A DYNAMICAL MODEL FOR SIMULATING
A DEBATE OUTCOME
A. Imoussaten, J. Montmain
LGI2P, Ecole des Mines d'Alès, Site EERIE Parc Scientifique G. Bresse, 30035 Nîmes cedex, Alès, France
A. Rico, F. Rico
ERIC, Université Claude Bernard Lyon 1, 43 bld Du 11 novembre 69100, Villeurbanne, France
Keywords: Debate, Influence, Decisional power, Choquet integral, Control, Collective decision, Social network.
Abstract: A group of agents is faced with collective decisional problems. The corresponding debate is seen as a
dynamical process. A first theoretical model based upon a muticriteria decision framework was proposed in
(Rico et al., 2004) but without semantic justifications and explicit dynamical representation. A second
descriptive model was proposed in (Imoussaten et al., 2009) where social influences and argumentation
strategy govern the dynamics of the debate. This paper aims at justifying the equations introduced in (Rico
et al., 2004) with the semantics concepts reported in (Imoussaten et al., 2009) to provide a model of a debate
in the framework of control theory that explicitly exhibits dynamical aspects and offers further perspectives
for control purposes of the debate.
1 INTRODUCTION
A group of agents is faced with a collective decision.
A debate is organized to identify which alternative
appears to be the most relevant one after
deliberation. This study is limited to the binary but
common situation where two options
1±
are
involved. It is assumed that each agent has an
inclination to choose one of both alternatives
1±
which, due to influence of other agents, may be
different from the decision of the agent (Grabisch
and Rusinowska, 2008). More generally, it can be
considered that each time a speaker intervenes in the
debate, agents may change their preference due to
social influences in the group. When agents
preferences do not change anymore, the deliberation
process ends and a group decision is made. The aim
of the debate is that every agent knows the
arguments of all the others at the end of the
deliberation process and makes his final decision
with full knowledge of the facts.
The deliberation is seen as a dynamical process
with its own dynamics where beliefs and preferences
of agents evolve when arguments are exchanged.
The deliberation outcome thus depends on the order
the agents intervene in the debate to explain their
opinion and on the influence an agent may exert on a
social network.
Social influence is here related to statistical
notion of decisional power of an individual in a
social network as proposed in (Hoede and Bakker,
1982) and (Grabisch and Rusinowska, 2008).
One of the conclusions of (Grabisch and
Rusinowska, 2008) concerns the integration of
dynamical aspects in the influence model. Indeed,
the authors’ framework is a decision process after a
single step of mutual influence. In reality, the mutual
influence does not stop necessarily after one step but
may iterate. This paper proposes a possible
extension of (Grabisch and Rusinowska, 2008) in
the dynamical case. The evolutions of agents’ beliefs
during the debate change or reinforce the agents’
convictions relatively to their initial preference.
Intuitively, among others, the social influence of an
agent depends on the more or less marked
convictions of the other agents. Thus, the idea is to
define influence as a time-varying variable itself in
our model.
(Rico et al., 2004) introduces the concepts of
influence and conviction in the simulation of a
debate. This article follows prior works proposed in
(Bonnevay et al., 2003). In (Rico et al., 2004),
31
Imoussaten A., Montmain J., Rico A. and Rico F..
A DYNAMICAL MODEL FOR SIMULATING A DEBATE OUTCOME .
DOI: 10.5220/0003126400310040
In Proceedings of the 3rd International Conference on Agents and Artificial Intelligence (ICAART-2011), pages 31-40
ISBN: 978-989-8425-40-9
Copyright
c
2011 SCITEPRESS (Science and Technology Publications, Lda.)
coalitions of agents are modeled with capacities; the
change of conviction during the debate was
computed with a symmetric Choquet integral which
is an aggregation function usually used in
multicriteria decision making (Grabisch and
Labreuche, 2002). The main drawback in (Rico et
al. 2004) is its lack of semantic justifications.
Thus, (Grabisch and Rusinowska, 2008) provides
a formal framework to define the notion of influence
and (Rico et al., 2004) introduces the revision
equations of agents’ convictions and preferences.
Finally, (Imoussaten et al., 2009) suggests a
cybernetic interpretation to merge both models. This
paper is the continuation of (Rico et al., 2004) in the
light of (Imoussaten et al., 2009). The main
contribution of this paper is to propose the state
equations of the cybernetic interpretation to describe
the way agents’ convictions may evolve in time. To
achieve this goal, a capacity is introduced to model
the relative importance of agents in the debate that is
based upon the decisional power of agents using the
generalized Hoede-Bakker index (Grabisch and
Rusinowska, 2008), (Hoede and Bakker, 1982).
Hence some simulations are proposed to illustrate
the collective decision making process.
