On the Dynamics of Social and Political Conflicts
Looking for the Black Swan
Nicola [email protected]
Politecnico di Torino
http://calvino.polito.it/fismat/poli/
“Modelling Social problems and Health”
University of Durham, 13-14-September, 2012
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THANKS
THANKS:
Giulia Ajmone Marsan , OECD, Paris, France
http://www.oecd.org/gov/regionaldevelopment/
Miguel A. Herrero , Universidad Complutense, Madrid, Spain
http://www.mat.ucm.es/imi/People/Herrero–Garcia–MiguelAngel.htm
Andrea Tosin, I.A.C - National Research Council, Roma, Italy
http://www.iac.rm.cnr.it/ tosin/
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Lecture 1 - Index
1. Reasonings on Complex Systems
1.1. A brief historical excursus on complex systems
1.2. A Personal Bibliographic Search
1.3. Do Living, hence complex, systems exhibit common features?
1.4. What is the Black Swan?
1.5. Complexity through the ”Metamorphosis” by Escher
2. Mathematical Tools and Sources of Nonlinearity
3. Modeling Welfare Policy
4. Social Conflicts and Political Competition
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Lecture 1 - Reasonings on Complex Systems
1.1 A brief historical excursus on complex systems
E. Kant , (1790),Critique de la raison pure, Traduction Francaise, Press Univ. de
France, 1967
Living systems: Special structures organized and with the ability to chase a purpose.
E. Schrodinger, (1943),What is Life?
Living systems have the ability to extract entropy to keep their own at low levels.
R. May, (2003), Science
In the physical sciences, mathematical theory and experimental investigation have
always marched together. Mathematics has been less intrusive in the life sciences,
possibly because they have been until recently descriptive, lacking the invariance
principles and fundamental natural constants of physics.
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Lecture 1 - Reasonings on Complex Systems
1.1 A brief historical excursus on complex systems
APRIL 2011 UNRAVELING COMPLEX SYSTEMS
The American Mathematical Society, the American Statistical Association, the
Mathematical Association of America, and the Society for Industrial and Applied
Mathematics announce that the theme of Mathematics Awareness Month 2011 is
”Unraveling Complex Systems.”
From www.mathaware.org
We live in a complex world. Many familiar examples of complexsystems might be
found in very different entities at very different scales: from power grids, to
transportation systems, from financial markets, to the Internet, and even in the
underlying environment to cells in our bodies. Mathematicsand statistics can guide us
in unveiling, defining and understanding these systems,in order to enhance their
reliability and improve their performance.
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Lecture 1 - Reasonings on Complex Systems
1.1 A brief historical excursus on complex systems
N.B. H. Berestycki, F. Brezzi, and J.P. Nadal, Mathematics and Complexity in Life
and Human Sciences, Mathematical Models and Methods in Applied Sciences, 2010.
• The study of complex systems, namely systems of many individuals interacting in a
non-linear manner, has received in recent years a remarkable increase of interest
among applied mathematicians, physicists as well as researchers in various other
fields as economy or social sciences.
• Their collective overall behavior is determined by the dynamics of their interactions.
On the other hand, a traditional modeling of individual dynamics does not lead in a
straightforward way to a mathematical description of collective emerging behaviors.
In particular it is very difficult to understand and model these systems based on the
sole description of the dynamics and interactions of a few individual entities.
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Lecture 1 - Reasonings on Complex Systems
1.2 A Personal Bibliographic Search
• E. Mayr,What Evolution Is , (Basic Books, New York, 2001).
• A.L. BarabasiLinked. The New Science of Networks, (Perseus Publishing,
Cambridge Massachusetts, 2002.)
• F. Schweitzer,Brownian Agents and Active Particles, (Springer, Berlin, 2003).
• M. Talagrand,Spin Glasses, a Challenge to Mathematicians, Springer, (2003).
• M.A. Nowak,Evolutionary Dynamics, Princeton Univ. Press, (2006).
• N.N. Taleb,The Black Swan: The Impact of the Highly Improbable, 2007.
• N.Bellomo,Modelling Complex Living Systems. A Kinetic Theory andStochastic Game Approach, (Birkhauser-Springer, Boston, 2008).
• N.Bellomo, On the Difficult Interplay Between Life (Complexity) and Mathematical
Sciences, (2012), to be published.
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1 - Reasonings on Complex Systems
1.3 Do Living, hence complex, systems exhibit common features?
10 Selected Common Features
• 1. Ability to express a strategy:Living entities are capable to develop
specificstrategiesrelated to theirorganization abilitydepending on the state and
entities in their surrounding environment. These can be expressed without the
application of any principle imposed by the outer environment. In general, they
typically operateout-of-equilibrium. For example, a constant struggle with the outer
environment is developed to remain in a particular state, namely stay alive.
• 2. Heterogeneity: The said ability is not the same for all entities. Indeed,
heterogeneous behaviorscharacterize a great part of living systems. Namely, the
characteristics of interacting entities can even differ from an entity to another
belonging to the same structure. In biology, this is due to different phenotype
expressions generated by the same genotype. In some cases, these expressions induce
genetic diseases, or different evolution of epidemics.
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1 - Reasonings on Complex Systems
1.3 Do Living, hence complex, systems exhibit common features?
• 3. Large number of components: Complexity in living systems is
induced by a large number of variables, which are needed to describe their overall
state. Therefore, the number of equations needed for the modeling approach may be
too large to be practically treated. For instance, biological systems are different from
the physical systems analyzed by statistical mechanics, which typically deals with
systems containing many copies of a few interacting components, whereas cells
contain from millions to a few copies of each of thousands of different components,
each with specific interactions.
• 4. Interactions: Interactions are nonlinearly additive and involve immediate
neighbors, but in some cases also distant particles, as living systems have the ability to
communicate and may possibly choose different observationpaths. In some cases, the
topological distribution of a fixed number of neighbors can play a prominent role in the
development of the strategy and interactions. Living entitiesplay a game at each
interactionwith an output that is technically related to their strategyoften related to
surviving and adaptation ability.
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1 - Reasonings on Complex Systems
1.3 Do Living, hence complex, systems exhibit common features?
• 5. Stochastic games: Living entities at each interactionplay a gamewith an
output that is technically related to their strategy often related to surviving and
adaptation ability, namely to a personal search of fitness. The output of the game is not
generally deterministic even when a causality principle isidentified. This dynamics is
also related to the fact that they receive a feedback from their environments, which
modifies the strategy they express adapting it to the new environmental conditions.
The interaction modifies the outer environment, hence the dynamics of interactions
evolves in time.
• 6. Learning ability: Living systems have theability to learn from past
experience. Therefore their strategic ability and the characteristics of interactions
among living entities evolve in time. In fact, living systems receive inputs from their
environments and have the ability to adapt themselves to theenvironmental conditions
that also evolve in time.
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1 - Reasonings on Complex Systems
1.3 Do Living, hence complex, systems exhibit common features?
• 7. Darwinian selection and time as a key variable: All living
systems are evolutionary. For instance birth processes cangenerate individuals more
fitted to the environment, who in turn generate new individuals again more fitted to the
outer environment. Neglecting this aspect means that the time scale of observation and
modeling of the system itself might not be long enough to observe evolutionary events.
