=Paper=
{{Paper
|id=Vol-2404/paper12
|storemode=property
|title=An Agent-Based Model on Scale-Free Networks for Personal Finance Decisions
|pdfUrl=https://ceur-ws.org/Vol-2404/paper12.pdf
|volume=Vol-2404
|authors=Loretta Mastroeni,Maurizio Naldi,Pierluigi Vellucci
|dblpUrl=https://dblp.org/rec/conf/woa/MastroeniNV19
}}
==An Agent-Based Model on Scale-Free Networks for Personal Finance Decisions==
https://ceur-ws.org/Vol-2404/paper12.pdf
Workshop "From Objects to Agents" (WOA 2019)
An Agent-Based Model on Scale-Free Networks for
Personal Finance Decisions
Loretta Mastroeni Maurizio Naldi Pierluigi Vellucci
Dept. of Economics Dept. of Civil Engineering and Computer Science Dept. of Economics
Roma Tre University University of Rome Tor Vergata Roma Tre University
Rome, Italy Dept. of Law, Economics, Politics Rome, Italy
loretta.mastroeni@uniroma3.it and Modern languages pierluigi.vellucci@uniroma3.it
LUMSA University
Rome, Italy
maurizio.naldi@uniroma2.it
Abstract—Personal finance decisions emerge from a complex The society is modeled as an Agent-Based Model (ABM).
network of human connections, where the nodes or agents — Agent-based simulation is most commonly used to model indi-
banks, investors, financial advisors — take their choices on the vidual decision-making and social-organizational behavior [1],
basis of a variety of factors. All these agents form a society,
which we modeled as an Agent-Based Model (ABM) on a scale- [3]–[10]. It allows to investigate mutual and causal influences
free network. In this paper, we will consider: honest agents, of the micro-elements on the complex system development
regular agents, insincere agents, stubborn agents and skilled (or [11], by involving research areas seemingly distant such as
unskilled) agents. Honest agents report truthfully their opinion game theory and control theory (see e.g. [12] and [13]).
while insincere agents state an opinion which is different from In this paper, to explain the diverse opinion structures
their internal belief. Regular agents are characterized by the
same propensity to listen, contrary to what stubborn agents do within that kind of society, we extend the bounded confidence
because these agents evaluate the counterpart’s opinion but never model of continuous opinion formation introduced in [14], by
approaches to it. Skilled and unskilled agents are the result introducing Gaussian updating functions [15]. According to
of influence of the competence in the evolution of decisions in classical bounded confidence models, the agents interact with
multi-agent systems. We perform a social simulation to show each other only when their opinions are close enough. But
that, in particular, consensus, polarization, extremism or the
emergence of a disordered regime are possible outcomes, even in many real world situations, the strength of this interaction
without explicit introduction of stubborn agents. usually depends on the distance between opinions (the lower
Index Terms—Agent-based modeling, Multi-agent systems, the distance, the higher the strength). For this reason we
Opinion dynamics, Scale free networks considered a Gaussian updating function.
We will consider several categories of agents: honest agents,
I. I NTRODUCTION regular agents, insincere agents, stubborn agents and skilled
(or unskilled) agents. Honest agents truthfully report their
Personal finance decisions are taken by individuals on the opinion while insincere agents state an opinion that may
basis of a variety of factors, emerging from a complex network be different from their internal belief. Regular agents are
of human connections. All of these human connections involve characterized by a common propensity to listen, contrary to
several agents, many of them clustered into fixed categories: what stubborn agents do because these agents evaluate the
banks, financial advisors, investors. They form a society. counterpart’s opinion but never approaches it. Skilled and
Investors usually resort to financial advisors to improve their unskilled agents behave as described in in [4], [5], where the
investment process. The latter are paid by the banks, whose influence of the competence in the evolution of decisions in
aim is to steer the investors towards a particular investment multi-agent systems has been considered.
decision and it is the reason why they ask the collaboration After describing the model in Section II, in Sections III and
of financial advisors. IV we consider some special cases, where the composition
When we look at the connections, we realize that: i) the of this artificial society is made of the following classes of
interaction is not of the any-to-any kind [1], since an agent will agents:
be typically connected with some other agents rather than all • honest, regular agents vs one class of stubborn agents;
of them (e.g., we assume that investors connect with advisors • honest, regular agents vs two classes of stubborn agents;
but not with banks); ii) some agents have a large number of • the presence of insincere agents in a population of regular
connections to other individuals, whereas most of them just honest agents;
have a handful (e.g., an advisor may have many customers, but • skilled regular agents vs unskilled regular ones.
customers usually have only one advisor). Societies satisfying For the sake of simplicity, we will assume that all these subsets
such rules are the popular “scale-free” networks [2]. have empty intersections i.e., they form a partition of the set
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Workshop "From Objects to Agents" (WOA 2019)
of all agents. Moreover, we will focus only on scale free function (because of threshold) and the interaction is in general
networks. non symmetric (because ǫi and ǫj could differ).
