state of the world Ω ∈ *W* that the
decision-maker can face. For each of this possible states, there is a set of
weighted edges linking each state of the world Ω to the actions
*i* ∈ *S* that the agent can take. These
are nested models implying a sequential structure in the cognitive process: the
agent is supposed to first form her (possibly own) beliefs about the
(distribution of) states of the world and then optimize her action choice as a
response to this information.

Our work departs from this standard setting and proposes a model where a richer
networked structure of the decision-making mechanism represents an incessant
flow of the agent’s response-probabilities conditioned on the topology
of the problem. We propose that the decision-making process is a combination of
the comparison of utilities taking place *simultaneously* with the
elicitacion of beliefs and therefore removing the nested structure. The process
extends over an interval of time and due to the dissipative dynamics we compute
the unique stationary distribution of random walkers defining the behavioral
choice-probabilities.

An appropriate definition of the so-called dissipators–operators
*L*_{(m, n)} in Eq.
(2)–allows the quantum formalism to also contain any classical
random walk. The possible moves that the walker can make from each node can be
described by a network, such that each node represents observable states of the
system, and the edges account for the allowed transitions. This in turn relates
the transition matrix defining the dynamics of the stochastic process to the
structure of an underlying network. We prove in the Methods section how the
dissipators lead to a unique stationary solution if they are defined as
*L*_{(m,
n)} = |*m*〉〈*n|*,
with *γ*_{(m,
n)} = *c*_{mn} being
*c*_{mn} the entries of a *cognitive matrix
C*(*λ*, *φ*) formalized as the linear
combination of two matrices, Π(*λ*) and *B*,
associated to the profitability comparison between alternatives and the
formation of beliefs, respectively. The parameters *λ* and
*φ* become meaningful in the next section, together with the
definition of *C*(*λ*, *φ*) in Eq. (4).

### The cognitive matrix

As the starting point for defining the *cognitive matrix* in our model, we
consider one of the most basic yet meaningful formulations of probabilistic
choice theory: Luce’s choice axiom^{37,38,39}. In this
framework, given a choice-set *S* containing the available alternatives,
the system of choice probabilities is defined by ,
for every *i* ∈ *S*, with
*w*_{i} being a scalar measure of some salient
properties of the alternatives: a *weight* of each element within the set
of available options.

A natural parametrization for the salience of each alternative
*i* ∈ *S* is to define
*w*_{i} = *u*(*i*|Ω)^{λ},
where *u*(*i*|Ω) relates to the payoff the
decision-maker obtains from taking action *i* if the state of the world is
Ω. Because the terms *u*(·|·) have to be
non-negative, situations with negative payoffs can be included after a monotonic
transformation, the standard procedure in discrete choice theory. The exponent
*λ* ∈ [0, ∞) measures the
agent’s ability to discriminate the profitability among the different
options. When *λ* = 0, each element
*i* ∈ *S* has the same probability of
being chosen (1/*N*_{S} with *N*_{S} the
cardinality of the set *S*), and when
*λ* → ∞ only the dominant
alternative is chosen. If there is more than one option with the same maximum
valuation, then the probability of an option being chosen is uniform within the
restricted subset of the most preferred ones.

We now build the aforementioned matrix Π(*λ*) relying on the
response probabilities *p*_{S}(*i*) already defined.
Let the connected components of the graph be in a bijection with the set of
states of the world *W* such that each
Ω ∈ *W* is related to one and only one
connected component. The number of nodes *K*_{Ω} in the
connected component associated to each possible state of the world Ω is
the size of the corresponding action set. Let
*n*_{i}(Ω) be the node representing the event of the
decision-maker taking action *i* when considering the state of the world is
Ω. Then, every node *n*_{i}(Ω) has
*K*_{Ω} incoming flows of walkers, one from each of the
other nodes *n*_{j}(Ω)
(*j* ≠ *i*) and one self-edge. These links
are edges weighted in the spirit of Luce’s choice axiom,

Note that every node *n*_{i}(Ω) has
*K*_{Ω} outgoing edges
*e*_{Ω}(*i*, *j*), generally with
*K*_{Ω} different weights
*p*_{Ω}(*j*). See Fig. 2 for
a graphical example deriving the matrix Π(*λ*) from the
sequential tree.

We can define Π(*λ*) as a transition matrix where every
entry *π*_{ij}(Ω) is the probability that a
random walker switches from action *i* to *j* for a given state of the
world Ω. The navigation of random walkers along the network described by
Π(*λ*) accounts for the comparison between alternatives
for each given state of the world.

