Analysis of a herding model in social economics

Transcription

1 Analysis of a herding model in social economics Lara Trussardi 1 Ansgar Jüngel 1 C. Kühn 1 1 Technische Universität Wien Taormina - June 13, L. Trussardi, A. Jüngel, C. Kühn (TUW) Analysis of a herding model Taormina - June 13, / 15

2 Index 1 Introduction 2 Mathematical study 3 Bifurcation approach 4 Outlook L. Trussardi, A. Jüngel, C. Kühn (TUW) Analysis of a herding model Taormina - June 13, / 15

3 Hearding and aim Herd behavior: a large number of people acting in the same way at the same time L. Trussardi, A. Ju ngel, C. Ku hn (TUW) Stock market: greed in frenzied buying (named bubbles) and fear in selling (named crash) Analysis of a herding model Taormina - June 13, / 15

4 Hearding and aim Herd behavior: a large number of people acting in the same way at the same time Stock market: greed in frenzied buying (named bubbles) and fear in selling (named crash) Goal To model information herding in a macroscopic setting with mathematical analysis L. Trussardi, A. Ju ngel, C. Ku hn (TUW) Analysis of a herding model Taormina - June 13, / 15

5 The model Ω R d : bounded domain, e.g. Ω = ( 1, 1) d x Ω : multidimensional information variable (political opinion, wealth of individual or company... ) u(x, t) : number of people having information x at time t, 0 u 1 v(x, t) : influence potential ( v: influence field), v R opinion distribution u diffusion+source drift influence potential v L. Trussardi, A. Jüngel, C. Kühn (TUW) Analysis of a herding model Taormina - June 13, / 15

6 Opinion distribution u We require a balance between the opinions which is represented by a diffusion term u The opinion distribution is subject to a certain influence, that means we need a drift term -div(g(u) v) We suppose that individuals with an extreme opinion are more stable in their convictions, so we need a non linear g(u) for example g(u) = u(1 u) t u = u div(u(1 u) v), in Ω u ν Ω = 0 u(, 0) = u 0 L. Trussardi, A. Jüngel, C. Kühn (TUW) Analysis of a herding model Taormina - June 13, / 15

7 Influence potential v We require averaging i.e. diffusion term with a positive constant κ v, κ > 0 We need a relaxation term that is represented with a linear term in v with a positive constant αv, α > 0 L. Trussardi, A. Jüngel, C. Kühn (TUW) Analysis of a herding model Taormina - June 13, / 15

8 Influence potential v We require averaging i.e. diffusion term with a positive constant κ v, κ > 0 We need a relaxation term that is represented with a linear term in v with a positive constant αv, α > 0 There is a herding effect that is modelled with a source term f (u) = u(1 u) We need a term as a regularization of the equation, enabling us to derive some entropy structure. This is represented with a diffusion in u δ u for small δ t v = δ u + κ v + u(1 u) αv, in Ω v ν Ω = 0 v(, 0) = v 0 Remark: δ 0 useful for mathematical analysis L. Trussardi, A. Jüngel, C. Kühn (TUW) Analysis of a herding model Taormina - June 13, / 15

9 The model Goal: existence of solutions, behaviour for t, stability, bifurcation analysis Cross-diffusion model t u = div( u g(u) v), in Ω t v = δ u + κ v + f (u) αv, in Ω ( u g(u) v) ν = (δ u + κ v) ν = 0, on Ω, t > 0 u(, 0) = u 0, v(, 0) = v 0 ( ) 1 g(u) Main difficulties: the diffusion matrix is not positive δ κ definite Main idea: entropy method (for δ 0!) L. Trussardi, A. Jüngel, C. Kühn (TUW) Analysis of a herding model Taormina - June 13, / 15

10 Entropy approach For δ > 0 the system possesses a logarithmic entropy H(t) = [u log u + (1 u) log(1 u) + v 2 ] dx 2δ Entropy inequality: Entropy variables Ω dh dt + (y, w) = Ω ( u 2 u(1 u) + v 2 ) dx C δ ( h u, h ) ( ( u ) = log, v ) v 1 u δ We get ( ) ( ( u u(1 u) δu(1 u) t = div v δu(1 u) δκ This matrix is positive definite for δ > 0. ) ( ) ) ( y + w 0 f (u) αv L. Trussardi, A. Jüngel, C. Kühn (TUW) Analysis of a herding model Taormina - June 13, / 15 )

11 Main result Global existence theorem Let Ω R 2 be a bounded domain with smooth boundary, δ > 0, 0 u 0, v 0 L 1 (Ω) such that H(u 0, v 0 ) 0, a weak solution (u, v) with u 0 in Ω (0, ) to { t u = div( u u(1 u) v) t v = δ u + κ v + u(1 u) αv Idea of the proof: 1 approximate elliptic problem: for τ > 0 time discretisation and addition of the term ε 2 y + εw 2 Leray-Schauder fixed point theorem 3 estimates from entropy inequality for (τ, ε) 0 L. Trussardi, A. Jüngel, C. Kühn (TUW) Analysis of a herding model Taormina - June 13, / 15

12 Main result Global existence theorem Let Ω R 2 be a bounded domain with smooth boundary, δ > 0, 0 u 0, v 0 L 1 (Ω) such that H(u 0, v 0 ) 0, a weak solution (u, v) with u 0 in Ω (0, ) to { t u = div( u u(1 u) v) t v = δ u + κ v + u(1 u) αv Idea of the proof: 1 approximate elliptic problem: for τ > 0 time discretisation and addition of the term ε 2 y + εw 2 Leray-Schauder fixed point theorem 3 estimates from entropy inequality for (τ, ε) 0 The existence result can be extended to δ < 0 (not too negative) L. Trussardi, A. Jüngel, C. Kühn (TUW) Analysis of a herding model Taormina - June 13, / 15