In this paper we wish to extend the simulation experiments At this point two other agents classifications naturally
performed in [14] by considering different compositions of arise. While the first classification concerns the distinction
the society. Then we perform a social simulation to show between xR i (t) and xi (t), the second and the third concern the
that, in particular, consensus, polarization, extremism or the parameters ǫi and µ. The second classification is as follows:
emergence of a disordered regime are possible outcomes, even • A stubborn agent i has parameter values of
without explicit introduction of stubborn agents.
II. AGENT-BASED M ODEL ǫi = 0 ∨ µ = 0 ; (4)
We have examined the Bounded Confidence Model studied • A non-stubborn agent i has parameter values of
in [14] which we are going to describe below.
Let G = {V, E} be a graph that consists of a finite set ǫi > 0 ∧ µ > 0 . (5)
of agents i ∈ V = {1, 2, . . . , n} who are defined as nodes
on a network and connected to each other with a finite set The third classification concerns the definition of regular
of links E. Links between the agents indicate the communi- agents. The model studied in [14] assumes that threshold
cation channels through which opinions are exchanged and levels are equal across the population of regular agents, i.e.
the influence is imposed. Communication requires a direct ǫ1 = ǫ2 = · · · = ǫn .
link between the agents (i, j) ∈ E. In this model two agents Lastly, according to the taxonomy introduced in [1], the
always influence each other mutually, and hence we talk about updating frequency of this model is periodic because all the
a bilateral interaction, i.e. (i, j) ∈ E ⇔ (j, i) ∈ E. agents change their opinion at each time step.
In this paper each agent is characterized by the following The results enclosed in [14] concern the following cases:
triple: a couple of opinions, threshold level and a set of • honest, regular agents;
xi (t), xR
connections. Then: i = i (t) , ǫ i , N i . All the • honest, regular agents vs one class of stubborn agents
opinions fall in the range [0, 1] and are related to the decisions (the latter with the same opinion xS = 0);
to buy a security rather than a different security or other • honest, regular agents vs two classes of stubborn agents
financial instruments. At each time t, agent i selects a random (xS0 = 0 for the first class, xS1 = 1 for the second);
counterpart j from his neighborhood Ni = {j ∈ V|(j, i) ∈ E} • honest, regular agents vs insincere, regular agents.
and the two share their opinions xi (t) and xj (t). If xR i (t) is
the opinion that agent i reports to the selected counterpart, Lastly, the authors of [14] compare the results on different
then we have a first distinction between agents: network topologies: complete network, small world network
R and the scale free network.
• xi (t) 6= xi (t) in the case of insincere agents;
R In this paper we wish to extend the simulation experi-
• xi (t) = xi (t) in the case of honest agents.
ments performed in [14] by replacing the updating function
Hence, according to the notation introduced in [1], this model χ(−ǫj ,ǫj ) (di,j (t)) with
is continuous over a bounded interval because
2
e−(xi (t)−xj (t)) χ(−ǫi ,ǫi ) (di,j (t)) ;
R
xi (t) ∈ [0, 1] ∀i ∈ V , t > 0 . (1) (6)
It is also bilateral and pairwise. Then we examine different compositions of the society:
Threshold levels are assigned to each agent at t = 0, with
• we consider honest, regular agents vs one class of stub-
ǫi ∈ [0, 1].
born agents at varying α (the latter with the same opinion
Moreover, agents adjust their opinion upon the principle
xS = α ∈ [0, 1]);
of bounded confidence. If xR j (t) is the opinion that agent j • we consider honest, regular agents vs two classes of
reports to i, and
stubborn agents at varying α and β (xS0 = α for the
∆xi (t) = xi (t + 1) − xi (t) first class, xS1 = β for the second);
• we consider the presence of insincere agents in a popu-
∆xj (t) = xj (t + 1) − xj (t) (2)
lation of regular honest agents;
are the changes of i and j’ opinions, then • we consider skilled regular agents vs unskilled regular
ones.
