=Paper=
{{Paper
|id=Vol-1867/w8
|storemode=property
|title=An Analytic Model of the Impact of Skeptical Agents on the Dynamics of Compromise
|pdfUrl=https://ceur-ws.org/Vol-1867/w8.pdf
|volume=Vol-1867
|authors=Stefania Monica,Federico Bergenti
|dblpUrl=https://dblp.org/rec/conf/woa/MonicaB17
}}
==An Analytic Model of the Impact of Skeptical Agents on the Dynamics of Compromise==
https://ceur-ws.org/Vol-1867/w8.pdf
43
An Analytic Model of the Impact of Skeptical
Agents on the Dynamics of Compromise
Stefania Monica, Federico Bergenti
Dipartimento di Scienze Matematiche, Fisiche e Informatiche
Università degli Studi di Parma
Parco Area delle Scienze 53/A, 43124 Parma, Italy
Email: {stefania.monica,federico.bergenti}@unipr.it
Abstract—This paper studies analytically the dynamics of the the framework of kinetic theory of gases to study social
opinion in multi-agent systems where two classes of agents phenomena relies on a proper parallelism between molecules
coexist. The fact that the population of considered multi-agent and their collisions in gases, and agents and their interactions
systems is divided into two classes is meant to account for agents
with different propensity to change their opinions. Skeptical in multi-agent systems. Such an idea is not new and it
agents, which are agents that are not inclined to change their recently gave birth to a discipline called sociophysics (see,
opinions, can be modeled together with moderate agents, which e.g., [1]). Note that the details of collisions among molecules
are agents that are moderately open to take into account the are different from the details of interactions among agents
opinions of the others. The studied analytic model of opinion and, hence, analytic results obtained at the macroscopic level
dynamics involves only compromise, which describes how agents
change their opinions to try to reach consensus. The adopted for the dynamics of the properties of multi-agent systems
model of compromise is stochastic in order to give agents are significantly different from those of kinetic theory of
some level of autonomy in their decisions. Presented results gases. For the specific case of the study of opinion dynamics,
show, analytically, that after a sufficient number of interactions the major advantage of the proposed approach consists in
consensus is reached, regardless of the initial distribution of the the derivation of analytic results, in contrast with simulation
opinion. Analytic results are confirmed by simulations shown in
the last part of the paper. results that are typically studied in the literature on the subject.
While the validity of simulations depends on the specific tool
I. I NTRODUCTION adopted, and on the choice of the parameters of simulated
This paper discusses an analytic framework that can be models, analytic results are valid as long as the hypothesis
used to study collective and asymptotic properties of multi- used to derive them are valid.
agent systems. The properties of studied multi-agent systems We have already applied the proposed approach to the
evolve because of binary interactions among agents, where study of various models of opinion dynamics (see, [2]–[9]),
the term (binary) interaction is used here to denote a message and analytic results were always confirmed by independent
exchange among two agents. Each interaction identifies a simulations. In this paper, we enrich previous studies on the
single step of the evolution of the systems, regardless of how subject by considering multi-agent systems where two types of
often interactions occur. Note that studied multi-agent systems agents, grouped into two disjoint classes, coexist. Each class
are completely decentralized and that they involve no form is associated with different parameters, such as a different
of supervised coordination. In particular, in this paper we number of agents, a different initial distribution of the opinion,
assume that each agent is associated with a scalar property, and a different inclination of contained agents at changing
which changes because of interactions with other agents, and their opinions. Results concerning a similar model, where
we assume that such a property represents the opinion of the only deterministic interactions among agents are considered,
agent on a given topic. Notably, we remark that the proposed have already been presented in [10]. Here, we extend those
approach can be used to study other collective and asymptotic results by adding stochastic parameters in order to describe
properties of multi-agent systems, and that it is not limited to the behavior of agents that exhibit some level of autonomy.
the study of the opinion, even if the development of specific Moreover, we concentrate on the impact of the presence of
analytic results is needed if different properties are considered. skeptical agents in the multi-agent system. Note that, in order
We start by considering proper rules that describe the effects to account for the existence of two classes of agents, we take
of interactions among agents on their opinions at a microscopic inspiration from kinetic theory of gas mixtures, which is used
level. Then, the dynamics of macroscopic properties of the to study gases composed of different types of molecules.
