text:
\section{Introduction}\label{S1}
\paragraph*{Problem description and motivation}
The wisdom of crowds effect generally refers to the phenomenon of improvement in estimate accuracy by pooling many independent estimates. Numerous experiments and simulations have been contributed to understand as well as to explain this collective intelligence for decades. A very recent interest and debate with respect to social networks is the question of whether and, if so, how social influence improves or undermines the wisdom of crowds. As much research with empirical evidence, even in support of contrary conclusions, was provided, there is still a lack of rigorous mathematical formulation and analysis. A relevant mathematical model is the so-called na\"{\i}ve learning model proposed by Golub and Jackson~\cite{BG-MOJ:10} and further investigated in~\cite{FB-FF-BF:17l}. In this model, individuals update their estimates for an unknown truth according to the French-DeGroot (FD) opinion dynamics with their initial estimates disturbed by independent zero-mean noises. Thereby, the group is initially wise in the sense that the initial collective estimate converges to the truth as group size increases. If the group remains wise as the influence process converges, the influence system preserves the wisdom.

This paper addresses the question of how social influence improves or undermines the wisdom of finite-size groups in a general framework of weighted-average opinion dynamics based on the na\"{\i}ve learning setting. Individuals' initial estimates are assumed to be realizations of their knowledge or expertise for the unknown truth, and are described by a family of independent random variables with uniformly unbiased expectations and possibly different variances. We propose formal formulation of wisdom of crowds as well as definitions of improving, optimizing and undermining of the wisdom for a class of weighted-average opinion dynamics. Necessary and/or sufficient conditions for improving, optimizing and undermining of wisdom are derived. Then our theoretical results are applied to the FD opinion dynamics to verify empirical findings in the literature. We aim to provide a theoretical framework to understand what social processes do indeed promote or diminish the collective intelligence and how to organize a wiser group.

\paragraph*{Literature review}
As well-known as the idiom ``two heads are better than one", wisdom of
crowds has been a source of fascination to academia for
centuries~\cite{JSU:04}. A comparable concept in the area of animal
behavior is called the many wrongs principle~\cite{AMS:04}. In 1907,
Galton~\cite{FG:1907} reported an experiment in which averaging hundreds of
guesses for the weight of an ox yielded a collective estimate essentially
close to the true value. In the middle of the last century, a structured
communication technique, known as the Delphi method, was developed by the RAND Corporation to systematically exploit experts'
opinions~\cite{ND-OH:63}. Lots of experiments have been conducted on the
Delphi method; evidence shows that the Delphi method produces better
decisions than uncontrolled group discussion across many
domains~\cite{RJB:74}. The most significant finding of Delphi experiments
is that participants without strong convictions tend to change their
estimates, while those who feel they have a good argument for a deviant
estimate tend to retain and defend their original
estimates~\cite{OH:67}. In~\cite{NCD:68}, participants who did not change
their estimates are called holdouts, in contrast to swingers who changed their estimates. The holdouts were reported more
accurate than swingers and even than the entire group. The same phenomenon
was also reported in~\cite{GM-GGdP:15}and~\cite{JB-DB-DC:17} recently, while~\cite{GM-GGdP:15} named it the
\emph{Parent\'{e} and Anderson-Parent\'{e} (PAP) hypothesis}.

In a related area, the investigation of social networks focuses on the
formation and evolution of individuals' opinions and social power in the
presence of interpersonal influence. Classic opinion dynamics models
include the FD model~\cite{MHDG:74}, the Friedkin-Johnsen
model~\cite{NEF-ECJ:99, YT-LW:18}, the Hegselmann-Krause
model~\cite{VDB-JMH-JNT:09,VDB-JMH-JNT:10} and the Altafini
model~\cite{CA:13,DM-ZM-YH:19}, to name but a
few~\cite{DB-HT-TB:16,AVP-RT:17}. In this literature, the FD
model serves as a basis of others and is widely established. As the
research progresses, social power and network centrality turn out to be
crucial in determining the outcome of opinion
formation~\cite{PJ-AM-NEF-FB:13d,YT-PJ-AM-LW-NEF-FB:19c}, where social
power indicates the dominance of an individual's initial opinion on group's
final opinions, and network centrality measures the relative importance of
an individual in the network.

Naturally, there arises the question of whether social influence improves
or undermines the wisdom, and how. Lorenz et al.~\cite{JL-HR-FS-DH:11}
concluded, with empirical evidence, that even mild social influence
undermines the wisdom by diminishing the diversity of estimates. In
contrast, Becker et al.~\cite{JB-DB-DC:17} shown that social influence
improves the accuracy of collective estimate even as individuals' estimates
become more similar. In~\cite{AEM-JBS-RPL:14} and~\cite{GM-GGdP:15}, the
authors shown, respectively, that selecting individuals according to their
exerted average accuracy and their resistance to social influence makes
large improvements in group accuracy. In~\cite{AA-ANC-AA-PMK-MM-AP:20}, the
authors shown that social influence produces more accurate collective
estimates in the presence of plasticity and feedback.

\paragraph*{Contributions}
This paper investigates the effect of social influence on wisdom of crowds in influence networks. Different from the na\"{\i}ve learning model, we address the question of how social influence improves or undermines the wisdom in finite-size groups. First, we propose mathematical formulation for wisdom of crowds and formal definitions for improving, optimizing and undermining of the wisdom in influence networks modelled by general weighted-average opinion dynamics. We show that in our definitions, if individuals' variances are finite and the influence system improves the wisdom of finite-size group, the collective estimate converges to the truth in probability as group size increases. In other words, improving of the wisdom in finite-size groups implies preserving of the wisdom in large-size groups.

Second, we study how to improve and to optimize the wisdom. We show that at an influence system level, whether social influence improves or undermines the wisdom is decided by the social power allocation of the influence system. Several notions for consistency of an influence system's social power allocation and individuals' accuracy are defined. Necessary and/or sufficient conditions for improving and optimizing the wisdom are provided. We prove that social influence optimizes the wisdom if and only if the influence system's social power allocation is exactly proportional to individuals' accuracy. The wisdom is improved only if individuals' variances are not uniform; and if so, social influence improves the wisdom if the influence system allocates relatively more, but not too much, social power to more accurate individuals. Moreover, we prove that the select crowd strategy works under proper condition. We also derive a condition on individuals' variances under which all social power allocations in certain orderings improve the wisdom; an algorithm is further designed to find all these orderings for given individuals' variances and its complexity is analyzed.

Third, we study how to undermine the wisdom. We show that if the influence system allocates more social power to less accurate individuals, the wisdom is undermined. Moreover, we prove that if social influence improves the wisdom with all social power allocations in a certain ordering, then social influence undermines the wisdom with all social power allocations in the reverse ordering. Therefore, the aforementioned algorithm also finds all the orderings that all social power allocations in these orderings undermine the wisdom.

Finally, we apply our theoretical results to the FD opinion dynamics. We show that at a population level, individuals' susceptibilities to interpersonal influence and network centralities both play important roles. In general influence networks, the PAP hypothesis is neither sufficient nor necessary for improving the wisdom. Roughly speaking, social influence improves the wisdom if more accurate individuals have larger network centralities and are less susceptible to social influence. However, if the influence network is democratic, the PAP hypothesis is, to some degree, sufficient and/or necessary to improve or to optimize the wisdom. We also show that the wisdom can be improved or optimized even in autocratic influence networks.

