A Differential Game Model of Opinion Dynamics: Accord and Discord as Nash Equilibria

Abstract

This paper presents a noncooperative differential (dynamic) game model of opinion dynamics with open-loop information structure. In this game, the agents’ motives are shaped by their expectations of the nature of others’ opinions as well as how susceptible they are to get influenced by others, how stubborn they are, and how quick they are willing to change their opinions on a set of issues in a prescribed time interval. These motives are independently formed by all agents. The existence of a Nash equilibrium in the network means that a collective behavior emerges out of local interaction rules and these individual motives. We prove that a unique Nash equilibrium may exist in the game under quite different circumstances. It may exist not only if there is a harmony of perceptions among the agents of the network, but also when agents have different views about the correlation among issues. The first leads to an accord in the network usually expressed as a partial consensus, and the second to a discord in the form of oscillating opinions. In the case of an accord, the harmony in the network may be in the form of similarity in pairwise conceptions about the issues but may also be an agreement on the status of a “leader” in the network. A Nash equilibrium may fail to exist only if the network is in a state of discord.

Appendix

1.1 The Game of Opinion Dynamics is of the Linear Quadratic Type

To show that (2) is of the linear quadratic type [6, Definition 6.5], we consider

$$\begin{aligned} {\mathbf {z}}_i=[\,\varDelta _{i1}'\;\dots \;\varDelta _{i(i-1)}'\;\;\varDelta _{ii}'\;\dots \;\varDelta _{in}']' \in \mathbb {R}^{nd}, \end{aligned}$$

where

$$\begin{aligned} \varDelta _{ij}:=\left\{ \begin{array}{ll} {\mathbf {x}}_i-{\mathbf {x}}_j, &{} \text {if }i\ne j; \\ {\mathbf {x}}_i-{\mathbf {b}}_i, &{} \text {if }i=j; \end{array}\right. \end{aligned}$$

for $i,j=1,\dots ,n$. Then, we have ${\mathbf {z}}=[\,{\mathbf {z}}_1'\;\dots \;{\mathbf {z}}_n'\,]'\in \mathbb {R}^{n^2d}$ as the state vector, which satisfies

$$\begin{aligned} \dot{{\mathbf {z}}} = \sum _{i=1}^n B_i\mathbf {u_i}, \end{aligned}$$

(15)

where $B_i\in \mathbb {R}^{n^2d\times d}$ is given as

$$\begin{aligned} B_i=\left[ \begin{array}{ccccccc} -({\mathbf {e}}_i\otimes I_d)'&\cdots&-({\mathbf {e}}_i\otimes I_d)'&(\mathbf {1}_n\otimes I_d)'&-({\mathbf {e}}_i\otimes I_d)'&\cdots&-({\mathbf {e}}_i\otimes I_d)' \end{array}\right] ', \end{aligned}$$

(16)

where the matrix block $\mathbf {1}_n\otimes I_d$ is the ith block and ${\mathbf {e}}_i=[\,0\dots 0\;\;1\;\;0\dots 0\,]'\in \mathbb {R}^n$ is the standard ith basis vector of $\mathbb {R}^n$.

The cost functional (1) can now be written as

$$\begin{aligned} \mathscr {J}_i({\mathbf {z}},{\mathbf {u}}_i) = \frac{1}{2}\int _0^\tau \left( {\mathbf {z}}'L_i{\mathbf {z}} + {\mathbf {u}}_i'{\mathbf {u}}_i\right) dt, \end{aligned}$$

(17)

where $L_i=H_i\otimes K_i \in \mathbb {R}^{n^2d\times n^2d}$ with $H_i\in \mathbb {R}^{n\times n}$ whose jkth entry, for $j,k=1,\dots ,n$, is defined as

$$\begin{aligned}{}[H_i]_{jk} = \left\{ \begin{array}{ll} 1, &{} \text {if } j=k=i; \\ 0, &{} \text {otherwise}; \end{array}\right. \end{aligned}$$

and $K_i=\text {diag}[\,W_{i1},\dots ,W_{in}\,]\in \mathbb {R}^{nd\times nd}$. From the above formulation, it can be seen that the game described by (15) and (17) is a linear quadratic differential game. By Assumption 1, $L_{i}$ is positive semi-definite, which gives that (17), equivalently (1), is a strictly convex function of ${\mathbf {u}}_i(t)$. Hence, the Nash equilibrium, if it exists, is unique [6, Proof of Theorem 6.12]. Thus, we can skip a more familiar approach of coupled Riccati differential equations, which involve manipulations with large sparse matrices like $L_i$ above.