In biology such a time scale can be very short for cellular systems and very long for
vertebrates. In some cases, such as the generation of daughter cells from mother cells,
new cell phenotypes can have origin from random mistakes during replication.
• 8. Multiscale aspects: The study of complex living systems always needs a
multiscale approach. For instance in biology, the dynamics of a cell at the molecular
(genetic) level determines the ability of cells to express specific functions. Moreover,
the structure of macroscopic tissues depends on such a dynamics. Moreover, living
systems are generally constituted by alarge number of components. In
general, we can state that one only observation and representation scale is not
sufficient to describe the dynamics of living systems.
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1 - Reasonings on Complex Systems
1.3 Do Living, hence complex, systems exhibit common features?
• 9. Emerging behaviors: Large living systemsshow collective emerging
behaviorsthat are not directly related to the dynamics of a few interacting entities.
Generally, emerging behaviors are reproduced at a qualitative level given certain input
conditions, However, quantitative levels are not generally reproduced. In fact small
changes in the input conditions generate large deviations even when the qualitative
behavior is reproduced.
• 10. Large deviations: The observation of living systems should focus on the
emerging behaviors that appear in non-equilibrium conditions. Very similar input
conditions reproduce, in several cases, the qualitative behaviors. However, large
deviations can be observed corresponding to small changes in the input conditions. In
some cases, they break out the quality of the afore-said behaviors, namely substantial
modifications can be observed. Heterogeneity of individualstrategies, learning ability,
and interactions with the outer environment modify such phenomena.
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1 - Reasonings on Complex Systems
1.4 What is the Black Swan?
It is worth detailing a little more the expressionBlack Swan, introduced in the
specialized literature for indicating unpredictable events, which are far away from
those generally observed by repeated empirical evidence. According to the definition
by Taleb a Black Swan is specifically characterized as follows:
“A Black Swan is a highly improbable event with three principal characteristics: It is
unpredictable; it carries a massive impact; and, after the fact, we concoct an
explanation that makes it appear less random, and more predictable, than it was.”
N. N. Taleb,The Black Swan: The Impact of the Highly Improbable, Random House,
New York City, 2007.
Since it is very difficult to predict directly the onset of a black swan, it is useful
looking for the presence of early signals
M. Scheffer, J. Bascompte, W. A. Brock, V. Brovkin, S. R. Carpenter, V. Dakos,
H. Held, E. H. van Nes, M. Rietkerk, and G. Sugihara, Early-warning signals for
critical transitions,Nature, 461:53–59, 2009.
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1 - Reasonings on Complex Systems
1.5 The Metamorphosis by Escher
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1 - Reasonings on Complex Systems
1.5 The Metamorphosis by Escher
Let us imagine, with the help of our fantasy, the time is on thex-axis. Some free
(imaginative) interpretations are as follows:
i) The landscape evolves in time going first from a geometrical village made of
similarly looking houses (looking like boxes) to a real village with heterogeneous
distribution of the shape of houses;
ii) The evolution is related to interactions. Definitely multiple ones; but are they
nonlinear?
iii) Does the dynamics shows multiscale interactions and isthe evolution selective in
some weak Darwinian sense?
iv) Suddenly the landscape changes from a village it becomesa chess plate, the only
connection is a Bridge and a Tower. Can this sudden change be interpreted as a “Black
Swan”?
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1 - Reasonings on Complex Systems
1.5 The Metamorphosis by Escher
Since the village exists in reality some further considerations can be developed:
v) The Tower is not yet the one of the chess plate. Can it be, however, interpreted as an
early signal of what is happening?
vi) Does the Tower exists in reality? If the answer is equallyYes and Not the same
time, why Yes and why Not?
vii) Referring to a Village, may even be the real one, can our imagination provide a
conceivable interpretation of the physical meaning of the sudden change?
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Lecture 2 - Index
1. Reasonings on Complex Systems
2. 2. Mathematical Tools and Sources of Nonlinearity
2.1. Functional Subsystems and Representation
2.2. Stochastic Games
2.3. Mathematical Structures
2.4. Sources of Nonlinearity
3. Modeling Welfare Policy
4. Modeling Social Conflicts and Political Competition
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2 - Mathematical Tools and Sources of Nonlinearity
2.1 Functional Subsystems and Representation
Hallmarks of the kinetic theory of active particles:
• The overall system is subdivided intofunctional subsystemsconstituted by
entities, calledactive particles, whose individual state is calledactivity;
• The state of each functional subsystem is defined by a suitable, time dependent,
probability distribution over the activity variable;
• Interactions are modeled by games, more precisely stochastic games, where the
state of the interacting particles and the output of the interactions are known in
probability;
• Interactions are delocalized and nonlinearly additive;
• The evolution of the probability distribution is obtained by a balance of particles
within elementary volumes of the space of the microscopic states, where the
dynamics of inflow and outflow of particles is related to interactions at the
microscopic scale.
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2 - Mathematical Tools and Sources of Nonlinearity
2.1 Functional Subsystems and Representation
Let us consider a system constituted by a large number of interacting living entities,
calledactive particles. Their microscopic state includes, in addition to
geometrical and mechanical variables, also an additional variableu ∈ Du, called
activity, which is heterogeneously distributed.
Living systems are constituted by different types of active particles, which
express several different functions. This source of complexity can be reduced byby
decomposing the system into suitable functional subsystems,
where afunctional subsystem is a collection of active particles, which have the
ability to express cooperatively the sameactivity regarded as a scalar variable.
A particular case is when particles are distributed ina system of networks
and nodes in a way that the role of the space variable is not relevant in the nodes,
while transition from one node to the other has to be properlymodeled by migration
dynamics.
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2 - Mathematical Tools and Sources of Nonlinearity
2.1 Functional Subsystems and Representation
The description of the state of the system within each node isdelivered by a
probability distribution function over the microscopic state, labeled by the
indexi, which denotes the functional subsystem within each node.
fi = fi(t, u) , i = 1, . . . , n,
such thatfi(t, u) du denotes the number of active particles whose state, at timet, is in
the interval[u, u+ du]. If the number of active particles is constant in time, then the
distribution function can be normalized with respect to such a number and
consequently is aprobability density. Moreover, the physical meaning of the
activity variable differs for each subscript.
Consider now the wholenetwork, then the overall number of functional subsystems
is given by the sum of all of them in the nodes:
fij = fij(t, u) , i = 1, . . . , n, j = 1, . . . ,m, where functional subsystems
differ if are localized in different nodes although they express the same activity
variable.
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2 - Mathematical Tools and Sources of Nonlinearity
2.1 Functional Subsystems and Representation
Representation by Moments and Macroscopic Quantities
Macroscopic quantities can be computed by moments of the afore defined distribution
function. In particular, the number of particles in each functional subsystem is given
by:
nij [fij ](t) =
∫
Du
fij(t, u) du.
Higher order moments are computed as follows:
Epij [fij ](t) =
1
nij [fij ](t)
∫
Du
upfij(t, u) du.
Remark: The physical meaning of the moments depends on the specific system under
consideration.
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2 - Mathematical Tools and Sources of Nonlinearity
2.1 Functional Subsystems and Representation
Representation for space distributed systems in each node
Consider active particles in a node for functional subsystems labeled by the subscripti.