Our theoretical analysis reveals some findings of sociological significance and contributes to understanding the role social influence plays in the emergence of collective intelligence. First, social influence can both improve and undermine the wisdom of crowds, which mainly depends on individuals' accuracy and the social power allocation of the influence system. If the influence system assigns relatively larger social power to relatively more accurate individuals, it improves the wisdom; if the influence system assigns less social power to more accurate individuals, it undermines the wisdom. Moreover, there is a symmetric conclusion that if the influence system with a certain ordering of social power allocation improves the wisdom, then it undermines the wisdom with inversely ordered social power allocation. Lastly, in democratic influence network, if relatively more accurate individuals are relatively less susceptible to social influence, the wisdom is improved, which supports the PAP hypothesis. In general influence networks, how social influence takes effect is more complicated and is determined not only by individuals' accuracy and susceptibilities, but also by their network centralities. Simply put, if more accurate individuals have larger network centralities and are less susceptible to interpersonal influence, the wisdom is improved.
\paragraph*{Paper organization}
In \cref{S2}, we propose mathematizations for improving, optimizing and undermining of the wisdom in influence networks. \Cref{S3,S4} investigate how social influence improves, optimizes and undermines the wisdom at influence system level. In \cref{S5} we further apply our results to the FD opinion dynamics. \Cref{S6} concludes the paper.

\paragraph*{Notation}
$\mathbf{1}_{n}$ and $I_{n}$ denote the $n\times 1$ all-ones vector and the $n\times n$ identity matrix, respectively. $\mathbf{e}_{i}$ denotes the $i$-th standard basis vector with proper dimension. $\mathbb{R}$, $\mathbb{R}^{n}$ and $\mathbb{R}^{n\times n}$ denote, respectively, the sets of real number, $n$-dimensional real vector and $n\times n$ real matrix. For vector or square matrix $A^{(n)}$, the superscript $(n)$ denotes its dimension; we will omit the superscript if it is not necessary to clarify. Given $\delta\in\mathbb{R}^{n}$, $[\delta]=\diag(\delta)$ denotes a diagonal matrix with diagonal elements $\delta_{1}, \dots, \delta_{n}$. The $n$-simplex is denoted by $\Delta_{n}\!=\!\setdef{z\in\mathbb{R}^{n}}{z\geq 0, \mathbf{1}^{\top}_{n}z\!=\!1}$; $\operatorname{int}{\Delta_{n}}\!=\!\setdef{z\in\mathbb{R}^{n}}{z> 0, \mathbf{1}^{\top}_{n}z\!=\!1}$ denotes its interior. A nonnegative matrix is row-stochastic (column-stochastic) if its row (column) sums are $1$; it is doubly-stochastic if both its row and column sums are $1$. The weighted digraph $\mathcal{G}(W)$ associated with nonnegative matrix $W$ is defined as: the node set is $\until n$; there is a directed edge $(i,j)$ from nodes $i$ to $j$ if and only if $W_{ij}>0$. $\mathcal{G}(W)$ is a star topology if all its directed edges are either from or to a center node. A strongly connected component (SCC) of $\mathcal{G}(W)$ is a maximal strongly connected subgraph. A SCC is called sink SCC if there exists no directed edge from this SCC to others. For an irreducible matrix $W$, $\mathcal{G}(W)$ is called democratic if $W$ is doubly-stochastic; $\mathcal{G}(W)$ is called autocratic if $\mathcal{G}(W)$ is a star topology.

\section{Basic modeling and results}\label{S2}
\subsection{Wisdom of crowds without social influence}

Consider $n\geq 2$ individuals in an influence network interacting their
estimates for an unknown state with constant true value
$\mu\in\mathbb{R}$. Suppose that no individual knows the exact value of
$\mu$, but some of them may have fragmented knowledge or
clues. Specifically, we assume that the knowledge possessed by individual
$i$ is a random variable, denoted by $\xi_{i}$, with expectation
$\mathbb{E}[\xi_{i}]=\mu$ and variance $\textup{Var}[\xi_{i}]=\sigma_{i}^{2}\in
(0,\infty)$. Suppose $\xi_{1},\dots,\xi_{n}$ are independent. At
the beginning, each individual gives an estimate which is a realization of
its knowledge, that is, $y_{i}(0)\sim (\mu,\sigma_{i}^{2})$. In this paper,
we employ the assumption in~\cite{FG:1907} that the collective estimate,
denoted by $y_{\textup{col}}$, is aggregated by averaging all individuals'
estimates arithmetically. Let
\begin{equation*}
y_{\textup{col}}(0)=\textup{ave}(y(0))=\frac{1}{n}\sum_{i=1}^{n}y_{i}(0)
\end{equation*}
denote the initial collective estimate. Because of the independence of $\xi_{1},\dots,\xi_{n}$, we have $\mathbb{E}[y_{\textup{col}}(0)]=\mu$ and $\textup{Var}[y_{\textup{col}}(0)]=\frac{1}{n^2}\sum_{i=1}^{n}\sigma_{i}^{2}$. Since $\mathbb{E}[y_{i}(0)]=\mathbb{E}[y_{\textup{col}}(0)]=\mu$, a smaller variance indicates that the estimate has a smaller mean square error, thus tends to be closer to the truth. Denote $\sigma^{2}=(\sigma_{1}^{2},\dots,\sigma_{n}^{2})^{\top}$, $\sigma_{\max}^{2}=\max_{i}\sigma_{i}^{2}$ and $\sigma_{\min}^{2}=\min_{i}\sigma_{i}^{2}$. Note that $\textup{Var}[y_{\textup{col}}(0)]<\textup{ave}(\sigma^2)$, and $\textup{Var}[y_{\textup{col}}(0)]<\sigma_{\min}^{2}$ if $\sigma_{\max}^{2}<n\sigma_{\min}^{2}$. That is to say, the variance of the initial collective estimate is smaller than the average variance of individuals' initial estimates and can be smaller than the variance of any single individual's initial estimate. This condensation of variance exhibits that simply averaging a group of individuals' independent estimates yields a collective estimate which is often better than estimates of some individuals, and even outperforms that of the best individual, known as the wisdom of crowds effect.