The decision-maker faces simultaneously another cognitive activity: the formation
of her beliefs about the state of the world (either a forecast on some external
random event, or a prediction on the behavior of an interacting agent). We model
this process through the definition of the matrix *B* such that its entries
connect nodes of the form
*a*_{i}(Ω_{k}) to those of the
form *a*_{i}(Ω_{l}). Thus, *B*
allows the walker to introduce a *change of belief* about the state of the
world in the cognitive process by jumping from one connected component
associated to a particular state of the world
Ω_{k} ∈ *W* to the
connected component associated to another one
Ω_{l} ∈ *W*, while
keeping the action *i* fixed.

We denote the cognitive matrix by *C*(*λ*, *φ*),
which is defined as

where *φ* ∈ (0, 1) is a parameter assessing
the relevance of the formation of beliefs during the decision-making process.
The superscript ^{T} denotes the transpose matrix. We discuss
the reason for obtaining *C*(*λ*, *φ*) after the
transposition of the transition matrix in the Methods section. Combining
Π(*λ*) with *B* is crucial for the dynamics of the
process: *B* establishes connections between the
*N*_{W} (originally disjoint) connected components
described by Π(*λ*). Therefore, *C*(*λ*,
*φ*) describes a weighted (and oriented) graph with only one
connected component which contains now *N*_{W} strongly
connected components, one per each possible state of the world.

Typically, we may consider risky or uncertain situations to be objective if a
random move has to be realized (lotteries), subjective if the agent has to
evaluate probabilities based on her own judgment, or strategic if there is a
game-theoretic interaction with hidden or simultaneous move of the opponents. As
a consequence, there is a certain degree of arbitrariness in the way we can
define the entries {*b*_{kl}} for the matrix of belief
formation, as long as its linear combination with Π(*λ*)
guarantees existence and uniqueness of the stationary distribution
*ρ**. This is satisfied when the cognitive matrix fulfills the
Perron-Frobenius theorem, *i.e.*, *C*(*λ*,
*φ*) is irreducible and aperiodic^{40}.

In the following part of the paper, we propose a definition for the matrix
*B* in line with the standard models for strategic decision-making.
Nevertheless, even if there is no particular information given by the problem,
one can always define
*b*_{kl}(*i*) = 1/(*N*_{W} − 1),
with *N*_{W} being the cardinality of the set *W*. This
‘homogenous’ law for the change of beliefs is reminiscent of the
long-distance hopping matrix which has been fruitfully exploited in the study of
ranking problems through quantum navigation of networks^{41,42}.

### Analyzing the Prisoner’s Dilemma

The Prisoner’s Dilemma is widely considered to be the cognitive and game-theoretical task equivalent to the harmonic oscillator in physics: a well-defined problem with quite a simple formulation but still rich possibilities for both experimental and theoretical exploration, which qualifies this problem as the first system for which a new model should provide consistent explanation as a benchmark case study.

The symmetric Prisoner’s Dilemma is a game involving two players,
*A* and *B*. They can choose among two actions: cooperate
(*C*) or defect (*D*). Considering the game in its normal form, it
is defined by the following payoff matrix,

where
*d* > *a* > *c* > *b*
implies mutual cooperation is the Pareto optimal situation (maximum social
payoff). In standard game theory, defection is the dominant strategy for both
players, so mutual defection is the Nash equilibrium of this game (strategy
profile stable against unilateral deviations), which is not an efficient outcome
when compared against mutual cooperation. The rational prediction for the play
of this game is the choice of defection as action, together with the expectation
(belief) of facing also defection from the opponent, even though a fraction of
cooperation usually appears when humans play this game^{43}.

On the one hand, deviations from the purely rational (Nash) equilibrium in games
can actually be modelled with classical probability theory if we consider
stochastic choice-making with a finite value of the ‘rationality
exponent’ analogous to the parameter *λ* already defined in
Eq. (3)–see, *e.g.*, the concept of
quantal response equilibrium^{44}–but on the other hand,
deeper empirical findings challenge the validity of the axioms of classical
probability theory at their fundamental level: experiments with the
Prisoner’s Dilemma game can also be used to show how the Sure Thing
Principle (a direct consequence from the law of total probability in the
classical framework) is violated^{45}.

These two effects together lead Pothos *et al.*^{46} to
formulate a quantum walk outperforming the predictions from the classical model,
concluding that “human cognition can and should be modelled within a
probabilistic framework, but classical probability theory is too restrictive to
fully describe human cognition”. Their model incorporates a unitary
evolution (QW) originated from a Hamiltonian operator implementing cognitive
dissonances. Nevertheless, the unitary evolution lacks stationary solutions
unless a stopping time is exogenously incorporated into the process, which
raises also a fundamental concern about how to apply the class of models using
QWs both from a conceptual and a practical standpoint^{47}. We show
later how the present model of QSWs accounts for these violations of the Sure
Thing Principle in a natural way and that it has a stationary solution.