The description of the overall state of the system is delivered by thegeneralized
one-particle distribution function
fi = fi(t,x,v, u) : [0, T ] × Ω ×Dv ×Du →+,
such thatfi(t,x,v, u) dx dv du denotes the number of active particles whose state, at
time t, is in the interval[w,w + dw]. wherew = x,v, u is an element of thespace
of the microscopic states. Marginal densities are computed as follows:
fmi (t,x,v) =
∫
Du
fi(t,x,v, u) du , fai (t,x, u) =
∫
Dv
fi(t,x,v, u) dv.
and
fbi (t, u) =
∫
Dx×Dv
fi(t,x,v, u) dx dv.
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2 - Mathematical Tools and Sources of Nonlinearity
2.1 Functional Subsystems and Representation
Representation for space distributed systems in each node
Local density of theith functional subsystem is:
ν[fi](t,x) =
∫
Dv×Du
fi(t,x,v, u) dv du =
∫
Dv
fmi (t,x,v) dv.
Integration over the volumeDx containing the particles gives thetotal size of the
ith subsystem:
Ni[fi](t) =
∫
Dx
ν[fi](t,x) dx ,
which can depend on time due to the role of proliferative or destructive interactions.
First order moments provide eitherlinear mechanical macroscopic
quantities, orlinear activity macroscopic quantities.
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2 - Mathematical Tools and Sources of Nonlinearity
2.1 Functional Subsystems and Representation
• Theflux of particles, at the timet in the positionx, is given by
Q[fi](t,x) =
∫
Dv×Du
v fi(t,x,v, u) dv du =
∫
Dv
fmi (t,x,v) dv.
• Themass velocity of particles, at the timet in the positionx, is given by
U[fi](t,x) =1
νi[fi](t,x)
∫
Dv×Du
v fi(t,x,v, u) dv du.
Theactivity terms are computed as follows:
a[fi](t,x) =
∫
Dv×Du
ufi(t,x,v, u) dv du ,
while thelocal activation density is given by:
ad[fi](t,x) =
aj [fi](t,x)
νi[fi](t,x)=
1
νi[fi](t,x)
∫
Dv×Du
ufi(t,x,v, u) dv du.
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2 - Mathematical Tools and Sources of Nonlinearity
2.2 Stochastic Games
• Test particles with microscopic stateu, at the timet: fij = fij(t, u).
• Field particles with microscopic state,u∗ at the timet: fij = fij(t, u∗).
• Candidate particles with microscopic state,u∗ at the timet: fij = fij(t, u∗).
The interaction rule is as follows:
i) Thecandidate particle interacts withfield particles and acquires, in probability,
the state of thetest particle.Test particles interact with field particles and lose their
state.
ii) Interactions can: modify the microscopic state of particles; generate proliferation or
destruction of particles in their microscopic state; and can also generate a particle in a
new functional subsystem.
Loss of determinism: Modeling of systems of the inert matter is developed
within the framework of deterministic causality principles, which does not any longer
holds in the case of the living matter.
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2 - Mathematical Tools and Sources of Nonlinearity
2.2 Stochastic Games
C ΩT
F F
F
F
F
Figure 1: – Active particles interact with other particles in their action domain
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2 Mathematical Tools and Sources of Nonlinearity
2.2 Stochastic Games
Interactions with modification of activity and transition: Generation of particles
into a new functional subsystem occurs through pathways. Different paths can be
chosen according to the dynamics at the lower scale.
F F F
F C FF
T
T
FF FF FF T
i + 1
i
i − 1
Figure 2: – Active particles during proliferation move fromone functional subsystem
to the other through pathways.
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2 - Mathematical Tools and Sources of Nonlinearity
2.2 Stochastic Games
Cooperative/Competitive Games
Cooperative behaviorThe active particle with stateh < k improves its state by
gaining from the particlek , which cooperates loosing part of its state.
Competitive behavior The active particle with stateh < k decreases its state by
contributing to the particlek, which due to competition increases part of its state.
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2 - Mathematical Tools and Sources of Nonlinearity
2.3 Mathematical Structures
Balance within the space of microscopic states
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2 - Mathematical Tools and Sources of Nonlinearity
2.3 Mathematical Structures
The Spatially Homogeneous CaseThe evolution equation follows from a balance equation for net flow of particles in the
elementary volume of the space of the microscopic state by transport and interactions:
∂tfi(t, u) + Fi(t) ∂ufi(t, u) = Ci[f ](t, u) + Pi[f ](t, u),
where
Ci[f ] =
n∑
j=1
η0ij
∫
Du×Du
ψij [f ]Bij [f ] (u∗ → u|u∗, u∗)
− fi(t, u)n
∑
j=1
η0ij
∫
Du
ψij [f ] fj(t, u∗) du∗
.
and
Pi[f ] =n
∑
h=1
n∑
k=1
η0hk
∫
Du
∫
Du
ψhk[f ]µihk[f ](u∗, u
∗)fh(t, u∗)fk(t, u∗) du∗.
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Lecture 2 - Mathematical Tools and Sources of Nonlinearity
2.3 Mathematical Structures
Models with Space Dynamics
H.1. Candidate or test particles inx, interact with the field particles inx∗ ∈ Ω located
in the interaction domainΩ. Interactions are weighted by theinteraction rate
ηhk[f ](x∗) supposed to depend on the local density in the position of thefield
particles.
H.2. Candidate particle modifies its state according to the term defined as follows:
Ahk[f ](v∗ → v, u∗ → u|v∗,v∗, u∗, u
∗), which denotes the probability density that
a candidate particles of theh-subsystems with statev∗, u∗ reaches the statev, u after
an interaction with the field particlesk-subsystems with statev∗, u∗.
H.3. Candidate particle, inx, can proliferate, due to encounters with field particles in
x∗, with rateµihk, which denotes the proliferation rate into the functional subsystemi,
due the encounter of particles belonging the functional subsystemsh andk.
Destructive events can occur only within the same functional subsystem.
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Lecture 2 - Mathematical Tools and Sources of Nonlinearity
2.3 Mathematical Structures
(∂t + v · ∂x) fi(t,x,v, u) =
[ n∑
j=1
Cij [f ] +n
∑
h=1
n∑
k=1
Sihk[f ]
]
(t,x,v, u) ,
where
Cij [f ] =
∫
Ω×D2u×D2
v
ηij [f ](x∗)Ahk[f ](v∗ → v, u∗ → u|v∗,v
∗, u∗, u
∗)
× fi(t,x,v∗, u∗)fj(t,x∗,v
∗, u
∗) dv∗ dv∗du∗ du
∗dx
∗,
− fi(t,x,v)
∫
Ω×Du×Dv
ηij [f ](x∗) fj(t,x
∗,v
∗, u
∗) dv∗du
∗dx
∗
Sihk[f ] =
∫
Ω×D2u×Dv
ηhk[f ](x∗)µihk[f ](u∗, u
∗)
× fh(t,x,v, u∗)fk(t,x∗,v
∗, u
∗) dv∗du∗ du
∗dx
∗.