\subsection{Improving, optimizing and undermining of the wisdom in influence networks}
Suppose that after execution of a social influence process, individuals' estimates converge and are given by
\begin{equation}\label{eq1}
\lim_{k\to\infty}y(k)=Vy(0),
\end{equation}
where $V$ is the row-stochastic transition matrix of a weighted-average opinion dynamics. In other words, each individual's final estimate $\lim_{k\to\infty}y_{i}(k)=\sum_{j=1}^{n}V_{ij}y_{j}(0)$ is a weighted average of all the individuals' initial estimates. Then, the final collective estimate is
\begin{equation*}
\lim_{k\to\infty}y_{\textup{col}}(k)=\frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{n}V_{ij}y_{j}(0)=\sum_{i=1}^{n}x_{i}y_{i}(0),
\end{equation*}
where $x_{i}=\frac{1}{n}\sum_{j=1}^{n}V_{ji}$ indicates individual $i$'s social power exerted over the influence process and $x\in\Delta_{n}$ is the social power allocation of the influence system, as defined in~\cite{NEF:11}. Similarly, we have $\mathbb{E}[\lim_{k\to\infty}y_{\textup{col}}(k)]=\mu$ and $\textup{Var}[\lim_{k\to\infty}y_{\textup{col}}(k)]=\sum_{i=1}^{n}x_{i}^{2}\sigma_{i}^{2}$. Since $\mathbb{E}[\lim_{k\to\infty}y_{\textup{col}}(k)]=\mathbb{E}[y_{\textup{col}}(0)]$, $\textup{Var}[\lim_{k\to\infty}y_{\textup{col}}(k)]<\textup{Var}[y_{\textup{col}}(0)]$ means the influence process condenses the collective variance thus improves the collective estimate, while $\textup{Var}[\lim_{k\to\infty}y_{\textup{col}}(k)]>\textup{Var}[y_{\textup{col}}(0)]$ implies the influence process amplifies the collective variance thereby undermines the collective estimate.
\begin{definition}[Improving, optimizing, undermining wisdom and converging to the truth]\label{D1}
For social influence process~\eqref{eq1} with individuals' initial estimates $y_{i}(0)\sim (\mu,\sigma_{i}^{2})$, $i\in\until{n}$ and social power allocation $x\in\Delta_{n}$, we say social influence or system~\eqref{eq1} asymptotically
\begin{enumerate}

\item improves the wisdom if $\sum_{i=1}^{n}x_{i}^{2}\sigma_{i}^{2}<\frac{1}{n^2}\sum_{i=1}^{n}\sigma_{i}^{2}$; \label{D1-1}

\item optimizes the wisdom if $\sum_{i=1}^{n}x_{i}^{2}\sigma_{i}^{2}=\min_{z\in\Delta_{n}}\sum_{i=1}^{n}z_{i}^{2}\sigma_{i}^{2}$; \label{D1-2}

\item undermines the wisdom if $\sum_{i=1}^{n}x_{i}^{2}\sigma_{i}^{2}>\frac{1}{n^2}\sum_{i=1}^{n}\sigma_{i}^{2}$.

\setlength\parindent{-2em} Additionally, we say

\item the final collective estimate asymptotically converges to the truth in probability ({\it i.p.}) as group size increases if $\lim_{n\to\infty}\mathbb{P}[\lim_{k\to\infty}y^{(n)}_{\textup{col}}(k)=\mu]=1$. \label{D1-3}

\end{enumerate}
\end{definition}

\begin{remark}
Social influence process \cref{eq1} can model all opinion dynamics where individuals' opinions converge to convex combinations of their initial opinions, therefore, \cref{D1} is well-posed for influence processes such as the FD model, the Friedkin-Johnsen model and the Hegselmann-Krause model, etc.
\end{remark}
\subsection{Connections between improving/optimizing wisdom and converging to the truth}

\begin{lemma}[Improving wisdom implies converging to the truth]\label{L1}
For social influence process~\eqref{eq1}, if there exists $\beta>0$ such that $\sigma^{2}_{i}\leq\beta<\infty$ for all $i\in\until{n}$, and social influence asymptotically improves the wisdom for all $n\geq 2$, then the final collective estimate asymptotically converges to the truth {\it i.p.} as $n\to \infty$.
\end{lemma}

\Cref{L1} is proved in \cref{AP1}. In~\cref{L1}, if individuals' estimates achieve consensus for all $n\geq 2$, then all individuals' final estimates converge to the truth {\it i.p.} as $n\to\infty$. In~\cite{BG-MOJ:10}, it has been proved that for the FD model, if the influence matrix sequence satisfies balance and minimal out-dispersion conditions, individuals' final estimates converge to the truth {\it i.p.} as $n\to\infty$. \Cref{L1} provides a different condition from the perspective of condensing the collective variance of finite-size group, thereby all the results on improving the wisdom of finite-size group with finite variances imply the collective estimate converges to the truth as group size increases. To avoid trivialities, we shall not state the results about converging to the truth in the sequel.

\section{The role of social power in improving and optimizing the wisdom}\label{S3}
\subsection{Consistency notions}
According to \cref{D1}, whether an influence system improves or undermines wisdom depends exclusively on the relationship between its social power allocation and individuals' variances. Given individuals' variances $\sigma^{2}\in\mathbb{R}^{n}$, define $\map{\mathcal{E}_{\sigma^{2}}}{\Delta_{n}}{\mathbb{R}_{>0}}$ by $\mathcal{E}_{\sigma^{2}}(z)=\sum_{i=1}^{n}z_{i}^{2}\sigma_{i}^{2}$, then $\mathcal{E}_{\sigma^{2}}(x)$ represents the final collective variance of system \eqref{eq1} with social power allocation $x$. Let $\mathcal{A}_{\sigma^{2}}=\setdef{z\in \Delta_{n}}{\mathcal{E}_{\sigma^{2}}(z)<\mathcal{E}_{\sigma^{2}}(\mathbf{1}_{n}/n)}$ be the improvement region, i.e., the wisdom is asymptotically improved if and only if $x\in\mathcal{A}_{\sigma^{2}}$. In fact, $\mathcal{A}_{\sigma^{2}}$ is the interior of the intersection of the $n$-simplex $\Delta_{n}$ and the hyperellipsoid
\begin{equation*}
\sum_{i=1}^{n}\frac{z_{i}^2}{\frac{\sum_{j=1}^{n}\sigma_{j}^{2}}{n^{2}\sigma_{i}^{2}}}=1,
\end{equation*}
see \cref{fig1}. Intuitively, improving or optimizing the wisdom needs the influence system to allocate larger social power to more accurate individuals, that is, the ordering of influence system's social power allocation is consistent with the ordering of individuals' accuracy, where individual $i$'s accuracy is represented by $1/\sigma^{2}_{i}$. More specifically, we define the following notions of consistency progressively.

\begin{figure}[t]
\centering
 \subfloat[t][$\sigma^{2}=(1, 2, 3)^{\top}$]{
\begin{minipage}[t]{0.3\textwidth}\label{f1-a}
  \includegraphics[width=\hsize]{cc}\\
 \end{minipage}
}
\hspace{3ex}
\subfloat[t][$\sigma^{2}=(1, 3, 4)^{\top}$]{
\begin{minipage}[t]{0.3\textwidth}\label{f1-b}
  \includegraphics[width=\hsize]{bb}\\
  \end{minipage}
}
\hspace{0ex}

