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Generalized Kinetic Equations and 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 Dynamics Camden

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SLIDE 1

Generalized Kinetic Equations and 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 Dynamics Camden 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

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SLIDE 2

Complexity Features of Social Systems

Living → active entities

Andrea Tosin, IAC-CNR (Rome, Italy) Generalized Kinetic Equations and Stochastic Game Theory for Social Systems 2/8

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SLIDE 3

Complexity Features of Social Systems

Living → active entities Behavioral strategies, bounded rationality → randomness of human behaviors

Andrea Tosin, IAC-CNR (Rome, Italy) Generalized Kinetic Equations and Stochastic Game Theory for Social Systems 2/8

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SLIDE 4

Complexity Features of Social Systems

Living → active entities Behavioral strategies, bounded rationality → randomness of human behaviors Heterogeneous distribution of strategies

Andrea Tosin, IAC-CNR (Rome, Italy) Generalized Kinetic Equations and Stochastic Game Theory for Social Systems 2/8

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SLIDE 5

Complexity Features of Social Systems

Living → active entities Behavioral strategies, bounded rationality → randomness of human behaviors Heterogeneous distribution of strategies Behavioral strategies can change in time

Andrea Tosin, IAC-CNR (Rome, Italy) Generalized Kinetic Equations and Stochastic Game Theory for Social Systems 2/8

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SLIDE 6

Complexity Features of Social Systems

Living → active entities Behavioral strategies, bounded rationality → randomness of human behaviors 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 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]

Andrea Tosin, IAC-CNR (Rome, Italy) Generalized Kinetic Equations and Stochastic Game Theory for Social Systems 2/8

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SLIDE 7

Methods of the Generalized Kinetic Theory for Active Particles

u1 ui un v1 vr vm Social classes Political opinion

Andrea Tosin, IAC-CNR (Rome, Italy) Generalized Kinetic Equations and Stochastic Game Theory for Social Systems 3/8

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SLIDE 8

Methods of the Generalized Kinetic Theory for Active Particles

u1 ui un v1 vr vm Social classes Political opinion

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

Andrea Tosin, IAC-CNR (Rome, Italy) Generalized Kinetic Equations and Stochastic Game Theory for Social Systems 3/8

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SLIDE 9

Methods of the Generalized Kinetic Theory for Active Particles

u1 ui un v1 vr vm Social classes Political opinion

Social classes: (poor) u1 = −1, . . . , ui, . . . , un = 1 (wealthy) Political opinion: (dissensus) v1 = −1, . . . , vr, . . . , vm = 1 (consensus)

Andrea Tosin, IAC-CNR (Rome, Italy) Generalized Kinetic Equations and Stochastic Game Theory for Social Systems 3/8

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SLIDE 10

Methods of the Generalized Kinetic Theory for Active Particles

u1 ui un v1 vr vm Social classes Political opinion

Social classes: (poor) u1 = −1, . . . , ui, . . . , un = 1 (wealthy) Political opinion: (dissensus) v1 = −1, . . . , vr, . . . , vm = 1 (consensus) Distribution function: f r

i (t) = density of people in (ui, vr) at time t

Andrea Tosin, IAC-CNR (Rome, Italy) Generalized Kinetic Equations and Stochastic Game Theory for Social Systems 3/8

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SLIDE 11

Methods of the Generalized Kinetic Theory for Active Particles

u1 ui un v1 vr vm Social classes Political opinion

Social classes: (poor) u1 = −1, . . . , ui, . . . , un = 1 (wealthy) Political opinion: (dissensus) v1 = −1, . . . , vr, . . . , vm = 1 (consensus) Distribution function: f r

i (t) = density of people in (ui, vr) at time t

Average wealth status: U(t) = n

i=1

m

r=1 uif r i (t)

d f r

i

dt =

Andrea Tosin, IAC-CNR (Rome, Italy) Generalized Kinetic Equations and Stochastic Game Theory for Social Systems 3/8