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Lecture 2 - Mathematical Tools and Sources of Nonlinearity
Conjecture 1. Interactions modify the activity variable according to topological
stochastic games, however independently on the distribution of the velocity variable,
while modification of the velocity of the interacting particles depends also on the
activity variable.
Ahk(·) = Bhk(u∗ → u, |u∗, u∗) × Chk(v∗ → v |v∗,v
∗, u∗, u
∗) ,
whereA, B, andC are, for positive definedf , probability densities.
Conjecture 2. Stochastic velocity perturbation
(
∂t + v · ∇x
)
fi = νi Li[fi] + ηi Ci[f ] + ηi µi Si[f ],
where
Li[fi] =
∫
Dv
(
Ti(v∗ → v)fi(t,x,v
∗, u) − Ti(v → v
∗)fi(t,x,v, u))
dv∗,
and whereTi(v∗ → v) is, for theith subsystem, the probability kernel for the new
velocityv ∈ Dv assuming that the previous velocity wasv∗.
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2 - Mathematical Tools and Sources of Nonlinearity
2.4 Sources of Nonlinearity
Encounter rate: In general,ηij is supposed to decay with the distance between the
interacting active particles, which depends both on their state and on the distance of
their functional subsystems. The following sources of nonlinearity can be possibly
identified:
Hierarchic distance: The rate of interactions within the same functional subsystem
depends on the distance of the interacting microstates. Generally, it decays with the
distancedhd = |u∗ − u∗|. However, when two active particles belong to different
functional subsystems, the rate differs depending on the distance between the
functional states. A conceivable numbering criterion consists in selecting the first
subsystem by a certain selection (for instance in the animalworld the “dominant”) and
in numbering the others by increasing numbers depending on the decreasing rate.
Namely:k ↑⇒ ηhk ↓.
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2 - Mathematical Tools and Sources of Nonlinearity
2.4 Sources of Nonlinearity
Affinity distance: The distance between the distribution function interacting active
particles can be source of nonlinearity according to the general idea that two systems
with closed distribution areaffineand hence they interact with higher frequency. In
this case the distance isdad = ||fh − fk||, where a uniform, e.g.L∞, or mean
approximation, e.g.L1, can be adopted as the most appropriate according to the
physics of the system under consideration. The rate decays with increasingdd.
An example is:
ηhk = η
hk[f ](u∗, u∗) = η
0hk exp
− α|u∗ − u∗| − β|h− k| − γ||fh − fk||
,
whereη0hk andα, β, andγ are positive constants, and
||fh − fk||(t) =
∫
R+
|fh(t, u) − fk(t, u)| du.
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2 - Mathematical Tools and Sources of Nonlinearity
2.4 Sources of Nonlinearity
Transition probability density:
Threshold: The occurrence of a type of interaction with respect to the other is ruled,
in several models, by a threshold on the distance between theinteracting particles. On
the other hand, the threshold evolves in time and depends on the whole dynamics of of
the overall system, hence onf .
Domain of influence: The communication is effective only within a certain domainof
influence of the micro-state. In the simplest case it is fixed,more in general it depends
on the density of active particles which can be captured in the communication, namely
there exists a critical density that defines the maximum number of particles, which can
be included in the interaction (see Parisi’s conjecture andthe application to swarm
modeling).
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2 - Mathematical Tools and Sources of Nonlinearity
2.4 Sources of Nonlinearity
Not sufficient visibility zone: An additional difficulty occurs when the visibility zone
is reduced and the candidate particle does not receive sufficient information. In this
case, the output of the interaction might be different from the usual one.
Output of the interaction: An active particle interacts with all particles in its
interaction domain. The output of the interaction depends on f within the interaction
domain. For instance it can depend on the moments computed for the particles within
the interaction domain as shown again on the modeling of swarms. A simple case
occurs when it depends on low order moments:
Proliferation rate: Sources of nonlinearity in the case of proliferative and/or
destructive interactions, in addition to those induced by the interaction rate, are
generally the same affecting conservative interactions ruled by games.
On the Dynamics of Social and Political Conflicts Looking forthe Black Swan– p. 37/83
Lecture 3 - Index
1. Reasonings on Complex Systems
2. 2. Mathematical Tools and Sources of Nonlinearity
3. Modeling Welfare Policy
3.1 Bibliography and Preliminary Reasonings
3.2 Modeling of the Dynamics of Welfare Distribution
3.3 Simulations
4. Modeling Social Conflicts and Political Competition
On the Dynamics of Social and Political Conflicts Looking forthe Black Swan– p. 38/83
3 - Modeling Welfare Policy
3.1. Bibliography and Preliminary Reasonings
M.A. Nowak, Evolutionary Dynamics, Princeton Univ. Press, Princeton, (2006).
A.-L. Barabasi, R. Albert, and H. Jeong, Mean-field theory for scale-free random
networks,Physica A, 272(1–2):173–187, 1999.
F. Vega-Redondo, Complex Social Networks, Cambridge University Press,
Cambridge, (2007).
D. Helbing, Quantitative Sociodynamics. Stochastic Methods and Models of Social
Interaction Processes. Springer Berlin Heidelberg, 2nd edition, 2010.
D. Acemoglu and J. Robinson, Economic Origins of Dictatorship and Democracy,
Cambridge University Press, Cambridge, (2006).
D. Acemoglu, D. Ticchi, and A. Vindigni, A theory of military dictatorship.
American Economic Journal: Macroeconomics, 2(1):1–42, 2010.
D. Acemoglu, D. Ticchi, and A. Vindigni, Emergence and persistence of inefficient
states.J. Eur. Econ. Assoc., 9(2):177–208, 2011.
On the Dynamics of Social and Political Conflicts Looking forthe Black Swan– p. 39/83
3 - Modeling Welfare Policy
3.1. Bibliography and Preliminary Reasonings
G. Ajmone Marsan New Paradigms and Mathematical Methods for Complex Systems
in Behavioral Economics. PhD thesis, IMT Lucca, Italy, 2009.
G. Ajmone Marsan, N. Bellomo, and M. Egidi, Towards a mathematical theory of
complex socio-economical systems by functional subsystems representation.Kinet.
Relat. Models, 1(2):249–278, 2008.
N. Bellomo and B. Carbonaro, Towards a mathematical theory of living systems
focusing on developmental biology and evolution,Physics of Life Reviews, 8 (2011)
1-18.
N. Bellomo, M.A. Herrero, and A. Tosin, On the dynamics of social conflicts looking
for the black swan, submitted, (2012).
N.N. Taleb, The Black Swan: The Impact of the Highly Improbable, 2007.
K. Sigmund, The Calculus of Selfishness, Princeton Univ. Press, (2011).
On the Dynamics of Social and Political Conflicts Looking forthe Black Swan– p. 40/83
3 - Modeling Welfare Policy
3.1. Bibliography and Preliminary ReasoningsSome conceivable fields of
application
G. Aletti, G. Naldi, and G. Toscani, First-order continuous models of opinion
formation,SIAM J. on Applied Mathematics67, 837-853, (2007).
F. Bagarello and F. Olivieri, An operator description of interactions between
populations with application to migrations,Mathematical Models and Methods in
Applied Sciences, 22, (2012), to appear.