∆xi (t) = µ χ(−ǫi ,ǫi ) (di,j (t)) xR
j (t) − xi (t)
The latter point is inspired by the results obtained in [4], [5],
∆xj (t) = µ χ(−ǫj ,ǫj ) (dj,i (t)) xR
i (t) − xj (t) (3)
where the influence of competence in the evolution of deci-
where µ ∈ [0, 1] is the adoption rate, representing the propor- sions in multi-agent systems has been considered. Moreover,
tion of counterpart’s opinion an agent integrates into his prior, since we are interested in complex societies in which some
di,j (t) = xi (t) − xR
j (t) and χ(−ǫi ,ǫi ) (x) is the characteristic individuals have a large number of connections to other people
function of the interval (−ǫi , ǫi ). Then, according to the — whereas most individuals have just a handful — in this
notation in [1], this model adopts a non linear updating paper we will focus only on scale free network [2].
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Workshop "From Objects to Agents" (WOA 2019)
Fig. 1: Time evolution of the opinion dynamics in two selected Fig. 2: Time evolution of the opinion dynamics in two selected
runs. (Top plot): µ = 0.3; (bottom plot): µ = 0.1. Here ǫ = 0.3 runs. (Top plot): µ = 0.5; (bottom plot): µ = 0.05. Here
and n = 500 agents. ǫ = 0.3 and n = 500 agents.
III. O PINION DYNAMICS
After defining the model and formulating it in an easily
computable way through the paradigm of array programming,
in this section we apply it to examine the resulting dynamics
of agents, i.e. how their opinion changes over time. We
use a simulation approach to examine the impact of the
interaction coefficients. We developed an R code to perform
these calculations, employed on a Windows machine equipped
with a 2.80GHz Intel(R) Core(TM) i7 CPU and 16.0 GB
RAM. For T = 1000 iterations and n = 500 agents, the
simulations run for around 5 minutes.
Let us define the opinion vector as:
T
x(t) = (x1 (t), . . . , xn (t)) .
A. Opinion Formation with Regular, Honest Agents
We start with the simplest case, in which only regular agents
are present. The initial opinion vector x(0) has been drawn Fig. 3: Time evolution of the opinion dynamics in two selected
from a standard uniform distribution with [0, 1] support. All runs. (Top plot): ǫ = 0.7; (bottom plot): ǫ = 0.9. Here µ = 0.3
agents have the same threshold level ǫi = ǫ, the same adoption and n = 500 agents.
rate and xRi (t) = xi (t) for every i (i.e., alla agents are honest).
The plots in Figs. 1 and 2 were obtained by considering the
Gaussian updating function defined in (6). They show that the number of extreme opinions in which they are isolated from
process of opinion formation within an integrated society (i.e. the low central consensus. In such a way, the model settles
a society in which agents integrate opinions of others into their on a steady pattern and, as µ increases, high extreme opinions
own) tends to self-organization and that the outcomes depend become more and more distinguishable (see top plots of Figs. 1
upon the parameter values. In this situation, those agents that and 2). Actually, µ represents the proportion of counterpart’s
have an initial starting opinion below a certain threshold (from opinion an agent integrates into its own and the greater its
the top plots of Figs. 1 and 2 it seems to be between 0.5 value, the greater the number of opinions emerging after the
and 0.6) are rapidly drawn to a low central consensus — and transient.
µ speeds up the convergence to the low central consensus. Fig. 3 underlines this effect by fixing µ and considering
Moreover, those agents that instead fall outside the attractor increasing, high values of threshold ǫ. As ǫ rises, high extreme
defined by this threshold, quickly settle down to a larger opinions emerge.
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Workshop "From Objects to Agents" (WOA 2019)
Fig. 5: The effect of insincere agents. Time evolution of the
opinion dynamics for µ = 0.5, ǫ = 0.3, m = 250 and n =
500.
C. Opinion Formation with Regular and Stubborn Agents
We now relax the assumption on the regularity of agents.
We assume that the set of agents V is divided into two distinct
groups, R (regular agents) and S (stubborn agents), such that
V = S ∪ R and S ∩ R = ∅.
As defined in Section II, an agent i is called stubborn
if at least one of the two parameter values ǫi , µ is zero.