The paper is organized as follows. Section 2
recalls briefly the main concepts of models in (Rico
et al., 2004) and (Grabisch and Rusinowska, 2008).
Based upon this formal framework, section 3
establishes the state equations that model the
dynamical relationships between convictions and
influences when a pair of speaker-agent, listener-
agent is isolated. Section 4 associates the revision of
convictions and the changes of preferences. Section
5 proposes some illustrations. Finally, the
conclusion evocates the use of the model for debates
controlling purposes.
2 CONCEPTS AND NOTATIONS
2.1 Notion of Influence in a Debate
The assumption behind our model is that the
influence of an agent is related to his capacity to
alter the group decision. It evocates the concept of
«weight» of an agent’s choice in a collective vote
procedure. This «weight» cannot be a static
parameter, because it should evolve with the
preferences of agents that make the formation of
certain coalitions more probable than other ones. To
tackle this issue the definition of decisional power as
proposed in (Grabisch and Rusinowska, 2008) is
first summarized.
We consider a set of agents denoted
1
{ ,..., }
N
aa
or
{1,..., }N
to simplify the notations and the power set
is denoted
1
{ ,..., }
2
N
aa
. It is assumed that each agent has
an inclination to choose +1 or -1 which, due to
influence of other agents, may be different from the
decision of the agent. The point of departure is the
concept of the Hoede-Bakker indexthe notion
which computes the overall decisional ‘power’ of an
agent in a social network (
n
agents). This index was
provided in 1982 (Hoede and Bakker, 1982).
Definition:
the Hoede-Bakker index of agent
j
a
is
defined by:
1
{/ 1}
1
(, ) . ( )
2
j
a
j
a
N
ii
GHB B gd gd Bi
=+
=
(1)
i
is an inclinations vector in
{1, 1}
N
I =− +
that
models the agents’ inclinations, more precisely, we
have
1
( ,..., )
N
aa
ii i=
where
{1, 1}
j
a
i ∈− +
the
jth
coordinate of
i
is the inclination of the agent
j
a
.
:
B
II
is the influence function and for any
inclination vector
i
the decision vector
B
i
is a n-
vector consisting of ones and minus ones and
indicating the decisions made by all agents.
:() {1,1}gd B I →− +
is the group decision
function, having the value +1 if the group decision is
+1, and the value 1 if the group decision is 1.
The main drawback of the Hoede-Bakker index
is that it hides the actual role of the influence
function, analyzing only the final decision in terms
of success and failure. The decision is successful for
an agent as soon as his inclination matches the group
decision.
In (Grabisch and Rusinowska, 2008), the authors
separate the influence part from the group decision
part, and propose a first modified index of decisional
power where the decision of the agent must coincide
with the group decision to be a success for the agent.
Lastly, the authors provide a second modified
decisional power, which allows the inclinations
vectors to be unequally probable.
Definition:
Let
:[0,1]pI
be a probability
distribution,
()pi
is the probability
i
occurs. The
modified decisional power is then:
{/( ) 1}
{/( ) 1}
(, , ) (). ( )
(). ( ).
j
a
j
a
j
a
iBi
iBi
B
gd p p i gd Bi
pi gd Bi
φ
=+
=−
=
ICAART 2011 - 3rd International Conference on Agents and Artificial Intelligence
32
To conclude this summary, for each agent
j
a
the
probabilities of success and failure are reminded:
1
{ /() ()}
(, , ) ()
j
a
j
a
bI b gdb
SUC B gd p p B b
∈=
=
D
1
{/() ()}
(, ,) ()
j
a
j
a
bI b gdb
F
AIL B gd p p B b
∈=
=
D
Note that we have
(, ,) (, ,) (, ,)
jjj
aaa
B gd p SUC B gd p FAIL B gd p
φ
=−
2.2 Convictions and Preferences in a
Debate
This section presents the dynamical model of the
debate proposed in (Rico et al., 2004). The influence
an agent may have on the others is modeled by a
capacity over
1
{ ,..., }
2
N
aa
.
Definition: A capacity
υ
over
1
{ ,..., }
2
N
aa
is a set
function
1
{,..., }
:2 [0,1]
N
aa
υ
such
that
() 0
υ
∅=
,
1
({ ,.., }) 1
N
aa
υ
=
and
,'AA∀⊆
1
{,.., }
N
aa
,
'()(')
A
AAA
υυ
.
The profile of an agent
j
a
includes his preference,
his importance (his capacity
()
j
a
υ
), his conviction
[0,1]
j
a
c
related to his preference. It is stated as a
rule that agents speak in turns. The agent
a
(speaker-agent) who speaks and any agent
l
a
(listener-agent) are isolated which introduces a
capacity
,
s
l
aa
υ
upon the pair of agents
(, )
ls
aa
. More
precisely, the following capacity is defined
,
()
()
(, )
ls
l
aa l
ls
a
a
aa
υ
υ
υ
=
,
,
()
()
(, )
ls
s
aa s
ls
a
a
aa
υ
υ
υ
=
and
,
(, ) 1
ls
aa l s
aa
υ
=
.
