Comments on “Opinion Dynamics Driven by Various Ways of Averaging”

Abstract

The two main findings in Hegselmann and Krause (Comput Econ 25:381–405, 2005), the theorem on opinion stabilization and the corollary on consensus formation, are built on partial abstract means (PAMs) with bounded confidence sets that are assumed to be continuous. However, we show that any PAM with bounded confidence sets cannot be continuous. The discontinuity of such PAMs threatens the validity of the main findings in Hegselmann and Krause (Comput Econ 25:381–405, 2005). Moreover, the condition that the corollary on consensus formation considers necessary and sufficient, under which agents will approach a consensus, is, in fact, not a sufficient condition. To resolve these issues, we provide a sufficient condition for PAMs with bounded confidence sets under which the theorem on opinion stabilization becomes valid. We also show that under this condition the condition in the corollary on consensus formation is necessary condition for agents to approach a consensus.

Appendices

Appendix A

Recall that at opinion profile $x=\left( x_{1},\ldots ,x_{N}\right)$, $I_{i}\left( x\right) =\left\{ j\in I:\left| x_{j}-x_{i}\right| \le \epsilon \right\}$ is agent i’s confidence set and $B_{i}\left( x\right) =\left\{ x_{j}:j\in I_{i}\left( x\right) \right\}$ is agent i’s opinion confidence neighborhood. Then $I_{i}\left( x\right)$ can be written as $I_{i}\left( x\right) =\left\{ j\in I:x_{j}\in B_{i}\left( x\right) \right\}$.

To facilitate the proof for Proposition 1, we define a function $\varPhi _{i}:\mathcal {A}\rightarrow \left( 0,+\infty \right)$, where $\mathcal {A}=\mathop {\cup }\limits _{k=1}^{N}\left( 0,+\infty \right) ^{k}$, for each agent $i\in I$ as follows⁴:

$$\begin{aligned} \varPhi _{i}\left( B_{i}\left( x\right) \right) =f_{i}\left( x\right) , \ \forall x\in K. \end{aligned}$$

Thus, $x_{i}^{t+1}=\varPhi _{i}\left( B_{i}\left( x^{t}\right) \right)$ for every $i\in I$ at any period $t\ge 0$.

If $f_{i}$ satisfies A1–A3, then $\varPhi _{i}$ satisfies the following B1–B3:

• B1 For any $x\in \mathbb {K}$,

  $$\begin{aligned} \min B_{i}\left( x\right) \le \varPhi _{i}\left( B_{i} \left( x\right) \right) \le \max B_{i}\left( x\right) , \end{aligned}$$

  where “$=$” holds if and only if all the elements in $B_{i}\left( x\right)$ are equal. If $B_{i}\left( x\right)$ has at least two numerically different elements, then $\min B_{i}\left( x\right)<\varPhi _{i}\left( B_{i}\left( x\right) \right) <\max B_{i}\left( x\right)$.

• B2 If $x=\left( x_{1},\ldots ,x_{N}\right)$ and $x'=\left( x_{1}',\ldots ,x_{N}'\right)$ satisfy $I_{i} \left( x\right) =I_{i}\left( x'\right)$ and $x_{j}\le x_{j}'$ for all $j\in I_{i}\left( x\right)$ with at least one strict inequality, then $\varPhi _{i}\left( B_{i}\left( x\right) \right) <\varPhi _{i}\left( B_{i}\left( x'\right) \right)$.

• B3 For agent $i\in I$, there exist two continuous and strictly increasing functions $g_{i}:[0,+\infty )\rightarrow [0,+\infty )$ and $h_{i}:[0,+\infty )\rightarrow [0,+\infty )$ such that

  • (1) $g_{i}(0)=h_{i}(0)=0$ and $g_{i}(\eta ),h_{i}(\eta )\in (0,\eta )$ if $\eta >0$; and

  • (2) for all $x=\left( x_{1},\ldots ,x_{N}\right)$ and $x'=\left( x_{1}',\ldots ,x_{N}'\right)$ such that $I_{i}\left( x\right) =I_{i}\left( x'\right) \cup \left\{ j\right\}$, where $j\notin I_{i}\left( x'\right)$, and $x_{k}=x_{k}'$, $k\in I_{i}\left( x'\right)$,

    • (a) if $x_{j}\ge \max B_{i}\left( x\right)$, then $\varPhi _{i}\left( B_{i}\left( x\right) \right) \ge \varPhi _{i} (B_{i}(x'))+g_{i}\left( x_{j}-\max B_{i}\left( x\right) \right)$;

    • (b) if $x_{j}\le \min B_{i}\left( x\right)$, then $\varPhi _{i}\left( B_{i}\left( x\right) \right) \le \varPhi _{i} \left( B_{i}\left( x'\right) \right) -h_{i}(\min B_{i}(x)-x_{j})$.

Note that B3 implies that if $x_{j}>\max B_{i}\left( x'\right)$, then $\varPhi _{i}\left( B_{i}\left( x\right) \right) >\varPhi _{i}\left( B_{i} \left( x'\right) \right)$; and if $x_{j}<\min B_{i}\left( x'\right)$, then $\varPhi _{i}\left( B_{i}\left( x\right) \right) <\varPhi _{i}\left( B_{i} \left( x'\right) \right)$.

Define $g:[0,+\infty )\rightarrow [0,+\infty )$ and $h:[0,+\infty )\rightarrow [0,+\infty )$ as follows:

$$\begin{aligned} g(\gamma )=\min _{i\in I}g_{i}(\gamma )\ \text {and}\ h(\gamma ) =\min _{i\in I}h_{i}(\gamma ),\ \forall \gamma \ge 0. \end{aligned}$$

Since I is a finite set, $g(\cdot )$ and $h(\cdot )$ are well-defined. Note that $g(0)=h(0)=0$ and $g(\gamma ),h(\gamma )\in (0,\gamma )$ for every $\gamma \in (0,1)$.

Lemma 1

Suppose for every $i\in I$, $\varPhi _{i}$ satisfies B3. Then the two functions g and h defined above are continuous and strictly increasing. Moreover, for all $i\in I$ and any two sets $B,B'\in \mathcal {\mathcal {A}}$ such that $B=\left\{ b_{1},\ldots ,b_{K}\right\}$, $B'=B\cup \left\{ b_{K+1}\right\}$, and $b_{K+1}\notin B$,

• (1) if $b_{K+1}\ge \max B$, then $\varPhi _{i}\left( B'\right) \ge \varPhi _{i}\left( B\right) +g\left( b_{K+1}-\max B\right)$; and

• (2) if $b_{K+1}\le \min B$, then $\varPhi _{i}\left( B'\right) \le \varPhi _{i}\left( B\right) -h\left( \min B-b_{K+1}\right)$.

Proof

Consider a randomly selected $\eta \in [0,+\infty )$ and any sequence $\left\{ \eta _{k}\right\} _{k\in \mathbb {N}}$, where $\eta _{k}\ge 0$ for every $k\in \mathbb {N}$, $\underset{k\rightarrow \infty }{\lim } \eta _{k}=\eta$. We show that $\underset{k\rightarrow \infty }{\lim } g\left( \eta _{k}\right) =g(\eta )$, which implies that g is continuous at $\eta$. Then, since $\eta$ is randomly selected from $[0,+\infty )$, we know that g is continuous on $[0,+\infty )$.

Let $I_{\eta }=\left\{ i\in I:g_{i}(\eta )=g(\eta )\right\}$ and $I_{\eta }^{+}=\left\{ i\in I:g_{i}(\eta )>g(\eta )\right\}$. Then $I_{\eta }\cup I_{\eta }^{+}=I$ by the definition of $g(\eta )$. Since I is a finite set, there must be some $i\in I$ such that $g_{i}(\eta )=g(\eta )$, thus, $I_{\eta }\ne \emptyset$. If $I_{\eta }^{+}\ne \emptyset$, consider any $\delta >0$ such that $\delta <\frac{1}{2}\min \left\{ g_{j}(\eta )-g(\eta ):j\in I_{\eta }^{+}\right\}$; if $I_{\eta }^{+}=\emptyset$, then consider any $\delta >0$.