\subfloat[t][$\sigma^{2}=(1, 4, 9)^{\top}$]{
\begin{minipage}[t]{0.3\textwidth}\label{f1-c}
 \centering
  \includegraphics[width=\hsize]{aa}\\
 \end{minipage}
}
\hspace{3ex}
\subfloat[t][$\sigma^{2}=(2, 1, 16)^{\top}$]{
\begin{minipage}[t]{0.33\textwidth}\label{f1-d}
 \centering
  \includegraphics[width=\hsize]{dd}\\
 \end{minipage}
}
\caption{The improvement regions $\mathcal{A}_{\sigma^{2}}$ for different $\sigma^{2}$. }\label{fig1}
\end{figure}

\begin{definition}[Consistency notions]\label{D2}
Let $x_{i}$ and $\frac{1}{\sigma^{2}_{i}}$ be individual $i$'s social power and accuracy, respectively. We say system~\eqref{eq1} is
\begin{enumerate}
\item ordering-consistent if $x_{i}\leq x_{j}$ for all $\sigma^{2}_{i}\geq\sigma^{2}_{j}$, and $x=\frac{\mathbf{1}_{n}}{n}$ only if $\sigma^{2}\in\Span{\mathbf{1}_{n}}$;
\item gap-consistent if it is ordering-consistent and $x_{i}\sigma^{2}_{i}\geq x_{j}\sigma^{2}_{j}$ for all $\sigma^{2}_{i}\geq\sigma^{2}_{j}$; \label{D2-3}
\item maximally consistent if $x\in\operatorname{int}{\Delta_{n}}$ and $\frac{x_{i}}{x_{j}}=\frac{\sigma^{2}_{j}}{\sigma^{2}_{i}}$ for all $i,j\in\until{n}$.
\end{enumerate}
\end{definition}
\begin{remark}\label{r3}
In \cref{D2}~\ref{D2-3}, if there exist $i,j$ such that $x_{j}>x_{i}=0$, by the definition of gap-consistency we have $x_{j}\sigma_{j}^{2}\leq x_{i}\sigma_{i}^{2}=0$, which is contradicted with $x_{j}\sigma_{j}^{2}>0$. Hence $x\in\operatorname{int}{\Delta_{n}}$. That is, both gap-consistency and maximally consistency require the influence system allocates positive social power to all individuals.
\end{remark}

In \cref{D2}, ordering-consistency requires the influence system allocates more social power to more accurate individuals, gap-consistency additionally excludes the case that it assigns too much social power to more accurate individuals, while maximally consistency demands its social power allocation is exactly proportional to individuals' accuracy.
\subsection{The effects of gap-consistency and maximal consistency}\label{s3.2}
As shown in~\cref{fig1}, the improvement region $\mathcal{A}_{\sigma^{2}}$ is the interior of the intersection of the $n$-simplex and a hyperellipsoid, for which it is difficult to provide an explicit formula. The next lemma captures the properties of $\mathcal{A}_{\sigma^{2}}$.
\begin{lemma}[Properties of the improvement region]\label{L3}
For $\mathcal{E}_{\sigma^{2}}(z)=\sum_{i=1}^{n}z_{i}^{2}\sigma_{i}^{2}$ and $\mathcal{A}_{\sigma^{2}}=\setdef{z\in \Delta_{n}}{\mathcal{E}(z)<\mathcal{E}(\frac{\mathbf{1}_{n}}{n})}$ defined above, let $\mathcal{\tilde{A}}_{\sigma^{2}}=\setdef{z\in\operatorname{int}{ \Delta_{n}}\setminus \{\frac{\mathbf{1}_{n}}{n}\}}{1\leq\frac{z_{i}}{z_{i+1}}\leq \frac{\sigma_{i+1}^{2}}{\sigma_{i}^{2}}, i\in \until{n-1}}$. The following statements hold:
\begin{enumerate}
\item $\mathcal{E}_{\sigma^{2}}(z)$ is strictly convex on $\Delta_{n}$ and $\mathcal{A}_{\sigma^{2}}$ is convex; \label{L3-0}
\item $\mathcal{A}_{\sigma^{2}}\neq\emptyset$ if and only if $\sigma^{2}\notin\Span{\mathbf{1}_{n}}$; \label{L3-1}
\item $\mathcal{\tilde{A}}_{\sigma^{2}}\subset\mathcal{A}_{\sigma^{2}}$. \label{L3-2}
\end{enumerate}
\end{lemma}

\Cref{L3} is proved in~\cref{AP2}. \Cref{L3}~\ref{L3-0} shows that $\mathcal{E}_{\sigma^{2}}(z)$ is strictly convex on $\Delta_{n}$, which implies that if there exist $l\geq 2$ and $x^{i}\in\Delta_{n}$ for $i\in\until{l}$ such that system~\eqref{eq1} with social power allocation $x^{i}$ improves the wisdom, then for all $\omega\in\Delta_{l}$, system~\eqref{eq1} with social power allocation $\sum_{i=1}^{l}\omega_{i}x^{i}$ improves the wisdom.
\begin{theorem}[Consistent influence system improves/optimizes wisdom]\label{T2}
For social influence process \eqref{eq1} with social power allocation $x\in\Delta_{n}$ and individuals' variances $\sigma^{2}\in\mathbb{R}^{n}$, the following statements hold:
\begin{enumerate}
\item if social influence asymptotically improves the wisdom, then $\sigma^{2}\notin\Span{\mathbf{1}_{n}}$; \label{T2-1}

\item if $\sigma^{2}\notin\Span{\mathbf{1}_{n}}$ and system~\eqref{eq1} is gap-consistent, then social influence asymptotically improves the wisdom;\label{T2-2}

\item social influence asymptotically optimizes the wisdom if and only if system \eqref{eq1} is maximally consistent. And if so, the variance of final collective estimate achieves minimum $\mathcal{E}^{*}_{\sigma^{2}}=\frac{1}{\sum_{i=1}^{n}\frac{1}{\sigma_{i}^{2}}}$ with optimal social power allocation
\begin{equation*}
x^{*}=\frac{[\sigma^{2}]^{-1}\mathbf{1}_{n}}{\mathbf{1}^{\top}_{n}[\sigma^{2}]^{-1}\mathbf{1}_{n}}.
\end{equation*} \label{T2-3}
\end{enumerate}
\end{theorem}

\begin{proof}
Statement~\ref{T2-1} immediately follows from \cref{L3}~\ref{L3-1}. Regarding~\ref{T2-2}, since $\sigma^{2}\notin\Span{\mathbf{1}_{n}}$ and system~\eqref{eq1} is gap-consistent, by \cref{r3}, $x\in\operatorname{int}{\Delta_{n}}\setminus \{\frac{\mathbf{1}_{n}}{n}\}$. Let $P\in\mathbb{R}^{n}$ be the permutation matrix such that $\tilde{\sigma}^{2}=P\sigma^{2}$ satisfies $\tilde{\sigma}_{i}^{2}\leq \tilde{\sigma}_{i+1}^{2}$, $i\in\until{n-1}$, by the definition of gap-consistency, $\tilde{x}=Px$ satisfies
\begin{equation*}
1\leq\frac{\tilde{x}_{i}}{\tilde{x}_{i+1}}\leq\frac{\tilde{\sigma}_{i+1}^{2}}{\tilde{\sigma}_{i}^{2}}
\end{equation*}
for all $i\in\until{n-1}$, which means that $\tilde{x}\in\mathcal{\tilde{A}}_{\tilde{\sigma}^{2}}$. By \cref{L3}~\ref{L3-2}, $\mathcal{\tilde{A}}_{\sigma^{2}}\subset\mathcal{A}_{\sigma^{2}}$ for any given $\sigma^{2}$. Hence, we obtain $\tilde{x}\in\mathcal{A}_{\tilde{\sigma}^{2}}$, which is equivalent to $x\in\mathcal{A}_{\sigma^{2}}$. Therefore, social influence asymptotically improves the wisdom.