Stubborn agents can be described as individuals that are biased
towards their initial opinions. They have the ability to exert
their influence onto others but cannot be influenced by the rest
of society.
We assume a society V of n = 500 agents and that m of
Fig. 4: The effect of insincere agents. Time evolution of the
them are stubborn. Stubborn agents are assigned same initial
opinion dynamics in three selected runs: m = 30 (top plot);
opinion xS = α ∈ [0, 1]. In the following we assume that
m = 150 (central plot); m = 250 (bottom plot). Here µ = 0.1,
stubborn agents cannot be distinguished from other regular
ǫ = 0.3 and n = 500 agents.
agents. Hence they fall in the neighborhood of regular agents,
which cannot identify and avoid them. In this way stubborn
and regular agents usually interact.
B. Opinion Formation with Insincere Agents
In Fig. 6 we examined the impact of the total number of
In previous section we considered the case xR i (t) = xi (t), stubborn agents, m, on the opinion dynamics for α = 0. In the
an assumption that has been relaxed by a number of authors in plots we can spot the presence of the xi = 0 extreme opinion
the last years (see, among others, [12], [16] in addition to the of the stubborn agents and, with respect to Figs. 1 and 2, we
aforementioned [14]). In the following we assume that the i-th notice also that the low central consensus is vanished. More
insincere agent states an opinion xRi (t) drawn from a standard precisely, when m is not too big (m = 30), the low central
uniform distribution with [0, 1] support, regardless the value consensus deviates toward the position of stubborn agents but
of “true” or “internal” opinion xi (t). it disappears when the number of stubborn agents increases.
With the inclusion of the insincere agents, the society V In Fig. 7, we examined the impact of α on the time evolution
can be subdivided into two subsets, H (honest agents) and of opinion dynamics, by considering α = 0.3. In this case
I (insincere agents), such that V = H ∪ I and H ∩ I = ∅. we can spot the presence of a consensus on the position of
Anyway all the agents are regular, i.e. ǫ1 = ǫ2 = · · · = ǫn . the stubborn agents that persists with increasing number of
We assume a society V of n = 500 agents and that m stubborn agents. Then we can argue that a consensus can be
of them are insincere. In Fig. 4 we examined the impact stimulated by stubborn agents, but it resists to their excessive
of the total number of insincere agents, m, on the opinion proliferation if α is sufficiently distant from 0. A similar
dynamics. As we can see, the low central consensus of Figs. consideration can be done for the opposite position, as we
1 and 2 disappeared, and a disordered regime emerged in can see in Fig. 8. If α is sufficiently near to 1, the consensus
which opinions are in a constant state of change around a cannot be reached.
central opinion. Besides, increasing the willingness to listen
(by increasing µ e.g.) does not seem to improve the picture D. Opinion Formation with Two Groups of Stubborn Agents
(see Fig. 5): indeed, in the presence of insincere agents, We now extend the previous section by introducing another
a greater proportion of counterpart’s opinion that an agent group of stubborn agents. We assume a society V of n =
is willing to accept leads to a more pronounced disordered 1000 agents and that 2m of them are stubborn. Two classes
regime. of stubborn agents are assigned to the initial opinions xS1 =
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Workshop "From Objects to Agents" (WOA 2019)
Fig. 6: The effect of stubborn agents (α = 0). Time evolution Fig. 7: The effect of stubborn agents (α = 0.3). Time evolution
of the opinion dynamics in three selected runs: m = 30 (top of the opinion dynamics in three selected runs: m = 30 (top
plot); m = 150 (central plot); m = 250 (bottom plot). Here plot); m = 150 (central plot); m = 250 (bottom plot). Here
µ = 0.1, ǫ = 0.3 and n = 500 agents. µ = 0.1, ǫ = 0.3 and n = 500 agents.
α ∈ [0, 1] and xS2 = β ∈ [0, 1], with α 6= β. We assume that
the groups of stubborn agents are equally sized, consisting of
m = 150 stubborn agents each.
In Fig. 9 we fixed β = 1 while α varies from 0 to 0.5.
When α = 0 the extremism prevails and the society ends
in a complete polarisation of the opinion space; see top plot
of Fig. 9, where it is also possible to identify the isolated
position of S2 class of stubborn agents with xS2 = 1. When
α reaches 0.5 (bottom plot of Fig. 9) the majority of regular Fig. 8: The effect of stubborn agents (α = 0.75). Number
agents concentrate in the center, forming a large single opinion of stubborn agents: m = 150. Here µ = 0.1, ǫ = 0.3 and
class. If we denote this class by C, we have that for T → +∞, n = 500 agents.
xi (T ) → 0.5 ∀i ∈ C.