Thus, it is our intention to show how the more general formulation of quantum
stochastic walks (QSWs) interpolating between both extreme cases (CRW with
*α* = 1, and QW with
*α* = 0) is able to incorporate the positive
aspects of both models such as the relaxing dynamics towards a well-defined
stationary solution from the classical perspective, together with the
possibility for coupling and interference between populations and coherences
through the unitary evolution. Besides, we respect the standard usage of the
parameter *λ* as an upper-bound in the optimality of the solution,
while our networked definition of the problem introduces new effects which are
not reachable with the traditional representations of decision-making trees that
do not allow for simultaneous exploration through both spaces of preferences and
beliefs in parallel.

### Fine tuning the model

To illustrate the predictions of this class of models, we consider the payoff
matrix used in the experiments conducted by Shafir and Tversky^{45},
*a* = 75, *b* = 25,
*c* = 30, and *d* = 85. From the point
of view of any of the two players (this game is symmetric), the following
identification for the action set and the possible states of the world is
straightforward: *u*(*C*|*C*) = *a*,
*u*(*C*|*D*) = *b*,
*u*(*D*|*C*) = *d*,
*u*(*D*|*D*) = *c*. We have a
four-dimensional space of states because two
possible actions are associated to two possible states of the world. We choose
the basis of the system spanning the space of states to be . As an example, these pure states are defined such
that indicates the cognitive state in which the
player chooses to execute the action *C* and holds the expectation of the
opponent choosing *D*. The same definition holds for the other
combinations. Following Eq. (3) and the definition of
Π(*λ*) discussed in the introduction of the model, we
write

with , and . Then,
for a given set of payoff values, the weights for the dynamics in the
decision-making process are specific to the type of player given by the
rationality parameter *λ*_{a}, where the index
*a* just indicates that the parameter *λ* is
representative of the player comparing the actions. See panel (a) in Fig. 3 for its graphical representation as a network.

We are dealing with a situation of strategic uncertainty, so we define the matrix
of formation of beliefs to be dependent on the payoff entries corresponding to
the opponent, analogous to the definition of Π. In this case, the
frequency with which a stochastic walker will jump from the belief associated to
the state of the world *C* to the state of the world *D* is directly
the estimation of how the other player is weighting her action *C* versus
her action *D* during her deliberation on profitabilities, and then we
write

In a first step, *B* gets defined as a function of the rationality exponent
associated to the second player whose action-set defines the set of possible
states of the world faced by the first player. Because the game of the
Prisoner’s Dilemma is symmetric, we can assume a common value
*λ*_{a} = *λ*_{b} = *λ*
which simplifies the model.

Combining these two connectivity patterns (do not forget the operation of matrix
transposition) in the linear fashion defined in Eq. (4)
finally determines the *cognitive matrix C*(*λ*,
*φ*) given in Eq. (8),

and modelling human behavior in Prisoner’s Dilemma games together with the Hamiltonian

We use a simplified construction *H*_{ij} = 1
if the nodes are connected in Π(*λ*) and zero otherwise, in
agreement with other applications of quantum rankings successfully explored in
the literature about complex networks^{42}. See a more thorough
discussion on possible definitions of *H* in the corresponding section in
Methods.

### Behavioral aspects

The cognitive matrix *C*(*λ*, *φ*) in Eq. (8) and the Hamiltonian operator in Eq.
(9) define a three-parametric family of Lindbladian operators
for the play of Prisoner’s
Dilemma games. Since there exists a unique stationary solution *ρ**
for each evolution defined by the values of (*α*, *λ*,
*φ*) as we prove in the Methods section, then we have defined a
whole family of behaviors in the game which we analyze now as a function of the
parameters.

*λ* as a measure of bounded rationality

We should consider the level of rationality in the play in comparison to the
Nash equilibrium (*DD*), the reference of a rational outcome in the
game. When we introduce *λ* in the model in Eq.
(3), this parameter is a monotonic measure of the ability to
discriminate between the profitability of the different options, and as such
it has also a strictly monotonic influence on the level of rationality in
the equilibrium predictions: the higher the *λ*, *cet.
par.*, the higher is the probability of choosing the dominant action.
See Fig. 3-Panel (b).