The change of conviction is then modeled with
the symmetric Choquet integral also called Sipos
integral. The definition of Choquet integral and
Sipos integral are now provided.
Definition: Let
1
( ,..., ) [0,1]
N
N
aa
cc c=∈
be a vector of
convictions, () be a permutation on
{1,..., }N
such
that
(1) ( )
...
N
aa
cc≤≤
and
υ
be a capacity on
1
{ ,..., }
2
N
aa
.
The Choquet Integral of
c
with respect to
υ
is
defined by:
() ( 1)
1
({( ),..., ( )})() .
ii
N
aa
i
iNCc c c
υ
υ
=
⎡⎤
=−
⎣⎦
with
(0)
0
a
c =
Definition: Let
1
( ,..., ) [ 1,1]
N
N
aa
cc c=∈
be a vector
which can take negatives values, () be the
permutation on
{1,..., }N
such that
(1) ( ) ( 1) ( )
... 0 ...
pp N
aaa a
ccc c
+
≤≤ ≤≤
and
υ
be a
capacity on
1
{ ,..., }
2
N
aa
.
The symmetric Choquet Integral of
c
with respect to
υ
is defined by:
() ( 1) ( )
(1) () ( 1)
1
1
2
( ) ({(1),..., ( )}) ({( ),..., ( )})
({( 1),..., ( )}) ({( ),..., ( )})
ii p
p ii
p
aa a
i
N
aaa
ip
Cc c c i c i p
cp N cc iN
υ
υυ
υυ
+
+
=
=+
=− +
++ +
⎡⎤
⎣⎦
⎡⎤
⎣⎦
In this paper the Sipos integral is defined on the set
of agents
{, }
ls
aa
. It is denoted
,aa
s
l
C
υ
. The changes
of convictions proposed in (Rico et al., 2004) can be
summarized as follows with
,aa
s
l
C
υ
:
- If agents
l
a
and
a
have the same preference,
When
s
l
aa
cc>
the new conviction is:
,
,
(,) ( ). ()
aa s l l s l l s
ls
aa a a a aa s
Ccc ccc a
υ
υ
=+
When
ls
aa
cc>
the new conviction is:
,
,
(,) ( ). ()
aa s l s l s l s
ls
aa a a a aal
Ccc c cc a
υ
υ
=+
.
- If agents
l
a
and
a
do not have the same
preference, the new conviction is:
,
,,
(,) . () . ()
aa s l s l s l l s
ls
aa aaa s aaa l
Ccc c ac a
υ
υυ
=− +
The main drawback to this model is its lack of
semantics justifications with regard to capacity
υ
(influence is merely a normalized relative
importance), the concept of conviction is not
formally defined and the revision equations are not
provided in an appropriate formalism where time
would appear explicitly (dynamical aspects).
3 THE DYNAMICAL MODEL
This section presents our dynamical model for
simulating a debate outcome. To begin note that in
the framework of this paper, influence function
B
used in (Grabisch and Rusinowska, 2008) is
perceived as a disturbance function applied to the set
of all the possible inclination vectors.
A DYNAMICAL MODEL FOR SIMULATING A DEBATE OUTCOME
33
3.1 Decisional Power and Capacities
This section proposes to design a capacity based
upon the decisional power for the above model.
For any
iI
, the group decision is modeled by
()
g
dBi
and belongs to
{1, 1}−+
.
Furthermore,
(, , ) [1,1]
j
a
Bgd p
φ
∈−
.
- If the decisional power of an agent is close to
1
, it
means that the agent scarcely chooses the alternative
the collective finally chooses: he fails most of the
time (
FAIL
).
- In revenge, when his decisional power is close to 1,
the agent is most of the time successful
()SUCC
;
his decisional power is high.
For example, without further information, the
importance of an agent
j
a
, i.e., his capacity
()
j
a
υ
,
can be defined as:
.
11
() (, ,)
22
j
ja
aBgdp
υφ
=+
,
()[0,1]
j
a
υ
with
() 0
j
a
υ
=
if and only if
(, , ) 1
j
a
Bgd p
φ
=−
and
()1
j
a
υ
=
if and only if
(, ,) 1
j
a
Bgd p
φ
=
.
It thus defines a function
1
:{ ,.., } [0,1]
N
aa
υ
.