Since for every $i\in I$, $g_{i}$ is continuous on $(0,+\infty )$, there is $K_{i}^{\eta }\in \mathbb {N}$ such that

$$\begin{aligned} \left| g_{i}\left( \eta _{k}\right) -g_{i}(\eta )\right| <\delta ,\ \forall k\ge K_{i}^{\eta }\left( \delta \right) . \end{aligned}$$

Let $K^{\eta }\left( \delta \right) =\max \left\{ K_{i}^{\eta }\left( \delta \right) :i\in I\right\}$. Since I is finite, $K^{\eta }\left( \delta \right)$ exists and is finite. Then

$$\begin{aligned} g_{i}(\eta )-\delta<g_{i}\left( \eta _{k}\right) <g_{i}(\eta )+\delta ,\ \forall i\in I\ \text {and}\ \forall k\ge K^{\eta }\left( \delta \right) . \end{aligned}$$

(2)

As for any $k\ge K^{\eta }\left( \delta \right)$,

$$\begin{aligned} g_{m}\left( \eta _{k}\right) -g(\eta ) {\left\{ \begin{array}{ll}>g_{m}(\eta )-\delta -g(\eta )>\delta &{} \text {if}\ m\in I_{\eta }^{+}\\ =g_{m}\left( \eta _{k}\right) -g_{m}(\eta )<\delta &{} \text {if}\ m\in I_{\eta }, \end{array}\right. } \end{aligned}$$

we have

$$\begin{aligned} g_{i}\left( \eta _{k}\right)<g(\eta )+\delta <g_{j}\left( \eta _{k}\right) ,\ \forall i\in I_{\eta }\ \text {and}\ j\in I_{\eta }^{+}, \end{aligned}$$

which implies $g\left( \eta _{k}\right) =\min \left\{ g_{m}\left( \eta _{k}\right) :m\in I\right\} =\min \left\{ g_{i}\left( \eta _{k}\right) :i\in I_{\eta }\right\}$ when $k\ge K^{\eta }$. Thus, we only need to focus on $g_{i}$ such that $i\in I_{\eta }$ when discussing the continuity of g.

Since $g_{i}(\eta )=g(\eta )$ for all $i\in I_{\eta }$, by (2) we have

$$\begin{aligned} g(\eta )-\delta<g_{i}\left( \eta _{k}\right) <g(\eta )+\delta ,\ \forall i\in I_{\eta }\ \text {and}\ \forall k\ge K^{\eta }, \end{aligned}$$

which implies

$$\begin{aligned} g\left( \eta _{k}\right) -g(\eta ) {\left\{ \begin{array}{ll} =\underset{i\in I_{\eta }}{\min }\left( \eta _{k}\right) -g(\eta )>g(\eta )-\delta -g(\delta )=-\,\delta \\ \le \underset{i\in I_{\eta }}{\max }g_{i}\left( \eta _{k}\right) -g(\eta )<g(\eta )+\delta -g(\delta )=\delta , \end{array}\right. } \end{aligned}$$

then $\left| g\left( \eta _{k}\right) -g(\eta )\right| <\delta$ for every $k>K^{\eta }\left( \delta \right)$. Hence, $\underset{k\rightarrow \infty }{\lim }g\left( \eta _{k}\right) =g(\eta )$, which means that g is continuous at $\eta$. Since $\eta$ is randomly selected from $[0,+\infty )$, we know that g is continuous on $[0,+\infty )$.

g is strictly increasing because for any $\eta ,\eta '\in [0,+\infty )$ such that $\eta '>\eta$, we must have

$$\begin{aligned} g\left( \eta '\right) =\min _{i\in I}g_{i}\left( \eta '\right) >\min _{i\in I}g_{i}(\eta )=g(\eta ), \end{aligned}$$

as I is finite and $g_{i}\left( \eta '\right) >g_{i}(\eta )$ for any $i\in I$.

The proof for the continuity and monotonicity of h is analogous to that for g, so we omit it. $\square$

Lemma 2

Given $\eta >0$, there are $\overline{\sigma }\left( \eta \right) , \overline{\gamma }(\eta )\in \left( 0,\frac{\eta }{2}\right)$ and $\overline{\delta }\left( \eta \right) \in \left( 0,\overline{\sigma } (\eta )\right)$ uniquely defined by

• (1) $g\left( \eta -2\overline{\sigma }\left( \eta \right) \right) =2\overline{\sigma }\left( \eta \right)$ and $g\left( g\left( \eta -2 \overline{\delta }\left( \eta \right) \right) -2\overline{\delta } \left( \eta \right) \right) =2\overline{\delta }\left( \eta \right)$;

• (2) $h\left( \eta -2\overline{\gamma }\left( \eta \right) \right) =2\overline{\gamma }\left( \eta \right)$,

Moreover, if $\sigma \in \left( 0,\overline{\sigma }(\eta )\right)$, $\gamma \in \left( 0,\overline{\gamma }(\eta )\right)$, and $\delta \in \left( 0,\overline{\delta }(\eta )\right)$, then

$$\begin{aligned} g\left( \eta -2\sigma \right)>\sigma ,\ h\left( \eta -2\gamma \right)>2\gamma ,\ \text {and}\ g\left( g\left( \eta -2\delta \right) -\delta \right) >\delta . \end{aligned}$$

Proof

Recall that g is continuous and strictly increasing on $[0,+\infty )$, $g(0)=0$, and $g(\eta )\in (0,\eta )$ for any $\eta >0$. Since for any $\eta >0$,

$$\begin{aligned} \lim _{z\rightarrow 0}\left[ g(\eta -2z)-z\right] =g(\eta )>0\ \text {and}\ \lim _{z\rightarrow \frac{\eta }{2}}\left[ g(\eta -2z)-z\right] =-\,\frac{\eta }{2}<0, \end{aligned}$$

there is a unique $\overline{\sigma }(\eta )\in \left( 0, \frac{\eta }{2}\right)$ such that $g\left( \eta -2\overline{\sigma }\left( \eta \right) \right) =2\overline{\sigma }\left( \eta \right)$. Then, as g is strictly increasing on $\left( 0,\overline{\sigma }(\eta )\right)$, we have

$$\begin{aligned} g(\eta -2\sigma )>g\left( \eta -2\overline{\sigma }(\eta )\right) =2\overline{\sigma }(\eta )>\sigma ,\ \forall \sigma \in \left( 0,\overline{\sigma }(\eta )\right) . \end{aligned}$$

Similarly, as h is continuous and strictly increasing on $[0,+\infty )$, there is a unique $\overline{\gamma }(\eta )\in \left( 0,\frac{\eta }{2}\right)$ such that $h\left( \eta -2\overline{\gamma }\left( \eta \right) \right) =2\overline{\gamma } \left( \eta \right)$, and $h\left( \eta -2\gamma \right) >2\gamma$ for all $\gamma \in \left( 0,\overline{\gamma }(\eta )\right)$.

Since if $\eta >0$ then $g(\eta )\in (0,\eta )$, we have

$$\begin{aligned} \lim _{z\rightarrow 0}\left[ g\left( g(\eta -2z)-2z\right) -2z\right] =g\left( g(\eta )\right) >0 \end{aligned}$$

and

$$\begin{aligned} \lim _{z\rightarrow \overline{\sigma }(\eta )}\left[ g\left( g(\eta -2z) -2z\right) -2z\right] =-\,2\overline{\sigma }(\eta )<0 \end{aligned}$$

there is a unique $\overline{\delta }(\eta )\in \left( 0,\overline{\sigma }(\eta )\right)$ such that

$$\begin{aligned} g\left( g\left( \eta -2\overline{\delta }(\eta )\right) -2\overline{\delta }(\eta )\right) -2\overline{\delta }(\eta )=0. \end{aligned}$$

Then, as g is strictly increasing on $\left( 0,\overline{\delta }(\eta )\right)$, we have

$$\begin{aligned} g\left( g\left( \eta -2\delta \right) -2\delta \right)>g \left( g\left( \eta -2\overline{\delta }(\eta )\right) -2\overline{\delta }(\eta )\right) =2\overline{\delta }(\eta )>2\delta ,\ \forall \delta \in \left( 0,\overline{\delta }(\eta )\right) . \end{aligned}$$

$\square$

As in Hegselmann and Krause (2005), say that $J\subseteq I$ is absorbing for x if $I_{i}\left( x\right) \subseteq J$ for every $i\in J$.

Lemma 3

If $x^{t}\in \mathbb {K}$ is not a consensus and $J\subset I$ is absorbing for $x^{t}$, then

$$\begin{aligned} \min \left\{ x_{j}^{t'}:j\in J\right\} \ge \min \left\{ x_{j}^{t}:j\in J\right\} \ \text {and}\ \max \left\{ x_{j}^{t'}:j\in J\right\} \le \max \left\{ x_{j}^{t}:j\in J\right\} \end{aligned}$$

at any period $t'\ge t+1$.