considered multi-agent system are analytically derived, taking This paper is organized as follows. Section II describes the
inspiration from physical models. In detail, the proposed details of the proposed kinetic framework. Section III derives
framework is related to the kinetic theory of gases, a branch of relevant analytic results concerning the average opinion in
physics which studies the temporal evolution of macroscopic the multi-agent system. Section IV shows simulation results
properties of gases starting from the description of micro- that are used to confirm analytic results. Finally, Section V
scopic collisions among molecules. The idea of generalizing concludes the paper.
44
II. A K INETIC F RAMEWORK OF O PINION DYNAMICS compromise, we also aim at reproducing the fact that the post-
interaction opinion of an agent is closer to its pre-interaction
In kinetic theory of gases, each molecule of a gas is
opinion than to the pre-interaction opinion of the other agent.
associated with specific physical quantities, such as its po-
This phenomenon can be reproduced in terms of the following
sition and its velocity, and the macroscopic features of gases,
inequalities
concerning, for example, its temperature and pressure, are
|v ∗ − v| < |v ∗ − w|
derived on the basis of a proper balance equation known (3)
as Boltzmann equation. Similarly, in the context of opinion |w∗ − w| < |w∗ − v|.
dynamics, we assume that each agent is associated with a
Simple algebraic manipulations show that a sufficient condi-
single scalar value which models its opinion, and macroscopic
tion for (3) to hold is that the support of random variables
characteristics of the considered multi-agent system, such as
{Γsr }2s,r=1 is restricted to a subset of
the average opinion and the standard deviation of the opinion,
are studied analytically using a proper balance equation, whose
1
formulation is inspired by the Boltzmann equation. IΓ = 0, . (4)
2
Opinion is modeled as a continuous variable v, which is
defined in a closed interval Iv . Without loss of generality [11], This is the reason why, in the remaining of this paper, we
interval Iv is typically set to assume that the supports of random variables {Γsr }2s,r=1 are
subsets of IΓ .
Iv = [−1, 1] (1) The expected value of the sum of post-interaction opinions
of two interacting agents can be computed from (2) as
where values close to 0 represent moderate opinions while
values close to −1 or to +1 correspond to extremal opinions. E[v ∗ + w∗ ] = v + w + (Γrs − Γsr )(v − w) (5)
The first step to derive analytic results consists in the
definition of the microscopic rules that govern the effects of where Γsr denotes the average value of random variable Γsr ,
an interaction among two agents. As detailed in the introduc- and Γrs is the average value of random variable Γrs . Observe
tion, we assume that two classes of agents characterize the that (5) shows that the opinion is not conserved through single
considered multi-agent system. Let us consider an interaction interactions. Actually, the average value of the sum of the
between an agent of class s ∈ {1, 2} and an agent of class opinions of two interacting agents can increase or decrease
r ∈ {1, 2}. Denoting as v the pre-interaction opinion of the depending on the sign of (Γrs − Γsr )(v − w). Note that if
agent of class s, and as w the pre-interaction opinion of the the two random variables Γsr and Γrs have the same average
agent of class r, we assume that the post-interaction opinions value, then opinion is conserved, on average, after single
of the two interacting agents can be computed according to interactions. Similarly, the difference between post-interaction
the following rules opinions of two interacting agents is
(
v ∗ = v − Γsr (v − w) v ∗ − w∗ = εrs (v − w). (6)
(2)
w∗ = w − Γrs (w − v)
where εrs = 1−(Γrs +Γsr ) and, according to (4), εrs ∈ (0, 1).
where v and w are the opinions of the two agents after
∗ ∗ From (6) it can than be concluded that the difference be-
the interaction, and {Γsr }2s,r=1 are four mutually independent tween post-interaction opinions is smaller than the difference
random variables. between pre-interaction opinions. According to these consider-
Assuming that the support of random variables {Γsr }2s,r=1 ations, it is reasonable to expect that, after a sufficiently large
is a subset of (0, 1), then it is guaranteed that the post- number of interactions, all agents would eventually end up
interaction opinions v ∗ and w∗ still belong to Iv . In addi- with the same opinion, regardless of their classes.
