An Agent-Based Model on Scale-Free Networks for Personal Finance Decisions

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 indibanks, investors, financial advisors - take their choices on the vidual decision-making and social-organizational behavior [1], bwahsiicshowf eamvoadreieletyd aosf afnacAtogresn.tA-BllastehdesMeoadgeeln(tAsBfoMrm)ona asoscciaeltey-, [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 tshaemire pinrtoeprennaslitbyeltioefl.isRteeng,uclaorntraagreynttso awrehacthsatruabcbtoerrinzeadgebnyts tdhoe 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 tehmaetr,gienncpeaortficauldairs,orcdoenrseednsruesg,i mpoelaarriezaptioosns,ibelextroeumtciosmmeso,revtehne 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. INTRODUCTION 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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of all agents. Moreover, we will focus only on scale free networks.

In this paper we wish to extend the simulation experiments performed in [ 14 ] by considering different compositions of the society. Then we perform a social simulation to show that, in particular, consensus, polarization, extremism or the emergence of a disordered regime are possible outcomes, even without explicit introduction of stubborn agents.

II. AGENT-BASED MODEL

We have examined the Bounded Confidence Model studied in [ 14 ] which we are going to describe below.

Let G = {V, E } be a graph that consists of a finite set of agents i ∈ V = {1, 2, . . . , n} who are defined as nodes on a network and connected to each other with a finite set of links E . Links between the agents indicate the communication channels through which opinions are exchanged and the influence is imposed. Communication requires a direct link between the agents (i, j) ∈ E . In this model two agents always influence each other mutually, and hence we talk about a bilateral interaction, i.e. (i, j) ∈ E ⇔ (j, i) ∈ E .

In this paper each agent is characterized by the following triple: a couple of opinions, threshold level and a set of connections. Then: i = xi(t), xiR(t) , ǫi, Ni . All the opinions fall in the range [ 0, 1 ] and are related to the decisions to buy a security rather than a different security or other financial instruments. At each time t, agent i selects a random counterpart j from his neighborhood Ni = {j ∈ V|(j, i) ∈ E } and the two share their opinions xi(t) and xj (t). If xiR(t) is the opinion that agent i reports to the selected counterpart, then we have a first distinction between agents: • xiR(t) 6= xi(t) in the case of insincere agents; • xiR(t) = xi(t) in the case of honest agents.

Hence, according to the notation introduced in [ 1 ], this model is continuous over a bounded interval because xi(t) ∈ [ 0, 1 ] ∀i ∈ V , t > 0 . (1)

It is also bilateral and pairwise.

Threshold levels are assigned to each agent at t = 0, with ǫi ∈ [ 0, 1 ].

Moreover, agents adjust their opinion upon the principle of bounded confidence. If xjR(t) is the opinion that agent j reports to i, and Δxi(t) = xi(t + 1) − xi(t) Δxj (t) = xj (t + 1) − xj (t) are the changes of i and j’ opinions, then Δxi(t) = µ χ (−ǫi,ǫi) (di,j (t)) xjR(t) − xi(t) Δxj (t) = µ χ (−ǫj,ǫj) (dj,i(t)) xiR(t) − xj (t) where µ ∈ [ 0, 1 ] is the adoption rate, representing the proportion of counterpart’s opinion an agent integrates into his prior, di,j (t) = xi(t) − xjR(t) and χ(−ǫi,ǫi)(x) is the characteristic function of the interval (−ǫi, ǫi). Then, according to the notation in [ 1 ], this model adopts a non linear updating (2) (3) function (because of threshold) and the interaction is in general non symmetric (because ǫi and ǫj could differ).

At this point two other agents classifications naturally arise. While the first classification concerns the distinction between xiR(t) and xi(t), the second and the third concern the parameters ǫi and µ . The second classification is as follows: • A stubborn agent i has parameter values of • A non-stubborn agent i has parameter values of The third classification concerns the definition of regular agents. The model studied in [ 14 ] assumes that threshold levels are equal across the population of regular agents, i.e. ǫ1 = ǫ2 = · · · = ǫn.

Lastly, according to the taxonomy introduced in [ 1 ], the updating frequency of this model is periodic because all the agents change their opinion at each time step.