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SLIDE 12

Methods of the Generalized Kinetic Theory for Active Particles

u1 ui un v1 vr vm Social classes Political opinion

Social classes: (poor) u1 = −1, . . . , ui, . . . , un = 1 (wealthy) Political opinion: (dissensus) v1 = −1, . . . , vr, . . . , vm = 1 (consensus) Distribution function: f r

i (t) = density of people in (ui, vr) at time t

Average wealth status: U(t) = n

i=1

m

r=1 uif r i (t)

d f r

i

dt =

m

  • p, q=1

n

  • h, k=1

ηpq

hkBpq hk[γ, U](i, r)f p hf q k

  • Gain

Bpq

hk[γ, U](i, r):=Prob((uh, vp)→(ui, vr)|(uk, vq), γ, U) Andrea Tosin, IAC-CNR (Rome, Italy) Generalized Kinetic Equations and Stochastic Game Theory for Social Systems 3/8

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SLIDE 13

Methods of the Generalized Kinetic Theory for Active Particles

u1 ui un v1 vr vm Social classes Political opinion

Social classes: (poor) u1 = −1, . . . , ui, . . . , un = 1 (wealthy) Political opinion: (dissensus) v1 = −1, . . . , vr, . . . , vm = 1 (consensus) Distribution function: f r

i (t) = density of people in (ui, vr) at time t

Average wealth status: U(t) = n

i=1

m

r=1 uif r i (t)

d f r

i

dt =

m

  • p, q=1

n

  • h, k=1

ηpq

hkBpq hk[γ, U](i, r)f p hf q k

  • Gain

Bpq

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

− f r

i m

  • q=1

n

  • k=1

ηrq

ik f q k

  • Loss

Andrea Tosin, IAC-CNR (Rome, Italy) Generalized Kinetic Equations and Stochastic Game Theory for Social Systems 3/8

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SLIDE 14

Stochastic Games: Cooperation/Competition + Self-Conviction

Social dynamics: cooperation vs. competition

1 n h k h - 1 k + 1 class distance ≤ γ competition 1 n h k - 1 h + 1 k class distance > γ cooperation

1 2 3 4 5 6 7 8 9

  • 1
  • 0.8 -0.6 -0.4 -0.2

0.2 0.4 0.6 0.8 1 γ S γ0=3 γ0=7

Andrea Tosin, IAC-CNR (Rome, Italy) Generalized Kinetic Equations and Stochastic Game Theory for Social Systems 4/8

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SLIDE 15

Stochastic Games: Cooperation/Competition + Self-Conviction

Social dynamics: cooperation vs. competition

1 n h k h - 1 k + 1 class distance ≤ γ competition 1 n h k - 1 h + 1 k class distance > γ cooperation

1 2 3 4 5 6 7 8 9

  • 1
  • 0.8 -0.6 -0.4 -0.2

0.2 0.4 0.6 0.8 1 γ S γ0=3 γ0=7

Opinion dynamics: self-conviction

Andrea Tosin, IAC-CNR (Rome, Italy) Generalized Kinetic Equations and Stochastic Game Theory for Social Systems 4/8

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SLIDE 16

Stochastic Games: Cooperation/Competition + Self-Conviction

Social dynamics: cooperation vs. competition

1 n h k h - 1 k + 1 class distance ≤ γ competition 1 n h k - 1 h + 1 k class distance > γ cooperation

1 2 3 4 5 6 7 8 9

  • 1
  • 0.8 -0.6 -0.4 -0.2

0.2 0.4 0.6 0.8 1 γ S γ0=3 γ0=7

Opinion dynamics: self-conviction

Poor individuals in poor society → distrust

Andrea Tosin, IAC-CNR (Rome, Italy) Generalized Kinetic Equations and Stochastic Game Theory for Social Systems 4/8

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SLIDE 17

Stochastic Games: Cooperation/Competition + Self-Conviction

Social dynamics: cooperation vs. competition

1 n h k h - 1 k + 1 class distance ≤ γ competition 1 n h k - 1 h + 1 k class distance > γ cooperation

1 2 3 4 5 6 7 8 9

  • 1
  • 0.8 -0.6 -0.4 -0.2

0.2 0.4 0.6 0.8 1 γ S γ0=3 γ0=7

Opinion dynamics: self-conviction

Poor individuals in poor society → distrust Wealthy individuals in a wealthy society → trust

