Passivity-based Analysis and Design for Population Dynamics with Conformity Biases

This subsection introduces $\delta$-passivity [23] which is a similar concept to passive systems [35]. Suppose a system $\Sigma$ represented by the state space model

where $x\in\mathbb{R}^{n}$ is the state, $u\in\mathbb{R}^{n}$ is the input, and $f:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ is a function. Then, $\Sigma$ is called $\delta$-passive from $\dot{u}$ to $\dot{x}$ if there exists a positive semi-definite function $S:\mathbb{R}^{n}\times\mathbb{R}^{n}\to[0,\infty)$ and a scalar $\rho\geq 0$ such that

for all input $u$, all initial state $x(0)$ and all $t\geq 0$. The positive semi-definite function $S$ is particularly called a storage function. In addition, $\Sigma$ is called $\delta$-input-strictly-passive if the above inequality is satisfied with some positive scalar $\rho>0$. As widely known, if $S$ is differentiable, we can replace the above inequality with $\dot{S}(u(t),x(t))\leq\dot{x}^{\top}(t)\dot{u}(t)-\rho||\dot{u}(t)||^{2}$.

In the same way as passivity shortage [36], we define $\delta$-output-passivity-shortage. The system $\Sigma$ is called $\delta$-output-passivity-short from $\dot{u}$ to $\dot{x}$ if there exists a positive semi-definite function $S:\mathbb{R}^{n}\times\mathbb{R}^{n}\to[0,\infty)$, and a scalar $\gamma\geq 0$ such that

for all input $u$, all initial state $x(0)$ and all $t\geq 0$. We call the value $\gamma$ as an impact coefficient. If $S$ is differentiable, the above inequality is equivalent to $\dot{S}(u(t),x(t))\leq\dot{x}^{\top}(t)\dot{u}(t)+\gamma||\dot{x}(t)||^{2}$.

Feedback system composed of strictly passive component and passivity-short one is related to the Lyapunov stability [35, 37]. Consider a $\delta$-input-strictly-passive system whose input is $u_{1}$ and state is $x_{1}$, which satisfies $\dot{S}_{1}\leq\dot{x}_{1}^{\top}\dot{u}_{1}-\rho\|\dot{u}_{1}\|^{2}$ for a differentiable storage function $S_{1}$ and $\rho>0$. We also suppose a $\delta$-output-passivity-short system whose input is $u_{2}$ and state is $x_{2}$, which satisfies $\dot{S}_{2}\leq\dot{x}_{2}^{\top}\dot{u}_{2}+\gamma\|\dot{x}_{2}\|^{2}$ for a differentiable storage function $S_{2}$ and $\gamma>0$. Then, the feedback interconnection under $u_{1}=x_{2}$ and $u_{2}=-x_{1}$ provides

If $\gamma\leq\rho$ holds and $S_{1}+S_{2}$ is radially unbounded, the above inequality suggests stability of the feedback system in the sense of Lyapunov [35]. In other words, $\delta$-input-strictly-passive systems can stabilize $\delta$-output-passivity-short ones by negative feedback.

This subsection introduces the logit dynamics [19, 23] which is a well-used dynamical discrete choice model.

Consider the situation that decision makers choose a strategy from $n$ available strategies. We denote the strategy set as $\mathcal{A}:=\{1,2,\ldots,n\}$. Suppose that the population of the decision makers can be represented as a continuum value. We now define $\mathcal{S}:=\{x\in[0,1]^{n}\mid\textbf{1}_{n}^{\top}x=1\}$ as the strategy choice distribution set, where $\textbf{1}_{n}$ describes $n$ dimensional vector whose elements are all $1$. We denote the relative interior of the set $\mathcal{S}$ as $\textrm{int}(\mathcal{S})$. The population state $\pi\in\mathcal{S}$ implies the distribution of strategy choice. Specifically, the $k$-th element of $\pi$, denoted by $\pi_{k}\in[0,1]$, implies a fraction of the decision makers selecting strategy $k\in\mathcal{A}$.