1.2 Proof of Theorem 1 Completed

The necessary conditions ([6, Theorem 6.11], [27, Chapter 5]) for the existence of a minimum of the cost function (1) are

$$\begin{aligned} {\mathbf {u}}_i^*(t)= & {} \arg \min _{{\mathbf {u}}_i\in {\mathsf {S}}_i} \mathcal{{H}}_i(\mathbf {p}_i,{\mathbf {x}},{\mathbf {b}}_i,{\mathbf {u}}_i), \nonumber \\ \dot{\mathbf {x}}_i(t)= & {} \frac{\partial \mathcal{{H}}_i}{\partial \mathbf {p}_i}, \ \ \dot{\mathbf {p}}_i(t) = -\frac{\partial \mathcal{{H}}_i}{\partial {\mathbf {x}}_i}, \nonumber \\ {\mathbf {x}}_i(0)= & {} {\mathbf {b}}_i, \ \ \mathbf {p}_i(\tau ) = 0, \ \ i\in {\mathsf {N}}, \end{aligned}$$

(18)

where $\mathbf {p}_i:[0,\tau ] \rightarrow {\mathbb {R}}^{nd}$ is a costate function and $\mathcal{{H}}_i$ is the Hamiltonian given by

$$\begin{aligned} \mathcal{{H}}_i= & {} \frac{1}{2}\left\{ \left[ \sum _{j\in {\mathsf {N}}_i} \varDelta _{ij}' W_{ij} \varDelta _{ij} \right] + \varDelta _{ii}' W_{ii} \varDelta _{ii} + {\mathbf {u}}_i' {\mathbf {u}}_i \right\} +\,\mathbf {p}_i' {\mathbf {u}}_i. \end{aligned}$$

Let $\mathbf {p} = [\mathbf {p}_{1}'\;\cdots \;\mathbf {p}_{n}']'$ and note that the combined state and costate equations are, by (18),

$$\begin{aligned} \left[ \begin{array}{c} \dot{\mathbf {x}} \\ \dot{\mathbf {p}} \end{array} \right] = \left[ \begin{array}{cc} 0 &{} -I \\ -Q &{} 0 \end{array} \right] \left[ \begin{array}{c} {\mathbf {x}} \\ \mathbf {p} \end{array} \right] + \left[ \begin{array}{cc} 0 &{} 0 \\ W &{} 0 \end{array} \right] \left[ \begin{array}{c} {\mathbf {b}} \\ \mathbf {p}(0) \end{array} \right] , \end{aligned}$$

which has the solution of the form

$$\begin{aligned} \left[ \begin{array}{c} {\mathbf {x}} \\ \mathbf {p} \end{array} \right] = \left( \varPhi (t) + \varPsi (t)B \right) \left[ \begin{array}{c} {\mathbf {b}} \\ \mathbf {p}(0) \end{array} \right] , \end{aligned}$$

(19)

where

$$\begin{aligned} \varPhi (t)= & {} e^{At} = \left[ \begin{array}{cc} \phi _{11}(t) &{} \phi _{12}(t) \\ \phi _{21}(t) &{} \phi _{22}(t) \end{array} \right] , \\ \varPsi (t)= & {} \int _0^t e^{A(t-\mu )} d\mu = \left[ \begin{array}{cc} \psi _{11}(t) &{} \psi _{12}(t) \\ \psi _{21}(t) &{} \psi _{22}(t) \end{array} \right] , \\ A= & {} \left[ \begin{array}{cc} 0 &{} -I \\ -Q &{} 0 \end{array} \right] , \qquad B = \left[ \begin{array}{cc} 0 &{} 0 \\ W &{} 0 \end{array} \right] . \end{aligned}$$

Note that

$$\begin{aligned} \begin{array}{lll} \varPhi (t) &{}=&{} {\mathscr {L}}^{-1}\{(sI - A)^{-1}\} \\ &{}=&{} {\mathscr {L}}^{-1} \bigg \{ { \left[ \begin{array}{cc} s(s^2I - Q)^{-1} &{} -(s^2I - Q)^{-1} \\ -Q(s^2I - Q)^{-1} &{} s(s^2I - Q)^{-1} \end{array} \right] } \bigg \}. \end{array} \end{aligned}$$