tion, this choice of the support of random variables allows We now describe the kinetic framework which allows deriv-
using interaction rules (2) to model compromise, which is ing analytic results starting from microscopic rules (2). As in
the sociological phenomenon that describes the tendency of kinetic theory of gas mixtures, we need a distribution function
agents to change their opinions towards those of the agents fs (v, t) for each class s ∈ {1, 2}, where fs (v, t)dv represents
they interact with. In fact, considering, for instance, the first the number of agents of class s with opinion in (v, v + dv).
rule, it can be observed that if the value of Γsr is close to 0, Observe that, using this notation, the number of agents of class
then the post-interaction opinion v ∗ of the first agent is close s at time t, denoted as ns (t), can be computed as
to its pre-interaction opinion v. At the opposite, if the value Z
of Γsr is close to 1, then the post-interaction opinion v ∗ of ns (t) = fs (v, t)dv s ∈ {1, 2}. (7)
the first agent is close to the pre-interaction opinion w of the Iv
second agent. Hence, it can be concluded that Γsr measures the We also denote the total number of agents at time t as n(t),
propensity of an agent of type s to change its opinion in favor which can be computed as
of that of an agent of type r after an interaction. In addition,
in order to properly model the sociological characteristics of n(t) = n1 (t) + n2 (t). (8)
45
In analogy with the kinetic theory of gas mixtures, the tem- III. A NALYTIC S TUDY OF M ACROSCOPIC P ROPERTIES
poral evolution of each distribution function is described by In this section, we show how the weak form of the
∂fs Boltzmann equation can be used to derive collective and
(v, t) = Is s ∈ {1, 2} (9) asymptotic properties of considered multi-agent systems. First,
∂t
we set φ(v) = 1, so that the weak form of the Boltzmann
where Is can be computed as
equation with respect to such a test function becomes
2
X Z
d
Is = Qsr (fs , fr ) s ∈ {1, 2}, (10) fs (v, t)dv = 0 s ∈ {1, 2}. (14)
r=1 dt Iv
and Qsr (fs , fr ) is a proper operator that depends on distribu- Recalling (7), it is possible to observe that the left-hand side of
tion functions fs (v, t) and fr (v, t). Equation (9) is a balance (14) represents the time derivative of the number of agents of
equation that plays in the described framework the same role class s. Hence, it can be concluded that the number of agents
that the Boltzmann equation plays in kinetic theory of gas of any class s ∈ {1, 2} is constant, and, therefore, also the
mixtures. For this reason, we use the same nomenclature and total number of agents in the system is constant.
we call such an equation Boltzmann equation, and its right- Let us now consider the test function φ(v) = v in (12),
hand size collisional operator relative to class s. However, which leads to simplify the right-hand side of (12) as
note that the explicit expression of the collisional operator X2 Z Z Z Z
in kinetic theory of gas mixtures depends on the details βΘsr (Γsr )Θrs (Γrs )
of interactions among molecules, which are different from r=1 Iv Iv Bsr Brs
(15)
the details of interactions among agents, expressed in (2). · fs (v)fr (w)Γsr (w − v)dvdwdΓrs dΓsr .
Hence, the explicit expression of the collisional operator (10)
Observe that (15) can be used to study the temporal evolution
is different from that of kinetic theory, leading to different
of the average opinion of agents of class s. Actually, recalling
developments of analytic results.