In our model, the parameter *λ* really plays the role of
upper-*binding* the level of rationality in the process and not
just a point-prediction. It determines the maximum probability of playing
defection, while for a given *λ* different probability outcomes
are achieved depending on the tradeoff between *α* and
*φ*. See in Figs 3-Panel (c) and
4-Panel (a) how the weight on the belief formation
in the dynamical process shapes the smoothness/steepness of the transition
from pure randomization to the bounded level of rationality as a function of
*α*, with the behavior getting closer to the allowed
maximum when the process becomes more classical
(*α* → 1).

We see in Fig. 4-Panel (b) how the finite limit in the
level of rationality *λ* translates into a one-to-one
correspondence with the expectation on the level of defection (black solid
line), which remains basically constant and independent of the values of
*α* and *φ*. Therefore, experimental results
on belief elicitation can be used to adjust the numerical value of
*λ*.

#### Believing the same to act different

This model based on the connected topology for the dynamical process
combining simultaneously the formation of beliefs and the comparison of
actions reveals an interesting effect: even for fixed values of
*λ* (and then also fixed expectation on the rival’s
move), it is possible to obtain different choice probabilities as a result
of the different weights assigned to each of the two cognitive processes
through *φ*.

We see in Fig. 4-Panel (b) how the probability of
choosing defection as action (orange solid line) is decreasing on
*φ* (for each possible value of *α*). This
effect is very intuitive since higher *φ* implies less focusing
on the discrimination between the profitability of own actions of the
player. The dynamical process of decision-making incorporates this effect as
a consequence of higher values of *λ* generating lower weights
in the connections of the cognitive network *C*(*λ*,
*φ*) inherited from the matrix Π(*λ*).
This effect is hardly obtainable with standard models based on
decision-making trees, since their sequential structure does not allow for
the interaction between nodes belonging to different states of the
world.

#### Relaxation time

By definition, *α* is the parameter interpolating between the
unitary evolution (*α* = 0) which is a process
of continuous oscillation without stationary solution, and the Markovian
evolution (*α* = 1) which is dissipative and
has a stationary solution. Thus, *α* is expected to play an
important role in the determination of the relaxation time of our networked
quantum stochastic model for decision-making. We denote it by
*τ*, and we discuss its definition in the Methods section.
As a cautious remark, let us say that the relaxation time of the dynamics
may not always be the endogenously determined decision time of a subject on
a given trial. The decision time will likely be a random variable with a
distribution of stopping times. For further elaboration on this issue, see
*e.g.*, Busemeyer *et al.*^{30} and Fuss and
Navarro^{48}.

Regarding the two cognitive parameters *λ* and *φ*,
we observe no influence of *λ* on the magnitude of the
relaxation time. We see in Fig. 4-Panel (c) how
*τ* depends on *φ* with a clear minimum
*τ*_{Min}. The curve asymptotically diverges for
*φ* → 0, while it remains finite
for *φ* > 0 if
*α* ∈ (0, 1], unless
*α* = 1 and
*φ* = 1, when *τ* diverges as
well. As *α* approaches 1, *τ* vs.
*φ* becomes very large for high values of *φ*,
resulting in a U-shaped curve (inset of Fig. 4-Panel
c). This comes from the tradeoff in the dynamics between the
cognitive matrix *C*(*λ*, *φ*) and our choice
of the Hamiltonian (Eq. 9). As the reader can see in
the Methods sections, existence of the stationary solution requires the
network to be connected such that no node is isolated, and the cognitive
network represented by the matrix *C*(*λ*,
*φ*) becomes disjoint if
*φ* = 0, when there is no transition allowed
between the components associated to the two states of the world. Thus, we
can say that the presence of deliberation about the possible states of the
world is crucial for the existence of a stationary solution, and therefore
the process of construction of the belief is a key aspect in the convergence
towards a stationary state.

In Fig. 4-Panel (d) we analyze
*τ*_{Min} and *φ*_{Min} as a
function of *α*. Considering *τ* as a function
*τ*(*α*, *φ*), we implicitly
define *φ*_{Min}(*α*) as the value of
*φ* for which *τ* is minimum, for each
possible *α*. This figure clearly shows how the relaxation time
remains finite for non-zero values of *α*, and decreasing the
higher is the influence of the Markovian aspect of the dynamics. The abrupt
step we observe in the relationship between *φ*_{Min}
and the values of the parameter *α* is due to the breakdown of
degeneracies in the spectrum of the Lindbladian superoperator at that point.
Note that for a fully classical case , which
would imply a fully homogeneous combination of
Π(*λ*_{b}) and
*B*(*λ*_{b}), in agreement with the
symmetry of the Prisoner’s Dilemma problem and the choice of
parametrization
*λ*_{a} = *λ*_{b} = *λ*.