From this function, a capacity
υ
can be generated
over
1
{ ,.., }
2
N
aa
, with constraints
,'AA∀⊆
1
{,.., }
N
aa
,
'()(')
A
AAA
υυ
. Without further
knowledge, it can be chosen:
1
() max( ), {,.., }
j
jN
aA
A
aAaa
υυ
=∀
and
1
({ ,.., }) 1
N
aa
υ
=
Note that this definition does not necessarily
imply that there exists an agent whose capacity is
equal to 1.
In the following and to simplify notations, such a
capacity is denoted
φ
υ
for a decisional power
(, ,)Bgd p
φ
. The decisional power of individuals
j
a
on which
1
{ ,.., }
:2 [0,1]
N
aa
φ
υ
is based measures the
cases where the final decision of
j
a
matches the
group decision. An agent with a high decisional
power is expected to bring several agents round and
thus the decisional power is considered as an
estimation of his “influence” in the group; although
it is not an influence index in the sense of (Grabisch
and Rusinowska, 2008).
3.2 Time-varying Probabilities
Note that this subsection is dedicated to the design
of probability
p
as a time-varying function. It is
thus supposed that convictions vectors
()ck
(the
convictions vector of the agents w.r.t alternative
1+
at time
k
) and
'( )ck
(the convictions vector of the
agents w.r.t alternative
1
at time
k
) are known at
k. Their computation is provided in the next section.
The model proposed in this paper is based upon
the extended decisional power in (Grabisch and
Rusinowska, 2008) that allows the inclinations
vectors to be unequally probable. The definition of
the associated probability distribution
:[0,1]pI
is now required (see section 2.1). This paper
proposes to base the probability computation upon
the convictions of agents with regard to the
alternatives.
The conviction of an agent regarding an
alternative is related to the probability this agent
chooses this alternative, i.e., the probability of his
inclination as defined in (Grabisch and Rusinowska,
2008). As stated above, convictions evolve in time
during the deliberation process.
()
1
() (),.. , (), ..., ()
jN
aa a
ck c k c k c k=
where
()
j
a
ck
is the conviction of agent
j
a
w.r.t alternative
1+
at
time
k
.
1
'' '
'( ) (c ( ),... ,c ( ), ...,c ( ))
jN
aa a
ck k k k=
where
'
()
j
a
ck
is the conviction of agent
j
a
w.r.t alternative
1
at
time
k
.
Let
iI
be an inclinations vector, and let define
() [0,1]
i
ck
as an “average” conviction at time
k
for
i
: this value summarizes the distributions of
agents’ convictions in
i
at
k
.
()
i
ck
is an
«aggregated conviction» of the group of agents for
i
. This aggregation should take into account relative
importance of agents and their interactions.
The probability is built by recurrence on k.
At time k = 0:
()
1
(0) (0),... , (0), ..., (0)
jN
aaa
cc c c=
is the a priori convictions vector of agents.
(0) [0,1], 1..
j
a
cjN∈=
is the a priori convictions
of
j
a
, and it is also the probability of his conviction.
Initially (
0k =
), if
j
a
i
is the preference of
j
a
then
the probabilities of the agent
j
a
regarding his
preference and the other alternative are:
(()[ 0] 0)
jj j
aa a
pi k c==
and
( )[0] 1 (0)
jj j
aa a
pi c−=
Before the debate starts, the inclination of each agent
ICAART 2011 - 3rd International Conference on Agents and Artificial Intelligence
34
does not depend on the social network. Then, the
probability distribution associated to a priori
probabilities is the product of the individual
probabilities
j
a
p
at
0k =
:
1
,()[0] ( )[0]
jj
N
jaa
iIpi pi
=
∀∈ =Π
. It is thus possible:
-
computing the decisional power for any agent
j
a
at
0k =
,
(, ,[0])
j
a
Bgd p
φ
;
- computing the capacity
[0]
φ
υ
over
1
{ ,.., }
2
N
aa
,
for
1k =
,
as proposed in subsection 3.1.
At time k = 1
Capacity
[0]
φ
υ
allows computing
(1)
i
c
, the
aggregated conviction at
1k =
for the inclinations
vector
i
. The relative importance of agents and of
their coalitions is taken into account in the
aggregation model of
i
c
through a Choquet integral.
Let
iI
be an inclinations vector. Each
coordinate
j
a
i
is one of the alternative
1
or
1+
. For
agent
j
a
,
(1)
j
a
c
or
'
(1)
j
a
c
is the conviction
associated to his preference. A conviction vector
()
1
(1) (1) ... (1)
n
aa
cc c=
is associated to each
inclinations vector
i
, where for any
j
,
(1)
j
a
c
is
(1)
j
a
c
or
'
(1)
j
a
c
. The synthetic conviction of the
group for inclination
i
at time
1k =
is computed
with a Choquet integral defined upon
[0]
φ
υ
.