the definition of the distribution function fs (v, t), the average
In order to obtain analytic results, the weak form of the
opinion of agents of class s at time t can be computed as
Boltzmann equation needs to be considered. Just like in kinetic Z
theory of gas mixtures, the weak form of the Boltzmann 1
us (t) = fs (v, t)vdv s ∈ {1, 2}. (16)
equation is obtained by multiplying (9) by a test function φ(v), n s Iv
a smooth function with compact support, and by integrating Note that the average opinion of the entire multi-agent system
the result with respect to v (see, e.g., [12]). Hence, the weak is computed as the following weighed sum of average opinions
form of Boltzmann equation (9) is
1
Z X 2 Z u(t) = (n1 u1 (t) + n2 u2 (t)) . (17)
∂fs n
φ(v)dv = Qsr (fs , fr )φ(v)dv. (11)
Iv ∂t Using (15) and (16), we can write
r=1 Iv
X 2 Z
By generalizing the results in [1] and in [10], the right-hand d
ns us (t) = β Γsr fs (v)fr (w)(w − v)dvdw. (18)
side of (11) can be rewritten as dt Iv 2
r=1
X 2 Z Z Z Z
From (18), it is evident that also its right-hand side can
βΘsr (Γsr )Θrs (Γrs )
(12) be expressed in terms of average opinions. Simple algebraic
r=1 Iv Iv Bsr Brs
manipulations show that the following equalities hold
· fs (v)fr (w)(φ(v ∗ (v, w)) − φ(v))dvdwdΓrs dΓsr
X2
where d
us (t) = β Γsr nr (ur (t) − us (t)) s ∈ {1, 2}. (19)
dt
1) Θsr (·) and Θrs (·) are the distributions of random vari- r=1
ables Γsr and Γrs ; The two equations (19) form a homogeneous system of
2) Bsr and Brs are the support of Γsr and Γrs , respec- first-order linear differential equations, which can be solved
tively; and analytically. In order to simplify notation, let us introduce the
3) β is the probability that two agents interact. following parameters
Note that using the fact that the integral with respect to v and
a1 = βΓ12 n2 a2 = βΓ21 n1 . (20)
the derivative with respect to t commute, the left-hand side of
(11) can be rewritten as Then, (19) can be written as
Z (
d u̇1 (t) = −a1 (u1 (t) − u2 (t))
fs (v, t)φ(v)dv. (13) (21)
dt Iv u̇2 (t) = a2 (u1 (t) − u2 (t)).
Proper choices of the test function φ(v) can be used to
Introducing the auxiliary function
study macroscopic properties of the system, as shown in the
following section. x(t) = u1 (t) − u2 (t) (22)
46
and subtracting the second equation from the first equation in TABLE I
(21), we obtain the following differential equation T HE CONSIDERED VALUES OF THE PARAMETERS FOR THE TWO CLASSES
OF AGENTS IN SIMULATIONS : NUMBER OF AGENTS , n1 AND n2 ; INITIAL
DISTRIBUTIONS OF THE OPINION , f1 (v, 0) AND f2 (v, 0); AND
ẋ(t) = −(a1 + a2 )x(t) (23) DISTRIBUTION OF RANDOM VARIABLES , Γ12 AND Γ21 .
whose solution is n1 n2 f1 (v, 0) f2 (v, 0) Γ12 Γ21
x(t) = Ce−(a1 +a2 )t (24) 900 100 U(−1,1) U(3/4,1) U(0,2/10) U(0,2/100)
990 10 U(−1,1) U(3/4,1) U(0,2/10) U(0,2/100)
where C is a constant. Recalling (22), the following relation
between u1 (t) and u2 (t) can be found
IV. V ERIFICATION OF R ESULTS BY S IMULATION
u1 (t) = u2 (t) + Ce−(a1 +a2 )t . (25)
In this section, we show analytic results obtained according
Using this result in the second equation of system (21), the to the framework outlined in previous sections for proper
following differential equation for u2 (t) is found choices of parameters. In order to confirm the validity of
such results, we compare them against those obtained by
u̇2 (t) = Ca2 e−(a1 +a2 )t (26) simulating a system composed of 103 agents, which interact
according to (2). We remark that simulations are performed by
and, hence, u2 (t) can be expressed as randomly choosing two interacting agents at each step and by
implementing interaction rules (2), independently of analytic
a2 results. Table I shows the values of the parameters relative to
u2 (t) = −C e−(a1 +a2 )t + K, (27)
a1 + a2 the two classes of agents that are considered to derive analytic
and simulation results. In particular, different values of the
where K is a constant. Finally, inserting the expression of parameters are considered for:
u2 (t) in (25), we obtain
1) The number of agents {ns }2s=1 ;
a1 2) The initial distribution of the opinion; and
u1 (t) = C e−(a1 +a2 )t + K (28) 3) The distribution of random variables {Γsr }2s,r=1 .