#### Stationary solution

We prove the existence and uniqueness of the stationary solution for our
class of quantum stochastic walks for decision-making in the Methods
section. Moreover, the solution is analytically defined and can be computed
by exact diagonalization of the Lindbladian superoperator without the need
for numerical simulations. Nevertheless, and only for illustrative purposes,
we show the explicit evolution of the component *P*_{DD}
for several initial conditions in Fig. 4-Panel (e),
and the joint evolution of the four components starting from one extreme
initial condition of full cooperation in Fig. 4-Panel
(f).

Finally, let us emphasize the connection between the three parameters of the
model and their observable counterparts. The bounded rationality parameter
*λ* can be related to the result of the formation of
beliefs about the opponent’s move. For a given *λ*, the
choice probabilities in the space of actions and the relaxation time of the
dynamical process are both governed by the pair (*α*,
*φ*). Thus, we provide a model with three parameters to be
estimated by the appropriate measurement of three observables. The presence
of a distribution in stopping times can be proxied through a distribution of
values for *τ* as a consequence of having a population of
players with heterogeneous values of the parameters, such as different
weights in the formation of expectations.

### Violation of the Sure Thing Principle

In the spirit of Pothos and Busemeyer^{46}, we turn now to the
application of this quantum model in explaining the so-called violations of the
‘Sure Thing’ Principle often observed in human behavior. This
principle dates back to the work by Savage^{49} and can be
understood as follows. Let a decision-maker decide between two options (*A*
or *B*) when the actual state of the world (may it be the choice of an
opponent, an objective lottery, or any other setting with uncertainty) is
unknown, but the decision-maker knows that it can be either *X* or
*Y*. Then, as a consequence of the (classical) law of total probability
applied to modelling human behavior, if a decision-maker prefers *A* over
*B* if the state of the world was known to be *X* and also prefers
*A* over *B* if the state was known to be *Y*, she should
also choose *A* when the state of the world is uknown because *A* is
superior to *B* for every expectation on the realization of
*X*/*Y*. Nevertheless, this principle was already refuted in an
experiment by Tversky and Shafir^{50}, an observation which has been
regularly reproduced afterwards.

Busemeyer *et al.*^{51} and Pothos and Busemeyer^{46} provide a further review of empirical evidence on this issue and also show
how quantum-inspired models can account for this effect, outperforming the
classical ones. They explicitly compare models based on unitary evolution of the
decision-making probabilities versus their Markovian counterparts. Despite of
the (qualitative and quantitative) success of these quantum-like models, they
are subject to the already mentioned criticism of lacking stationary solutions
defined endogenously. We want to briefly show here how the stationary states of
the quantum stochastic walks that we have defined in this paper can model the
violations of the Savage’s Principle in a parsimonious manner.
Furthermore, this effect is available only if the model is not restricted to its
classical part (*α* = 1) but applied in its general
way (0 < *α* < 1),
emphasizing the synergies from combining both the quantum and the classical term
in this dynamics.

In order to make our case, we reproduce the experimental results in Busemeyer
*et al.*^{51}. The entries of the payoff matrix are
*a* = 20, *b* = 5,
*c* = 10, and *d* = 25. Their results
show a defection rate of 91% when the subjects know their opponent will defect,
and of 84% when they know the rival’s action is to cooperate. The Sure
Thing Principle is violated in this experiment because the defection rate when
the choice of the opponents is unknown drops to 66%. See model fit to this data
in Fig. 5-Panel (a).

First, we consider the two defection rates when the state of the world is known,
and use them to obtain the best fit of the model under the constraint
*φ* = 0, because in these two situations the
decision-maker does not need to allocate any effort to build an expectation
about the rival’s move since it is fixed by default. The dynamics are
solved numerically, and we choose the density matrices with diagonal elements
and as
initial points for the two scenarios (the rival defects or cooperates) such that
the system is confined to the subspace of each announced state of the world. We
obtain the best fit for the parameter values
*λ* = 10.495 and
*α* = 0.812, yielding predictions of 0.911 and
0.839 for the two defection rates in the sure situations.

