Opinion Dynamics and Stubbornness Via Multi-Population Mean-Field Games

Abstract

This paper studies opinion dynamics for a set of heterogeneous populations of individuals pursuing two conflicting goals: to seek consensus and to be coherent with their initial opinions. The multi-population game under investigation is characterized by (i) rational agents who behave strategically, (ii) heterogeneous populations, and (iii) opinions evolving in response to local interactions. The main contribution of this paper is to encompass all of these aspects under the unified framework of mean-field game theory. We show that, assuming initial Gaussian density functions and affine control policies, the Fokker–Planck–Kolmogorov equation preserves Gaussianity over time. This fact is then used to explicitly derive expressions for the optimal control strategies when the players are myopic. We then explore consensus formation depending on the stubbornness of the involved populations: We identify conditions that lead to some elementary patterns, such as consensus, polarization, or plurality of opinions. Finally, under the baseline example of the presence of a stubborn population and a most gregarious one, we study the behavior of the model with a finite number of players, describing the dynamics of the average opinion, which is now a stochastic process. We also provide numerical simulations to show how the parameters impact the equilibrium formation.

Appendix

The optimization problem introduced in Sect. 2 can be turned into a Multi-Population Mean-Field Game. Preliminary to the derivation of a mean-field game is the definition of a value function as commonly done also in differential game theory or optimal control. The value function is the value of the optimization problem, carried out by each single player k, starting at time t from state $x^k$ and for given densities m(t). As we will show, the value function only depends on population’s characteristics (apart from the initial state $x^k(t)$).

Proposition 6.1

Consider a generic population i and any agent k such that $i(k)=i$. Define the value function for agent k as

$$\begin{aligned} v_{i}(x^k(t),t): = \sup _{u^k(\cdot )} {\mathbb {E}}\left\{ \int _t^{\infty }e^{-\rho \tau } c^k(x^k(\tau ), u^k(\tau ), m(\tau )) d \tau )\right\} . \end{aligned}$$

Then, the mean-field system is described by the equations

$$\begin{aligned} \left\{ \begin{array}{l} \partial _t v_i(x^k(t),t) + (1-\alpha _i)\sum _{j\in I} \nu _{ij} \ln (m_j(x^k(t),t)) - \alpha _i (x^k(t) - \mu _{i}(0))^2 + \\ \quad +\frac{1}{2 \beta } (\partial _x v_i(x^k(t),t))^2 + \frac{\xi _i^2}{2} \partial _{xx}^2 v_i(x^k(t),t) -\rho v_i(x^k(t),t)=0, \\ \quad \partial _t m_i(x^k(t),t) +\partial _x\Big [m_i(x^k(t),t) \Big (- \frac{ 1 }{2 \beta } \partial _x v_i(x^k(t),t) \Big ) \Big ] - \frac{1}{2}\xi _i^2 \partial ^2_{xx} m_i(x^k(t),t)\\ \quad =0, \end{array}\right. \nonumber \\ \end{aligned}$$

(36)

for some initial population state distribution $m_{i}(0)$ for all $i\in I$. Furthermore, the optimal control is of the form

$$\begin{aligned} u_i^*(x^k(t),t)= - \frac{ 1 }{2 \beta } \partial _x v_i(x^k(t),t). \end{aligned}$$

(37)

Proof

From dynamic programming, the value function can be obtained from a corresponding maximized Hamiltonian function $H^k$ involving an adjoint variable $p_i$, called the ith co-state, and given by

$$\begin{aligned} H^k(x,p_i,m)=\sup _{u_i}\left\{ c^k(x,u_i,m)+p_i u_i \right\} . \end{aligned}$$

From [13], the mean-field system associated with the mean-field game introduced in Sect. 2 is given by

$$\begin{aligned}&\partial _t v_i(x,t)+ {H^k}(x,\partial _x v_i(x,t),m) + \frac{1}{2}\xi _i^2 \partial ^2_{xx} v_i(x,t) - \rho v_i(x,t)=0,\nonumber \\&\partial _t m_i(x,t)+\partial _x\Big (m_i(x,t)\partial _p H^k(x,\partial _x v_i(x,t),m) \Big ) - \frac{1}{2}\xi _i^2 \partial ^2_{xx} m_i(x,t)=0,\qquad \end{aligned}$$

(38)

where $m_i(x,0)=m_{0i}(x)$ for all $i \in I$ are the initial distributions and where $x=x^k(t)$.