U. Bastolla, M.A. Fortuna, A. Pacual-Garcia, A. Ferrera, B.Luque, and
J. Bascompte, The architecture of mutualistic networks minimizes competition and
increases biodiversity,Nature, 458:1018–1021, 2009.
M. Olson, Dictatorship, Democracy, and Development,The American Political Science
Review, 87, 3, 567-576, (1993).
G. Mulone and B. Straughan, A note on heroin epidemics,Mathematical
Biosciences, 218, (2009), 138-141.
G. Mulone and B. Straughan, Modeling binge drinking,Int. J. Biomathematics5,
(2012), no.1250005On the Dynamics of Social and Political Conflicts Looking forthe Black Swan– p. 41/83
3 - Modeling Welfare Policy
3.1. Bibliography and Preliminary Reasonings
The dynamics of social and economic systems are necessarilybased on individual
behaviors, by which single subjects express, either consciously or unconsciously, a
particular strategy, which is heterogeneously distributed. The latter is often based not
only on their own individual purposes, but also on those theyattribute to other agents.
However, the sheer complexity of such systems makes it oftendifficult to ascertain the
impact of personal decisions on the resulting collective dynamics. In particular,
interactions need not have an additive, linear character. As a consequence, the global
impact of a given number of entities (“field entities”) over asingle one (“test entity”)
cannot be assumed to merely consist in the linear superposition of any single field
entity action.
This nonlinear feature represents a serious conceptual difficulty to the derivation,and subsequent analysis, of mathematical models for that type of systems.
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3 - Modeling Welfare Policy
3.1. Bibliography and Preliminary Reasonings
In the last few years, a radical philosophical change has been undertaken in social and
economic disciplines. An interplay among Economics, Psychology, and Sociology has
taken place, thanks to a new cognitive approach no longer grounded on the traditional
assumption of rational socio-economic behavior. Startingfrom the concept of bounded
rationality, the idea of Economics as a subject highly affected by individual (rational or
irrational) behaviors, reactions, and interactions has begun to impose itself. In this
frame, the contribution of mathematical methods to a deeperunderstanding of the
relationships between individual behaviors and collective outcomes may be
fundamental.
A key experimental feature of such systems is that interaction among
heterogeneous individuals often produces unexpected outcomes, which wereabsent at the individual level, and are commonly termed emergent behaviors.
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3 - Modeling Welfare Policy
3.1. Bibliography and Preliminary Reasonings
In fields ranging from Economics to Sociology and Ecology, the last decades have
witnessed an increasing interest for the introduction of quantitative mathematical
methods that could account for individual, not necessarilyrational, behaviors. Terms
as game theory, bounded rationality, evolutionary dynamics are often used in that
context and clearly illustrate the continuous search for techniques able to provide
mathematical models than can describe, and predict, livingbehaviors. The new point
of view promoted the image of Economics as an evolving complex system, where
interactions among heterogeneous individuals produce unpredictable emerging
outcomes. In this context, setting up a mathematical description able to capture the
evolving features of socio-economic systems is a challenging, however difficult, task,
which calls for a proper interaction between mathematics and social sciences.
Mathematical models might focus, in particular, on the prediction of the so called
Black Swan. The latter is defined to be a rare event, showing up as an irrationalcollective trend generated by possibly rational individual behaviors.
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3 - Modeling Welfare Policy
3.2 Modeling of the Dynamics of Welfare Distribution
Socio-economic systems can be described as ensembles of several living entities able
to developbehavioral strategies, by which they interact with each other. For this
reason, such entities are also calledactive particles. Typically, their strategies are
heterogeneously distributedandchange in timein consequence of the interactions,
since active particles can update them bylearning from past experiences.Therefore the mathematical representation calls fordistribution functions over the
strategies, whose time evolution depends on the dynamics ofinteractions. According
to the KTAP approach, the latter are regarded asstochastic games, whose payoff is the
new strategy that each player will use in the next interaction.
Remarkably, payoffs are supposed to be known only in probability because irrational
behaviors in individual choices cannot be excluded. Deterministic payoffs would
imply instead a kind of rational behavior of active particles, as if they reacted the same
when placed in the same conditions.
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3 - Modeling Welfare Policy
3.2 Modeling of the Dynamics of Welfare Distribution
Let u be a variable denoting the strategy expressed by active particles, that in the
present context can be identified with their wealth status. For the kind of applications
we are interested in it is customary to speak ofwealth classes, which implies that the
strategyu is a discrete variable taking values in a lattice
Iu = u1, . . . , ui, . . . , un.
The number of individuals that, at a certain timet, are expressing the strategyui, or, in
other words, that belong to thei-th wealth class, is given by a distribution function
fi = fi(t) : [0, Tmax] →+, i = 1, . . . , n,
Tmax > 0 being a certain final time (possibly+∞). Notice that, up to normalizing with
respect to the total number of active particles,f = (f1, . . . , fi, . . . , fn) is a
time-evolving discrete probability distribution over thelatticeIu.
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3 - Modeling Welfare Policy
3.2 Modeling of the Dynamics of Welfare Distribution
• The testparticle, with strategyui, is the representative entity of the system.
Studying interactions means studying how the test particlecan loose its strategy or
other particles can gain it.
• Candidateparticles, with strategyuh, are the particles which can gain the test
strategyui in consequence of the interactions.
• Field particles, with strategyuk, are the particles whose presence triggers the
interactions of the candidate particles.
• The interaction rateηhk, modeling the frequency of interaction between auh
candidate particle and auk field particle.
• The transition probabilities, denoted byBhk(i), expressing the probability that a
candidate particle with strategyuh changes the latter toui after interacting with a
field particle with strategyuk.
n∑
i=1
Bhk(i) = 1, ∀h, k = 1, . . . , n.
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3 - Modeling Welfare Policy
3.2 Modeling of the Dynamics of Welfare Distribution
By letting all particles play together for a time∆t > 0, the expected number of
particles shifting to and losing, respectively, the strategy ui during the time interval
[t, t+ ∆t] can be estimated, at first order, as:
∆tn
∑
h, k=1
ηhkBhk(i)fh(t)fk(t) + o(∆t), ∆tfi(t)n
∑
k=1
ηikfk(t) + o(∆t).
The variationfi(t+ ∆t) − fi(t) of the number of active particles expressing the
strategyui can then be equated with the net balance of these two terms, which, in the
limit ∆t→ 0+, leads to the equation:
dfi
dt=
n∑
h, k=1
ηhkBhk(i)fhfk − fi
n∑
k=1
ηikfk, i = 1, . . . , n. (1)
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3 - Modeling Welfare Policy
3.2 Modeling of the Dynamics of Welfare Distribution
The modeling approach adopts the following qualitative paradigm of
consensus/dissensus dynamics:
• Consensus– The candidate particle sees its state either increased, byprofiting
from a field particle with a higher state, or decreased, by pandering to a field
particle with a lower state. After mutual interaction, the states of the particles
become closer than before the interaction.
• Dissensus– The candidate particle sees its state either further decreased, by facing
a field particle with a higher state, or further increased, byfacing a field particle
with a lower state. After mutual interaction, the states of the particles become
farther than before the interaction.
A critical distance triggers either cooperation or competition among the classes. If the
distance is lower than the critical one then a competition takes place, which causes a
further enrichment of the wealthier class and a further impoverishment of the poorer
one. Conversely, if the actual distance is greater than the critical one then the social
organization forces cooperation.