Second, we take these values for (*α*, *λ*) as fixed
and study the impact of introducing uncertainty in the decision-making process.
This is modelled by the parameter *φ* > 0, which
means that the decision-maker has to assign some effort to the ‘guessing
task’. We see in Panel (a) that the quantum stochastic walk naturally
includes violations of the Sure Thing Principle in this setting when the weight
of the matrix *B* in the dynamics becomes more relevant. We obtain
as the critical value of *φ*
for which the predicted outcome for this experiment lies below the defection
rate of 84%, and the value
*φ*_{exp} = 0.898 models the experimental
result of only 66% of defection in the uncertain situation. Finally, Fig. 5-Panel (b) illustrates how this effect is not
available when only the classical term is considered (by fixing
*α* = 1). It is straightforward to see how in such
a classical case, the prediction (for any value of *λ*) is
independent of the parameter *φ*. One can understand this by
noticing that several type of transitions are not present in the dynamics when
only the CRW applies (see Fig. 1-Panel (b) once
again).

## Discussion

Understanding how us humans process the information that we retrieve from our
environment and how this affects our ability to make decisions is of major relevance
in the analysis of individuals’ behavior under circumstances of risk and
uncertainty, and we consider that the interplay between quantum and classical random
walks may be a promising attempt to incorporate human-like reasoning biases in the
formulation of stochastic decision-making dynamics. Furthemore, the quantum nature
of this algorithm does not imply any quantum functioning of the physical substrate
(the *brain*) in which the decision-making process is embedded at all, in the
same way that quantum navigation of networks for ranking their nodes does not
require for any quantum hardware and outperforms the classical ranking
techniques^{42}.

We have proposed a new way to model the deliberation process undergoing any decision-making mechanism via the navigation of small cognitive networks with quantum walkers. Our class of models can extend the dynamic-stochastic theory of decision-making to the quantum domain, incorporating coherences in a random walk which occurs along an otherwise classical set of nodes. This hybrid dynamics defines a unique and stationary distribution for the stochastic behavior. In our illustrative example, we build the cognitive network to perform only the two cognitive operations required in the Prisoner’s Dilemma game: the comparison between the payoffs of one’s own actions, and the estimation of other players’ moves. Of course, the definition is able to contain the weighted combination of any finite number of tasks, with the linear coefficients representing the decision-maker’s allocation of relative efforts.

The application of these quantum stochastic walks on networks for decision-making
shares the building blocks of the renowned decision field theory^{52},
already formulated as a connectivist model^{53}. We consider this is a
promising avenue of research in order to bring the successful stochastic-dynamic
family of cognitive models into the quantum domain. This generalization is a natural
step given the latest evidence available^{54}, especially the experiment
by Busemeyer *et al.*^{55}, designed to prove wrong the belief that
quantum models fit better just because they are more complex.

## Methods

### Hilbert space and density matrix

For finite dimensional systems, a Hilbert space is
simply a linear space endowed with a scalar product . Its elements (or states) are denoted by . We consider only states with non-vanishing norm . If the state of the system is we say it is in a *pure state*.

The projector , an operator acting on as , has a bijective
relation with |*ψ*〉, so we can describe the state
|*ψ*〉 in terms of
*P*_{ψ}.

A density matrix *ρ* is an operator acting on , with the following
properties: (*i*) it is Hermitian:
*ρ*^{†} = *ρ*,
(*ii*) it has trace one: Tr(*ρ*) = 1, and
(*iii*) it is positive semi-definite:
.

Any *ρ* can be written as , with
. Notice that if
*p*_{1} = 1 and
*p*_{n} = 0 for
*n* > 1, then , so
*ρ* describes the pure state
|*ψ*_{1}〉. In general
*p*_{n} > 0 ∀*n* and
*ρ* is not a projector. In such a case, *ρ*
describes a situation where we have some uncertainty about the state of the
system with the probability of the system being in
|*ψ*_{n}〉 given by
*p*_{n}, and we say the system is in a *mixed
state*.

### Relationship between quantum and classical random walks

A comprehensive approach to quantum stochastic dynamics (QSWs) can be achieved by considering the classical random walk in discrete time as the basic setup, grounding the path towards the more sophisticated quantum formulation. Here, we first define the CRW in discrete time, and later extend its formulation naturally to the continuous time domain. Second, we introduce the QW directly in continuous time and bring both of them together, supporting the formulation of Eq. (2) already stated.

Let us consider a classical random walk in its discrete time version for which
there is a certain set of *N* possible states of the system. At each time
step *t*, the system may transit to state *i* from state *j*
according to the relations defined in a
*N* × *N* transition matrix
*T* = {*T*_{ij}}. The state of the
system is described by a vector
*p*(*t*) ∈ Δ^{Ν}
(Δ^{Ν} is the *N*-dimensional simplex, such
that every component
*p*_{i}(*t*) ≥ 0 and
, ∀*t*). If the state of the
system at time *t* − 1 is
*p*(*t* − 1), the state of the system at the
following time step *t* is and then, if the
initial condition is *p*(0), the state of the system after *t* steps
is given by
*p*(*t*) = *T*^{t}*p*(0),
where *T*^{t} is the *t*-th power of the transition
matrix.