Regarding~\ref{T2-3}, define $\map{L}{\Delta_{n}\times \mathbb{R}}{\mathbb{R}}$ by $L(z,\lambda)=\sum_{i=1}^{n}z_{i}^{2}\sigma_{i}^{2}+\lambda (\mathbf{1}^{\top}_{n}z-1)$. Then, the optimal solution $z^{*}$ satisfies $2z_{i}^{*}\sigma_{i}^{2}+\lambda=0$ for all $i\in\until{n}$ and $\sum_{j=1}^{n}z_{j}^{*}=1$.
That is, the minimum collective estimate is achieved by
\begin{equation*}
x_{i}^{*}=\frac{\frac{1}{\sigma_{i}^{2}}}{\sum_{j=1}^{n}\frac{1}{\sigma_{j}^{2}}},
\end{equation*}
which is equivalent to $\frac{x_{i}^{*}}{x_{j}^{*}}=\frac{\sigma_{j}^{2}}{\sigma_{i}^{2}}$ for all $i,j\in\until{n}$. Moreover, $\mathcal{E}^{*}_{\sigma^{2}}=\mathcal{E}_{\sigma^2}(x^{*})=\frac{1}{\sum_{i=1}^{n}\frac{1}{\sigma_{i}^{2}}}$. Thus, social influence asymptotically optimizes the wisdom if and only if system~\eqref{eq1} is maximally consistent.
\end{proof}

\Cref{T2} provides necessary and/or sufficient conditions for improving and optimizing the wisdom based on the consistency notions defined in \cref{D2}. It suggests that the wisdom can be improved only if individuals' variances are non-uniform; and if so, social influence improves the wisdom if system~\eqref{eq1} allocates relatively larger, but not too much, social power to relatively more accurate individuals; social influence optimizes the wisdom if and only if social power allocation of system~\eqref{eq1} is exactly proportional to individuals' accuracy. In \cref{fig2}, quadrilaterals $DFOH$ (excluding the points $D$) and points $O$ depict the gap-consistency regions and the maximal consistency points of social power allocation in the $3$-simplex for different distributions of variances.
\begin{figure}
\centering
 \subfloat[t][$\sigma^{2}=(1, 2, 3)^{\top}$]{
\begin{minipage}[t]{0.3\textwidth}\label{f2-a}
  \includegraphics[width=\hsize]{ee}\\
 \end{minipage}
}
\hspace{3ex}
\subfloat[t][$\sigma^{2}=(3, 4, 1)^{\top}$]{
\begin{minipage}[t]{0.3\textwidth}\label{f2-b}
  \includegraphics[width=\hsize]{ff}\\
  \end{minipage}
}
\hspace{0ex}

\subfloat[t][$\sigma^{2}=(1, 4, 9)^{\top}$]{
\begin{minipage}[t]{0.3\textwidth}\label{f2-c}
 \centering
  \includegraphics[width=\hsize]{gg}\\
 \end{minipage}
}
\hspace{3ex}
\subfloat[t][$\sigma^{2}=(2, 1, 16)^{\top}$]{
\begin{minipage}[t]{0.31\textwidth}\label{f2-d}
 \centering
  \includegraphics[width=\hsize]{hh}\\
 \end{minipage}
}
\caption{The improvement region, the undermining region and the consistency regions depicted in the $3$-simplex with the point $D=\mathbf{1}_{n}/3$. For each $\sigma^{2}$, blue area is the improvement region, point $O$ is the maximal consistency point, quadrilateral $DFOH$ (excluding point $D$) is the gap-consistency region and the undermining region includes gray and orange areas. The triangles $DCE$ and $DBE$ (excluding the points $D$) in \cref{f2-c,f2-d} are the ordering-consistency regions.}\label{fig2}
\end{figure}

\subsection{The effects of select crowd and ordering-consistency}
For any given individuals' variances $\sigma^{2}\notin\Span{\mathbf{1}_{n}}$, there always exist a gap-consistency region and a maximal consistency point of social power allocation such that the wisdom is improved or optimized. Yet we notice that on one hand, both gap-consistency and maximal consistency require the social power of each individual to be positive, as discussed in \cref{r3}; on the other hand, whether the influence system is gap-consistent or maximally consistent depends quantitatively upon individuals' social power and variances, while ordering-consistency, as a criterion, seems more explicit and more intuitive. This subsection focuses on the effects of select crowd and ordering-consistency in improving the wisdom, where select crowd means that not every individual is assigned positive social power.

Note that there exist $\frac{n(n-1)}{2}$ hyperplanes $\{z\in\mathbb{R}^{n}\mid z_{i}=z_{j}\}$, by which the $n$-simplex $\Delta_{n}$ can be partitioned into $n!$ hypertriangles, thereby the orderings of components of $z\in\Delta_{n}$ in different hypertriangles are different, as depicted in \cref{f2-d}. Let $\Delta^{\tau}_{n}=\setdef{z\in\Delta_{n}\setminus\{ \frac{\mathbf{1}_{n}}{n}\}}{z_{\tau_{i}}\geq z_{\tau_{i+1}}, i\in\until{n-1}}$ be the hypertriangle corresponding to $\tau$, where $\tau=(\tau_{1}, \dots,\tau_{n})$ is a permutation of $\tau^{0}=(1, \dots,n)$. Let $\mathcal{T}$ be the set of all permutations of $\tau^{0}$, then clearly $\bigcup_{\tau\in\mathcal{T}}\Delta^{\tau}_{n}=\Delta_{n}\setminus\{ \frac{\mathbf{1}_{n}}{n}\}$. For $z\in\mathbb{R}^{n}$, we denote $z_{\tau}=(z_{\tau_{1}},\dots,z_{\tau_{n}})^{\top}$.

\begin{lemma}[The properties of the hypertriangle]\label{L4}
For the hypertriangle $\Delta^{\tau^{0}}_{n}=\setdef{z\in \Delta_{n}\setminus \{\frac{\mathbf{1}_{n}}{n}\}}{z_{i}\geq z_{i+1}, i\in \until{n-1}}$ associated with the identity permutation $\tau^{0}=(1, \dots,n)$, let $\mathcal{\hat{A}}_{m}=\setdef{z\in \Delta^{\tau^{0}}_{n}}{z_{m}>z_{m+1}=0}$ with $m<n$, then
\begin{enumerate}
\item $\Delta^{\tau^{0}}_{n}\subset \mathcal{A}_{\sigma^{2}}$ if and only if for all $j\in\until{n-1}$, there holds\label{L4-1}
\begin{equation}\label{eq2}
\frac{1}{j^2}\sum_{r=1}^{j}\sigma_{r}^{2}<\frac{1}{n^2}\sum_{i=1}^{n}\sigma_{i}^{2};
\end{equation}

\item $\mathcal{\hat{A}}_{m}\subset \mathcal{A}_{\sigma^{2}}$ if and only if~\eqref{eq2} holds for all $j\in\until{m}$. \label{L4-2}
\end{enumerate}
\end{lemma}
\Cref{L4} is proved in~\cref{AP3}.