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3 - Modeling Welfare Policy
Modeling Consesus/dissensus Games
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3 - Modeling Social Conflicts and Political Competition
3.2 Modeling of the Dynamics of Welfare Distribution
The characteristics of the present framework are summarized as follows.
• Interaction rate. Two different rates of interaction are considered, corresponding
to competitive and cooperative interactions, respectively.
• Strategy leading to the transition probabilities. When interacting with other
particles, each active particle plays a game with stochastic output. If the difference
of wealth class between the interacting particles is lower than a critical distance
γ[f ] (where, here and henceforth, square brackets indicate a functional
dependence on the probability distributionf ) then the particles compete in such a
way that those with higher wealth increase their state against those with lower
wealth. Conversely, if the difference of wealth class is higher thanγ[f ] then the
opposite occurs. The critical distance evolves in time according to the global
wealth distribution over wealthy and poor particles.
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3 - Modeling Welfare Policy
3.2 Modeling of the Dynamics of Welfare Distribution
1. Interaction between equally wealthy particles, viz. h = k:
Bhh(h) = 1
Bhh(i) = 0, ∀ i 6= h(2)
2. Cooperative interaction, viz. h 6= k and|h− k| > γ[f ]:
h < k
Bhk(h) = 1 − αhk
Bhk(h+ 1) = αhk
Bhk(i) = 0, ∀ i 6= h, h+ 1
h > k
Bhk(h− 1) = αhk
Bhk(h) = 1 − αhk
Bhk(i) = 0, ∀ i 6= h− 1, h,
(3)
where it is assumed that interactions within the same class produce no effect.
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3 - Modeling Welfare Policy
3. Competitive interaction, viz. h 6= k and|h− k| ≤ γ[f ]:
h = 1, n
Bhk(h) = 1
Bhk(i) = 0, ∀ i 6= h
h 6= 1, n
h < k
k < n
Bhk(h− 1) = αhk
Bhk(h) = 1 − αhk
Bhk(i) = 0, ∀ i 6= h− 1, h
k = n
Bhn(h) = 1
Bhn(i) = 0, ∀ i 6= h
h > k
k = 1
Bh1(h) = 1
Bh1(i) = 0, ∀ i 6= h
k > 1
Bhk(h) = 1 − αhk
Bhk(h+ 1) = αhk
Bhk(i) = 0, ∀ i 6= h, h+ 1
(4)
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3 - Modeling Welfare Policy
3.2 Modeling of the Dynamics of Welfare Distribution
The parameterαhk ∈ [0, 1] has the following meaning:
• In case of competition, it is the probability that the candidate particle further
increases (or decreases) its wealth if it is richer (or poorer) than the field particle;
• In case of cooperation, it is the probability that the candidate particle gains (or
transfers) part of its wealth if it is poorer (or richer) thanthe field particle.
This probability may be constant or may depend on the wealth classes, e.g.,
αhk =|k − h|
n− 1,
in such a way that the larger the distance between the interacting classes the more
stressed the effect of cooperation or competition. Moreover, for wealth conservation
purposes, the transition probabilities are such that the extreme classes never take part
nor trigger social competition.
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3 - Modeling Welfare Policy
3.2 Modeling of the Dynamics of Welfare Distribution
Among the possible choices, we select a uniformly spaced wealth grid in the interval
[−1, 1] with odd numbern of classes:
Iu = u1 = −1, . . . , un+1
2
= 0, . . . , un = 1,
ui =2
n− 1i−
n+ 1
n− 1, i = 1, . . . , n,
agreeing thatui < 0 identifies a poor class whereasui > 0 a wealthy one.
We next assume that theinteraction rate ηhk ≥ 0 is piecewise constant over the
wealth classes:
ηhk =
η0 if |k − h| ≤ γ[f ] (competition),
µη0 if |k − h| > γ[f ] (cooperation),
whereη0 > 0 is a constant to be hidden in the time scale and0 < µ ≤ 1.
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3 - Modeling Welfare Policy
3.2 Modeling of the Dynamics of Welfare Distribution
Thetransition probabilities Bhk(i) ∈ [0, 1] are required to satisfy additional
conditions such as conservation of the average wealth status of the population:
n∑
i=1
uifi(t) = constant, ∀ t ≥ 0. (5)
This means that the interaction dynamics cause globally neither production nor loss of
wealth, but simply its redistribution among the classes.
U0 :=
n∑
i=1
uifi(0).
It turns out that sufficient conditions for the fulfillment ofthis condition are:
symmetric encounter rate, i.e.,ηhk = ηkh, ∀h, k = 1, . . . , n; and
n∑
i=1
uiBhk(i) = uh + σhk, ∀h, k = 1, . . . , n,
whereσhk is an antisymmetric tensor, i.e.,σhk = −σkh, ∀h, k = 1, . . . , n.
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3 - Modeling Welfare Policy
Welfare Policy and Political Competition
Thecritical distance γ[f ] is here assumed to depend on the instantaneous distribution
of the active particles over the wealth classes, such that the time evolution ofγ[f ]
should translate the following phenomenology of social competition:
• in general,γ[f ] grows with the number of poor active particles, thus causinglarger
and larger gaps of social competition. Few wealthy active particles insist on
maintaining, and possibly improving, their benefits;
• in a population constituted almost exclusively by poor active particlesγ[f ] attains
a value such that cooperation is inhibited, for individualstend to be involved in a
“battle of the have-nots”;
• conversely, in a population constituted almost exclusively by wealthy active
particlesγ[f ] attains a value such that competition is inhibited, because
individuals tend preferentially to cooperate for preserving their common benefits.
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3 - Modeling Welfare Policy
Welfare Policy and Political Competition
Introduce the number of poor and wealthy active particles attime t:
N−(t) =
n−1
2∑
i=1
fi(t), N+(t) =
n∑
i=n+3
2
fi(t).
Notice that, by excluding the middle classun+1
2
= 0 from bothN− andN+, we
implicitly regard it as economically “neutral”. Up to normalization over the total
number of active particles, we have0 ≤ N± ≤ 1 with alsoN− +N+ ≤ 1, hence the
quantity
S[f ] := N− −N
+,
which provides a macroscopic measure of thesocial gapin the population, is bounded
between−1 and1. Given that, we now look for a quadratic polynomial dependence of
γ[f ] onS[f ] taking into account the following conditions, which bring to a quantitative
level the previous qualitative arguments:
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3 - Modeling Welfare Policy
Welfare Policy and Political Competition
• S[f ] = S0 ⇒ γ[f ] = γ0, whereS0, γ0 are a reference social gap and the
corresponding reference critical distance, respectively;
• S[f ] = 1 ⇒ γ[f ] = n, which implies that when the population is composed by
poor particles only (N− = 1,N+ = 0) the socio-economic dynamics are of full
competition;
• S[f ] = −1 ⇒ γ[f ] = 0, which implies that, conversely, when the population is
composed by wealthy particles only (N− = 0,N+ = 1) the socio-economic
dynamics are of full cooperation.