A standard microfoundation for this process comes when picturing the system
evolution as the evolution of the distribution of a random walker hopping along
a network composed of *N* nodes (one per each possible state) and defined
such that its connectivity pattern
*A* = {*a*_{ij}} generates the
dynamics in *T*. The *i*-th component of the state-vector
*p*(*t*) accounts for the probability of the walker being found in
node *i* at time *t*. Given two consecutive instants
(*t* − 1, *t*), the distribution changes
according to . The edges of the network are
defined such that *a*_{ij} represents a link from node
*i* to node *j* (out-flow orientation), so it is straightforward
to observe how the transition matrix *T* (which denotes transitions in the
in-flow orientation) is related to the connectivity pattern *A* through the
operation of matrix transposition (and a possible operation of connectivity
normalization to preserve the total probability equal to one via the out-degree
of the nodes if the network is not weighted a
priori). Because our cognitive matrix *C*(*λ*,
*φ*) directly determines the stochastic evolution, the reader
can now see why *T* = *C*(*λ*,
*φ*) is defined from the transposed matrices
Π^{T} and *B*^{T} in
Eq. (4). These dynamics (and their continuous
counterpart) tends to a unique stationary solution if the network is
connected.

Provided the understanding of the discrete time transition process, the
continuous time CRW reads as , where , and is the
*N* × *N* identity matrix.

We now turn to the quantum case and introduce the
*N* × *N* quantum density operator
*ρ* playing the role analogous to the state-vector *p* in
the classical case. The relationship between quantum and classical random walks
is made through the occupation probabilities defined such that
*ρ*_{ii} = 〈*i*|*ρ*|*i*〉 = *p*_{i}.
From the discussion above, it follows that the Markovian master equation can be
written as (*δ*_{ij}
is the Kronecker symbol such that
*δ*_{ij} = 1 only if
*i* = *j*, and zero otherwise), and it can be
shown how this walk can be quantized identifying
*M*_{ij} = 〈*i*|*H*|*j*〉,
with *H* being an Hermitian operator (the Hamiltonian) ensuring that
*M* is a real matrix. *M* can be asymmetric for classical models,
and in this more general case, asymmetries can be incorporated via the Lindblad
operators. This approach takes the Schrödinger evolution as the building
block, and depending on certain properties of the system the task of
classical-to-quantum identification might not be straightforward^{41,42}, but as Whitfield *et al.*^{36} show, both
classical irreversibility and quantum coherence can be brought together applying
the Markovian master equation for density matrices we introduced in Eq. (2) of the main body of the paper. Using the definition
of the Lindbladian operators discussed in the main text, we directly obtain the
classical part in the evolution of the diagonal terms (populations) of the
density matrix given as .

### Defining the Hamiltonian

The Hamiltonian introduces the quantumness in the dynamics. For the case of
undirected networks it is usual to define .
Nevertheless, because we deal with a walk along a weighted and directed graph,
we simplify the definition to if the nodes are
connected in Π(*λ*) and zero otherwise, in agreement with
other applications of quantum rankings successfully explored in the literature
on complex networks^{42}. One may consider more intricate
definitions of this operator like as long as it
remains symmetric when restricted to the real domain
(*H* = *H*^{T}), or Hermitian
(*H* = *H*^{†}) in general.
Summarizing, the Hamiltonian couples diagonal and non-diagonal elements in the
dynamics (known as coherence). The classical term is responsible for the
exponential decayment of the non-diagonal elements of the density matrix over
time and for the existence of a steady state.

### Existence and uniqueness of the stationary solution

In order to prove that the stationary solution for our class of models exists and
is unique we draw upon Spohn’s 1977 theorem^{56}:

*Given a Lindbladian evolution –*
*Eq. (2)*
*–, the dynamic is relaxing (it tends to a stationary solution
ρ* for any initial condition) if*

*the set*{*L*_{(m, n)}}*is self-adjoint (this means the adjoint of L*_{(m, n)}*, denoted by**, belongs to the set), and**the only operator commuting with all the L*_{(m, n)}*is proportional to the identity.*

We show how our system fulfills both conditions.

#### Proof

The first condition is trivially satisfied. If
*c*_{mn} ≠ 0, we have
*L*_{(m,
n)} = |*m*〉〈*n*|.
Then , which is equal to
*L*_{(n, m)} if
*c*_{mn} ≠ 0. Due to the
definition of *C*(*λ*, *φ*), this holds
unless *λ* → ∞, so our system
follows the first condition of the theorem.