Andrea Tosin, IAC-CNR (Rome, Italy) Generalized Kinetic Equations and Stochastic Game Theory for Social Systems 4/8

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SLIDE 18

Stochastic Games: Cooperation/Competition + Self-Conviction

Social dynamics: cooperation vs. competition

1 n h k h - 1 k + 1 class distance ≤ γ competition 1 n h k - 1 h + 1 k class distance > γ cooperation

1 2 3 4 5 6 7 8 9

  • 1
  • 0.8 -0.6 -0.4 -0.2

0.2 0.4 0.6 0.8 1 γ S γ0=3 γ0=7

Opinion dynamics: self-conviction

Poor individuals in poor society → distrust Wealthy individuals in a wealthy society → trust Poor individuals in a wealthy society Wealthy individuals in a poor society

  • → most uncertain behavior

Andrea Tosin, IAC-CNR (Rome, Italy) Generalized Kinetic Equations and Stochastic Game Theory for Social Systems 4/8

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SLIDE 19

An example of transition probabilities

Ansatz: Bpq

hk[γ, U](r, i) = ¯

Bhk[γ](i)

  • social

dynamics

· ˆ Bp

h[U](r)

  • pinion

dynamics

Andrea Tosin, IAC-CNR (Rome, Italy) Generalized Kinetic Equations and Stochastic Game Theory for Social Systems 5/8

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SLIDE 20

An example of transition probabilities

Ansatz: Bpq

hk[γ, U](r, i) = ¯

Bhk[γ](i)

  • social

dynamics

· ˆ Bp

h[U](r)

  • pinion

dynamics

Social dynamics Cooperation: |k − h| > γ If h ≤ k: ¯ Bhk[γ](i) =        1 − |k−h|

n−1

if i = h

|k−h| n−1

if i = h + 1

  • therwise

If h > k: ¯ Bhk[γ](i) =       

|k−h| n−1

if i = h − 1 1 − |k−h|

n−1

if i = h

  • therwise

Andrea Tosin, IAC-CNR (Rome, Italy) Generalized Kinetic Equations and Stochastic Game Theory for Social Systems 5/8

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SLIDE 21

An example of transition probabilities

Ansatz: Bpq

hk[γ, U](r, i) = ¯

Bhk[γ](i)

  • social

dynamics

· ˆ Bp

h[U](r)

  • pinion

dynamics

Social dynamics Cooperation: |k − h| > γ If h ≤ k: ¯ Bhk[γ](i) =        1 − |k−h|

n−1

if i = h

|k−h| n−1

if i = h + 1

  • therwise

If h > k: ¯ Bhk[γ](i) =       

|k−h| n−1

if i = h − 1 1 − |k−h|

n−1

if i = h

  • therwise

Opinion dynamics Self-conviction Wealthy individual (uh ≥ 0) in a poor society (U < 0) Poor individual (uh < 0) in a wealthy society (U ≥ 0) ˆ Bp

h[U](r) =

         β if r = p − 1 1 − 2β if r = p β if r = p + 1

  • therwise

0 ≤ β ≤ 1 2

Andrea Tosin, IAC-CNR (Rome, Italy) Generalized Kinetic Equations and Stochastic Game Theory for Social Systems 5/8

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SLIDE 22

Case Studies

Initial conditions

  • 1 -0.75
  • 0.5
  • 0.25 0 0.25 0.5 0.75 1

Social classes

  • 1
  • 0.75
  • 0.5
  • 0.25

0.25 0.5 0.75 1

Political

  • pinion

0.05

Society “neutral” on average Mean wealth: 0

  • 1 -0.75
  • 0.5
  • 0.25 0 0.25 0.5 0.75 1

Social classes

  • 1
  • 0.75
  • 0.5
  • 0.25

0.25 0.5 0.75 1

Political

  • pinion

0.05

Society poor on average Mean wealth: −0.4

Andrea Tosin, IAC-CNR (Rome, Italy) Generalized Kinetic Equations and Stochastic Game Theory for Social Systems 6/8

