Generalized Kinetic Equations and Stochastic Game Theory for Social - PowerPoint PPT Presentation

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

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

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

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

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

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

Methods of the Generalized Kinetic Theory for Active Particles v m Political opinion v r v 1 u 1 u i u n Social classes Andrea Tosin, IAC-CNR (Rome, Italy) Generalized Kinetic Equations and Stochastic Game Theory for Social Systems 3/8

Methods of the Generalized Kinetic Theory for Active Particles v m Political opinion v r v 1 u 1 u i u n Social classes Social classes: (poor) u 1 = − 1 , . . . , u i , . . . , u n = 1 (wealthy) Andrea Tosin, IAC-CNR (Rome, Italy) Generalized Kinetic Equations and Stochastic Game Theory for Social Systems 3/8

Methods of the Generalized Kinetic Theory for Active Particles v m Political opinion v r v 1 u 1 u i u n Social classes Social classes: (poor) u 1 = − 1 , . . . , u i , . . . , u n = 1 (wealthy) Political opinion: (dissensus) v 1 = − 1 , . . . , v r , . . . , v m = 1 (consensus) Andrea Tosin, IAC-CNR (Rome, Italy) Generalized Kinetic Equations and Stochastic Game Theory for Social Systems 3/8

Methods of the Generalized Kinetic Theory for Active Particles v m Political opinion v r v 1 u 1 u i u n Social classes Social classes: (poor) u 1 = − 1 , . . . , u i , . . . , u n = 1 (wealthy) Political opinion: (dissensus) v 1 = − 1 , . . . , v r , . . . , v m = 1 (consensus) Distribution function: f r i ( t ) = density of people in ( u i , v r ) at time t Andrea Tosin, IAC-CNR (Rome, Italy) Generalized Kinetic Equations and Stochastic Game Theory for Social Systems 3/8

Methods of the Generalized Kinetic Theory for Active Particles v m Political opinion v r v 1 u 1 u i u n Social classes Social classes: (poor) u 1 = − 1 , . . . , u i , . . . , u n = 1 (wealthy) Political opinion: (dissensus) v 1 = − 1 , . . . , v r , . . . , v m = 1 (consensus) Distribution function: f r i ( t ) = density of people in ( u i , v r ) at time t Average wealth status: U ( t ) = � n � m r =1 u i f r i ( t ) i =1 f r d i dt = Andrea Tosin, IAC-CNR (Rome, Italy) Generalized Kinetic Equations and Stochastic Game Theory for Social Systems 3/8

Methods of the Generalized Kinetic Theory for Active Particles v m Political opinion v r v 1 u 1 u i u n Social classes Social classes: (poor) u 1 = − 1 , . . . , u i , . . . , u n = 1 (wealthy) Political opinion: (dissensus) v 1 = − 1 , . . . , v r , . . . , v m = 1 (consensus) Distribution function: f r i ( t ) = density of people in ( u i , v r ) at time t Average wealth status: U ( t ) = � n � m r =1 u i f r i ( t ) i =1 m n f r d � � i η pq hk B pq hk [ γ, U ]( i, r ) f p h f q dt = k p, q =1 h, k =1 � �� � Gain B pq hk [ γ, U ]( i, r ):=Prob(( u h , v p ) → ( u i , v r ) | ( u k , v q ) , γ, U ) Andrea Tosin, IAC-CNR (Rome, Italy) Generalized Kinetic Equations and Stochastic Game Theory for Social Systems 3/8

Methods of the Generalized Kinetic Theory for Active Particles v m Political opinion v r v 1 u 1 u i u n Social classes Social classes: (poor) u 1 = − 1 , . . . , u i , . . . , u n = 1 (wealthy) Political opinion: (dissensus) v 1 = − 1 , . . . , v r , . . . , v m = 1 (consensus) Distribution function: f r i ( t ) = density of people in ( u i , v r ) at time t Average wealth status: U ( t ) = � n � m r =1 u i f r i ( t ) i =1 m n m n f r d � � � � i η pq hk B pq hk [ γ, U ]( i, r ) f p h f q − f r η rq ik f q dt = i k k p, q =1 q =1 h, k =1 k =1 � �� � � �� � Loss Gain B pq hk [ γ, U ]( i, r ):=Prob(( u h , v p ) → ( u i , v r ) | ( u k , v q ) , γ, U ) Andrea Tosin, IAC-CNR (Rome, Italy) Generalized Kinetic Equations and Stochastic Game Theory for Social Systems 3/8

