Generalized Kinetic Equations and Stochastic Game Theory ...Generalized Kinetic Equations and...

Generalized Kinetic Equationsand Stochastic Game Theory for Social Systems

Andrea Tosin∗

Istituto per le Applicazioni del Calcolo “M. Picone”Consiglio Nazionale delle Ricerche

Rome, Italy

Modeling and Control in Social DynamicsCamden NJ, USA, October 6-9, 2014

∗Joint work with G. Ajmone-Marsan, N. Bellomo, M. A. Herrero

Andrea Tosin, IAC-CNR (Rome, Italy) Generalized Kinetic Equations and Stochastic Game Theory for Social Systems 1/8

Complexity Features of Social Systems

Living → active entities

Behavioral strategies, bounded rationality → randomness of humanbehaviors

Heterogeneous distribution of strategies

Behavioral strategies can change in time

Self-organized collective behavior can emerge spontaneously:

A Black Swan is a highly improbable event with threeprincipal characteristics: It is unpredictable; it carries a massiveimpact; and, after the fact, we concoct an explanation thatmakes 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]

Methods of the Generalized Kinetic Theory for Active Particles

u1 ui un

Social classes

Social classes: (poor) u1 = −1, . . . , ui, . . . , un = 1 (wealthy)

Political opinion: (dissensus) v1 = −1, . . . , vr, . . . , vm = 1 (consensus)

Distribution function: fri (t) = density of people in (ui, vr) at time t

Average wealth status: U(t) =∑ni=1

∑mr=1 uif

ri (t)

m∑p, q=1

n∑h, k=1

ηpqhkBpqhk[γ, U ](i, r)fphf

qk︸︷︷︸

[γ, U ](i, r):=Prob((uh, vp)→(ui, vr)|(uk, vq), γ, U)

− frim∑q=1

n∑k=1

ηrqik fqk︸︷︷︸

u1 ui un

Social classes

∑mr=1 uif

ri (t)

m∑p, q=1

n∑h, k=1

qk︸︷︷︸

− frim∑q=1

n∑k=1

u1 ui un

Social classes

∑mr=1 uif

ri (t)

m∑p, q=1

n∑h, k=1

qk︸︷︷︸

− frim∑q=1

n∑k=1

u1 ui un

Social classes

∑mr=1 uif

ri (t)

m∑p, q=1

n∑h, k=1

qk︸︷︷︸

− frim∑q=1

n∑k=1

u1 ui un

Social classes

∑mr=1 uif

ri (t)

dfridt

m∑p, q=1

n∑h, k=1

qk︸︷︷︸

− frim∑q=1

n∑k=1

u1 ui un

Social classes

∑mr=1 uif

ri (t)

dfridt

p, q=1

n∑h, k=1

qk︸︷︷︸

− frim∑q=1

n∑k=1

u1 ui un

Social classes

∑mr=1 uif

ri (t)

dfridt

p, q=1

n∑h, k=1

qk︸︷︷︸

− frim∑q=1

n∑k=1

Stochastic Games: Cooperation/Competition + Self-Conviction

Social dynamics: cooperation vs. competition

h kh - 1 k + 1

class distance ≤ γ

competition

h k - 1h + 1 k

class distance > γ

cooperation

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Opinion dynamics: self-conviction

Poor individuals in poor society → distrust

Wealthy individuals in a wealthy society → trust

Poor individuals in a wealthy societyWealthy individuals in a poor society

}→ most uncertain behavior

h kh - 1 k + 1

competition

h k - 1h + 1 k

class distance > γ

cooperation

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

h kh - 1 k + 1

competition

h k - 1h + 1 k

class distance > γ

cooperation

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

h kh - 1 k + 1

competition

h k - 1h + 1 k

class distance > γ

cooperation

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

h kh - 1 k + 1

competition

h k - 1h + 1 k

class distance > γ

cooperation

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

An example of transition probabilities

Ansatz:

Bpqhk[γ, U ](r, i) = B̄hk[γ](i)︸︷︷︸social

dynamics

· B̂ph[U ](r)︸︷︷︸opinion

dynamics

Social dynamicsCooperation: |k − h| > γ

If h ≤ k:

B̄hk[γ](i) =

1− |k−h|

n−1 if i = h

|k−h|n−1 if i = h+ 1

0 otherwise

If h > k:

B̄hk[γ](i) =

|k−h|n−1 if i = h− 1

1− |k−h|n−1 if i = h

0 otherwise

Opinion dynamicsSelf-conviction

Wealthy individual (uh ≥ 0) in a poorsociety (U < 0)

Poor individual (uh < 0) in a wealthysociety (U ≥ 0)

B̂ph[U ](r) =

β if r = p− 1

1− 2β if r = p

β if r = p+ 1

0 otherwise

0 ≤ β ≤1

Ansatz:

dynamics

If h ≤ k:

B̄hk[γ](i) =

1− |k−h|

n−1 if i = h

|k−h|n−1 if i = h+ 1

0 otherwise

If h > k:

B̄hk[γ](i) =

|k−h|n−1 if i = h− 1

1− |k−h|n−1 if i = h

0 otherwise

B̂ph[U ](r) =

β if r = p− 1

1− 2β if r = p

β if r = p+ 1

0 otherwise

0 ≤ β ≤1

Ansatz:

dynamics

If h ≤ k:

B̄hk[γ](i) =

1− |k−h|

n−1 if i = h

|k−h|n−1 if i = h+ 1

0 otherwise

If h > k:

B̄hk[γ](i) =

|k−h|n−1 if i = h− 1

1− |k−h|n−1 if i = h

0 otherwise

B̂ph[U ](r) =

β if r = p− 1

1− 2β if r = p

β if r = p+ 1

0 otherwise

0 ≤ β ≤1

Case Studies

Initial conditions

-1-0.75

-0.5-0.25

0 0.25

0.5 0.75

1Social classes -1

-0.75-0.5

-0.25 0 0.25

0.5 0.75

Political opinion

Society “neutral” on averageMean wealth: 0

-1-0.75

-0.5-0.25

0 0.25

0.5 0.75

1Social classes -1

-0.75-0.5

-0.25 0 0.25

0.5 0.75

Political opinion

Society poor on averageMean wealth: −0.4

Case Studies

Society which is “economically neutral” on average

γ = 3 γ = 7cooperative competitive

-1-0.75

-0.5-0.25

0 0.25

0.5 0.75

1Social classes -1

-0.75-0.5

-0.25 0 0.25

0.5 0.75

Political opinion

-1-0.75

-0.5-0.25

0 0.25

0.5 0.75

1Social classes -1

-0.75-0.5

-0.25 0 0.25

0.5 0.75

Political opinion

Case Studies

Society which is poor on average

γ = 3 γ = 7cooperative competitive

constant γ

-1-0.75

-0.5-0.25

0 0.25

0.5 0.75

1Social classes -1

-0.75-0.5

-0.25 0 0.25

0.5 0.75

Political opinion

-1-0.75

-0.5-0.25

0 0.25

0.5 0.75

1Social classes -1

-0.75-0.5

-0.25 0 0.25

0.5 0.75

Political opinion

variable γ

-1-0.75

-0.5-0.25

0 0.25

0.5 0.75

1Social classes -1

-0.75-0.5

-0.25 0 0.25

0.5 0.75

Political opinion

-1-0.75

-0.5-0.25

0 0.25

0.5 0.75

1Social classes -1

-0.75-0.5

-0.25 0 0.25

0.5 0.75

Political opinion

0 0.05

0.1 0.15

0.2 0.25

0.3 0.35

0.4 0.45

0.5 0.55

0.6 0.65

What About Black Swans?

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.

Need for macroscopically detectable indicators of sudden changes

Identify early-warning signals preceding radicalization

Example

Let f̃ be a phenomenologically guessed/expected asymptotic distribution:

dBS(t) := ‖f̃ − f(t)‖Rn×m

e.g.= max

1≤r≤m

n∑i=1

|f̃ri − fri (t)|.

Example

e.g.= max

1≤r≤m

n∑i=1

Example

e.g.= max

1≤r≤m

n∑i=1

Example

e.g.= max

1≤r≤m

n∑i=1

Example

dBS(t) := ‖f̃ − f(t)‖Rn×me.g.= max

1≤r≤m

n∑i=1

0 Tmax

γ =3γ =7

0 Tmax

γ =3γ =7

tipping points

References

G. Ajmone Marsan, N. Bellomo, A. Tosin.Complex Systems and Society – Modeling and Simulation.SpringerBriefs in Mathematics. Springer, 2013.

N. Bellomo, F. Colasuonno, D. Knopoff, J. Soler.

From systems theory of sociology to modeling the onset and evolution of criminality.In preparation.

N. Bellomo, M. A. Herrero, A. Tosin.

On the dynamics of social conflicts looking for the Black Swan.Kinet. Relat. Models, 6(3):459-479, 2013.

M. Delitala, T. Lorenzi.

A mathematical model for value estimation with public information and herding.Kinet. Relat. Models, 7(1):29-44, 2014.

M. Dolfin, M. Lachowicz.

Modeling altruism and selfishness in welfare dynamics: The role of nonlinear interactions.Math. Models Methods Appl. Sci., 24(12), 2014.

A. Tosin.

Kinetic equations and stochastic game theory for social systems.In A. Celletti, U. Locatelli, T. Ruggeri, E. Strickland, Eds., Mathematical Models andMethods for Planet Earth.Springer INdAM Series, vol. 6, pp. 37-57, Springer International Publishing, 2014.

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