Second, we show that the only operator commuting with *L*_{(m,
n)} is proportional to the identity. This is, if
, then .
First, we consider a generic operator . We
take the commmutator of *A* with , and we
impose that it vanishes:

Computing the matrix elements of this commutator
〈*l*|[*A*, *L*_{(m,
n)}]|*k*〉,

Taking *l* = *m* and
*k* ≠ *n* we obtain
*A*_{nk} = 0, and taking
*l* ≠ *m* and
*k* = *n*, then
*A*_{lm} = 0. We have just shown that
the matrix *A* is diagonal. In order to figure out how the diagonal
elements are, we fix *m* = *m*_{1} and
define such that . Now we consider , in Eq. (11), .

In this way, we prove that the submatrix of *A* corresponding to node
*m*_{1} and all the nodes directly linked to it is
proportional to the identity matrix. By repeating the same procedure with
*m* = *m*_{2} such that (this means *m*_{2} is linked to
*m*_{1}), we show that the submatrix related to
*m*_{2} and the nodes linked to it is also proportional to
the identity, but the proportionality constant must be the same as the one
for the submatrix of nodes connected to *m*_{1}, because
*m*_{2} is linked to *m*_{1}. As our network
is connected, we eventually reach for
*φ* ∈ (0, 1) in an iterative
manner.

We have proven here that the second condition is also fulfilled, so the stationary solution of our system does exist and is unique.

We denote the stationary distribution by *ρ**, and we give
further details about its computation below.

### Vectorization of *ρ*

In order to solve the Lindblad-Kossakowski equation (Eq. 2 in the main text), we need to rewrite it as a matrix equation:

where is the vector with *N*^{2}
components vectorizing the density matrix *ρ*, (*i.e.*, a
column vector formed by the columns of *ρ* arranged one after
another), and is the superoperator in its
*N*^{2} × *N*^{2}
matrix form. To that end, we insert the identity operator into the
Lindblad-Kossakowski equation:

At this point we need to introduce the following tensor identity^{57,58}, , with *X*, *Y*,
and *Z* being matrices, and the
aforementioned vectorization of *Y*. Then, we obtain in Eq.
(13),

The formal solution of Eq. (14) for any given initial condition is Once we have vectorized the Lindblad-Kossakowski equation, we solve it by means of exact diagonalization of .

### Stationary solution and relaxation time

The full spectrum of the Lindbladian provides all
the information about the system. However, a lot can be known by partial
knowledge of it. It can be shown that any
fulfilling the conditions of the theorem above can be decomposed as a direct sum
of Jordan forms^{35}. There exists a matrix *S* such that
, where ,
and the others are

with being the eigenvalues of .

The evolution superoperator becomes , with
*N*_{k} being nilpotent matrices. As
Re(*λ*_{k}) < 0 because of
the existence and uniqueness theorem, the only term surviving for
*t* → ∞ is the one corresponding to
. Hence, the eigenvector associated to the
eigenvalue 0 is the vectorized form of the stationary solution
*ρ**. Besides, if we order the eigenvalues such that
0 > Re(*λ*_{1}) > Re(*λ*_{2}) > … > Re(*λ*_{K}),
the relaxation time is given by
*τ* = −1/Re(*λ*_{1}).
This is the definition we use throughout the paper.

## Additional Information

**How to cite this article**: Martínez-Martínez, I. and
Sánchez-Burillo, E. Quantum stochastic walks on networks for
decision-making. *Sci. Rep.*
**6**, 23812; doi: 10.1038/srep23812 (2016).

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## Author information

## Affiliations

### Düsseldorf Institute for Competition Economics (DICE), Heinrich Heine Universität Düsseldorf, Universitätsstraße 1, D-40225 Düsseldorf, Germany

- Ismael Martínez-Martínez

### Instituto de Ciencia de Materiales de Aragón (ICMA), CSIC-Universidad de Zaragoza and Departamento de Física de la Materia Condensada, Universidad de Zaragoza, E-50009 Zaragoza, Spain

- Eduardo Sánchez-Burillo

## Authors

### Search for Ismael Martínez-Martínez in:

### Search for Eduardo Sánchez-Burillo in:

### Contributions

Both authors (I.M.M. and E.S.B.) equally contributed to the conception and development of this theoretical work, and jointly wrote the manuscript.

### Competing interests

The authors declare no competing financial interests.

## Corresponding author

Correspondence to Ismael Martínez-Martínez.

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