1
[0]
(1) ( (1), ..., (1)) [0,1]
N
i
aa
cCc c
φ
υ
=∈
A probability at
1k =
can then be defined:
(1)
()[1]
(1)
i
t
tI
c
pi
c
=
It is now possible:
- Computing the decisional power of any agent
j
a
at
1k =
,
(, ,[1])
j
a
Bgd p
φ
;
- Computing the capacity
[1]
φ
υ
over
1
{ ,.., }
2
N
aa
,
for
2k =
,
as proposed in subsection 3.1.
More generally, at time k + 1:
Capacity
[]k
φ
υ
computed at
k
allows
computing
(1)
i
ck+
, the aggregated conviction at
1k +
for inclinations vector
i
with the Choquet
integral:
1
[]
( 1) ( ( 1),..., ( 1))
N
i
ka a
ck C c k c k
φ
υ
+= + +
.
Then, probability at time
1k +
is defined as:
(1)
()[ 1]
(1)
i
t
tI
ck
pi k
ck
+
+=
+
It is now possible computing
(, ,[ 1])
j
a
Bgd pk
φ
+
and
[1]k
φ
υ
+
.
The probability required by the extended model
of decisional power has been designed as a time
varying variable because it evolves with the agents’
convictions. Therefore,
(, ,[])Bgd pk
φ
evolves in
time too. This principle seems rather intuitive
because it corresponds to the idea that the social
influence of an agent depends on the more or less
marked convictions of the other agents when he
speaks.
3.3 Conviction State Equations
The aim of this section is to establish the state
equations that model the dynamical relationship
between convictions and influences. Let consider a
pair of listener-agent, speaker-agent denoted
l
a
and
a
. Their convictions for the alternative
1+
are
()
l
a
ck
and
()
s
a
ck
, respectively,
'
()
l
a
ck
and
'
()
s
a
ck
for the alternative
1
.
Two variables are necessary to model the
rhetoric quantity that is exchanged between both
agents
l
a
and
a
:
-
The difference of convictions between both agents;
- The relative importances of agents
l
a
and
a
modeled by capacities
[]( )
s
ka
φ
υ
and
[]( )
l
ka
φ
υ
.
Four rhetoric exchanges are distinguished. These
four situations are presented in the case when
l
a
prefers alternative
1+
. Then, there exist two sub
cases for agent
a
: his favorite alternative is the one
of
a
or the opposite one. Each case can be divided
again into two sub cases:
a
’s conviction is greater
than (respectively lower than)
l
a
’s conviction.
When agent
l
a
prefers the alternative
1
,
convictions
'c
take the place of convictions
c
in the
formula: the equations that appear in the
computation of
(1)
l
a
ck+
when both agents have the
same preference are the same ones to compute
'
(1)
l
a
ck+
in case of opposite preferences and vice
versa.
Synergic Exchange
It is the case when the preference of the agent
l
a
is
reinforced by the intervention of the agent
a
who
resolutely looks on the same alternative in favor.
The conviction of the agent
l
a
increases. The
increase is proportional to the difference between
A DYNAMICAL MODEL FOR SIMULATING A DEBATE OUTCOME
35
both convictions and to the capacity of speaker
a
.
This situation corresponds to the case when
l
a
and
a
have the same preference and moreover
s
l
aa
cc>
. The intuitive difference equation is then
(Figure 1):
( 1) ( ) ( ( ) ( )). [ ]( )
llsl
aaaa s
ck ck ck ck ka
φ
υ
+− =
or
( 1) () ( () ()). []( )
llsl
aaaa s
ck ck ck ck ka
φ
υ
+= +
+
-
()
s
a
ck
()
l
a
ck
[]( )
s
ka
φ
υ
[]( )
s
ka
φ
υ
Figure 1: Synergic Exchange.
Revisionist Exchange
The agent
l
a
understands the argument of the
agent
a
, who has the same preference but more
moderately.
a
appears to speak with restraint from
l
a
point of view
and
l
a
’s
doubt appears.
l
a
’s
conviction is thus mitigated by
a
intervention. This
situation corresponds to the case when
l
a
and
a
have the same preference and moreover
s
l
aa
cc>
.
The intuitive difference equation is then (Figure 2):
( 1) () ( () ()).(1 []( ))
llls
aaaa l
ck ck ck ck ka
φ
υ
+− =
or
( 1) () ( () ()). []( )
lsls
aaaa l
ck ck ck ck ka
φ
υ
+= +
The agent
l
a
observes the indecision
of
agent
a
who nevertheless shares his opinion:
a
contributes to
l
a
’s
doubt. The conviction
decreases due to
a
’s intervention that is
proportional to
(1 [ ]( ))
l
ka
φ
υ
on one hand (lack of
assurance of
l
a
related to his social position in the
group) and to the difference between both
convictions of agents
a
and
l
a
on the other hand.