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SLIDE 23

Case Studies

Society which is “economically neutral” on average γ = 3 γ = 7 cooperative competitive

  • 1 -0.75
  • 0.5
  • 0.25 0 0.25 0.5 0.75 1

Social classes

  • 1
  • 0.75
  • 0.5
  • 0.25

0.25 0.5 0.75 1

Political

  • pinion

0.05 0.1 0.15

  • 1 -0.75
  • 0.5
  • 0.25 0 0.25 0.5 0.75 1

Social classes

  • 1
  • 0.75
  • 0.5
  • 0.25

0.25 0.5 0.75 1

Political

  • pinion

0.05 0.1 0.15 0.2 0.25 0.3 0.35

Andrea Tosin, IAC-CNR (Rome, Italy) Generalized Kinetic Equations and Stochastic Game Theory for Social Systems 6/8

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SLIDE 24

Case Studies

Society which is poor on average γ = 3 γ = 7 cooperative competitive constant γ

  • 1 -0.75
  • 0.5
  • 0.25 0 0.25 0.5 0.75 1

Social classes

  • 1
  • 0.75
  • 0.5
  • 0.25

0.25 0.5 0.75 1

Political

  • pinion

0.05 0.1 0.15 0.2 0.25

  • 1 -0.75
  • 0.5
  • 0.25 0 0.25 0.5 0.75 1

Social classes

  • 1
  • 0.75
  • 0.5
  • 0.25

0.25 0.5 0.75 1

Political

  • pinion

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55

variable γ

  • 1 -0.75
  • 0.5
  • 0.25 0 0.25 0.5 0.75 1

Social classes

  • 1
  • 0.75
  • 0.5
  • 0.25

0.25 0.5 0.75 1

Political

  • pinion

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

  • 1 -0.75
  • 0.5
  • 0.25 0 0.25 0.5 0.75 1

Social classes

  • 1
  • 0.75
  • 0.5
  • 0.25

0.25 0.5 0.75 1

Political

  • pinion

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

Andrea Tosin, IAC-CNR (Rome, Italy) Generalized Kinetic Equations and Stochastic Game Theory for Social Systems 6/8

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SLIDE 25

What About Black Swans?

Andrea Tosin, IAC-CNR (Rome, Italy) Generalized Kinetic Equations and Stochastic Game Theory for Social Systems 7/8

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SLIDE 26

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.

Andrea Tosin, IAC-CNR (Rome, Italy) Generalized Kinetic Equations and Stochastic Game Theory for Social Systems 7/8

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SLIDE 27

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

Andrea Tosin, IAC-CNR (Rome, Italy) Generalized Kinetic Equations and Stochastic Game Theory for Social Systems 7/8

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SLIDE 28

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 ˜ f be a phenomenologically guessed/expected asymptotic distribution: dBS(t) := ˜ f − f(t)Rn×m

Andrea Tosin, IAC-CNR (Rome, Italy) Generalized Kinetic Equations and Stochastic Game Theory for Social Systems 7/8

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SLIDE 29

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 ˜ f be a phenomenologically guessed/expected asymptotic distribution: dBS(t) := ˜ f − f(t)Rn×m

e.g.

= max

1≤r≤m n

  • i=1

| ˜ f r

i − f r i (t)|.

Andrea Tosin, IAC-CNR (Rome, Italy) Generalized Kinetic Equations and Stochastic Game Theory for Social Systems 7/8

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SLIDE 30

What About Black Swans?

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Tmax dBS t γ =3 γ =7 Andrea Tosin, IAC-CNR (Rome, Italy) Generalized Kinetic Equations and Stochastic Game Theory for Social Systems 7/8

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SLIDE 31

What About Black Swans?

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Tmax dBS t γ =3 γ =7

tipping points

Andrea Tosin, IAC-CNR (Rome, Italy) Generalized Kinetic Equations and Stochastic Game Theory for Social Systems 7/8

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SLIDE 32

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 and Methods for Planet Earth. Springer INdAM Series, vol. 6, pp. 37-57, Springer International Publishing, 2014.

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