(20)

The state transition matrix $\varPhi (t)$ and the matrix $\varPsi (t)$ are calculated using the formal power series in $s^{-1}$ of each block in (20) and, with (3) in view, are given by

$$\begin{aligned} \varPhi (t) = \left[ \begin{array}{cc} \phi _{11}(t) &{} \phi _{12}(t) \\ \phi _{21}(t) &{} \phi _{22}(t) \end{array} \right] = \left[ \begin{array}{cc} f_{Q}(t) &{} -g_{Q}(t) \\ -Qg_{Q}(t) &{} f_{Q}(t) \end{array} \right] , \\ \varPsi (t) = \left[ \begin{array}{cc} \psi _{11}(t) &{} \psi _{12}(t) \\ \psi _{21}(t) &{} \psi _{22}(t) \end{array} \right] = \left[ \begin{array}{cc} g_{Q}(t) &{} -h_{Q}(t) \\ -Qh_{Q}(t) &{} g_{Q}(t) \end{array} \right] . \end{aligned}$$

From (19), we have

$$\begin{aligned} {\mathbf {x}}(t)=[\phi _{11}(t)+\psi _{12}(t)W] {\mathbf {b}}+\phi _{12}(t)\mathbf {p}(0). \end{aligned}$$

Also,

$$\begin{aligned} \mathbf {p}(t)=~[\phi _{21}(t)+\psi _{22}(t)W]{\mathbf {b}}+\phi _{22}(t)\mathbf {p}(0). \end{aligned}$$

Evaluating at $t=\tau$ and employing the boundary condition $\mathbf {p}(\tau ) = 0$, we obtain

$$\begin{aligned} \phi _{22}(\tau ) \mathbf {p}(0)= -[\phi _{21}(\tau ) + \psi _{22}(\tau )W] {\mathbf {b}}, \end{aligned}$$

or substituting the expressions in terms of $f_{Q}(t), g_{Q}(t), h_{Q}(t)$ above,

$$\begin{aligned} f_{Q}(\tau )\mathbf {p}(0)= g_{Q}(\tau )(Q-W) {\mathbf {b}} \end{aligned}$$

(21)

and

$$\begin{aligned} {\mathbf {x}}(t)={\mathbf {b}}+h_{Q}(t)(Q-W){\mathbf {b}}-g_{Q}(t)\mathbf {p}(0). \end{aligned}$$

(22)

If $f_{Q}(\tau )$ is nonsingular, then a unique solution $\mathbf {p}(0)$ to (21) exists for all ${\mathbf {b}}$. If $f_{Q}(\tau )$ is singular, then there is a negative eigenvalue $-r^2$ of Q. Let $\mathbf {v}$ be a left eigenvector associated with $-r^2$ and note that

$$\begin{aligned} \begin{array}{l} \mathbf {v}' Q= -r^2 \mathbf {v}', \mathbf {v}' f_{Q}(\tau )= \cos (r\tau ) \mathbf {v}', \mathbf {v}' g_{Q}(\tau )= r^{-1}\,\sin (r\tau ) \mathbf {v}'. \end{array} \end{aligned}$$

It follows, by multiplying (21) by $\mathbf {v}'$ on the left, that $\cos (r\tau )\mathbf {v}'\mathbf {p}(0)=-r^{-1}\sin (r\tau )\mathbf {v}'(r^2 I+W){\mathbf {b}}$. If $r\tau$ is an odd multiple of $\pi /2$, then the left-hand side is zero and the right-hand side is nonzero for at least ${\mathbf {b}}=\mathbf {v}$ as $r^2 I+W$ is symmetric, positive definite. It follows that $f_{Q}(\tau )$ must be nonsingular for a solution to exist for all ${\mathbf {b}}$.