+
-
1[]()
l
ka
φ
υ
1[]()
l
ka
φ
υ
()
l
a
ck
()
s
a
ck
Figure 2: Revisionist exchange.
Antagonist Exchange
Both agents do not share the same preference;
agent
l
a
nevertheless understands the advantages of
a
preference. A convincing intervention of
a
may
contribute to make
l
a
doubtful whereas a non
persuasive intervention may strengthen his
preference on the contrary.
'
(1 ( ))
s
a
ck
is a measure of
a
‘s hesitation and
provides
l
a
with an estimation of the strength of
a
’s
opposition. According to the strength of this
hesitation, the previous difference equations are
usable with
'
(1 ( ))
s
a
ck
and two situations are to be
distinguished (Figure 3).
A too weakly marked preference of
a
means a
weak opposition from
l
a
point of view and
reinforces
l
a
‘s opinion.
l
a
‘s conviction should then
increase.
The intuitive difference equation is then
(synergic exchange with
'
(1 ( ))
s
a
ck
):
Case 1:
'
1
s
l
aa
cc−≥
'
(1)()((1())()).[]()
ll sl
aa aa s
ck ck ck ck ka
φ
υ
+− =
or
'
(1) ()(1 () ()).[]()
ll sl
aa aa s
ck ck ck ck ka
φ
υ
+= +
In the second case,
l
a
‘s conviction decreases after
a
’s intervention (revisionist exchange with
'
(1 ( ))
s
a
ck
).
Case 2:
'
1
s
l
aa
cc−<
'
( 1) () ( () (1 ())).(1 []( ))
lll s
aaa a l
ck ck ck ck ka
φ
υ
+− = −−
or
''
(1)(1 ())(() ()1).[]()
lsls
aaaa l
ck ck ck ck ka
φ
υ
+= + +
All these different types of exchanges can be
synthesized with a Sipos integral as follows:
The agents
a
and
l
a
have got the same preference:
( 1) [ ]( ( ), ( ))
lsl
aaa
ck Ckckck
φ
υ
+=
The agents
a
and
l
a
do not share the same
preference:
'
( 1) [ ]((1 ( )), ( ))
lsl
aaa
ck Ck ck ck
φ
υ
+=
To conclude this part, the decisional power
φ
provides a semantic interpretation for the capacity
υ
in the recurrence equations in (Rico et al., 2004),
conviction is here related to the probability an agent
will choose an alternative rather than the other one
(probability distribution over inclinations vectors).
ICAART 2011 - 3rd International Conference on Agents and Artificial Intelligence
36
Thus, the model in (Rico et al., 2004) becomes
interpretable in games theory framework (Grabisch
and Rusinowska, 2008). Revision equations of
conviction appear as inputs-outputs balances
according to alternatives assessment. Introducing
time in the equations of (Rico et al., 2004) implies
that revision equations of conviction are now seen as
state equations of agents’ mental perception. This
new interpretation provides a semantics for the
model of a debate in (Rico et al., 2004): it is related
to the notions of influence and decisional power as
proposed in (Grabisch and Rusinowska, 2008) with
a formalism close to the one of dynamical models in
control theory as suggested in (Imoussaten et al.,
2009).
+
-
()
l
a
ck
[]( )
s
ka
φ
υ
[]( )
s
ka
φ
υ
'
(1 ( ) )
s
a
ck
Cas 1
+
-
1[]()
l
ka
φ
υ
1[]()
l
ka
φ
υ
()
l
a
ck
Cas 2
'
(1 ( ) )
s
a
ck
Figure 3: Antagonist exchange.
4 PREFERENCES CALCULUS
This section presents how to compute the preference
during the debate.
Initially each agent
j
a
assesses both alternatives
1+
and
1
with a score in
[0,1]
. These assessments
are noted
1
j
a
n
+
and
1
j
a
n
. It is then possible to build
initial preferences and convictions:
-
j
a
prefers alternative
a
with the highest score,
j
a
’s conviction related to alternative
a
is
()
jjj
aaa
aaa
nnn+
and
j
a
’s conviction related to the
other alternative
a
is
()
jjj
aaa
aaa
nnn+
.
Preferences changes depend on the way convictions
evolve in time. For any agent
j
a
, it is supposed
there exists a threshold
0
j
a
ε
>
such that when the
difference between two convictions is below this
threshold then the agent
j
a
cannot have a
preference. The threshold value may be
characteristic of each agent. To summarize:
- When
'
jj j
aa a
cc
ε
−<
, then
has no preference;
- When
'
jj j
aa a
cc
ε
−≥
,
prefers the alternative
with the highest conviction.
Finally, an agent without preference cannot
intervene is stated as a rule of the debate.