Proof

By Lemma 1 in Hegselmann and Krause (2005), we know that at any period $t\ge 0$,

$$\begin{aligned} \min \left\{ x_{j}^{t+1}:j\in J\right\} =\min \left\{ f_{j}\left( x^{t}\right) :j\in J\right\} \ge \min \left\{ x_{j}^{t}:j\in J\right\} \end{aligned}$$

and

$$\begin{aligned} \max \left\{ x_{j}^{t+1}:j\in J\right\} =\max \left\{ f_{j}\left( x^{t}\right) :j\in J\right\} \le \max \left\{ x_{j}^{t}:j\in J\right\} . \end{aligned}$$

By induction we have

$$\begin{aligned} \min \left\{ x_{j}^{t'}:j\in J\right\} \ge \min \left\{ x_{j}^{t}:j\in J\right\} \ \text {and}\ \max \left\{ x_{j}^{t'}:j\in J\right\} \le \max \left\{ x_{j}^{t}:j\in J\right\} \end{aligned}$$

whenever $t'\ge t+1$. $\square$

Lemma 4

Consider two non-empty sets D and $D'$ in which the elements are positive real numbers. Suppose $\max D<\min D'$. If agents’ revision functions satisfy B3, then for every $i\in I$,

$$\begin{aligned} \varPhi _{i}\left( D\cup D'\right) {\left\{ \begin{array}{ll} \ge \varPhi _{i}\left( D\right) +g\left( \min D'-\max D\right) >\varPhi _{i}(D)\\ \le \varPhi _{i}\left( D'\right) -h\left( \min D'-\max D\right) <\phi _{i} \left( D'\right) . \end{array}\right. } \end{aligned}$$

Proof

Since $\min D'-\max D>0$, we know

$$\begin{aligned} g\left( \min D'-\max D\right)>0\ \text {and}\ h\left( \min D'-\max D\right) >0. \end{aligned}$$

Without loss of generality, suppose D has K elements, denoted as $d_{1},\ldots ,d_{K}$, such that $d_{k}\le d_{k+1}$, $k=1,\ldots K-1$; $D'$ has M elements, denoted as $d_{1}',\ldots ,d_{M}'$, such that $d_{m}'\le d_{m+1}'$, $m=1,\ldots ,M-1$. Then $d_{1}=\min D'$ and $d_{M}=\max D'$.

Since $\varPhi _{i}$ satisfies B3 and $d_{m}'\ge d_{m-1}'>\max D$ for any $m=1,\ldots ,M-1$, we have

$$\begin{aligned} \varPhi _{i}\left( D\cup D'\right)&=\varPhi _{i}\left( D\cup \left\{ d_{m}':1 \le m\le M\right\} \right) \\&\ge \varPhi _{i}\left( D\cup \left\{ d_{m}':1\le m\le M-1\right\} \right) \\&\vdots \\&\ge \varPhi _{i}\left( D\cup \left\{ d_{m}':1\le m\le 1\right\} \right) \\&=\varPhi _{i}\left( D\cup \left\{ \min D'\right\} \right) \\&\ge \varPhi _{i}\left( D\right) +g\left( \min D'-\max D\right) \\&>\varPhi _{i}\left( D\right) \end{aligned}$$

and

$$\begin{aligned} \varPhi _{i}\left( D\cup D'\right)&=\varPhi _{i}\left( D'\cup \left\{ d_{k}:1 \le k\le K\right\} \right) \\&\le \varPhi _{i}\left( D'\cup \left\{ d_{k}:2\le k\le K\right\} \right) \\&\vdots \\&\le \varPhi _{i}\left( D'\cup \left\{ d_{k}:K\le k\le K\right\} \right) \\&=\varPhi _{i}\left( D'\cup \left\{ \max D\right\} \right) \\&<\varPhi _{i}\left( D'\right) -h\left( \min D'-\max D\right) \\&<\varPhi _{i}\left( D'\right) , \end{aligned}$$

as was to be shown. $\square$

Lemma 5

Suppose $\left\{ x^{t}\right\} _{t\ge 0}$ is a sequence of $\epsilon$-profiles on $J\subseteq I$. If $\left\{ x^{t}\right\} _{t\ge 0}$ has a convergent subsequence, denoted as $\left\{ x^{t_{\tau }}\right\} _{\tau \ge 0}$, such that $\underset{t_{\tau }\rightarrow \infty }{\lim }x_{j}^{t_{\tau }}=c_{j}$, $j\in J$, then at any period $t\ge 0$,

$$\begin{aligned} \min \left\{ x_{i}^{t}:j\in J\right\} \le \min \left\{ c_{j}:j\in J\right\} . \end{aligned}$$

Proof

Since $J\subseteq I$ is finite, there must be some $k\in J$ such that $\underset{t\rightarrow \infty }{\lim }x_{k}^{t}=\min \left\{ c_{j}:j\in J\right\}$.

By the definition of $\epsilon$-profiles on J, we know that $\left| x_{j}^{t}-x_{i}^{t}\right| >\epsilon$ for all $j\in J$ and $i\in I\backslash J$, which means that at any period $t\ge 0$, $B_{j}\left( x^{t}\right) =\left\{ x_{m}^{t}:\left| x_{m}^{t}-x_{j}^{t}\right| \le \epsilon \right\} \subseteq \left\{ x_{m}^{t}:m\in J\right\}$ for all $j\in J$, which means that J is absorbing for all $x^{t}$ where $t\ge 0$.

Suppose that there is some period $T>0$ at which $\min \left\{ x_{i}^{T}:j\in J\right\} >\min \left\{ c_{j}:j\in J\right\}$. By Lemma 3 we know that at any $t>T$,

$$\begin{aligned} \min \left\{ x_{j}^{t}:j\in J\right\} \ge \min \left\{ x_{j}^{T}:j\in J\right\} , \end{aligned}$$

which means that $\underset{t\rightarrow \infty }{\lim }x_{j}^{t}\ge \min \left\{ x_{j}^{T}:j\in J\right\} >\min \left\{ c_{j}:j\in J\right\}$ for all $j\in J$, contradicting the notion that there must be some $k\in J$ such that $\underset{t\rightarrow \infty }{\lim }x_{k}^{t}=\min \left\{ c_{j}:j\in J\right\}$. $\square$

Proof of Proposition 1

Since $\left\{ x^{t}\right\} _{t\ge 0}$ is a sequence of $\epsilon$-profiles on J, $I_{j}\left( x^{t}\right) \subseteq J$ for every agent $j\in J$ at every period t, which means that each agent $j\in J$ never takes into account the opinions of agents who are not in J when revising her opinion. In other words, $B_{j}\left( x^{t}\right) \subseteq \left\{ x_{i}^{t}:i\in J\right\}$ for all $j\in I$ and $t\ge 0$.

By Lemma 3 we can easily see that at any period $t\ge 0$, $x^{t}\left( J\right)$ is in the compact set $\left[ m\left( x^{0}\left( J\right) \right) ,M\left( x^{0} \left( J\right) \right) \right] ^{N}$, which means that $\left\{ x^{t}\right\} _{t\ge 0}$ must have a convergent subsequence on J. Let $\left\{ x^{t_{\tau }}\right\} _{t_{\tau }\ge 0}$ denote this convergent subsequence on J and let $c_{j}=\underset{t_{\tau }\rightarrow \infty }{\lim }x_{j}^{t_{\tau }}$, $j\in J$. Note that for any $t_{\tau }\ge 0$, $x^{t_{\tau }}$ must be an $\epsilon$-profile on J.

We prove this proposition by showing that if there are two different $j,k\in J$ such that $\underset{t\rightarrow \infty }{\lim }x_{j}^{t} \ne \underset{t\rightarrow \infty }{\lim }x_{k}^{t}$, then there must be some period $T\ge 0$ such that at any period $t\ge T$, $\min \left\{ x_{j}^{t}:j\in J\right\} >\min \left\{ c_{j}:j\in J\right\}$, contradicting Lemma 5 that $\min \left\{ x_{j}^{t}:j\in J\right\}$ must be lower than $\min \left\{ c_{j}:j\in J\right\}$.

To facilitate the discussion, we use the function $\varPhi _{i}:\mathcal {A}\rightarrow \left( 0,+\infty \right)$, where $\mathcal {A}=\mathop {\cup }\limits _{k=1}^{N}\left( 0,+\infty \right) ^{k}$, defined at the beginning of “Appendix A”, that is,

$$\begin{aligned} \varPhi _{i}\left( B_{i}\left( x\right) \right) =f_{i}\left( x\right) ,\ \forall x\in \mathbb {K}. \end{aligned}$$

Thus, $x_{i}^{t+1}=f_{i}\left( x^{t}\right) =\varPhi _{i}\left( B_{i} \left( x^{t}\right) \right)$ for every $i\in I$ at any period $t\ge 0$.