\begin{theorem}[Allocating social power to select individuals improves wisdom]\label{T3}
For social influence process~\eqref{eq1} with social power allocation $x\in\Delta_{n}$ and individuals' variances $\sigma^{2}\in\mathbb{R}^{n}$, if there exist $m<n$ and a permutation $\tau$ of $(1,\dots,n)$, such that $x_{\tau}\in\mathcal{\hat{A}}_{m}=\setdef{z\in \Delta^{\tau^{0}}_{n}}{z_{m}>z_{m+1}=0}$ and
\begin{equation}\label{e3}
\sigma_{\tau_{j}}^{2}<\frac{2j-1}{n^2}\sum_{i=1}^{n}\sigma_{i}^{2} \quad \text{for all} \quad j\in\until{m},
\end{equation}
then social influence asymptotically improves the wisdom.
\end{theorem}

\begin{proof}
Let $P$ be the permutation matrix associated with $\tau$, then $x_{\tau}=Px$, $\sigma_{\tau}^{2}=P\sigma^{2}$, and $x\in\mathcal{A}_{\sigma^{2}}$ if and only if $x_{\tau}\in\mathcal{A}_{\sigma_{\tau}^{2}}$.
Next, we prove that
\begin{equation}\label{e4}
\frac{1}{j^2}\sum_{r=1}^{j}\sigma_{\tau_{r}}^{2}<\frac{1}{n^2}\sum_{i=1}^{n}\sigma_{i}^{2} \quad \text{for all} \quad j\in\until{m}
\end{equation}
if~\eqref{e3} holds. Using induction, for $j=1$, by~\eqref{e3} we have $\sigma_{\tau_{1}}^{2}<\frac{1}{n^2}\sum_{i=1}^{n}\sigma_{i}^{2}$. Suppose that for $h-1\in\until{m-1}$,
\begin{equation*}
\frac{1}{(h-1)^2}\sum_{r=1}^{h-1}\sigma_{\tau_{r}}^{2}<\frac{1}{n^2}\sum_{i=1}^{n}\sigma_{i}^{2},
\end{equation*}
then
\begin{equation*}
\begin{split}
\frac{1}{h^2}\sum_{r=1}^{h}\sigma_{\tau_{r}}^{2}=&\frac{1}{h^2}\sigma_{\tau_{h}}^{2}+\frac{1}{h^2}\sum_{r=1}^{h-1}\sigma_{\tau_{r}}^{2}
<\frac{1}{h^2}\sigma_{\tau_{h}}^{2}+\frac{(h-1)^2}{h^{2}n^{2}}\sum_{i=1}^{n}\sigma_{i}^{2}
<\frac{1}{n^2}\sum_{i=1}^{n}\sigma_{i}^{2},
\end{split}
\end{equation*}
where the last inequality is implied by $\sigma_{\tau_{h}}^{2}<\frac{2h-1}{n^2}\sum_{i=1}^{n}\sigma_{i}^{2}$. Therefore, \eqref{e4} holds. By \cref{L4}~\ref{L4-2}, we have $x_{\tau}\in\mathcal{A}_{\sigma_{\tau}^{2}}$, i.e., $x\in\mathcal{A}_{\sigma^{2}}$, which means social influence improves the wisdom asymptotically.
\end{proof}

Note that
$\frac{2j-1}{n^2}\sum_{i=1}^{n}\sigma_{i}^{2}=\frac{2j-1}{n}\textup{ave}(\sigma^{2})$. \Cref{T3}
shows that selecting a subgroup of individuals whose variances are upper
bounded, respectively, by specific fractions of individuals' average
variance, and allocating social power to the select individuals according
to the ordering of the upper bounds improves the wisdom. This is similar
with the select-crowd strategy proposed in~\cite{AEM-JBS-RPL:14}. In
practice, social influence systems in which not everyone has positive
social power are ubiquitous, such as administrative systems or
bureaucracies. \Cref{T3} suggests that in these systems, if more powerful
people have smaller upper bounds of variance, in other words, are more
expert, then the wisdom is improved, regardless of the expertise of people
who have no power. Moreover, the condition in \cref{T3} does not necessarily imply ordering-consistency as required
by \cref{T2}. For example, consider social power allocation $x=(x_{1}, 0, 0, x_{4})^{\top}$ where $x_{1}\geq
x_{4}>0$ and $x_{1}+ x_{4}=1$, then $\sigma^{2}=(3,24,20,2)^{\top}$
satisfies condition~\eqref{e3} with $m=2$ and $\tau=(1, 4, 2, 3)$.

Next, we focus on whether and when we can determine that social influence improves the wisdom by only checking the ordering of social power allocation. Without loss of generality, individuals are labelled in ascending order according to their variances. In addition, for permutation $\tau\neq\tau^{0}$, we say $(\tau_{p},\tau_{q})$ is an inversion if $\tau_{p}>\tau_{q}$ and $p<q$. For permutations $\tau$ and $\tau^{\prime}$, we denote $\tau^{\prime}\prec\tau$ if $\tau^{\prime}$ can be obtained from $\tau$ by swapping inversions one by one, and $\tau^{\prime}\preceq\tau$ if $\tau^{\prime}\prec\tau$ or $\tau^{\prime}=\tau$.
\begin{algorithm}
 \caption{\textbf{Modified permutation generation (MPG) algorithm}}
 \label{Alg1}
\begin{algorithmic}[1]
\item[\algorithmicrequire] $\sigma^{2}$ with $\sigma_{i}^{2}\leq\sigma_{i+1}^{2}$ for all $i\in\until{n-1}$
\item[\algorithmicensure] $\mathcal{T}_{\sigma^{2}}$ such that $\Delta^{\tau}_{n}\subset \mathcal{A}_{\sigma^{2}}$ for all $\tau\in\mathcal{T}_{\sigma^{2}}$ and $\Delta^{\tau}_{n}\not\subset \mathcal{A}_{\sigma^{2}}$ for all $\tau\in\mathcal{T}\setminus\mathcal{T}_{\sigma^{2}}$ where $\mathcal{T}$ is the set of all permutations of $(1,\dots,n)$
\STATE initialize $\mathcal{T}_{\sigma^{2}}:=\emptyset$; $s_{0}:=0$, $u_{0}:=\sigma^{2}_{1}$; $s_{i}:=\sum_{r=1}^{i}\sigma_{r}^{2}$ and $u_{i}:=\frac{i^2}{n^2}s_{n}-s_{i-1}$ for all $i\in\until{n}$
\STATE find $f_{i}=\max\setdef{j\in\{0,\dots,i-1\}}{\sigma^{2}_{i}\geq u_{j}}$ for all $i\in\until{n}$
\STATE
$j:=0$
\FOR{$i$ from $1$ to $n$}
 \IF{$f_{i}=i-1$}
\STATE
$j:=j+1$\\
$h_{j}:=i$
\ENDIF
\ENDFOR
\STATE
$d:=j$, $h_{d+1}:=n+1$
\FOR{$i$ from $1$ to $d$}
\STATE generate all permutations of segment $(h_{i},\dots,h_{i+1}-1)$ using permutation generation algorithms (e.g., Heap's algorithm provided in~\cite{RS:77})
\ENDFOR
\STATE concatenate permutations of all segments, obtain $N=\prod_{i=1}^{d}(h_{i+1}-h_{i})!$ different permutations of $(1,\dots,n)$, denoted by $\tau^{1},\dots,\tau^{N}$
\FOR{$i$ from $1$ to $N$}
\IF{$\sigma^{2}_{\tau^{i}}$ satisfies~\cref{e3.1}}
\STATE$\mathcal{T}_{\sigma^{2}}:=\mathcal{T}_{\sigma^{2}}\cup\{\tau^{i}\}$
\ENDIF
\ENDFOR
\RETURN $\mathcal{T}_{\sigma^{2}}$
\end{algorithmic}
\end{algorithm}
\begin{theorem}[Sufficiency of ordering-consistency in improving the wisdom]\label{T4}
For social influence process~\eqref{eq1} with social power allocation $x\in\Delta_{n}$ and individuals' variances $\sigma^{2}\in\mathbb{R}^{n}$, suppose that $\sigma_{i}^{2}\leq\sigma_{i+1}^{2}$ for all $i\in\until{n-1}$, $\mathcal{T}$ is the set of all permutations of $\tau^{0}=(1,\dots,n)$ and $\Delta^{\tau}_{n}=\setdef{z\in\Delta_{n}\setminus\{ \frac{\mathbf{1}_{n}}{n}\}}{z_{\tau_{i}}\geq z_{\tau_{i+1}}, i\in\until{n-1}}$. Then,