5 ILLUSTRATION
5.1 Simulations of the Debate’s
Outcome
In order to illustrate the principle of the above
dynamical representation of a debate, the four
following elementary models for influence function
B
have been implemented:
B
is the identity that is to say for any inclination
vector i we have
B
ii=
B
is the opposite of identity, for any inclination
vector i we have
B
ii=−
B
is a mass psychology effect function. More
precisely, if we denote
{/}
k
ikNi
ε
ε
=∈ =
,
B
satisfy : for each
iI
:
,()itthenBi i
εεε
>⊇
where
[1, ]tn
and
1
ε
.
B
is a majority
influence function models behaviors of type: if a
majority of agents has an inclination +1, then all
agents decide +1; if not, all agents decide 1
For the four cases, the group decision function
g
d
is a mere majority and a basic capacity is designed
as proposed in section 1.3.
Let consider a group of
8N =
agents. The initial
convictions of agents relatively to both alternatives
are considered as variates: 50 random drawings of
these 8 initial probabilities are carried out (Figure 4).
For each of these 50 initial convictions vectors the
order the agents intervene in the debate is then
considered: 200 permutations are randomly selected
(among the
8!
possible rankings) for each initial
convictions vector.
Each of the four elementary illustrations is
plotted in figure 4 (one for each
B
function). For
each of the 50 initial convictions vectors randomly
selected, a bar represents the number of outcomes
1±
(light-grey for
1+
and dark-grey for
1
).
To each of these figures is associated the
maximal number of rounds that have been necessary
to achieve the ground decision for each initial
convictions vector. In the proposed simulations this
number does not exceed 8 rounds in any
B
-case.
A DYNAMICAL MODEL FOR SIMULATING A DEBATE OUTCOME
37
The indifference threshold is
0.01
ε
=
for any
agent. Agents speak in turns according to the order
induced by the 200 permutations on condition they
have a clear opinion: an agent
j
a
can speak if
'
jj
aa
cc
ε
−≥
.
For a same initial convictions vector it can be
observed that for each function
B
, the outcome of
the debate may depend on the order the agents
intervene. This type of situation can be interpreted as
weakly marked preferential contexts where any
perturbation can change the debate’s outcome. From
this point of view, influence function
B
is a
disturbance function in this dynamical model of a
debate. As a consequence, simulations allow
checking that the order the agents intervene in the
debate and their influence are decisive variables with
regard to the convergence of conviction state
equations.
The social influence of an agent may thus be
considered as a disturbance in the deliberation
process except if it is relevantly used by the debate
manager to govern the discussion. Indeed, in this
later case, social influence can be envisaged as an
actuator that enables controlling the outcome of the
debate or at least accelerating its convergency. For
example, when the outcome of the debate is quasi
certain (the bar is almost completely light or dark
grey), then the simplest control could consist in
choosing the order the agents intervene that
minimizes the maximal number of rounds. More
complex control can be clearly envisaged but the
aim of this paper was merely to propose a dynamical
model of the debate in a framework close to control
theory representations, then control techniques
should be naturally implemented in the future.
5.2 Debate as a Decision Making
Process
This part presents a potential application of the
presented dynamical model. The aim is to use it as
a vote system. More precisely, in this example, both
alternatives
1 and 1+ are not considered to be
equivalent: +1 is the right decision while -1 is
associated to an error. This situation may occur in
classification problems when the agents are
competitive classification algorithms.
The agents are expected to provide the right
answer most of the time but they usually disagree on
singular cases. A common solution is to use a voting
process to achieve a group decision. For example, let
the agents be 7 different classification algorithms
whose success rates are respectively: 0.6, 0.7, 0.8,
0.8, 0.6, 0.7 and 0.6; then, the group success rate
using normal vote is 0.86. Even when a weighted
vote is introduced, the same rate is obtained because
of the value of the Shapley–Shubik power index
(Shapley, 1953) which is equal to 1/7 for any
classification algorithms. Indeed as probability are
hardly, bigger than 50% for each classification
algorithms, for the normal vote as for the weighted
vote the chosen value is the one which is chosen by
at least four agents. This effect does not take place
in the proposed method because the least agents are
also the ones who change most easily his point of
view. More precisely, this issue can also be tackled
with our debate model with identity as
B
function,
and success rates for convictions. It is supposed that
7 competitive classification algorithms are available
and that the right solution is supposed to be
alternative +1. The initial probability of the 7
algorithms to choose the alternative +1 are: 0.6, 0.7,
0.8, 0.8, 0.6, 0.7 and 0.6. Moreover it is supposed
that 10 000 cases are studied by each agent. For each
case, the answer of the agent is inferred according to
his probability to be right ( this is one method to
model the aggregation function corresponding to the
different classification algorithm. Then, for each of
the 10000 cases, the choice with a majority vote
procedure and the collective decision achieved with
our model when convictions at the start are the
initial probabilities are both computed. The program
stops when all the classification algorithms do agree.