Only integer values ofγ[f ] are meaningful, for so are the distances between pairs of
indexes of wealth classes, the resulting analytical expression ofγ[f ] yields
γ[f ] =2γ0(S[f ]2 − 1) − n(S0 + 1)(S[f ]2 − S0)
2(S20 − 1)
+n
2S[f ],
where· denotes integer part (floor).
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3 - Modeling Welfare Policy
Simulations
0
1
2
3
4
5
6
7
8
9
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
γ
S
γ0=3
γ0=7
Figure 3: The critical distanceγ[f ] vs. the macroscopic social gapS[f ] for the two
casesγ0 = 3, 7 andn = 9 social classes. Empty bullets indicate the reference value
γ0 corresponding to the reference social gapS0 = 0. Filled bullets indicate instead the
actual initial critical distanceγ[f ](t = 0) corresponding to the actual initial social gap
S[f ](t = 0). Dotted lines are plotted for visual reference.
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3 - Modeling Welfare Policy
Simulations
The evolution of the system predicted by the model depends essentially on the four
parametersn (the number of wealth classes),µ (the relative encounter rate for
cooperation,U0 (the average wealth of the population), andγ0 (the reference critical
distance). In more detail:
• n = 9 andµ = 0.3 are selected;
• two case studies forU0 are addressed, namelyU0 = −0.4 < 0 andU0 = 0, in
order to compare, respectively, the economic dynamics of a society in which poor
classes dominate with those of a society in which the initialdistribution of active
particles encompasses uniformly poor and rich classes;
• in addition, in each of the case studies above the asymptoticconfigurations for
both constant and variableγ[f ] are investigated, assuming, for duly comparison,
that in the former the critical distance coincides withγ0. Particularly,γ0 = 3,
corresponding to a mainly cooperative attitude, andγ0 = 7, corresponding instead
to a strongly competitive attitude, are chosen.
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3 - Modeling Welfare Policy
Simulations
On the Dynamics of Social and Political Conflicts Looking forthe Black Swan– p. 62/83
3 - Modeling Welfare Policy
Simulations
!
"#$!
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Lecture 4 - Index
1. Reasonings on Complex Systems
2. Mathematical Tools and Sources of Nonlinearity
3. Modeling Welfare Policy
4. Modeling Social Conflicts and Political Competition
4.1 Welfare Policy and Political Competition
4.2 Simulations
4.3 Early Signals of a Black Swan
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4 - Modeling Social Conflicts and Political Competition
Welfare Policy and Political Competition
The mathematical framework presented does not account for apossible interplay
between different interaction dynamics due to concomitantsocial and economic issues.
Let us consider the case of a multiple strategy(u, v), with u representing the wealth
status of the active particles andv their level of support/opposition to a Government
policy. Let us investigate how the welfare dynamics can induce changes of personal
opinions in terms of support/opposition to a certain political regime.
Assume that alsov is a discrete variable taking values in a lattice
Iv = v1 = −1, . . . , vm+1
2
= 0, . . . , vm = 1,
vr =2
m− 1r −
m+ 1
m− 1, r = 1, . . . , m,
agreeing thatv1 = −1 corresponds to the strongest opposition whereasvm = 1 to the
maximum support.
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4 - Modeling Social Conflicts and Political Competition
Welfare Policy and Political Competition
The mathematical structure is as follows:
dfri
dt=
m∑
p, q=1
n∑
h, k=1
ηpq
hkBpq
hk(i, r)fp
hfq
k − fri
m∑
q=1
n∑
k=1
ηrq
ik fq
k ,
for i = 1, . . . , n andr = 1, . . . , m, and where
fri = f
ri (t) : [0, Tmax] → IR +
is the number of individuals that, at timet, are expressing the strategy(ui, vr), or, in
other words, that belong to the wealth classui and to the opinion classvr. In addition:
• ηpq
hk is theinteraction rate between candidate particles with strategy(uh, vp) and
field particles with strategy(uk, vq). The same model fur the welfare policy is used,
according to the assumption that interactions among activeparticle are mainly driven
by the wealth status rather than by the difference of political opinion. Thusηpq
hk is
independent of the opinion classes that candidate and field particles belong to,
ηpq
hk = ηhk.
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4 - Modeling Social Conflicts and Political Competition
Welfare Policy and Political Competition
• Bpq
hk(i, r) is theprobability density that a candidate particle changes its strategy to
the test one(ui, vr) after interacting with a field particle. Analogously to the one
dimensional case, it is required to fulfill the probability density property:
m∑
r=1
n∑
i=1
Bpq
hk(i, r) = 1, ∀h, k = 1, . . . , n, ∀ p, q = 1, . . . , m.
The following factorization on the output test state(ui, vr) is proposed, relying
simply on intuition:Bpq
hk(i, r) = Bpq
hk(i)Bpq
hk(r),
• Bpq
hk(i) encodes the transitions of wealth class, which are further supposed to be
independent of the political feelings of the interacting pairs: Bpq
hk(i) = Bhk(i).
• Bpq
hk(r) encodes the changes of political opinion resulting from interactions.
Coherently with the observation made above that political persuasion is neglected,
so that political feelings originate in the individuals in consequence of their own
wealth condition, this term is assumed to depend on the wealth status and political
opinion of the candidate particle only:Bpq
hk(r) = Bp
h(r).
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4 - Modeling Social Conflicts and Political Competition
Welfare Policy and Political Competition
In view of the special structure
Bpq
hk(i, r) = Bhk(i)Bp
h(r),
it turns out that sufficient conditions ensuring the conservation in time of both the total
number of active particles and the average wealth status of the system are:
∑n
i=1Bhk(i) = 1, ∀h, k = 1, . . . , n
∑n
i=1uiBhk(i) = uh + σhk, ∀h, k = 1, . . . , n, σhk antisymmetric
∑m
r=1B
p
h(r) = 1, ∀h = 1, . . . , n, ∀ p = 1, . . . , m.
Mathematical are obtained by prescribing the interaction rateηpq
hk and the transition
probabilitiesBpq
hk(i, r).
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4 - Modeling Social Conflicts and Political Competition
Welfare Policy and Political Competition
As far as the modeling ofBp
h(r) is concerned, the following set of transition
probabilities is proposed:
1. Poor individual (uh < 0) in a poor society (U0 < 0):
p = 1
B1h(1) = 1
B1h(r) = 0, ∀ r 6= 1
p > 1
Bp
h(p− 1) = 2β
Bp
h(p) = 1 − 2β
Bp
h(r) = 0, ∀ r 6= p− 1, p
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4 - Modeling Social Conflicts and Political Competition
Welfare Policy and Political Competition 2. Wealthy individual (uh ≥ 0) in a poor
society (U0 < 0) or poor individual (uh < 0) in a wealthy society (U0 ≥ 0):
p = 1
B1h(1) = 1 − β
B1h(2) = β
B1h(r) = 0, ∀ r 6= 1, 2
p 6= 1, m
Bp
h(p− 1) = β
Bp
h(p) = 1 − 2β
Bp
h(p+ 1) = β
Bp
h(r) = 0, ∀ r 6= p− 1, p, p+ 1
p = m
Bmh (m− 1) = β
Bmh (m) = 1 − β
Bmh (r) = 0, ∀ r 6= m− 1, m
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4 - Modeling Social Conflicts and Political Competition
Welfare Policy and Political Competition
3. Wealthy individual (uh ≥ 0) in a wealthy society (U0 ≥ 0):
p < m
Bp
h(p) = 1 − 2β
Bp
h(p+ 1) = 2β
Bp
h(r) = 0, ∀ r 6= p, p+ 1
p = m
Bmh (m) = 1
Bmh (r) = 0 ∀ r 6= m,
whereβ ∈ [0, 1
2] is a parameter expressing the basic probability of changingpolitical
opinion.