\begin{enumerate}
\item ordering-consistency is sufficient for system~\eqref{eq1} asymptotically improving the wisdom if and only if \label{T4-1}
\begin{equation}\label{e3.2}
\frac{1}{i^2}\sum_{r=1}^{i}\sigma_{r}^{2}<\frac{1}{n^2}\sum_{j=1}^{n}\sigma_{j}^{2} \quad \text{for all} \quad i\in\until{n-1};
\end{equation}
\item if there exists $\tau\neq\tau^{0}$ such that
\begin{equation}\label{e3.1}
\frac{1}{i^2}\sum_{r=1}^{i}\sigma_{\tau_{r}}^{2}<\frac{1}{n^2}\sum_{j=1}^{n}\sigma_{j}^{2} \quad \text{for all} \quad i\in\until{n-1},
\end{equation}
system~\eqref{eq1} with all social power allocation $x\in\bigcup_{\tau^{\prime}\in\setdef{\bar{\tau}\in\mathcal{T}}{\bar{\tau}\preceq\tau}}\Delta^{\tau^{\prime}}_{n}$ asymptotically improves the wisdom.\label{T4-2}

\item For~\cref{Alg1} and its output $\mathcal{T}_{\sigma^2}$,\label{T4-3}
\begin{enumerate}
\item system~\eqref{eq1} with all social power allocation $x\in\bigcup_{\tau\in\mathcal{T}_{\sigma^{2}}}\Delta^{\tau}_{n}$ asymptotically improves the wisdom;\label{T4-3.1}
\item for all $\tau\in\mathcal{T}\setminus\mathcal{T}_{\sigma^{2}}$, there exists $x\in\Delta^{\tau}_{n}$ such that system~\eqref{eq1} with social power allocation $x$ does not asymptotically improve the wisdom;\label{T4-3.2}
\item the running time of \cref{Alg1} is between $\mathcal{O}(n^2)$ and the running time of the enumeration method, which is at least $\mathcal{O}(n\times n!)$.\label{T4-3.3}
\end{enumerate}
\end{enumerate}
\end{theorem}

\begin{proof}
Regarding~\ref{T4-1}, ordering-consistency is sufficient for improving the wisdom means that system~\eqref{eq1} asymptotically improves the wisdom as long as it is ordering-consistent. Since $\sigma_{i}^{2}\leq\sigma_{i+1}^{2}$ for all $i\in\until{n-1}$, system~\eqref{eq1} is ordering-consistent if and only if $x\in\Delta^{\tau^{0}}_{n}$. By \cref{L4}~\ref{L4-1}, $\Delta^{\tau^{0}}_{n}\subset \mathcal{A}_{\sigma^{2}}$ if and only if~\eqref{e3.2} holds, which implies that ordering-consistency is sufficient for system~\eqref{eq1} asymptotically improving the wisdom if and only if~\eqref{e3.2} holds.

Regarding~\ref{T4-2}, let $P$ be the permutation matrix corresponding to $\tau$, then $Px\in\Delta_{n}^{\tau^{0}}$ and $P\sigma^{2}=\sigma^{2}_{\tau}$. Since~\eqref{e3.1} holds, \cref{L4}~\ref{L4-1} suggests that $\Delta_{n}^{\tau^{0}}\subset\mathcal{A}_{\sigma^{2}_{\tau}}$, which is equivalent to $\Delta_{n}^{\tau}\subset\mathcal{A}_{\sigma^{2}}$. Assume $\tau_{p}>\tau_{q}$ for $p<q$, let $\tilde{\tau}_{p}=\tau_{q}$, $\tilde{\tau}_{q}=\tau_{p}$, and $\tilde{\tau}_{l}=\tau_{l}$ for $l\in\until{n}\setminus\{p,q\}$, then $\tilde{\tau}\prec\tau$. Since $\tau_{l}=\tilde{\tau}_{l}$ for all $l\neq p,q$, we have $\sum_{r=1}^{i}\sigma_{\tilde{\tau}_{r}}^{2}=\sum_{r=1}^{i}\sigma_{\tau_{r}}^{2}$ for all $i\in\until{p-1}\cup\{q,\dots,n-1\}$. For $i\in\{p,\dots,q-1\}$, we have
\begin{equation*}
\sum_{r=1}^{i}\sigma_{\tilde{\tau}_{r}}^{2}=\sum_{r=1}^{p-1}\sigma_{\tau_{r}}^{2}+\sigma_{\tilde{\tau}_{p}}^{2}+\sum_{r=p+1}^{i}\sigma_{\tau_{r}}^{2}\leq\sum_{r=1}^{i}\sigma_{\tau_{r}}^{2}<\frac{i^2}{n^2}\sum_{j=1}^{n}\sigma_{j}^{2}
\end{equation*}
due to $\sigma_{\tilde{\tau}_{p}}^{2}=\sigma_{\tau_{q}}^{2}\leq\sigma_{\tau_{p}}^{2}$. That is, \eqref{e3.1} holds for $\tilde{\tau}$, which implies $\Delta_{n}^{\tilde{\tau}}\subset\mathcal{A}_{\sigma^{2}}$ and consequently $\Delta_{n}^{\tilde{\tau}}\subset\mathcal{A}_{\sigma^{2}}$ for all $\tilde{\tau}\preceq\tau$. In conclusion, system~\eqref{eq1} asymptotically improves the wisdom with any social power allocation $x\in\bigcup_{\tau^{\prime}\in\setdef{\bar{\tau}\in\mathcal{T}}{\bar{\tau}\preceq\tau}}\Delta^{\tau^{\prime}}_{n}$.