We first prove condition (37). To this end, let us write the Hamiltonian as:

$$\begin{aligned} {H^k}(x,\partial _x v_i(x,t),m)= & {} \sup _{u_i} \Big \{(1-\alpha _i) \sum _{j \in I} \nu _{ij} \ln (m_j(x,t)) \nonumber \\&- \alpha _i (x - \mu _{0i})^2 - \beta u^2 + \partial _x v_i(x,t) u_i\Big \}=0. \end{aligned}$$

(39)

By differentiating with respect to $u_i$, we obtain

$$\begin{aligned} 2 \beta u_i(x,t) + \partial _x v_i(x,t) = 0, \end{aligned}$$

(40)

which yields (37). Note that convexity of the cost functional guarantees sufficiency of the above first-order condition.

We now prove (36). Concerning the first equation, which is a PDE corresponding to the Hamilton–Jacobi–Bellman equation, let us replace $u_i$ in the Hamiltonian (39) by its expression (37), i.e.,

$$\begin{aligned} {H^k}(x,\partial _x v_i(x,t),m)&= (1-\alpha _i)\sum _{j \in I} \nu _{ij}\ln (m_j(x,t)) - \alpha _i (x - \mu _{0i})^2\\&\quad - \beta (u_i^*(x,t))^2 + \partial _x v_i(x,t) u_i^*(x,t) \\&= (1-\alpha _i)\sum _{j \in I} \nu _{ij}\ln (m_j(x,t)) - \alpha _i (x - \mu _{0i})^2 \\&\quad +\frac{1}{2 \beta } (\partial _x v_i(x,t))^2. \end{aligned}$$

Using the above expression of the Hamiltonian in the first equation in (38), we obtain the Hamilton–Jacobi–Bellman equation in (36).

To prove the second equation, which is a PDE representing the Fokker–Planck–Kolmogorov equation, we simply substitute (37) into the second equation in (38), and this concludes the proof. $\square$

The significance of the above result is that to find the optimal controls, we need to solve the set of coupled PDEs defined in (36) with given boundary conditions. This can done by iteratively solving the Hamilton–Jacobi–Bellman equation for fixed $m_i$ and by entering the optimal $u_i$ obtained from (37) in the Fokker–Planck–Kolmogorov equation, until a fixed point in $v_i$ and $m_i$ is reached [24]. In other words, it must be proved that such a map is a contraction, and to do this, we rely on compactness of the map itself and on the Schauder fixed point theorem [15]. Note that, in Proposition 6.1, we do not consider a stationary control or a stationary population density distribution, although we deal with a discounted objective function over an infinite horizon. In fact, we are interested in determining the evolution of the population density distribution function over time under the general hypothesis that at time 0 the population is not distributed according to the long-term equilibrium density distribution.

A solution of (36) is said Nash mean-field equilibrium as it involves a set $\{(m^*_i,u^*_i): i \in I\}$ of functions defined for all times $t\ge 0$ such that

$$\begin{aligned} (m^*_i,u^*_i)= & {} \text {arg}\sup _{m_i(.),u_i(.)} {\mathbb {E}}\left\{ \int _0^{\infty }e^{-\rho t} c^k(x^k, u_i, m) \hbox {d}t\Big |m_j = m^*_j, \forall j \in I \setminus \{i\}\right\} \\ \forall i\in & {} I. \end{aligned}$$

In other words, any player of population i does not benefit from changing its control policy $u_i^*$ if the control policies, and therefore also the distributions, of the other populations are fixed to, respectively, $u_j^*$ and $m_j^*$ for all $j \in I \setminus \{i\}$. As a consequence, also the trajectory over time of the distribution $m_i^*$ is unchanged.