a1 + a2
First, we consider the parameters shown in the first row
where K is the same constant used in (27). The two constants of Table I. In this case, n1 = 900 and n2 = 100, meaning
C and K can be found by imposing that initial conditions are that 90% of the agents belong to class 1 and only 10% of
satisfied. Simple algebraic manipulations show that the agents belong to class 2. Initial opinions of agents of
class 1 are uniformly distributed in interval Iv , as shown
C = u1 (0) − u2 (0) in the third column. Therefore, the initial average opinion
a2 a1 (29)
K = u1 (0) + u2 (0) of agents of class 1 is 0. Initial opinions of agents of class
a1 + a2 a1 + a2 2, instead, are uniformly distributed in the smaller interval
(3/4; 1). This choice implies that agents of class 2 have
where {us (0)}2s=1 are the initial average values of the opinions
extremal opinions and that their initial average opinion is
of the two classes of agents.
u2 (0) = 7/8. Another feature that distinguishes the agents in
Therefore, it can be concluded that the solution of (21) can
the two classes concerns their inclination to change opinion.
be expressed in closed form as
The distributions of random variables Γ12 and Γ21 are related
a1 to such an inclination. As shown in Table I, we assume that Γ12
u1 (t) = C e−(a1 +a2 )t + K
a1 + a2 has uniform distribution in interval (0, 2/10), corresponding
a2 (30) to an average value Γ̄12 of 1/10, and that Γ21 has uniform
u2 (t) = −C e−(a1 +a2 )t + K
a1 + a2 distribution in interval (0, 2/100), corresponding to an average
value Γ̄21 of 1/100. According to such choices, Γ̄12 = 10Γ̄21 ,
where C and K are computed in (29) using the initial which means that the propensity of agents of class 2 to change
distributions of the opinion of the two classes of agents. From their opinions in favor of those of agents of class 1 is much
(30), it can be observed that since a1 > 0 and a2 > 0, the lower than the propensity of agents of class 1 to change
following equalities hold their opinions in favor of those of agents of class 2. For
this reason, agents of class 2 can be considered skeptical.
lim u1 (t) = lim u2 (t) = K. (31) Fig. 1 shows the average opinion u1 (t) of the agents of class
t→+∞ t→+∞
1 (blue line) and the average opinion u2 (t) of the agents of
Observe that, according to (20) and (29), K depends on the class 2 (red line). As expected from (31), u1 (t) and u2 (t)
average initial opinions {us (0)}2s=1 , on the number of agents converge to the same value, which, according to this choice
{ns }2s=1 in each class, and on Γ12 and Γ21 . of parameters, corresponds to K ' 0.46. Fig. 1 also shows
47
0.9 0.9
0.8 0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1 0
0 −0.1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 2 2.5 3
t 5
x 10 t 5
x 10
Fig. 1. Average opinions u1 (t) (blue line) and u2 (t) (red line) derived Fig. 2. Average opinions u1 (t) (blue line) and u2 (t) (red line) derived
analytically, and average opinions ũ1 (t) and ũ2 (t) (dotted black lines) analytically, and average opinions ũ1 (t) and ũ2 (t) (dotted black lines)
obtained by simulation, all computed with values in the first row of Table I. obtained by simulation, all computed with values in the second row of Table I.
the values of {ũs (t)}2s=1 obtained by simulation (dashed black 3) Fig. 3(c) shows the distributions fs (v, t) after 1.5 · 105
lines), showing that analytic results are in agreement with interactions, which corresponds to 300 interactions per
those obtained by simulation. agent on average; and
We now consider the parameters shown in the second row 4) Fig. 3(d) shows the distributions fs (v, t) after 2 · 105
of Table I, which differ from those in the first row only for interactions, which corresponds to 400 interactions per
{ns }2s=1 . The initial distributions of opinion {fs (v, t)}2s=1 agent on average.