Without loss of generality, suppose that there are L agents in J. Here $2\le L\le N$. Suppose there are at least two different $i,j\in J$ such that $\underset{t\rightarrow \infty }{\lim }x_{i}^{t} \ne \underset{t\rightarrow \infty }{\lim }x_{j}^{t}$, or equivalently, $c_{i}\ne c_{j}$. Then we can partition $\left\{ c_{j}:j\in J\right\}$ into $c_{(1)},\ldots ,c_{(K)}$ such that

• $2\le K\le L$.

• $c_{(k)}<c_{(k+1)}$, $\forall k=1,\ldots ,K-1$.

• $c_{(1)}=\min \left\{ c_{j}:j\in J\right\}$ and $c_{(K)}=\max \left\{ c_{j}:j\in J\right\}$.

• For any $j\in J$, there is a $k\in \left\{ 1,\ldots ,K\right\}$ such that $c_{j}=c_{(k)}$.

Let $J_{(k)}=\left\{ j\in J:\underset{t_{\tau }\rightarrow \infty }{\lim }x_{j}^{t_{\tau }}=c_{(k)}\right\} =\left\{ j\in J:c_{j}=c_{(k)}\right\}$. Note that $J_{(1)}\ne \emptyset$, $J_{(2)}\ne \emptyset$, and $J_{(k)}\cap J_{\left( k'\right) }=\emptyset$ if $k\ne k'$.

First note that $c_{(k+1)}-c_{(k)}\le \epsilon$ for every $k\in \{1,\ldots ,K-1\}$. Suppose not, then there is a $k_{\epsilon }\in \{1,\ldots ,K-1\}$ such that $c_{\left( k_{\epsilon } +1\right) }-c_{\left( k_{\epsilon }\right) }>\epsilon$. Consider $\theta >0$ such that

$$\begin{aligned} \theta<\min \left\{ \frac{c_{(k+1)}-c_{(k)}}{2}:k=1,\ldots ,K-1\right\} \ \text {and}\ \theta <\frac{c_{\left( k_{\epsilon }+1\right) }-c_{\left( k_{\epsilon }\right) }-\epsilon }{2}. \end{aligned}$$

Since for every $j\in J$, $\underset{t_{\tau }\rightarrow \infty }{\lim }x_{j}^{t_{\tau }}=c_{j}$, there must be some $T_{\theta }\ge 0$ such that at any $t_{\tau }\ge T_{\theta }$, $\left| x_{j}^{t_{\tau }}-c_{j}\right| <\theta$ for every $j\in J$. Then, at any period $t_{\tau }\ge T_{\theta }$, if $i\in J_{(k)}$ and $j\in J_{\left( k'\right) }$ where $k'\ge k_{\epsilon }+1>k_{\epsilon }\ge k$, we must have

$$\begin{aligned} x_{j}^{t_{\tau }}-x_{i}^{t_{\tau }}&>c_{\left( k'\right) }-\theta -\left( c_{(k)}+\theta \right) \nonumber \\&\ge c_{\left( k_{\epsilon }+1\right) }-\theta -\left( c_{\left( k_{\epsilon }\right) }+\theta \right) .\nonumber \\&=c_{\left( k_{\epsilon }+1\right) }-c_{\left( k_{\epsilon }\right) }-2\theta \nonumber \\&>\epsilon \end{aligned}$$

(3)

(3) implies that, for any permutation of $\left\{ x_{j}^{t_{\tau }}:j\in J\right\}$, denoted as $x_{(j_{1})}^{t_{\tau }},\ldots ,x_{(j_{L})}^{t_{\tau }}$, such that $x_{\left( j_{1}\right) }^{t_{\tau }}\le x_{\left( j_{2}\right) }^{t_{\tau }}\le \ldots \le x_{\left( j_{L}\right) }^{t_{\tau }}$, we can always find some $m\in \left\{ 1,\ldots ,L\right\}$ such that $x_{\left( j_{m+1}\right) }^{t_{\tau }}-x_{\left( j_{m}\right) }^{t_{\tau }} >\epsilon$. Thus, $x^{t_{\tau }}$ cannot be an $\epsilon$-profile on J at any period $t_{\tau }\ge T_{\theta }$, contradicting the assumption that $x^{t_{\tau }}$ is an $\epsilon$-profile on J. Hence, $c_{(k+1)}-c_{(k)}\le \epsilon$ for every $k\in \{1,\ldots ,K-1\}$.

Next, we show that if there are two different $j,k\in J$ such that $c_{j}\ne c_{k}$, then there must be some $T\ge 0$ such that at any $t\ge T$, $\min \left\{ x_{j}^{T}:j\in J\right\} >\min \left\{ c_{j}:j\in J\right\}$. Functions g and h in the rest of this proof are defined as at the beginning of “Appendix A”, that is, $g(\delta )=\min _{i\in I}g_{i}(\delta )$ and $h(\delta )=\min _{i\in I}h_{i}(\delta )$ for all $\delta >0$. Recall that g and h are strictly increasing and continuous on $[0,+\infty )$.

Case 1$c_{(2)}-c_{(1)}<\epsilon$.

For the ease of notation, let $\eta =c_{(2)}-c_{(1)}$. By Lemma 2 we know that there is a unique $\overline{\sigma }(\eta )\in \left( 0,\frac{\eta }{2}\right)$ defined by $g\left( \eta -2\overline{\sigma }\left( \eta \right) \right) =2\overline{\sigma }\left( \eta \right)$. Consider $\sigma >0$ such that $\sigma <\min \left\{ \frac{\epsilon -\eta }{2},\overline{\sigma } (\eta )\right\}$ and $\sigma <\min \left\{ \frac{c_{(k+1)}-c_{(k)}}{2}:k=1,\ldots ,K-1\right\}$. Then $c_{(k+1)}-\sigma >c_{(k)}+\sigma$, $k=1,\ldots ,K-1$. By Lemma 2 we know that $g\left( \eta -2\sigma \right) >2\sigma$ as $\sigma <\overline{\sigma }(\eta )$.

Since for every $j\in J$, $\underset{t_{\tau }\rightarrow \infty }{\lim } x_{j}^{t_{\tau }}=c_{j}$, there must be some $T_{\sigma }\ge 0$ such that at any $t_{\tau }\ge T_{\sigma }$, $\left| x_{j}^{t_{\tau }}-c_{j}\right| <\sigma$ for every $j\in J$, then for any $k,k'\in \{1,\ldots ,K\}$ such that $k'>k$, we have

$$\begin{aligned} \min \left\{ x_{j}^{T_{\sigma }}:j\in J_{\left( k')\right) }\right\}>c_{\left( k'\right) }-\sigma \ge c_{(k+1)}-\sigma>c_{(k)}+\sigma >\max \left\{ x_{j}^{T_{\sigma }}:j\in J_{(k)}\right\} . \end{aligned}$$

Moreover, for every $k=1,\ldots ,K$, and $j\in J_{(k)}$,

$$\begin{aligned} \left\{ x_{i}^{t_{\tau }}:i\in J_{(k)}\right\}&\subseteq \left\{ x_{i}^{t_{\tau }}:\left| x_{i}^{t_{\tau }}-x_{j}^{t_{\tau }}\right| \le 2\sigma ,i\in I\right\} \\&\subseteq \left\{ x_{i}^{t_{\tau }}: \left| x_{i}^{t_{\tau }}-x_{j}^{t_{\tau }}\right| \le \epsilon \right\} =B_{j}\left( x^{t_{\tau }}\right) , \end{aligned}$$

as $\sigma<\frac{\epsilon -\eta }{2}<\frac{\epsilon }{2}$.

