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).

Source : https://www.nature.com/articles/srep23812

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