Opinion dynamics, stubbornness and mean-field games Alexander - PowerPoint PPT Presentation

Opinion dynamics, stubbornness and mean-field games Alexander Aurell October 29, 2015

Introduction: what is modeled? Opinion propagation: Dynamics that describe the evolution of the opinions in a large population as a result of repeated interactions on the individual level. The level of stubbornness amongst the individuals vary. Phenomenon that occure in opinion propagation: Herd behaviour - convergence towards one (consensus) or multiple (polarization/plurality) opinion values.

Introduction: model set-up A set of populations is considered, each made up of uniform agents characterized by a given level of stubbornness. Individuals are partially stubborn while interested in reaching a consensus with as many other agents as possible. In Sweden you need 4% of the election votes to get seats in the parliament.

Introduction: main contribution Affine controls preserve the Gaussian distribution of population under the considered model.

Model set-up: dynamics State process for a generic member of population i ∈ I follows: � dx i = u i dt + ξ i dW i x i (0) = x 0 i where u i is a control function which may depend on time t , state x i and the density m ( x , t ) and ξ i is a real number.

Model set-up: cost Player i wants to maximize the functional �� ∞ � e − ρ t c i ( x i , u i , m ) dt J i ( u i , x 0 i ) = E 0 where � c i ( x i , u i , m ) = (1 − α i ) ( ν j ln( m j ( x i ( t )) , t )) j ∈ I m 0 i ) 2 − α i ( x i ( t ) − ¯ − β u 2 i .

Model set-up: cost Lets look at the terms in the running cost... ◮ � j ∈ I ν j ln ( m j ( x i ( . ) , . )): player i wants to share its opinion with as many other players as possible. m 0 i ) 2 : player i dislikes departing from the initial mean ◮ ( x i ( . ) − ¯ of its population. ◮ β u 2 i : the usual energy penalization. The parameter α i is determining the level of stubbornness of the players in population i .

Model set-up: problem statement The problem to solve, in u i , for each population is �� ∞ � e − ρ t c i ( x i , u i , m ) dt Maximize J i ( u i ; x 0 i ) = E (P i ) 0 Subjec to dx i = u i dt + ξ i dW i x i (0) = x 0 i u i admissible

The mean-field equations Theorem 1 The mean-field system corresponding to (P i ) is described by the equations: for all i ∈ I , � m 0 i ) 2 ∂ t v i ( x i , t ) + (1 − α i ) ν j ln( m j ( x i , t )) − α i ( x i − ¯ j ∈ I 2 β ( ∂ x v i ( x i , t )) 2 + ξ 2 1 2 ∂ 2 i + xx v i ( x i , t ) − ρ v i ( x i , t ) = 0 (1) � � �� − 1 ∂ t m i ( x i , t ) + ∂ x m i ( x i , t ) 2 β ∂ x v i ( x i , t ) − ξ 2 2 ∂ 2 i xx m i ( x i , t ) = 0 (2) m i ( x i , 0) = m 0 i (3)

The mean-field equations The optimal control to in (P i ) is given by i ( x i , t ) = − 1 u ∗ 2 β ∂ x v i ( x i , t ) So far so good. What happens if we restrics ourselves to linear strategies only?

Inverse Fokker-Planck problem Consider the Fokker-Planck problem in one dimension: ∂ t m i ( x i , t ) − ξ 2 2 ∂ 2 i xx m i ( x i , t ) + ∂ x ( u i ( x i , t ) m i ( x i , t )) . If we assume that m i : S → R , where S ⊂ R 2 , that m i ∈ C 2 ( S ) and that m i is a probability density function for each t which is positive for all ( x , t ) ∈ S . Then u i is the solution to � � � x i C ( t ) + ξ 2 1 i u i ( x i , t ) = 2 ∂ x m i ( x i , t ) − ∂ t m i ( x , t ) dx m i ( x i , t ) x 0 i where C ( t ) is an arbitrary function.

Gaussian Distribution Preserving Strategies Assume that population i has initial density − ( x i − µ 0 i ) 2 1 2 σ 2 m i ( x i , 0) = √ e 0 i 2 π σ 0 i and that the agents in this population implement a linear control strategy p ∈ C 1 ( R + ) u i ( x , t ) = ˆ p i ( t ) x + ˆ q ( t ) , ˆ q , ˆ Then � � � t � t x i ( t ) = e ˆ e − ˆ e − ˆ P i ( t ) P i ( τ ) q i ( τ ) d τ + ξ i P i ( τ ) dW i x 0 i + 0 0 � t where ˆ P i ( t ) = ˆ p i ( τ ) d τ . 0

Gaussian Distribution Preserving Strategies Furthermore, the density of population i is for each time t equal to − ( x i − µ i ( t )) 2 1 2 σ 2 i ( t ) m i ( x i , t ) = √ e σ i ( t ) 2 π where � � � t i ( t ) = e 2ˆ e − 2ˆ P i ( t ) P i ( τ ) d τ σ 2 σ 2 0 i + ξ 2 , i 0 � � t � µ i ( t ) = e ˆ e − ˆ P i ( t ) P i ( τ ) q i ( τ ) d τ µ 0 i + 0 Note: The implemented u i is not necessarily optimal.