and, hence, the initial values of the average opinion of the From Fig. 3 it can be observed that not only the average
two classes of agents, are the same as in the first scenario. opinions us (t) converge to the same value K ' 0.46, as
This means that agents of class 2 still have extremal opinions. already shown in Fig. 1, but also that consensus among agents
Also the distributions of random variable Γ12 and Γ21 are the is reached, since the opinions of all agents tend to the same
same as in the first scenario, so that agents of class 2 can value, as expected from the analytic model. Similar diagrams
still be considered skeptical. The only difference is that, in can be drawn using the parameters in the second row of
this case, skeptical agents are only 1%. In fact, 990 agents, Table I, obtaining similar results.
corresponding to 99% of the total number of agents, belong
to class 1, and only 10 agents, corresponding to 1% of the V. C ONCLUSIONS
total number of agents, belong to class 2. Fig. 2 shows the This paper presented an analytic model of opinion dynamics
average opinion u1 (t) of the agents of class 1 (blue line) and which assumes that the agents in the studied multi-agent
the average opinion u2 (t) of the agents of class 2 (red line). system can be grouped into two classes on the basis of their
As expected from (31), the values of u1 (t) and u2 (t) converge characteristic parameters. Such classes can be used to model
to the same value, which now corresponds to K ' 0.08. The agents with different propensity to change their opinions after
values of ũ1 (t) and ũ2 (t) (dotted black lines) obtained by interactions, and they are used here to model the presence
simulation with the parameters in the second row of Table I of skeptical agents in the multi-agent system. Among the
are also shown. As in the first scenario, simulation results are sociological phenomena that can be used to describe the
in agreement with analytic ones. dynamics of the opinion, the presented model considers only
In order to improve the analysis of considered scenarios, compromise, which is the phenomenon that describes interac-
Fig. 3 shows the distributions f1 (v, t) (solid blue lines) and tions that tend to consensus. The characteristic autonomy of
f2 (v, t) (dashed red lines) of the opinions of the two classes agents is modeled in terms of random variables used at the
of agents obtained by simulating the multi-agent system with microscopic level to perturb the classic model of compromise,
the parameters in the first row of Table I. In detail: which is deterministic. Analytic results ensure that the average
1) Fig. 3(a) shows the distributions fs (v, t) after 5 · 104 opinion of the multi-agent system is conserved, and that
interactions, which corresponds to 100 interactions per consensus is always reached for a sufficiently large number of
agent on average; interactions. Simulations shown in the last part of the paper
2) Fig. 3(b) shows the distributions fs (v, t) after 105 inter- confirm such properties.
actions, which corresponds to 200 interactions per agent Ongoing research involves the extension of the presented
on average; model to account for major sociological phenomena (see,
48
10 20
8
15
6
10
4
5
2
0 0
−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1
v v
(a) (b)
40 50
40
30
30
20
20
10
10
0 0
−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1
v v
(c) (d)
Fig. 3. The distributions f1 (v, t) (blue line) and f2 (v, t) (red line) obtained by simulation using the parameters in the first row of Table I for: (a) t = 5 · 104 ;
(b) t = 105 ; (c) t = 1.5 · 105 ; (d) t = 2 · 105 .
e.g., [8]). Among considered phenomena, ongoing research [5] S. Monica and F. Bergenti, “Kinetic description of opinion evolution in
includes stochastic models for: multi-agent systems: Analytic model and simulations,” in PRIMA 2015:
Principles and Practice of Multi-Agent Systems, Q. Chen, P. Torroni,
• Diffusion, the phenomenon according to which the opin- S. Villata, J. Hsu, and A. Omicini, Eds. Springer, 2015, pp. 483–491.
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• Homophily, the process according to which agents interact
Distributed Computing and Artificial Intelligence (DCAI 2016), Sevilla,
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• Negative influence, the idea according to which agents [7] S. Monica and F. Bergenti, “An analytic study of opinion dynamics
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Advances in Artificial Intelligence: XVth International Conference of
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vol. 73, no. 10, pp. 2272–2284, 2017.
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