• (1) Consider $i\in J_{(1)}$. Since $\overline{\sigma }(\eta )<\frac{\eta }{2}$ and $0<\sigma <\min \left\{ \frac{\epsilon -\eta }{2},\overline{\sigma }(\eta )\right\}$, we have

  $$\begin{aligned} x_{j}^{T_{\sigma }}-x_{i}^{T_{\sigma }}>c_{(2)}-\sigma -\left( c_{(1)}+\sigma \right) =\eta -2\sigma >0,\ \forall j\in J_{(2)} \end{aligned}$$

  and

  $$\begin{aligned} x_{j}^{T_{\sigma }}-x_{i}^{T_{\sigma }}<c_{(2)}+\sigma -\left( c_{(1)} -\sigma \right) =\eta +2\sigma <\epsilon ,\ \forall j\in J_{(2)}, \end{aligned}$$

  which implies that for all $i\in J_{(1)}$,

  $$\begin{aligned} \left\{ x_{j}^{T_{\sigma }}:j\in J_{(2)}\right\} \subseteq \left\{ x_{m}^{T_{\sigma }}:\left| x_{m}^{T_{\sigma }}-x_{i}^{T_{\sigma }}\right| \le \epsilon ,m\in I\right\} =B_{i}\left( x^{t_{\tau }}\right) . \end{aligned}$$

  Hence, $\left\{ x_{m}^{T_{\sigma }}:m\in J_{(1)}\cup J_{(2)}\right\} \subseteq B_{i}\left( x^{T_{\sigma }}\right)$, which means

  $$\begin{aligned} B_{i}\left( x^{T_{\sigma }}\right) \cap \left\{ x_{m}^{T_{\sigma }}:m\in J_{(1)}\cup J_{(2)}\right\} =\left\{ x_{m}^{T_{\sigma }}:m\in J_{(1)}\cup J_{(2)}\right\} . \end{aligned}$$

  Let $D_{i}=B_{i}\left( x^{T_{\sigma }}\right) \cap \left\{ x_{m}^{T_{\sigma }}:m\in J_{(1)}\cup J_{(2)}\right\} =\left\{ x_{m}^{T_{\sigma }}:m\in J_{(1)}\cup J_{(2)}\right\}$ and $D_{i}'=B_{i}\left( x^{T_{\sigma }}\right) \cap \left\{ x_{s}^{T_{\sigma }}:s\in \mathop {\cup }\limits _{k=3}^{K}J_{(k)}\right\}$. Then, $B_{i}\left( x^{T_{\sigma }}\right) =D_{i}\cup D_{i}'$. Note that if $D_{i}'$ is non-empty, then

  $$\begin{aligned} \min D_{i}'&\ge \min \left\{ x_{s}^{T_{\sigma }}:s\in \mathop {\cup }\limits _{k=3}^{K}J_{(k)}\right\} \\&>c_{(3)}-\sigma \\&>c_{(2)}+\sigma \\&>\max \left\{ x_{m}^{T_{\sigma }}:m\in J_{(1)}\cup J_{(2)}\right\} \\&\ge \max D_{i}. \end{aligned}$$

  Since both $J_{(1)}$ and $J_{(2)}$ are non-empty and $\min \left\{ x_{j}^{t}:i\in J_{(2)}\right\} >\max \{ x_{i}^{t}:i\in J_{(1)}\}$, we have

  $$\begin{aligned} \varPhi _{i}\left( D_{i}\right)&= \varPhi _{i}\left( \left\{ x_{m}^{T_{\sigma }}:m\in J_{(1)}\cup J_{(2)}\right\} \right) \\ \ge&\varPhi _{i}\left( \left\{ x_{m}^{T_{\sigma }}:m\in J_{(1)}\right\} \right) \\&+g\left( \min \left\{ x_{j}^{t}:i\in J_{(2)}\right\} -\max \left\{ x_{i}^{t}:i\in J_{(1)}\right\} \right) \\ \ge&\min \left\{ x_{i}^{T_{\sigma }}:i\in J_{(1)}\right\} +g\left( c_{(2)}-\sigma -\left( c_{(1)}+\sigma \right) \right) \\>&c_{(1)}-\sigma +g\left( \eta -2\sigma \right) \\ >&c_{(1)} \end{aligned}$$

  If $D_{i}'=\emptyset$, then $\varPhi _{i}\left( B_{i} \left( x^{T_{\sigma }}\right) \right) =\varPhi _{i}\left( D_{i}\right)$; if $D_{i}'\ne \emptyset$, then by Lemma 4 we have

  $$\begin{aligned} \varPhi _{i}\left( B_{i}\left( x^{T_{\sigma }}\right) \right) =\varPhi _{i}\left( D_{i}\cap D_{i}'\right) >\varPhi _{i}\left( D_{i}\right) . \end{aligned}$$

  Hence, $x_{i}^{T_{\sigma }}=\varPhi _{i}\left( B_{i}\left( x^{T_{\sigma }} \right) \right)>\varPhi _{i}\left( D_{i}\right) >c_{(1)}$.

• (2) Consider $j\in \mathop {\cup }\limits _{k=2}^{K}J_{(k)}$. Let $D_{j}=B_{j}\left( x^{T_{\sigma }}\right) \cap \left\{ x_{m}^{T_{\sigma }}:m\in J_{(1)}\right\}$ and $D_{j}'=B_{j}\left( x^{T_{\sigma }}\right) \cap \left\{ x_{s}^{T_{\sigma }}:s\in \mathop {\cup }\limits _{k=2}^{K}J_{(k)}\right\}$. Note that $D_{j}'$ is non-empty as $x_{j}^{T_{\sigma }}$ must be in this set.

  • (a) If $D_{j}=\emptyset$, then $B_{j}\left( x^{T_{\sigma }}\right) =D_{j}'$, and hence,

    $$\begin{aligned} x_{j}^{T_{\sigma }+1}=\varPhi _{j}\left( B_{j}\left( x^{T_{\sigma }} \right) \right)>c_{(1)}+\sigma >c_{(1)} \end{aligned}$$

    as $\min D_{j}\ge \min \left\{ x_{s}^{T_{\sigma }}:s\in \mathop {\cup }\limits _{k=2}^{K}J_{(k)}\right\}>c_{(2)}-\sigma >c_{(1)}+\sigma$.

  • (b) Suppose $D_{j}=B_{j}\left( x^{T_{\sigma }}\right) \cap \left\{ x_{m}^{T_{\sigma }}:m\in J_{(1)}\right\} \ne \emptyset$, then

    $$\begin{aligned} \max D_{j}&\le \max \left\{ x_{m}^{T_{\sigma }}:m\in J_{(1)}\right\} \\&<c_{(1)}+\sigma \\&<c_{(2)}-\sigma \\&<\min \left\{ x_{s}^{T_{\sigma }}:s\in \mathop {\cup }\limits _{k=2}^{K}J_{(k)}\right\} \\&\le \min D_{j}'. \end{aligned}$$

    By B3 and Lemma 4 we have

    $$\begin{aligned} x_{j}^{T_{\sigma }+1}&= \varPhi _{j}\left( B_{j}\left( x^{T_{\sigma }}\right) \right) \\&= \varPhi _{j}\left( D_{j}\cup D_{j}'\right) \\ \ge&\varPhi _{i}\left( D_{j}\right) +g\left( \min D_{j}'-\max D_{j}\right) \\>&\min \left\{ x_{m}^{T_{\sigma }}:m\in J_{(1)}\right\} +g\left( c_{(2)}-\sigma -\left( c_{(1)}+\sigma \right) \right) \\>&\, c_{(1)}-\sigma +g\left( \eta -2\sigma \right) \\ >&\, c_{(1)}, \end{aligned}$$

    as $g\left( \eta -2\sigma \right) >2\sigma$.

From above we see that if $c_{(2)}-c_{(1)}<\epsilon$, then $x_{j}^{T_{\sigma }+1}>c_{(1)}$ for every $j\in J$. Hence, $\min \left\{ x_{j}^{T_{\sigma }+1}:j\in I\right\} >c_{(1)}=\min \left\{ c_{j}:j\in J\right\}$.

Case 2$c_{(2)}-c_{(1)}=\epsilon$.

By Lemma 2, we know that there are unique $\overline{\delta }(\epsilon ),\overline{\gamma }(\epsilon ) \in \left( 0,\epsilon \right)$ such that $h\left( \epsilon -2\overline{\gamma }\left( \epsilon \right) \right) =2\overline{\gamma }\left( \epsilon \right)$ and $g\left( g\left( \epsilon -2\overline{\delta }(\epsilon )\right) -2\overline{\delta }(\epsilon )\right) =2\overline{\delta }(\epsilon )$. Also by Lemma 2 we know that $\overline{\delta }(\epsilon )<\overline{\sigma }(\epsilon )$ where $\overline{\sigma }(\epsilon )\in \left( 0,\frac{\epsilon }{2}\right)$ is uniquely defined by $g\left( \epsilon -2\overline{\sigma }(\epsilon ) \right) =\overline{\sigma }(\epsilon )$.