Gaussian Distribution Preserving Strategies

Gaussian Distribution Preserving Strategies

Gaussian Distribution Preserving Strategies

Gaussian Distribution Preserving Strategies

Gaussian Distribution Preserving Strategies

Two crowd seeking populations A detailed example with two populations. If the agents of two populations apply linear strategies then the utility function becomes: �� ∞ e − ρ t � J i ( x 0 i ) = sup (1 − α i ) E u i 0 � � 2 � m j ( t )) 2 − v j j ( t )) + ( x i ( t ) − ¯ ln(2 πσ 2 = σ 2 2 j ( t ) j =1 � � m 0 i ) 2 − β u 2 = − α i ( x i ( t ) − ¯ dt i The populations follow the same dynamics as previously.

Two crowd seeking populations Two assuptions are made on the crowds: A1. At time 0 the two popilations have a Gaussian distribution A2. The agens adopt linear strategies that track a weighted sum: u i ( x , t ) = d i ( a i ¯ m i ( t ) + b i ¯ m j ( t ) + c i ¯ m 0 i − x i ) (4) d i > 0 1 = a i + b i + c i Note: Assumption 1 together with the dynamics of the population implies that the strategies (4) are GDPS.

Extending the state space By introducing the dynamics of ¯ m i , i = 1 , 2, into the model it is possible to characterize an optimal control. The extended state space equations are: for i = 1 , 2, j � = i , dx i = u i dt + ξ i dW i x i (0) = x 0 i ˙ m i ( t ) = d i ( b i ¯ ¯ m j ( t ) + c i ¯ m 0 i − ( b i + c i ) ¯ m i ( t )) m i (0) = µ 0 i ¯

Extending the state space Time for some rewriting... Denote by γ i = − d i ( b i + c i ). Then the state space equations can be written as       dx i 0 0 0 x i  =      dt d ¯ 0 ¯ m i γ i d i b i m i d ¯ m j 0 d j b j γ j m j ¯       1 0 0 u i ξ i     dt +   dW i + 0 c i 0 m 0 i ¯ 0 0 0 c j m 0 j ¯ 0

Extending the state space An even more compact notation is � − γ i � � ¯ � � c i � � ¯ � m i ( t ) 0 d i b i m 0 i ˙ m ( t ) = ¯ + d j b j γ j m j ( t ) ¯ 0 c j m 0 j ¯ � �� � � �� � � �� � � �� � m ¯ m 0 ¯ M C Adding the constant vector ¯ m 0 to the state vector, the problem becomes �� ∞ � e − ρ t � sup c i ( x , u , m , θ ) dt E u 0           dx i 0 0 0 x i 1 ξ i   =     dt +   u i dt +   dW i d ¯ 0 ¯ 0 0 m M C m d ¯ m 0 0 0 0 m 0 ¯ 0 0

Extending the state space � � T we get the LQ problem Finally, by letting X i = x i m ¯ m 0 ¯ �� ∞ � e − ρ t � � X T i � QX i + β u 2 inf E dt , i u 0 dX i = ( FX i + GU i ) dt + LdW i . The solution to an LQ problem of this kind is well known. Consider the new value function V i ( X i , t ) that solves ∂ t V t ( X i , t ) + H ( X i , ∂ X i V i ( X i , t )) + 1 2 ∂ 2 i ∂ 2 xx V i ( X i , t ) = 0 . If we assume that the value function is quadratic V i ( X i , t ) = X T i P ( t ) X i , then P ( t ) is the solution to the Riccati equation � GR − 1 G T � P ( t ) − ρ P ( t )+ P ( t ) F + F T P ( t ) − P ( t ) ˙ P ( t )+ � Q + W = 0

Extending the state space If P solves the Riccati equation then the optimal control is given by i ( t ) = − 1 β G T P ( t ) X i ˆ u ∗ = 1 β ( P 11 ( t ) x i ( t ) + P 12 ( t ) ¯ m i + P 13 ( t ) ¯ m j + P 14 ( t ) ¯ m 0 i + P 15 ( t ) ¯ m 0 j ) Conclusion: The extended state space model allows us to characterize an optimal control under assumptions (A1)-(A2).

Model behavior The mean ¯ m i ( t ) converges to a finite value. i ( t ) converges exponentially to ξ 2 The variance σ 2 i . The term 2 d i ( x i ( t ) − µ 2 j ( t )) ln(2 πσ 2 j ( t )) + therefore grows to infinity if there is no 2 σ 2 j ( t ) disturbance in population j , i.e. ξ j = 0. A consequence is that an optimal strategy (in the no disturbance case) must satisfy ρ − 2 d i > 0 or guarantee that all states converge to a single consensus.

Model behavior Recall that under A2, u i = d i ( a i ¯ m i ( t ) + b i ¯ m j ( t ) + c i ¯ m 0 i − x i ) where a i + b i + c i = 1. The mean of population i converges to m si = ( c i c j + b j c i ) ¯ m 0 i + b i c j ¯ m 0 j ¯ c i c j + b j c i + b i c j Note: If c i , c j � = 0 then ¯ m si � = ¯ m sj .

Model behavior A population with a i = b i = 0 is called hard core stubborn. A population with c i = 0 is called most gregarious. Some extreme cases: ◮ If population i is hard core stubborn, then ¯ m i ( t ) = ¯ m 0 i . If b i = 0 but a i � = 0, ¯ m si = ¯ m 0 i . ◮ If both populations are most gregarious, consensus is reached at m sj = b i d i ¯ m j + b j d j ¯ m i m si = ¯ ¯ . b i d i + b j d j ◮ If population i is most gregarious and population j is not, then ¯ m si = ¯ m sj = ¯ m 0 j . However, if population j is not hard core stubborn then ¯ m j ( t ) � = ¯ m 0 j .