In Fig. 10, we examined the impact of ǫ on the opinion
formation process with regular agents and two groups of those best disposed to dialogue. According to this view of
stubborn agents, for which we assumed xS1 = 0 and xS2 = 1. competence-opinion relation, an agent with an attitude to listen
When ǫ < 0.3 the agents whose opinion is approximately other people is characterized by a high competence, while an
in the range [0, 0.6] move towards either a central consensus individual unwilling to listen and dialogue is usually marked
or the position of S1 (however there is room for alternative by a lower level of the described trait. Hence we postulate that
opinions at the upper bound of opinion range). Anyway, as ǫ the threshold of Gaussian bounded confidence model depends
rises, the central consensus vanishes and only xS1 = 0 remains on the degree of competence, e.g. replacing Eq. (6) with:
in addition to the high extreme opinions. 2
e−(xi (t)−xj (t)) χ(−ǫi,j ,ǫi,j ) (di,j (t)) , (7)
IV. S KILLED R EGULAR , H ONEST AGENTS
Although it is not a strict rule, we have a tendency to where
ǫ
think that more well-educated and competent people are also ǫi,j = ,c ≫ 1 (8)
1 + ec(yj −yi )
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Workshop "From Objects to Agents" (WOA 2019)
Fig. 9: The effect of two classes of stubborn agents (fixed β =
1). Time evolution of the opinion dynamics in two selected
runs: α = 0 (top plot); α = 0.5 (bottom plot). Here µ = 0.1,
ǫ = 0.3.
and
T
y = (y1 , . . . , yn )
is the competence vector, which is supposed to be constant
in time. In this way we are assuming that each agent i is
characterized by two variables, (xi (t), yi ). Eq. (8) has been
considered in [5] in order to model the so-called equality bias
effect (see also [4]).
The competence vector y has been drawn from a standard
uniform distribution with: [0, 1] support for the first m agents,
[10, 15] support for the remaining ones. For simplicity, initial
opinion vector x(0) has been arranged in such a way its
Fig. 10: Opinion dynamics with regular agents and two groups
elements are in ascending order, i.e. x1 (0) < x2 (0) < · · · <
of stubborn agents (α = 0 and β = 1). Selected single runs
xn (0).
for the given parameter values are displayed. From the top to
In Fig. 11 the system evolves toward two clusters, charac-
the bottom, respectively: ǫ = 0.2, ǫ = 0.45, ǫ = 0.55 and
terizing two subpopulations with different decisions driven by
ǫ = 0.7. Parameter µ = 0.3.
the most competent agents (upper part of the plot) and the
less skilled ones (lower part). We can spot the presence of
a region in which regular skilled agents continuously change
their opinions, in the upper part of the plot, and the presence • With the inclusion of the insincere agents, the low central
of a lower consensus for the unskilled people. consensus disappeared, and a disordered regime in which
opinions are in a constant state of change around a central
V. C ONCLUSIONS AND F UTURE P ERSPECTIVES opinion, emerged by varying the number of insincere
We have built and simulated an Agent-Based Model (ABM) agents. Anyway, a greater proportion of counterpart’s
for opinion dynamics in personal finance decisions. We em- opinion, that an agent integrates into his prior, leads to a
ployed a Gaussian bounded confidence with pairwise random more pronounced disordered regime.
meetings to examine the role of different categories of agents • When we relax the assumption on the regularity of agents,
in opinion formation. The model was simulated on a scale free in presence of stubborn agents, if the number of these
network. Our findings can be summarized as follows. agents is not too big, the low central consensus deviates
• When only regular, honest agents are present those agents toward the position of stubborn agents but it disappears
with an initial starting opinion that is below a certain with the increase of this number.
threshold are rapidly drawn to a low central consensus; • When another population of stubborn agents is added, the
µ speeds up the convergence to the low central consensus. extremism prevails and the society ends in a complete
Moreover, as ǫ rises, high extreme opinions emerge. polarisation of the opinion space.
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Workshop "From Objects to Agents" (WOA 2019)
social networks, e.g. Twitter, and the analysis of the dynamic
sentiments of users to investigate realistic opinion evolution,
as proposed in [17].
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