Let us define a cost vector $\tau\in\mathbb{R}^{n}$ of which $k$-th element $\tau_{k}\in\mathbb{R}$ corresponds to the cost for choosing strategy $k\in\mathcal{A}$. Then, the logit dynamics [19, 23] is represented as

where $\eta$ is a positive constant which indicates the update rate. The function $Q:\mathbb{R}^{n}\to\textrm{int}(\mathcal{S})$ corresponds to the steady state of (1) and follows

where $Q_{k}:\mathbb{R}^{n}\to(0,1)$ implies $k$-th element of $Q$, and $\beta>0$ is a constant. The system (1) is known to guarantee $\pi(t)\in\textrm{int}(\mathcal{S})$ for any time $t\geq 0$. Remarking that $Q_{k}(\tau)>Q_{i}(\tau)$ for any pair $k\neq i$ satisfying $\tau_{k}<\tau_{i}$, the logit dynamics (1) can be interpreted as an approximated model of best response, i.e., rational decision making [18].

The logit dynamics (1) is also known to satisfy $\delta$-passivity as below.

Remark that the storage function $S(\tau,\pi)$ has a kind of radial unboundedness. See Appendix 6 for the details.

The logit model (1) corresponds to an approximation of rational strategy choice. Meanwhile, in behavioral economics field, it has been pointed out that human’s decision process is sometimes biased relying on population state. This subsection introduces two models of conformity bias. In particular, we represent these bias models as the ones to affect the cost $\tau$. Hereafter, we refer to $\tau$ as a biased cost. Alternatively, the actual cost is denoted by $T\in\mathbb{R}^{n}$, which is basically supposed to be non-negative.

First, we introduce the bias model which is a generalization of interactions model [33]. In this model, the biased cost $\tau$ is given by

where $B:\mathcal{S}\to\mathbb{R}^{n}$ is the bias function, whose $k$-th element obeys $b_{k}(\pi_{k})$. The function $b_{k}:[0,1]\to\mathbb{R}$ is assumed to satisfy the following:

The item (ii) in Assumption 1 implies that if $k\in\mathcal{A}$ is a majority strategy, the decision makers get a lower impression of cost than the actual cost $T_{k}$. Hence, $B(\pi)$ has the tendency that the strategy chosen by many people intensifies its own popularity, which corresponds to conformity bias. The first derivative $\nabla b_{k}$ implies the dependency of $b_{k}$ on $\pi_{k}$, i.e., this corresponds to the strength of the bias. In particular, $c^{H}$, which is the maximum of $|\nabla b_{k}(\pi_{k})|$, can be interpreted as the maximal bias-strength. In the sequel, we call the decision making model composed of (1) and (4) as Model 1. The block diagram of Model 1 is illustrated in Fig. 1.

In this paper, we address another model of conformity bias, presented in [25, 34]. For given $T$, in this model, the biased cost $\tau$ obeys

where $W(\pi)=\textrm{diag}(w_{1}(\pi_{1}),w_{2}(\pi_{2}),\ldots,w_{n}(\pi_{n}))$ is the bias matrix. The $k$-th diagonal element $w_{k}:[0,1]\to\mathbb{R}$ ($k\in\mathcal{A}$) is the bias function for strategy $k$. Throughout this paper, we set the following assumption for the bias function $w_{k}$.

In the subsequent discussion, we call the model composed of (1) and (5) as Model 2. Under Assumption 2, the biased cost for strategy $k$ decreases when $\pi_{k}$ gets large. In other words, the majority strategy tends to be well chosen. Since $|\nabla w_{k}|$ is the dependency of $w_{k}$ on $\pi_{k}$, this implies the strength of the bias and its maximum value $v^{H}$ represents the maximal bias-strength of Model 2. The block diagram of Model 2 is illustrated in Fig. 2.

In this subsection, we consider a mechanism for Model 1. The structure of the mechanism is illustrated in Fig. 5. The gray part in Fig. 5 implies Model 1, and the block $\Sigma_{1}$ is the mechanism. Notice that the system enclosed by the red line is $\delta$-output-passivity-short, as proved in Lemma 2. In terms of passivity theory, positive energy generated from an output-passivity-short system can be canceled out by negative feedback of an input-strictly-passive system. Hence, we can expect to implement a stable mechanism by designing $\Sigma_{1}$ so as to satisfy input-strict-passivity of the system enclosed by blue line.

Based on the above concept, we propose the mechanism inspired by Proportional-Integral controller, as below:

where $\rho>0$ and $\kappa>0$ is a design parameter. The mechanism (10) satisfies the following lemma.

The result in Lemma 4.4 suggests that the system from $\dot{\pi}$ to $\dot{T}$, enclosed by the blue line in Fig. 5, is $\delta$-input-strictly-passive if we apply (10) to $\Sigma_{1}$. Thus, it is expected that the mechanism (10) cancels out the positive energy in (7) and induces the population state $\pi$ to the desired state $\pi^{*}$.