Stochastic Games: Cooperation/Competition + Self-Conviction Social dynamics: cooperation vs. competition γ 0 =3 competition γ 0 =7 9 h - 1 h k k + 1 8 7 1 n 6 5 class distance ≤ γ γ 4 cooperation 3 h h + 1 k - 1 k 2 1 1 n 0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 class distance > γ S Andrea Tosin, IAC-CNR (Rome, Italy) Generalized Kinetic Equations and Stochastic Game Theory for Social Systems 4/8

Stochastic Games: Cooperation/Competition + Self-Conviction Social dynamics: cooperation vs. competition γ 0 =3 competition γ 0 =7 9 h - 1 h k k + 1 8 7 1 n 6 5 class distance ≤ γ γ 4 cooperation 3 h h + 1 k - 1 k 2 1 1 n 0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 class distance > γ S Opinion dynamics: self-conviction Andrea Tosin, IAC-CNR (Rome, Italy) Generalized Kinetic Equations and Stochastic Game Theory for Social Systems 4/8

Stochastic Games: Cooperation/Competition + Self-Conviction Social dynamics: cooperation vs. competition γ 0 =3 competition γ 0 =7 9 h - 1 h k k + 1 8 7 1 n 6 5 class distance ≤ γ γ 4 cooperation 3 h h + 1 k - 1 k 2 1 1 n 0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 class distance > γ S 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

Stochastic Games: Cooperation/Competition + Self-Conviction Social dynamics: cooperation vs. competition γ 0 =3 competition γ 0 =7 9 h - 1 h k k + 1 8 7 1 n 6 5 class distance ≤ γ γ 4 cooperation 3 h h + 1 k - 1 k 2 1 1 n 0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 class distance > γ S 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

Stochastic Games: Cooperation/Competition + Self-Conviction Social dynamics: cooperation vs. competition γ 0 =3 competition γ 0 =7 9 h - 1 h k k + 1 8 7 1 n 6 5 class distance ≤ γ γ 4 cooperation 3 h h + 1 k - 1 k 2 1 1 n 0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 class distance > γ S Opinion dynamics: self-conviction Poor individuals in poor society → distrust Wealthy individuals in a wealthy society → trust Poor individuals in a wealthy society � → most uncertain behavior Wealthy individuals in a poor society Andrea Tosin, IAC-CNR (Rome, Italy) Generalized Kinetic Equations and Stochastic Game Theory for Social Systems 4/8

An example of transition probabilities Ansatz: hk [ γ, U ]( r, i ) = ¯ · ˆ B pq B p B hk [ γ ]( i ) h [ U ]( r ) � �� � � �� � social opinion dynamics dynamics Andrea Tosin, IAC-CNR (Rome, Italy) Generalized Kinetic Equations and Stochastic Game Theory for Social Systems 5/8

An example of transition probabilities Ansatz: hk [ γ, U ]( r, i ) = ¯ · ˆ B pq B p B hk [ γ ]( i ) h [ U ]( r ) � �� � � �� � social opinion dynamics dynamics Social dynamics Cooperation: | k − h | > γ If h ≤ k : 1 − | k − h |  if i = h n − 1    ¯ | k − h | B hk [ γ ]( i ) = if i = h + 1 n − 1    0 otherwise If h > k : | k − h |  if i = h − 1 n − 1    ¯ 1 − | k − h | B hk [ γ ]( i ) = if i = h n − 1   0 otherwise  Andrea Tosin, IAC-CNR (Rome, Italy) Generalized Kinetic Equations and Stochastic Game Theory for Social Systems 5/8