Consider $\delta >0$ such that $\delta <\min \big \{\overline{\delta } (\epsilon ),\overline{\gamma }\left( \epsilon \right) \big \}$ and $\delta <\min \big \{ \frac{c_{(k+1)}-c_{(k)}}{2}:k=1,\ldots ,K\big \}$. By Lemma 2, we know $g\left( \epsilon -2\delta \right) >2\delta$, $g\left( g(\epsilon -2\delta )\right) >2\delta$, and $h\left( \epsilon -2\delta \right) >2\delta$. $\delta \le \frac{c_{(2)}-c_{(1)}}{2}=\frac{\epsilon }{2}$ implies that for any $j_{(k)}$, $k=1,\ldots ,K$ and all $j\in c_{(k)}$, then

$$\begin{aligned} \left\{ x_{m}^{T_{\delta }}:m\in c_{(k)}\right\} \subseteq \left\{ x_{i}^{T_{\delta }}:\left| x_{i}^{T_{\delta }}-x_{j}^{T_{\delta }}\right| \le \epsilon \right\} =B_{j}\left( x^{T_{\delta }}\right) . \end{aligned}$$

Since for every $j\in J$, $\underset{t_{\tau }\rightarrow \infty }{\lim }x_{j}^{t_{\tau }}=c_{j}$, there is $T_{\delta }\ge 0$ such that at any $t_{\tau }\ge T_{\delta }$,

$$\begin{aligned} \left| x_{i}^{t_{\tau }}-c_{(k)}\right| <\delta ,\quad \forall i\in J_{(k)},\ k=1,\ldots ,K, \end{aligned}$$

(4)

then $J_{(k)}=\left\{ i\in J:\left| x_{i}^{T_{\delta }}-c_{(k)}\right| <\delta \right\}$. Note that (4) implies that for any $i\in J_{(k)}$ and $j\in J_{\left( k'\right) }$ where $k'\ge k+1$,

$$\begin{aligned} x_{j}^{T_{\delta }}-x_{i}^{T_{\delta }}>c_{\left( k'\right) } -\delta -\left( c_{(k)}+\delta \right) =c_{\left( k'\right) }-c_{(k)}-2\delta \ge c_{\left( k+1\right) }-c_{(k)}-2\delta >0. \end{aligned}$$

Then, $\min \left\{ x_{j}^{T_{\delta }}:j\in \mathop {\cup }\limits _{k=m+1}^{K} J_{(k)}\right\} >\max \left\{ x_{j}^{T_{\delta }}:j\in \mathop {\cup }\limits _{k=1}^{m}J_{(k)}\right\}$ for all $m\in \left\{ 1,\ldots ,K-1\right\}$. Moreover, $\delta \le \frac{c_{(2)}-c_{(1)}}{2}=\frac{\epsilon }{2}$ implies that if $j\in c_{(k)}$, then

$$\begin{aligned} \left\{ x_{m}^{T_{\delta }}:m\in c_{(k)}\right\} \subseteq \left\{ x_{i}^{T_{\delta }}:\left| x_{i}^{T_{\delta }}-x_{j}^{T_{\delta }}\right| \le \epsilon \right\} =B_{j}\left( x^{T_{\delta }}\right) . \end{aligned}$$

We partition J into the following four mutually exclusive subsets:

• $P_{H}=\mathop {\cup }\limits _{k=2}^{K}J_{(k)}$

• $P_{M}=\left\{ i\in J_{(1)}:\min \left\{ x_{i}^{T_{\delta }}:j\in \mathop {\cup }\limits _{k=2}^{K}J_{(k)}\right\} -x_{i}^{T_{\delta }}\le \epsilon \right\}$

• $P_{L}=\left\{ i\in J_{(1)}:\min \left\{ x_{i}^{T_{\delta }}:j\in \mathop {\cup }\limits _{k=2}^{K}J_{(k)}\right\} -x_{i}^{T_{\delta }}>\epsilon \right\}$

Note that $P_{M}$ must be non-empty as $x^{T_{\delta }}$ is an $\epsilon$-profile on J. Next we show that at period $T_{\delta }+2$, all agents’ opinions are strictly greater than $c_{(1)}$.

Agents’ Opinions at Period $T_{\delta }+1$

• (1) Consider $l\in P_{H}$. Let

  $$\begin{aligned} D_{l}=B_{l}\left( x^{T_{\delta }}\right) \cap \left\{ x_{i}^{T_{\delta }}:i\in J_{(1)}\right\} \ \text {and}\ D_{l}'=B_{l}\left( x^{T_{\delta }}\right) \cap \left\{ x_{j}^{T_{\delta }}:j\in \mathop {\cup }\limits _{k=2}^{K}J_{(k)}\right\} . \end{aligned}$$

  Then, $B_{l}\left( x^{T_{\delta }}\right) =D_{l}\cup D_{l}'$. Since

  $$\begin{aligned} \min D_{l}'&\ge \min \left\{ x_{j}^{T_{\delta }}:j\in \mathop {\cup }\limits _{k=2}^{K}J_{(k)}\right\} \\&>c_{(2)}-\sigma \\&>c_{(1)}+\sigma \\&>\max \left\{ x_{i}^{T_{\delta }}:i\in J_{(1)}\right\} \\&\ge \max D_{l}, \end{aligned}$$

  by Lemma 4 we have

  $$\begin{aligned} x_{l}^{T_{\delta +1}}&= \varPhi _{l}\left( B_{l}\left( x^{T_{\delta }}\right) \right) \\&= \varPhi _{l}\left( D_{l}\cup D_{l}'\right) \\&\ge \varPhi _{l}\left( D_{l}\right) +g\left( \min D_{l}'-\max D_{l}\right) \\&> \min D_{l}+g\left( c_{(2)}-\delta -\left( c_{(1)}+\delta \right) \right) \\&\ge \min \left\{ x_{i}^{T_{\delta }}:i\in J_{(1)}\right\} +g\left( \epsilon -2\delta \right) \\&> c_{(1)}-\delta +g\left( \epsilon -2\delta \right) \\&> c_{(1)}+\delta , \end{aligned}$$

  where the first weak inequality holds with “$=$” if $D_{l}=\emptyset$.

• (2) Consider $i\in P_{M}$. Let

  $$\begin{aligned} D_{i}= & {} B_{i}\left( x^{T_{\delta }}\right) \cap \left\{ x_{m}^{T_{\delta }}:m\in J_{(1)}\right\} \ \text {and}\ D_{i}'\\= & {} B_{i}\left( x^{T_{\delta }}\right) \cap \left\{ x_{j}^{T_{\delta }}:j\in \mathop {\cup }\limits _{k=2}^{K}J_{(k)}\right\} . \end{aligned}$$

  Then, $B_{i}\left( x^{T_{\delta }}\right) =D_{i}\cup D_{i}'$. Recall that $\left\{ x_{m}^{T_{\delta }}:m\in J_{(1)}\right\} \subseteq B_{i}\left( x^{T_{\delta }}\right)$ as $i\in J_{(1)}$ and $\delta <\frac{\epsilon }{2}$, then $D_{i}=\left\{ x_{m}^{T_{\delta }}:m\in J_{(1)}\right\}$.

  Since for every $m\in \mathop {\cup }\limits _{k=3}^{K}J_{(k)}$,

  $$\begin{aligned} x_{m}^{T_{\delta }}-x_{i}^{T_{\delta }}>c_{(3)}-\delta -\left( c_{(1)}+\delta \right)= & {} c_{(3)}-c_{(2)}-2\delta +c_{(2)}-c_{(1)}>c_{(2)}-c_{(1)}\\= & {} \epsilon \end{aligned}$$

  as $\delta <\frac{c_{(3)}-c_{(2)}}{2}$, $B_{i}\left( x^{T_{\delta }}\right) \cap \left\{ x_{j}^{T_{\delta }}:j\in \mathop {\cup }\limits _{k=3}^{K}J_{(k)}\right\} =\emptyset$, which implies

  $$\begin{aligned} D_{i}'=B_{i}\left( x^{T_{\delta }}\right) \cap \left\{ x_{j}^{T_{\delta }}:j\in \mathop {\cup }\limits _{k=2}^{K}J_{(k)}\right\} =B_{i}\left( x^{T_{\delta }}\right) \cap \left\{ x_{j}^{T_{\delta }}:j\in J_{(2)}\right\} , \end{aligned}$$