While simple and weighted majorities obtain the
right answer with a rate of 86 %, our method rate is
94 %.
For 7 agents, several values of probability to
make the right decision are randomly generated and
3 rates are computed:
- the rate of success of the vote,
-the rate of success the weighted vote,
- the rate of success of our debate.
The rate of the weighted vote and of our debate
according to the rate of simple vote are plotted in
figure 5. Note that the same rate for the simple vote
can be obtained with very different sets of
probabilities. That is why the rate of success of the
weighted vote is somewhat equal to the simple vote
for very particular sets of probabilities where several
agents (algorithms) are much better than the others.
The debate always gives a better rate but its
preferences change according to the profile of
involved probabilities.
ICAART 2011 - 3rd International Conference on Agents and Artificial Intelligence
38
B := Identity B := -Identity
0
50
100
150
200
250
135791113151719212325272931333537394143454749
"+1"
"-1"
0
2
4
6
8
10
1 3 5 7 9 1113151719212325272931333537394143454749
0
50
100
150
200
250
1 3 5 7 9 1113151719212325272931333537394143454749
"+1"
"-1"
0
2
4
6
8
10
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
B := Majority B := Mass Psychology Effect
0
50
100
150
200
250
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
"+1"
"-1"
0
2
4
6
8
10
1 3 5 7 9 1113151719212325272931333537394143454749
0
50
100
150
200
250
135791113151719212325272931333537394143454749
"+1"
"-1"
0
2
4
6
8
135791113151719212325272931333537394143454749
Figure 4: Simulations of the debate outcomes (50 initial convictions; 200 permutations).
Figure 5: Simulationsweighted vote and debate.
6 CONCLUSIONS
AND PERSPECTIVES
The state equations that have been established in this
paper allow simulating macroscopically the outcome
of a debate according to the initial inclinations of
agents and the social influences in the group (the
influence function is a priori known). The
deliberation outcome depends on the order the
agents intervene in the debate to explain their
opinion and on the influence an agent may exert on a
social network.
The formalism of the model that is proposed in
this paper is close to the one used in control theory
to model dynamical behaviors of technical systems.
Governing a debate could then be seen as a control
problem whose aim could be, for example, how to
A DYNAMICAL MODEL FOR SIMULATING A DEBATE OUTCOME
39
reach as quick as possible a consensus or how to
reinforce one alternative rather than the other one,
etc.
A debate is thus seen as continuous dynamical
system: a state equations representation has been
preferred to the muticriteria decision-making
framework in (Rico et al., 2004) because time
explicitly appears in revision of convictions. The
model semantic is also inspired of games theory
concepts proposed in (Grabisch and Rusinowska,
2008): influence and decisional power in a social
network. In our dynamical extension, the decisional
power is a time-varying variable itself and can be
used as the actuator signal in the control loop of
debate. The state equations system established in
this paper allow stochastically simulating the
outcome of a debate and effects of a control strategy
on this issue.
One possible application of this model is
obviously simulating a debate’s outcome in order to
obtain some indications about the final collective
decision. When simulations are performed for a
great number of initial agents’ convictions and of
speaker intervention rankings, the probability the
outcome is
1± can be estimated. Hence, the
dynamical influence model can be used to make the
debate outcome more certain (it may appear as a
dishonest method when agents are human beings but
as a relevant technique when agents are artificial
agents such as sensors or classifiers) or modify the
convergence dynamics of the debate.
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D., 2003, A multiagent system to aggregate
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Systems, Man and Cybernetics, pages 545-550,
Washington, USA.
Grabisch, M. and Labreuche, C., 2002. The symmetric and
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Grabisch, M. and Rusinowska, A., 2008. A model of
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Economics. s.l.: Springer Netherlands.
Grabisch, M. and Rusinowska, A., 2009. Measuring
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Welfare 33: 177-209.
Hoede, C, Bakker, R., 1982. A theory of decisional power.
Journal of Mathematical Sociology, 8:309-322.
Imoussaten, A., Montmain, J. et Rigaud, E., 2009.
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Process: Disturbances or Control Variables?.
COGIS’2009, Paris.
Rico, A., Bonnevay, S., Lamure, M., Tounissoux, D. A,
2004. Debat modelisation with the Sipos integral.
LFA’2004, Nantes.
Rusinowska, A. and De Swart, 2007. On some properties
of the Hoede-Bakker index. Journal of Mathematical
Sociology 31: 267-293.
Shapley, L. S., 1953. A Value for n-Person Games. in H.
Kuhn and A. Tucker (Eds.), Contribution to the
Theory of Games, vol. II, number 28 in Annals of
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