Summarizing, in order to be able to solve (21) uniquely for $\mathbf {p}(0)$ for any initial state ${\mathbf {b}}$ it is necessary and sufficient that $f_{Q}(\tau )$ is invertible. This establishes the necessity and sufficiency of the condition (C1) of Theorem 1(i) for the existence of a unique Nash solution. If $f_{Q}(\tau )$ is invertible, we then obtain $\mathbf {p}(0)=f_{Q}(\tau )^{-1}g_{Q}(\tau )(Q-W){\mathbf {b}}$. Substituting in ${\mathbf {x}}(t)=[f_{Q}(t)-h_{Q}(t)W] {\mathbf {b}}-g_{Q}(t)\mathbf {p}(0)$, we obtain

$$\begin{aligned} {\mathbf {x}}(t) = [f_{Q}(t)-h_{Q}(t)W] {\mathbf {b}}-g_{Q}(t)f_{Q}(\tau )^{-1}g_{Q}(\tau )(Q-W){\mathbf {b}}, \end{aligned}$$

which gives (7) upon employing $f_{Q}(t)=I+Qh_{Q}(t)$ and noting again that functions of Q commute. Taking the derivative with respect to t and using $\frac{d}{dt}h_{Q}(t)=g_{Q}(t)$, $\frac{d}{dt}g_{Q}(t)=f_{Q}(t)$, we also obtain (6). This completes the proof of Theorem 1. $\square$

Remark 4

(Non-unique equilibria) Here, we briefly characterize non-unique Nash profiles that may arise. Suppose $f_{Q}(\tau )$ is singular, so that (C2) of Theorem 1 holds. Then, with $J_{Q}=P^{-1}QP$ denoting the Jordan form of Q, Eq. (21) is equivalent to

$$\begin{aligned} \left[ \begin{array}{cc}f_{J_{r}}(\tau )&{}0\\ 0&{}f_{J}(\tau )\end{array}\right] \left[ \begin{array}{c}\mathbf {p}_{1}\\ \mathbf {p}_{2}\end{array}\right] = \left[ \begin{array}{c}G_{1}(\tau )\\ G_{2}(\tau )\end{array}\right] (Q-W){\mathbf {b}}, \end{aligned}$$

where $\text {diag}[J_{r}, J] = J_{Q}$, $[G_{1}(\tau )'\;G_{2}(\tau )']' = P^{-1}g_{Q}(\tau )$, $[\mathbf {p}_{1}'\;\mathbf {p}_{2}']' = \mathbf {p}(0)$ with $J_{r}$ denoting the Jordan blocks associated with the negative eigenvalue $-r^2$ of Q; and J, associated with the other eigenvalues. It follows that, if $\tau$ is an odd multiple of $\pi /(2r)$, then $f_{J_{r}}(\tau )$ is singular and $f_{J}(\tau )$ is nonsingular. In this case, $\mathbf {p}(0)$ and ${\mathbf {b}}$ satisfy (21) if and only if $\mathbf {p}_{1}$ is in the null space of $f_{J_{r}}(\tau )$, $\mathbf {p}_{2}=[f_{J}(\tau )]^{-1}G_{2}(Q-W){\mathbf {b}}$, and $G_{1}(Q-W){\mathbf {b}}=0$. (It is easy to see that $\mathbf {p}_{1}$ is in the null space of $f_{J_{r}}(\tau )$ if and only if $\mathbf {v}'f_{Q}(\tau )\mathbf {p}(0)=0$ for all left eigenvectors of Q associated with $-r^2$.) Since the null space of $f_{J_{r}}(\tau )$ for $\tau$ that are odd multiples of $\pi /(2r)$ is nontrivial, $\mathbf {p}_{1}\ne 0$ results in infinitely many different ${\mathbf {x}}(t)$ in (22). Among ${\mathbf {b}}$ such that $G_{1}(Q-W){\mathbf {b}}=0$, one can distinguish between ${\mathbf {b}}=\tilde{\mathbf {b}}$ that are in the null space of $Q-W$ and those with nonzero components $\hat{\mathbf {b}}$ in the range space of $(Q-W)'$, see Remark 1(c). In the first case, the choice $\mathbf {p}_{2}=0$ is necessary and for $\mathbf {p}_{1}=0$, (22) gives the constant profile ${\mathbf {x}}(t)={\mathbf {b}}$, while any ${\mathbf {p}}_{1}\ne 0$ in the null space of $f_{J_{r}}(\tau )$ gives ${\mathbf {x}}(t)={\mathbf {b}}-g_{Q}(t)\mathbf {p}(0)$ as another profile that is not constant since $g_{Q}(t)$ is nonsingular for almost all t, by Remark 1(d). In the second, $(Q-W){\mathbf {b}}$ $=(Q-W){\hat{\mathbf {b}}}$ gives $\mathbf {p}_{2}=[f_{J}(\tau )]^{-1}G_{2}(Q-W) {\hat{\mathbf {b}}}$ and there are again infinitely many profiles (22) that result from different choices of $\mathbf {p}_{1}$ in the null space of $f_{J_{r}}(\tau )$. This characterizes all non-unique solutions to (21). We emphasize that constant profiles that are in the null space of $Q-W$ exist whether $f_{Q}(\tau )$ is singular or not, but they lead to unique opinion profiles only if $f_{Q}(\tau )$ is nonsingular. $\triangle$