  Since

  $$\begin{aligned} \min D_{i}'&\ge \min \left\{ x_{j}^{T_{\delta }}:j\in J_{(2)}\right\} \\&>c_{(2)}-\delta \\&>c_{(1)}+\delta \\&>\max \left\{ x_{m}^{T_{\delta }}:m\in J_{(1)}\right\} \\&\ge \max D_{i}, \end{aligned}$$

  by Lemma 4 we have

  $$\begin{aligned} x_{i}^{T_{\delta }+1}&= \varPhi _{i}\left( B_{i}\left( x^{T_{\delta }}\right) \right) \\&= \varPhi _{i}\left( D_{i}\cup D_{i}'\right) \\&\ge \varPhi _{i}\left( D_{i}\right) +g\left( \min D_{i}'-\max D_{i}\right) \\&> \min D_{i}-g\left( c_{(2)}-\delta -\left( c_{(1)}+\delta \right) \right) \\&\ge \min \left\{ x_{m}^{T_{\delta }}:m\in J_{(1)}\right\} +g\left( \epsilon -2\delta \right) \\&> c_{(1)}-\delta +g\left( \epsilon -2\delta \right) \\&> c_{(1)}+\delta \end{aligned}$$

  as $g(\epsilon -2\delta )>2\delta$ and

  $$\begin{aligned} x_{i}^{T_{\delta }+1}&= \varPhi _{i}\left( B_{i}\left( x^{T_{\delta }}\right) \right) \\&= \varPhi _{i}\left( D_{i}\cup D_{i}\right) \\&\le \varPhi _{i}\left( D_{i}'\right) -h\left( \min D_{i}'-\max D_{i}\right) \\&\le \max D_{i}'-h\left( c_{(2)}-\delta -\left( c_{(1)}+\delta \right) \right) \\&< \max \left\{ x_{m}^{T_{\delta }}:m\in J_{(2)}\right\} -h\left( \epsilon -2\delta \right) \\&< c_{(2)}+\delta -h\left( \epsilon -2\delta \right) \\&< c_{(2)}-\delta \end{aligned}$$

  as $h\left( \epsilon -2\delta \right) >2\delta$.

• (3) Consider $j\in P_{L}$. Since $\min \left\{ x_{m}^{T_{\delta }}:m\in \mathop {\cup }\limits _{k=2}^{K}J_{(k)}\right\} -x_{j}^{T_{\delta }}>\epsilon$, we have

  $$\begin{aligned} B_{j}\left( x^{T_{\delta }}\right) \cap \left\{ x_{m}^{T_{\delta }}:m\in \mathop {\cup }\limits _{k=2}^{K}J_{(k)}\right\} =\emptyset , \end{aligned}$$

  which implies $B_{j}\left( x^{T_{\delta }}\right) \subseteq \left\{ x_{i}^{T_{\delta }}:i\in J_{(1)}\right\}$. Thus,

  $$\begin{aligned} x_{j}^{T_{\delta }+1}=\varPhi _{j}\left( B_{i}\left( x^{T_{\delta }}\right) \right) \le \max B_{j}\left( x^{T_{\delta }}\right) \le \max \left\{ x_{i}^{T_{\delta }}:i\in J_{(1)}\right\} <c_{(1)}+\delta \end{aligned}$$

  and

  $$\begin{aligned} x_{j}^{T_{\delta }+1}=\varPhi _{j}\left( B_{i}\left( x^{T_{\delta }}\right) \right) \ge \min B_{j}\left( x^{T_{\delta }}\right) \le \min \left\{ x_{i}^{T_{\delta }}:i\in J_{(1)}\right\} >c_{(1)}-\delta . \end{aligned}$$

Agents’ Opinions at Period $T_{\delta }+2$

From above we know that $x_{i}^{T_{\delta }+1}>c_{(1)} -\delta +g\left( \epsilon -2\delta \right)>c_{(1)}+\delta >x_{j}^{T_{\delta }+1}$ for all $i\in P_{H}\cup P_{M}$ and all $j\in P_{L}$, then

$$\begin{aligned} \min \left\{ x_{i}^{T_{\delta }+1}:i\in P_{H}\cup P_{M}\right\}&>c_{(1)}-\delta +g\left( \epsilon -2\delta \right) \nonumber \\&>c_{(1)}+\delta \nonumber \\&>\max \left\{ x_{j}^{T_{\delta }+1}:j\in P_{L}\right\} . \end{aligned}$$

(5)

• (1) Consider $i\in P_{H}\cup P_{M}$. Let

  $$\begin{aligned} D_{i}=B_{i}\left( x^{T_{\delta }+1}\right) \cap \left\{ x_{j}^{T_{\delta }+1}:j\in P_{L}\right\} \end{aligned}$$

  and

  $$\begin{aligned} D_{i}'=B_{i}\left( x^{T_{\delta }+1}\right) \cap \left\{ x_{m}^{T_{\delta }+1}:m\in P_{H}\cup P_{M}\right\} \end{aligned}$$

  Then, $B_{i}\left( x^{T_{\delta }+1}\right) =D_{i}\cup D_{i}'$.

  If $D_{i}=\emptyset$, then $B_{i}\left( x^{T_{\delta }+1}\right) =D_{i}'\subseteq \left\{ x_{m}^{T_{\delta }+1}:m\in P_{H}\cup P_{M}\right\}$, which means

  $$\begin{aligned} \varPhi _{i}\left( B_{i}\left( x^{T_{\delta }+1}\right) \right)= & {} \varPhi _{i}\left( D_{i}'\right) \\\ge & {} \min D_{i}'\ge \min \left\{ x_{m}^{T_{\delta }+1}:m\in P_{H}\cup P_{M}\right\} >c_{(1)}+\delta . \end{aligned}$$

  If $D_{i}\ne \emptyset$, then

  $$\begin{aligned} \max D_{i}\le & {} \left\{ x_{j}^{T_{\delta }+1}:j\in P_{L}\right\} \\< & {} c_{(1)}+\delta <\min \left\{ x_{m}^{T_{\delta }+1}:m\in P_{H}\cup P_{M}\right\} \le \min D_{i}'. \end{aligned}$$

  By Lemma 4 and (5) we have

  $$\begin{aligned} \varPhi _{i}\left( B_{i}\left( x^{T_{\delta }+1}\right) \right)&= \varPhi _{i}\left( D_{i}\cup D_{i}'\right) \\&\ge \varPhi _{i}\left( D_{i}\right) +g\left( \min D_{i}'-\max D_{i}\right) \\&> \min D_{i}+g\left( c_{(2)}-\delta -\left( c_{(1)}+\delta \right) \right) \\&\ge \min \left\{ x_{j}^{T_{\delta }+1}:j\in P_{L}\right\} +g\left( c_{(2)}-c_{(1)}-2\delta \right) \\&> c_{(1)}-\delta +g\left( g\left( \epsilon -2\delta \right) -2\delta \right) \\&> c_{(1)} \end{aligned}$$

  as $g\left( g\left( \epsilon -2\delta \right) -2\delta \right) >2\delta$. Therefore, $x_{i}^{T_{\delta }+2}=\varPhi _{i}\left( B_{i}\left( x^{T_{\delta }+1} \right) \right) >c_{(1)}$.

• (2) Consider $j\in P_{L}$. As discussed earlier, for every agent $i\in P_{M}$,

  $$\begin{aligned} c_{(1)}-\delta +g\left( \epsilon -2\delta \right)<x_{i}^{T_{\delta }+1} <c_{(2)}-\delta . \end{aligned}$$

  then

  $$\begin{aligned} x_{i}^{T_{\delta }+1}-x_{j}^{T_{\delta }+1}{\left\{ \begin{array}{ll}<c_{(2)}-\delta -\left( c_{(1)}+\delta \right) =\epsilon -2\delta <\epsilon \\>c_{(1)}-\delta +g\left( \epsilon -2\delta \right) -\left( c_{(1)}+\delta \right) =g\left( \epsilon -2\delta \right) -2\delta >0, \end{array}\right. } \end{aligned}$$

  which implies $\left\{ x_{i}^{T_{\delta }+1}:i\in P_{M}\right\} \subseteq B_{j}\left( x^{T_{\delta }+1}\right)$.

  Let

  $$\begin{aligned} D_{j}=B_{j}\left( x^{T_{\delta }+1}\right) \cap \left\{ x_{k}^{T_{\delta }+1}:k\in P_{L}\right\} \end{aligned}$$

  and

  $$\begin{aligned} D_{j}'=B_{j}\left( x^{T_{\delta }+1}\right) \cap \left\{ x_{m}^{T_{\delta }+1}:m\in P_{H}\cup P_{M}\right\} . \end{aligned}$$

  Since

  $$\begin{aligned} \min D_{j}'&\ge \min \left\{ x_{m}^{T_{\delta }+1}:m\in P_{H}\cup P_{M}\right\} \\&>c_{(1)}-\delta +g\left( \epsilon -2\delta \right) \\&>c_{(1)}+\delta \\&>\max \left\{ x_{k}^{T_{\delta }+1}:k\in P_{L}\right\} \\&\ge \max D_{i}, \end{aligned}$$

  by Lemma 4 and (5) we have

  $$\begin{aligned} \varPhi _{j}\left( B_{j}\left( x^{T_{\delta }+1}\right) \right)&= \varPhi _{j}\left( D_{j}\cup D_{j}'\right) \\&\ge \varPhi _{j}\left( D_{j}\right) +g\left( \min D_{j}'-\max D_{j}\right) \\&> \min D_{j}+g\left( c_{(1)}-\delta +g\left( \epsilon -2\delta \right) -\left( c_{(1)}+\delta \right) \right) \\&\ge \min \left\{ x_{k}^{T_{\delta }+1}:k\in P_{L}\right\} +g\left( g\left( \epsilon -2\delta \right) -2\delta \right) \\&> c_{(1)}-\delta +g\left( g\left( \epsilon -2\delta \right) -2\delta \right) \\&> c_{(1)} \end{aligned}$$

  as $g\left( g\left( \epsilon -2\delta \right) -2\delta \right) >2\delta$. Hence, $x_{i}^{T_{\delta }+2}=>c_{(1)}$.