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4 - Modeling Social Conflicts and Political Competition
Welfare Policy and Political Competition
Therefore, changes of political opinion are triggered jointly by the individual wealth
status, and the average collective one of the population:
• poor individuals in a poor society tend to distrust markedlythe Government
policy, sticking in the limit at the strongest opposition;
• wealthy individuals in a poor society and poor individuals in a wealthy society
exhibit, in general, the most random behavior. In fact, theymay trust the
Government policy either because of their own wealthiness,regardless of the
possibly poor general condition, or because of the collective affluence, in spite of
their own poor economic status. On the other hand, they may also distrust the
Government policy either because of the poor general condition, in spite of their
individual wealthiness, or because of their own poor economic status, regardless
of the collective affluence;
• wealthy individuals in a wealthy society tend instead to trust earnestly the
Government policy, sticking in the limit at the maximum support.
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4 - Modeling Social Conflicts and Political Competition
Simulations
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Simulations
This Figure refers to the caseU0 = 0, and shows that:
• In an economically neutral society with uniform wealth distribution not only do
wealthy classes stick at an earnest support to the Government policy, but also poor
ones do not completely distrust them, especially in a context of prevalent
cooperation among the classes (γ0 = 3).
• Therefore, this example does not suggest the development ofsignificant
polarization in that society, although a greater polarization is observed for higher
values ofγ.
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4 - Modeling Social Conflicts and Political Competition
Simulations
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4 - Modeling Social Conflicts and Political Competition
Simulations
The Figure refers to the caseU0 = −4, and shows that:
• A strong radicalization of the opposition. The model predicts indeed that, in such
a poor society, poor classes stick asymptotically at the strongest opposition,
whereas wealthy classes spread over the whole range of political orientations,
however with a mild tendency toward opposition for the moderately rich ones
(say,u5 = 0, u6 = 0.25, andu7 = 0.5.
• The growth of political aversion is especially emphasized under variable critical
distanceγ[f ], when the marked clustering of the population in the lowest wealth
classes, due to a more competitive spontaneous attitude, entails in turn a clustering
in the highest distrust of the regime.
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4 - Modeling Social Conflicts and Political Competition
Simulations
Remark 1. The case studies discussed above indicate that an effectiveinterpretation
of the social phenomena under consideration requires a careful examination of the
probability distribution over the microscopic states. Indeed, show entirely different
scenarios, that might not be completely caught simply by average macroscopic
quantities. An important role is given by the initial wealthin addition to the
distribution.
Remark 2. As a general concluding remark, we notice that, even with a variable
critical distance, none of the asymptotic configurations ofthe system seems to be
properly identifiable as a Black Swan. On the other hand, the simple case studies
addressed here show that radical situations can be generated from the complex
interplay between these two social aspects so that that Black Swans are mostly
expected to arise. Therefore it is worth looking for early signals.
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4 - Modeling Social Conflicts and Political Competition
Early Signals of a Black Swan
Let us details the expressionBlack Swan, introduced in the specialized literature for
indicating unpredictable events, which are far away from those generally observed by
repeated empirical evidence. A Black Swan is specifically characterized as follows:
(H. Taleb) “A Black Swan is a highly improbable event with three
principal characteristics: It is unpredictable; it carries a massive impact;
and, after the fact, we concoct an explanation that makes it appear less
random, and more predictable, than it was.”
In the author’s opinion, mathematical models usually rely on what is already known,
thus failing to predict what is instead unknown. It is a rare example of research moving
against the main stream of the traditional approaches, generally focused on
well-predictable events. His book had an important impact on the search of new
research perspectives: for instance, it motivated appliedmathematicians to propose
formal approaches to study the Black Swan, in an attempt to forecast conditions for its
onset. Individual behavioral rules and strategies are not,in most cases, constant in time
due to the evolutionary characteristics of living complex system.
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4 - Modeling Social Conflicts and Political Competition
Early Signals of a Black Swan
Bearing in mind the previous remarks, we now provide some suggestions for the
possible detection of a Black Swan within our current mathematical framework.
Let us assume that a specific model, derived from the mathematical structures
presented above, has a trend to an asymptotic configuration described by stationary
distributionsfri i=1, ..., n
r=1, ..., m:
limt→+∞
‖fr· − f
r· (t)‖ = 0, ∀ r = 1, . . . , m,
where‖ · ‖ is a suitable norm inn over the activityu ∈ Iu, for instance the1-norm:
‖fr· (t)‖1 =
n∑
i=1
|fri |(t).
Alternative metrics can also be introduced depending on thephenomenology of the
system at hand, which may need, for instance, either uniformor averaged ways of
measuring the distance between different configurations.
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Early Signals of the Black Swan
In addition, let us assume that the modeled system is expected to exhibit a stationary
trend described by some phenomenologically guessed distributionsfri i=1, ..., n
r=1, ..., m
. In
principle, such expected distribution have to be determined heuristically for each
specific case study, as we will see in the following.
Accordingly, we define the following time-evolving distancedBS (the subscript “BS”
standing for Black Swan):
dBS(t) := maxr=1, ..., m
‖fr· − f
r· (t)‖,
which, however, will generally not approach zero as time goes by for the heuristic
asymptotic distribution does not translate the actual trend of the system. This function
can be possibly regarded as one of theearly-warning signalsfor the emergence of
critical transitions to rare events, because it may highlight the onset of strong
deviations from expectations.
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Early Signals of the Black Swan
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 Tmax
dB
S
t
γ0=3
γ0=7
The mappingt 7→ dBS(t) computed in the case studies with variableγ, taking as
phenomenological guess the corresponding asymptotic distributions obtained with
constantγ.
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4 - Modeling Social Conflicts and Political Competition
Early Signals of the Black Swan
Remark 3. Specific applications may suggest other distances. For instance, linear or
quadratic moments might be taken into account. We consider that extreme events are
likely to be generated by the interplay of different types ofdynamics, a fact that should
be reflected in the choice of appropriate metrics.
Remark 4. The Figure shows the (common) qualitative trends of the mapping
t 7→ dBS(t): an initial decrease of the distance, which may suggest a convergence to
the guessed distribution, is then followed by a sudden increase (notice the singular
point in the graph ofdBS) toward a nonzero steady value, which ultimately indicatesa
deviation from the expected outcome. Such a turnround is possibly a macroscopic
signal that a Black Swan is about to appear. However, in orderto get a complete
picture, the average gross information delivered bydBS(t) has to be supplemented by
the detailed knowledge of the probability distribution over the microscopic states,
which is the only one able to properly distinguish between lower (γ0 = 3) and higher
(γ0 = 7) radicalization of political feelings when welfare dynamics are ruled by a
variable critical thresholdγ[f ].
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