1.3 Proof of Theorem 2(iii)

By Section 2.1, $H_{p}=H_{+}-H_{-}$ for a square root $H=H_{+}+H_{-}$ of Q so that

$$\begin{aligned} \begin{array}{lll} &{}&{}\displaystyle \lim _{\tau \rightarrow \infty } \cosh [H(\tau -t)]\cosh (H\tau )^{-1}\\ &{}&{}\quad = \displaystyle \lim _{\tau \rightarrow \infty } \left\{ \begin{array}{l} \left\{ \exp [(H_{+}+H_{-})(\tau -t)]+\exp [-(H_{+}+H_{-})(\tau -t) \right\} \\ \times \left\{ \exp [(H_{+}+H_{-})\tau ]+\exp [-(H_{+}+H_{-})\tau ] \right\} ^{-1} \end{array} \right\} \\ &{}&{}\quad = \displaystyle \exp (-H_{+}t)\exp (H_{-}t)\lim _{\tau \rightarrow \infty } \left\{ \begin{array}{l} \left\{ \exp [2H_{-}(\tau -t)]+\exp [-2H_{+}(\tau -t)]\right\} \\ \times [\exp (2H_{-}\tau )+\exp (-2H_{+}\tau )]^{-1} \end{array} \right\} \\ &{}&{}\quad = \exp (-H_{p}t), \end{array} \end{aligned}$$

where the last equality follows by

$$\begin{aligned} \lim _{\tau \rightarrow \infty }\{\exp [2H_{-}(\tau -t)]+\exp [-2H_{+}(\tau -t)]\}=I \end{aligned}$$

and by

$$\begin{aligned} \lim _{\tau \rightarrow \infty }[\exp (2H_{-}\tau )+\exp (-2H_{+}\tau ]]=I. \end{aligned}$$

Therefore, applying the limit to (8), the expression for ${\mathbf {x}}^{\text {inf}}(t$) in (9) is obtained. This completes the proof of Theorem 2. $\square$

Remark 5

(Connection with the infinite horizon game) If Q is free of any real negative eigenvalue, then it has real square roots H, all of which have eigenvalues with nonzero real parts including a square root $H_{p}$ with eigenvalues of all positive real parts. Consider the state and costate vectors given by

$$\begin{aligned} \begin{array}{l} {\mathbf {x}}(t) = [Q^{-1}W +\exp (-H_{p}t)\,(I-Q^{-1}W)] {\mathbf {b}},\\ \mathbf {p}(t) = H_{p}\exp (-H_{p}t)\,(I-Q^{-1}W) {\mathbf {b}}. \end{array} \end{aligned}$$

(23)

It is straightforward to verify that state–costate equation above is satisfied for any initial state ${\mathbf {x}}(0)={\mathbf {b}}$. Moreover, the final condition $\lim _{t\rightarrow \infty }\mathbf {p}(t)=\mathbf {0}$ also holds, due to the fact that $H_{p}$ has all its eigenvalues with positive real parts. It follows that ${\mathbf {x}}(t)={\mathbf {x}}^{\text {inf}}(t)$ satisfies the necessary conditions for the infinite horizon game $\tau \rightarrow \infty$. In order for these to constitute the unique solution for that game, we may revert to counterparts for n players of Theorems 7.16 and 7.10 in [13] and show that the corresponding matrix “M” has a strongly stable solution. The fact that this can be done in a number of special cases suggests that the limiting Nash solution of Theorem 2(iii) is also the unique solution of the infinite horizon game.