The above discussion shows that at $T_{\delta }+2$, every agent’s opinion is strictly greater than $c_{(1)}$.

From above we know that, if there are two different $j,k\in J$ such that $c_{j}\ne c_{k}$, then there will be some period T at which all agents’ opinions are strictly greater than $c_{(1)}=\min \left\{ c_{j}:j\in J\right\}$, contradicting $\min \left\{ x_{j}^{t}:j\in J\right\} \le \min \left\{ c_{j}:j\in J\right\}$ at any period $t\ge 0$ in Lemma 5. Hence, $c_{j}=c_{k}$ for any $j,k\in J$; in other words, $\underset{t\rightarrow \infty }{\lim } x_{j}^{t}=\underset{t\rightarrow \infty }{\lim }x_{k}^{t}$ for all $j,k\in J$. $\square$

Note that we include the proof for Proposition 2 here for readers’ convenience. A discussion for why $\epsilon <\underline{\epsilon }^{*}\left( f,x^{0}\right)$ is a necessary condition for agents to approach a consensus is implicit in the proof for Theorem on Opinion Stabilization in Hegselmann and Krause (2005), though it is not clearly stated in the proof for the Corollary on Consensus Formation in that paper.

Proof of Proposition 2

Recall that $\underline{\epsilon }^{*}\left( f,x^{0}\right) =\inf E\left( x^{0}\right)$, where $E\left( x^{0}\right)$ is the set of all $\epsilon$ such that $x^{t}=f^{t}\left( x^{0}\right)$ is an $\epsilon$-profile on I at every period t. We focus on the case in which $\underline{\epsilon }^{*}\left( f,x^{0}\right) >0$, as if $\underline{\epsilon }^{*}\left( f,x^{0}\right) =0$, then $x^{0}$ must be a consensus itself.

If $\epsilon <\underline{\epsilon }^{*}\left( f,x^{0}\right)$, then there must be some period T at which $x^{T}$ is not an $\epsilon$-profile on I. That is, for any re-ordering of agents’ opinions, denoted as $x_{i_{1}}^{T},\ldots ,x_{i_{N}}^{T}$, such that $x_{i_{n}}^{T}\le x_{i_{n+1}}^{T}$, $n=1,\ldots ,N-1$, there must be some $m\in \{1,\ldots ,N-1\}$ such that $x_{i_{m+1}}^{T}-x_{i_{m}}^{T}>\epsilon$.

Consider any $j,k\in \{1,\ldots ,N-1\}$ such that $j\le m<m+1\le k$, then

$$\begin{aligned} x_{i_{k}}^{T}-x_{i_{j}}^{T}\ge x_{i_{m+1}}^{T}-x_{i_{m}}^{T}>\epsilon , \end{aligned}$$

which means that

$$\begin{aligned} I_{i_{j}}\left( x^{T}\right) \cap I_{i_{k}}\left( x^{T}\right) =\left\{ i\in I:\left| x_{i}^{T}-x_{i_{j}}^{T}\right| \le \epsilon \right\} \cap \left\{ j\in I:\left| x_{j}^{T}-x_{i_{k}}^{T}\right| \le \epsilon \right\} =\emptyset . \end{aligned}$$

Thus, $I_{i_{j}}\left( x^{T}\right) \subseteq \left\{ i\in I:x_{i}^{T}\le x_{i_{m}}^{T}\right\}$ and $I_{i_{k}}\left( x^{T}\right) \subseteq \left\{ j\in I:x_{j}^{T}\ge x_{i_{m+1}}^{T}\right\}$. Then, as

$$\begin{aligned} x_{i_{j}}^{T+1}=f_{i_{j}}\left( x^{T}\right) \le \max \left\{ x_{i}^{T}:i\in I_{i_{j}}\left( x^{T}\right) \right\} \le \max \left\{ x_{i}^{T}:x_{i}^{T}\le x_{i_{m}}^{T}\right\} =x_{i_{m}}^{T} \end{aligned}$$

and

$$\begin{aligned} x_{i_{k}}^{T+1}=f_{i_{k}}\left( x^{T}\right) \ge \min \left\{ x_{j}^{T}:j\in I_{i_{k}}\left( x^{T}\right) \right\} \ge \min \left\{ x_{j}^{T}:x_{j}^{T}\ge x_{i_{m+1}}^{T}\right\} =x_{i_{m+1}}^{T}, \end{aligned}$$

we have $x_{i_{k}}^{T+1}-x_{i_{j}}^{T+1}\ge x_{i_{m+1}}^{T}-x_{i_{m}}^{T}>\epsilon$. By induction we know that $x_{i_{k}}^{t}-x_{i_{j}}^{t}>\epsilon$ at any period $t\ge T+1$. Thus, agents will never approach a consensus, as some agents’ opinions differ more than $\epsilon$ at all periods following period T. $\square$

Appendix B: Examples of PAMs Satisfying A1–A3

Many functions satisfy A1–A3. Here we list two examples.

Example 3

Consider $x=\left( x_{1},\ldots ,x_{N}\right)$ and $x'=\left( x_{1}',\ldots ,x_{N}'\right)$ such that $I_{i}\left( x\right) =I_{i}\left( x'\right) \cup \left\{ j\right\}$, where $j\notin I_{i}\left( x'\right)$, and $x_{k}=x_{k}'$ for all $k\in I_{i}\left( x'\right)$.

• (1) Weighted mean

  Agent $i\in I$ assigns “strength” or “importance” parameter $q_{ij}\in (0,1)$ to each agent $j\in I$ and revises her opinion using the following opinion revision function

  $$\begin{aligned} f_{i}\left( x\right) =\sum _{k\in I_{i}\left( x\right) }\frac{q_{ik}}{\sum _{k\in I_{i}\left( x\right) }q_{ik}}x_{k}. \end{aligned}$$

  It is easy to see that $f_{i}$ satisfies A1 and A2. This function satisfies A3 as

  • (a) if $x_{j}>\max B_{i}\left( x'\right)$, then

    $$\begin{aligned} f_{i}\left( x\right) >f_{i}\left( x'\right) +\frac{\underline{q}}{N\overline{q}+\underline{q}}\left[ x_{j} -\max B_{i}\left( x'\right) \right] ,\ \forall i\in I; \end{aligned}$$

  • (b) if $x_{j}<\min B_{i}\left( x'\right)$, then

    $$\begin{aligned} f_{i}\left( x\right) <f_{i}\left( x'\right) -\frac{\underline{q}}{N\overline{q}+\underline{q}}\left[ \min B_{i}\left( x'\right) -x_{j}\right] ,\ \forall i\in I. \end{aligned}$$

  Here $\overline{q}=\max \left\{ q_{ij}:i,j\in I\right\}$ and $\underline{q}=\min \left\{ q_{ij}:i,j\in I\right\}$.

• (2) Harmonic mean

  Agent $n\in I$ revises her opinion using the harmonic mean

  $$\begin{aligned} f_{n}\left( x\right) =\frac{\#I_{n}\left( x\right) }{\underset{k\in I_{n}\left( x\right) }{\sum }\frac{1}{x_{k}}}. \end{aligned}$$

  Here $\#I_{n}\left( x\right)$ represents the number of agents in $I_{n}\left( x\right)$. It is easy to see that $f_{n}$ satisfies A1 and A2. $f_{n}$ satisfies A3 as

  • (a) if $x_{j}>\max B_{n}\left( x'\right)$, then $f_{n}\left( x\right) >f_{n}\left( x'\right) +\frac{1}{1+N}\cdot \left[ x_{j}-\max B_{n}\left( x'\right) \right]$;

  • (b) if $x_{j}<\min B_{n}\left( x'\right)$, then $f_{n}\left( x\right) <f_{n}\left( x'\right) -\frac{1}{1+N}\cdot \left[ \min B_{n}\left( x'\right) -x_{j}\right]$.