The results enclosed in [ 14 ] concern the following cases: • honest, regular agents; • honest, regular agents vs one class of stubborn agents (the latter with the same opinion xS = 0); • honest, regular agents vs two classes of stubborn agents (xS0 = 0 for the first class, xS1 = 1 for the second); • honest, regular agents vs insincere, regular agents.

Lastly, the authors of [ 14 ] compare the results on different network topologies: complete network, small world network and the scale free network.

In this paper we wish to extend the simulation experiments performed in [ 14 ] by replacing the updating function χ(−ǫj,ǫj) (di,j (t)) with e−(xi(t)−xjR(t))2 χ(−ǫi,ǫi) (di,j (t)) ; (6) Then we examine different compositions of the society: • we consider honest, regular agents vs one class of stubborn agents at varying α (the latter with the same opinion xS = α ∈ [ 0, 1 ]); • we consider honest, regular agents vs two classes of stubborn agents at varying α and β (xS0 = α for the first class, xS1 = β for the second); • we consider the presence of insincere agents in a population of regular honest agents; • we consider skilled regular agents vs unskilled regular ones.

The latter point is inspired by the results obtained in [ 4 ], [ 5 ], where the influence of competence in the evolution of decisions in multi-agent systems has been considered. Moreover, since we are interested in complex societies in which some individuals have a large number of connections to other people — whereas most individuals have just a handful — in this paper we will focus only on scale free network [ 2 ]. Fig. 1: Time evolution of the opinion dynamics in two selected runs. (Top plot): µ = 0.3; (bottom plot): µ = 0.1. Here ǫ = 0.3 and n = 500 agents.

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:

x(t) = (x1(t), . . . , xn(t))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 from a standard uniform distribution with [ 0, 1 ] support. All agents have the same threshold level ǫi = ǫ, the same adoption rate and xiR(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 process of opinion formation within an integrated society (i.e. a society in which agents integrate opinions of others into their own) tends to self-organization and that the outcomes depend upon the parameter values. In this situation, those agents that have an initial starting opinion below a certain threshold (from the top plots of Figs. 1 and 2 it seems to be between 0.5 and 0.6) are rapidly drawn to a low central consensus — and µ speeds up the convergence to the low central consensus. Moreover, those agents that instead fall outside the attractor defined by this threshold, quickly settle down to a larger

Fig. 3: Time evolution of the opinion dynamics in two selected runs. (Top plot): ǫ = 0.7; (bottom plot): ǫ = 0.9. Here µ = 0.3 and n = 500 agents. number of extreme opinions in which they are isolated from the low central consensus. In such a way, the model settles on a steady pattern and, as µ increases, high extreme opinions become more and more distinguishable (see top plots of Figs. 1 and 2). Actually, µ represents the proportion of counterpart’s opinion an agent integrates into its own and the greater its value, the greater the number of opinions emerging after the transient.

B. Opinion Formation with Insincere Agents

In previous section we considered the case xiR(t) = xi(t), an assumption that has been relaxed by a number of authors in the last years (see, among others, [ 12 ], [ 16 ] in addition to the aforementioned [ 14 ]). In the following we assume that the i-th insincere agent states an opinion xiR(t) drawn from a standard uniform distribution with [ 0, 1 ] support, regardless the value of “true” or “internal” opinion xi(t).

With the inclusion of the insincere agents, the society V can be subdivided into two subsets, H (honest agents) and I (insincere agents), such that V = H ∪ I and H ∩ I = ∅. Anyway all the agents are regular, i.e. ǫ1 = ǫ2 = · · · = ǫn.

We assume a society V of n = 500 agents and that m of them are insincere. In Fig. 4 we examined the impact of the total number of insincere agents, m, on the opinion dynamics. As we can see, the low central consensus of Figs. 1 and 2 disappeared, and a disordered regime emerged in which opinions are in a constant state of change around a central opinion. Besides, increasing the willingness to listen (by increasing µ e.g.) does not seem to improve the picture (see Fig. 5): indeed, in the presence of insincere agents, a greater proportion of counterpart’s opinion that an agent is willing to accept leads to a more pronounced disordered regime.

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 them are stubborn. Stubborn agents are assigned same initial opinion xS = α ∈ [ 0, 1 ]. In the following we assume that stubborn agents cannot be distinguished from other regular 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.