13 Steady state Goal: study under which condition there is congestion of opinion { div( u u (1 u ) v ) = 0 δ u + κ v + u (1 u ) αv = 0 Possible steady states (u, v ): 1 constants (u, v ) =(ū, ū(1 ū) α ) with ū [0, 1] 2 1 non constant (u, v ) = (, φ) with φ = v 1+e φ c ( ( u )) 0 = u u (1 u ) v = v log 1 u and φ solves φ = F (φ, φ) L. Trussardi, A. Jüngel, C. Kühn (TUW) Analysis of a herding model Taormina - June 13, / 15

14 Long time behaviour Goal: study the exponential decay of the weak solution to the homogeneous steady state (u, v ) Long time behaviour by studying the relative entropy H(u, v) = [ u log( u ) + (1 u) log( 1 u ] ) + (v v ) 2 dx u 1 u 2δ Ω L. Trussardi, A. Jüngel, C. Kühn (TUW) Analysis of a herding model Taormina - June 13, / 15

15 Long time behaviour Goal: study the exponential decay of the weak solution to the homogeneous steady state (u, v ) Long time behaviour by studying the relative entropy H(u, v) = Conclusion: [ u log( u ) + (1 u) log( 1 u ] ) + (v v ) 2 dx u 1 u 2δ Ω positive δ: decay to the constant steady state for t 0 u(t) u 2 L 2 (Ω) 0, v(t) v 2 L 2 (Ω) 0, negative δ: decay to the constant steady state IF AND ONLY IF δ > 4κ NO herding occurs if δ > 4κ L. Trussardi, A. Jüngel, C. Kühn (TUW) Analysis of a herding model Taormina - June 13, / 15

16 Bifurcation approach Goal: study the existence of non-constant steady states Idea: Choose δ as a bifurcation parameter We can apply bifurcation theory to show that the solutions may bifurcate from the constant steady state (u, v ) 1 linearization of the system around the constant steady state (u, v ) 2 study the eigenvalue problem L. Trussardi, A. Jüngel, C. Kühn (TUW) Analysis of a herding model Taormina - June 13, / 15

17 Crandall-Rabinowitz theorem We consider U = F(U, δ), with Neumann boundary condition and homogeneous steady state (w.l.o.g.) U = 0. The eigenvalue problem is: U (D U F)(0, δ)u = λu In our model: ( ) u U = v F(u, v, δ) = ( ) ( u + g(u) v) δ u κ v + αv f (u) We need to incorporate the mass constraint: Ω u(x)dx Ω u 0 L. Trussardi, A. Jüngel, C. Kühn (TUW) Analysis of a herding model Taormina - June 13, / 15

18 Crandall-Rabinowitz theorem We consider U = F(U, δ), with Neumann boundary condition and homogeneous steady state (w.l.o.g.) U = 0. The eigenvalue problem is: U (D U F)(0, δ)u = λu In our model: ( ) u U = v F(u, v, δ) = ( ) ( u + g(u) v) δ u κ v + αv f (u) We need to incorporate the mass constraint: Ω u(x)dx Ω u 0 Hypothesis: the Fredholm index of D U F(0, δ) is zero the null space N(D U F(0, δ)) {0}, in particular N(D U F(0, δ)) = span[û] D δu F(0, δ )(Û) / R(D U F(0, δ )) L. Trussardi, A. Jüngel, C. Kühn (TUW) Analysis of a herding model Taormina - June 13, / 15

19 Crandall-Rabinowitz theorem Theorem Assume the previous hypothesis, then there is a non trivial continuosly differentiable curve through (0, δ ) {(U(s), δ(s)) s ( σ, σ), (U(0), δ(0)) = (0, δ )} such that F(U(s), δ(s)) = 0 for s ( σ, σ) and all solutions of F(U, δ) = 0 in a neighborhood of (0, δ ) belong to this curve. The intersection (0, δ ) is called a bifurcation point. Main problems: prove that the Frechet derivatives of F is a Fredholm operator with index zero check the transversality condition δ (0) 0 study the derivatives of F L. Trussardi, A. Jüngel, C. Kühn (TUW) Analysis of a herding model Taormina - June 13, / 15

20 Crandall-Rabinowitz theorem Theorem Assume the previous hypothesis, then there is a non trivial continuosly differentiable curve through (0, δ ) {(U(s), δ(s)) s ( σ, σ), (U(0), δ(0)) = (0, δ )} such that F(U(s), δ(s)) = 0 for s ( σ, σ) and all solutions of F(U, δ) = 0 in a neighborhood of (0, δ ) belong to this curve. The intersection (0, δ ) is called a bifurcation point. Main problems: prove that the Frechet derivatives of F is a Fredholm operator with index zero check the transversality condition δ (0) 0 study the derivatives of F We expect a transcritical bifurcation L. Trussardi, A. Jüngel, C. Kühn (TUW) Analysis of a herding model Taormina - June 13, / 15

21 Further studies To study under which conditions this model describes herding Analysis and numerics for negative δ (kind of bifurcation and in which direction) Modelling of herding using a kinetic approach Possibly identification of this diffusion model as the mean-field limit of the kinetic equation Thanks for your attention L. Trussardi, A. Jüngel, C. Kühn (TUW) Analysis of a herding model Taormina - June 13, / 15