As a preparation for convergence analysis, we introduce the following lemma about the storage function $S$ under the mechanism (10).

We are now ready to prove the following theorem. In the proof, we use the notation $\mathcal{L}_{\infty}$ as the set of all signals $u:[0,\infty)\to\mathbb{R}^{n}$ satisfying $\sup_{t\in[0,\infty)}\|u(t)\|<\infty$.

Theorem 4.8 shows that the mechanism (10) can induce the decision makers of Model 1 to desired state if we design the parameter $\kappa$ large enough. The right hand side of the convergence condition $\kappa>c^{H}$ is given by the bias-strength. Hence, for the strongly biased decision makers, $\kappa$ is needed to be set large.

Let us consider to design a mechanism for Model 2. Its structure is illustrated in Fig. 6: The gray part is the decision makers, and $\Sigma_{2}$ is the mechanism. Similar to 4.1, we design $\Sigma_{2}$ to guarantee the passivity from $y$ to $\dot{T}$.

Applying the mechanism (10) to $\Sigma_{2}$, the following $\delta$-passivity is satisfied, that is proved in the previous work [34].

By using (10) for $\Sigma_{2}$, certainly, the whole system in Fig. 6 is composed of the feedback of a $\delta$-output-passivity-short system and a $\delta$-input-strictly-passive system. However, the analysis in [34] leaves an important issue about signal boundedness. In [34], the boundedness of the signal $T$ is given by an assumption, which is not theoretically guaranteed. In particular, to certify the upper bound of signal $T$, denoted by $T_{\max}$, is the most significant problem to cancel out the positive energy in (9). Therefore, we should redesign a mechanism for Model 2 to guarantee the existence of $T_{\max}$, with keeping passivity.

To solve the above issue, we propose the following mechanism.

where $\rho>0$ and $\kappa>0$ are design parameters, $\alpha>0$ is a parameter satisfying $\alpha>\frac{1}{2\kappa}$, and $\bar{T}>0$ is a constant. Remark that (13a) ensures $\mu\leq 0$ for all time. Thus, the system (13) guarantees the existence of $T_{\max}<\bar{T}+\kappa$ such that $T_{k}\leq T_{\max}\ \forall k\in\mathcal{A}$ holds for all time. About the time derivative of $T$, $\dot{T}=\dot{\mu}+\kappa\dot{\pi}$ holds from (13b).

For the passivity analysis of (13), we consider the primitive function of $w_{k}$. Under Assumption 2, $w_{k}$ is continuous. Hence, we can define the following function:

For $\pi_{k}\in(0,1)$, $\nabla^{2}F_{k}(\pi_{k})=w_{k}(\pi_{k})\geq w^{L}$ clearly holds and this reveals that $F_{k}$ is convex. We denote $F^{\star}_{k}:\mathbb{R}\to\mathbb{R}$ as the convex conjugate of $F_{k}$ [39]. Due to the convexity of $F_{k}$,

is satisfied [39]. By using these results, we can prove the following lemma.

Thanks to the result in Lemma 4.10, it is revealed that the proposed mechanism (13) guarantees both $\delta$-input-strict-passivity and the existence of $T_{\max}$. In other words, the remained issue in the previous method [34] can be cleared by (13). Thus, we can expected to achieve the passivity-based mechanism for Model 2.

Here we introduce the following lemma, which will be used in convergence analysis.

From Lemma 3.2, Lemma 4.10 and Lemma 4.12, we are now ready to analyze convergence of the proposed mechanism (13) for Model 2. We should remark that the LaSalle’s invariance principle [38] cannot be applied due to non-smoothness of the storage function $U(\mu,\pi)$. Alternatively, we discuss the convergence by using the invariance principle for non-smooth Lyapunov functions [40]. Then, the following theorem can be proven.

Thanks to the upper bound condition $T_{k}(t)\leq T_{\max}\ \forall k\in\mathcal{A},\forall t\geq 0$ ensured by (13), the stability and convergence of the feedback system in Fig. 6 can be exactly guaranteed under the gain condition $\kappa>\frac{2v^{H}T_{\max}}{w^{L}}$. Similar to Theorem 4.8, the result in Theorem 4.14 suggests the tendency that large gain $\kappa$ will be required for strongly biased decision makers. The quantitative inequality is calculated as a result of passivity-based analysis.