In Fig. 6 we examined the impact of the total number of stubborn agents, m, on the opinion dynamics for α = 0. In the plots we can spot the presence of the xi = 0 extreme opinion of the stubborn agents and, with respect to Figs. 1 and 2, we notice also that the low central consensus is vanished. More precisely, when m is not too big (m = 30), the low central consensus deviates toward the position of stubborn agents but it disappears when the number of stubborn agents increases. In Fig. 7, we examined the impact of α on the time evolution of opinion dynamics, by considering α = 0.3. In this case we can spot the presence of a consensus on the position of the stubborn agents that persists with increasing number of stubborn agents. Then we can argue that a consensus can be stimulated by stubborn agents, but it resists to their excessive proliferation if α is sufficiently distant from 0. A similar consideration can be done for the opposite position, as we can see in Fig. 8. If α is sufficiently near to 1, the consensus cannot be reached.

D. Opinion Formation with Two Groups of Stubborn Agents

We now extend the previous section by introducing another group of stubborn agents. We assume a society V of n = 1000 agents and that 2m of them are stubborn. Two classes of stubborn agents are assigned to the initial opinions xS1 = 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 agents concentrate in the center, forming a large single opinion class. If we denote this class by C, we have that for T → +∞, 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 stubborn agents, for which we assumed xS1 = 0 and xS2 = 1. When ǫ < 0.3 the agents whose opinion is approximately in the range [0, 0.6] move towards either a central consensus or the position of S1 (however there is room for alternative opinions at the upper bound of opinion range). Anyway, as ǫ rises, the central consensus vanishes and only xS1 = 0 remains in addition to the high extreme opinions.

IV. SKILLED REGULAR, HONEST AGENTS Although it is not a strict rule, we have a tendency to think that more well-educated and competent people are also

those best disposed to dialogue. According to this view of competence-opinion relation, an agent with an attitude to listen other people is characterized by a high competence, while an individual unwilling to listen and dialogue is usually marked by a lower level of the described trait. Hence we postulate that the threshold of Gaussian bounded confidence model depends on the degree of competence, e.g. replacing Eq. (6) with: where e−(xi(t)−xj(t))2 χ(−ǫi,j,ǫi,j) (di,j (t)) ,

ǫ ǫi,j = 1 + ec(yj−yi) , c ≫ 1 (7) (8)

y = (y1, . . . , yn)T 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 elements are in ascending order, i.e. x1(0) < x2(0) < · · · < xn(0).

In Fig. 11 the system evolves toward two clusters, characterizing two subpopulations with different decisions driven by 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 of a lower consensus for the unskilled people.

V. CONCLUSIONS AND FUTURE PERSPECTIVES

We have built and simulated an Agent-Based Model (ABM) for opinion dynamics in personal finance decisions. We employed a Gaussian bounded confidence with pairwise random meetings to examine the role of different categories of agents in opinion formation. The model was simulated on a scale free network. Our findings can be summarized as follows. • When only regular, honest agents are present those agents with an initial starting opinion that is below a certain threshold are rapidly drawn to a low central consensus; µ speeds up the convergence to the low central consensus. Moreover, as ǫ rises, high extreme opinions emerge.

Fig. 10: Opinion dynamics with regular agents and two groups of stubborn agents (α = 0 and β = 1). Selected single runs for the given parameter values are displayed. From the top to the bottom, respectively: ǫ = 0.2, ǫ = 0.45, ǫ = 0.55 and ǫ = 0.7. Parameter µ = 0.3.

• With the inclusion of the insincere agents, the low central consensus disappeared, and a disordered regime in which opinions are in a constant state of change around a central opinion, emerged by varying the number of insincere agents. Anyway, a greater proportion of counterpart’s opinion, that an agent integrates into his prior, leads to a more pronounced disordered regime. • When we relax the assumption on the regularity of agents, in presence of stubborn agents, if the number of these agents is not too big, the low central consensus deviates toward the position of stubborn agents but it disappears with the increase of this number. • When another population of stubborn agents is added, the extremism prevails and the society ends in a complete polarisation of the opinion space.

• The system of skilled and unskilled agents evolves toward two clusters; regular skilled agents continuously change their opinions, while the presence of a lower consensus is due to the unskilled people.

Since our ultimate goal is to achieve a better understanding of human decisions in personal finance in a real world context, we expect to validate our results concerning a stylized model of a real society. Future work therefore includes the collection of large amount of user interaction information from online 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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