Regarding~\ref{T4-3.1}, by steps $11,12,13$ of \cref{Alg1}, $\sigma^{2}_{\tau}$ satisfies~\eqref{e3.1} for any $\tau\in\mathcal{T}_{\sigma^{2}}$. Statement~\ref{T4-2} suggests system~\eqref{eq1} asymptotically improves the wisdom with any social power allocation $x\in\Delta_{n}^{\tau}$ for all $\tau\in\mathcal{T}_{\sigma^{2}}$. Regarding~\ref{T4-3.2}, suppose $\tau\in\mathcal{T}\setminus\mathcal{T}_{\sigma^{2}}$. There are two situations. One is that $\tau$ is obtained from step $10$, i.e., $\tau\in\{\tau^{1},\dots,\tau^{N}\}$, but is not added to $\mathcal{T}_{\sigma^{2}}$ in the following steps; the other is that $\tau\notin\{\tau^{1},\dots,\tau^{N}\}$. The first situation means that $\sigma^{2}_{\tau}$ does not satisfy~\eqref{e3.1}. For the second situation, note that from steps $3$ to $7$ the algorithm divides $(1,\dots,n)$ into $d$ segments, that is, $(h_{i},\dots,h_{i+1}-1)$ for $i\in\until{d}$, and $\{\tau^{1},\dots,\tau^{N}\}$ contains all the permutations of $(1,\dots,n)$ obtained by concatenating permutations of $d$ segments. Therefore, $\tau\notin\{\tau^{1},\dots,\tau^{N}\}$ implies that there exists $j^{*}\in\until{n}$ and $i^{*}\in\until{d}$ such that $j^{*}\in\{h_{i^{*}},\dots,h_{i^{*}+1}-1\}$ and $\tau_{j^{*}}\notin\{h_{i^{*}},\dots,h_{i^{*}+1}-1\}$. Without loss of generality, let $n\geq\tau_{j^{*}}\geq h_{i^{*}+1}>j^{*}$. Since $f_{h_{i^{*}+1}}=h_{i^{*}+1}-1$, we have $\sigma^{2}_{h_{i^{*}+1}}\geq u_{h_{i^{*}+1}-1}$, i.e., $s_{h_{i^{*}+1}-2}+\sigma^{2}_{h_{i^{*}+1}}\geq \frac{(h_{i^{*}+1}-1)^2}{n^2}s_{n}$.
Hence,
\begin{equation*}
\sum_{r=1}^{h_{i^{*}+1}-1}\sigma^{2}_{\tau_{r}}=\sum_{\substack{r=1 \\ r\neq j^{*}}}^{h_{i^{*}+1}-1}\sigma^{2}_{\tau_{r}}+\sigma^{2}_{\tau_{j^{*}}}\geq s_{h_{i^{*}+1}-2}+\sigma^{2}_{h_{i^{*}+1}}\geq \frac{(h_{i^{*}+1}-1)^2}{n^2}s_{n},
\end{equation*}
where the second last inequality is implied by $\sigma_{i}^{2}\leq\sigma_{i+1}^{2}$ for all $i\in\until{n-1}$. Therefore, $\sigma^{2}_{\tau}$ does not satisfy~\eqref{e3.1} for $i=h_{i^{*}+1}-1$. In conclusion, $\sigma^{2}_{\tau}$ does not satisfy~\eqref{e3.1} for any $\tau\in\mathcal{T}\setminus\mathcal{T}_{\sigma^{2}}$. By \cref{L4}~\ref{L4-1} and the proof of statement~\ref{T4-2}, $\Delta_{n}^{\tau}\not\subset\mathcal{A}_{\sigma^{2}}$ for all $\tau\in\mathcal{T}\setminus\mathcal{T}_{\sigma^{2}}$, which completes the proof. Regarding~\ref{T4-3.3}, by~\cite{RS:77}, the running time of permutation generation algorithm employed in step $9$ is at least $\mathcal{O}(m!)$ for generating all permutations of $m$ numbers. The time complexity of step $2$ is $\mathcal{O}(n^2)$. Steps $3$ to $7$ need to execute elementary operations $3n$ times. Steps $8$ and $9$ need to call the permutation generation algorithm $d$ times with total running time at least $\sum_{i=1}^{d}(h_{i+1}-h_{i})!$. Step $10$ needs to concatenate $\sum_{i=1}^{d}(h_{i+1}-h_{i})!$ segments and obtains $N=\prod_{i=1}^{d}(h_{i+1}-h_{i})!$ permutations of $(1,\dots,n)$. Hence, the running time of step $10$ is $N$. Moreover, the complexity of steps $11$ to $13$ is $\mathcal{O}(nN)$. In the case that $d=n$, i.e., $f_{i}=i-1$ for all $i\in\until{n}$, we have $\sum_{i=1}^{d}(h_{i+1}-h_{i})!=n$ and $N=1$. Thus, the running time of \cref{Alg1} is $\mathcal{O}(n^2)$. In the case that $d=1$, i.e., $f_{i}=i-1$ only for $i=1$, we have $\sum_{i=1}^{d}(h_{i+1}-h_{i})!=N=n!$ and \cref{Alg1} generates all $n!$ permutations of $(\sigma^{2}_{1},\dots,\sigma^{2}_{n})$ and runs steps $12$ and $13$ for each permutation, which is exactly the enumeration method with running time at least $\mathcal{O}(n\times n!)$. In conclusion, the running time of \cref{Alg1} is between $\mathcal{O}(n^2)$ and the running time of the enumeration method.
\end{proof}

\begin{remark}
It is worth noting that the existence of any permutation of $\sigma^{2}$ satisfying~\eqref{e3.1} is equivalent to the existence of any solution of the following $0$-$1$ integer programming without optimization:
\begin{equation}\label{e3.3}
\begin{aligned}
 &z\in\mathbb{R}^{n^2}\\
\textup{s.t.} \ \ \ &Az<b\\
&A^{\prime}z=\mathbf{1}_{2n},\\
&z_{i}\in\{0,1\} \ \textup{for all} \ i\in\until{n^{2}},
\end{aligned}
\end{equation}
where $z=[P_{1}  \  \dots \ P_{n}]^{\top}$ with $P_{i}\in\mathbb{R}^{1\times n}$ being the $i$-th row of the permutation matrix corresponding to $\tau$, $A=Q\otimes(\sigma^{2})^{\top}\in\mathbb{R}^{n-1\times n^{2}}$ with $Q\in\mathbb{R}^{n-1\times n}$, $Q_{ij}=1$ for $i\geq j$ and $Q_{ij}=0$ otherwise, $b\in\mathbb{R}^{n-1}$ with $b_{i}=\frac{i^2}{n^2}\sum_{r=1}^{n}\sigma_{r}^{2}$, $A^{\prime}=[I_{n}\otimes \mathbf{1}_{n} \ \ \mathbf{1}_{n}\otimes I_{n}]^{\top}\in\mathbb{R}^{2n\times n^{2}}$. Karp~\cite{RMK:72} proved that problem~\eqref{e3.3} is NP-complete. Equivalently, \cref{Alg1} finds all the feasible solutions of problem~\eqref{e3.3}. There may be other algorithms which can solve problem~\eqref{e3.3}, for example, the intlinprog function of MATLAB. However, the intlinprog function only provides one feasible solution.
\end{remark}

\Cref{T4} provides several results based on the idea of partitioning the $n$-simplex into hypertriangles such that social power allocation in different hypertriangles is ordered differently. \cref{T4}~\ref{T4-1} shows that ordering-consistency is sufficient for social influence improving wisdom if and only if individuals' variances satisfy~\eqref{e3.2}; and if so, \cref{T4}~\ref{T4-2} suggests there may exist other orderings of social power allocation such that the wisdom is also improved with social power allocation in those orderings. To address this issue, we propose \cref{Alg1}, which, for given individuals' variances, outputs all the orderings of social power allocation that improves the wisdom. In \cref{f2-c,f2-d}, triangles $DCE$ and $DBE$ (excluding the points $D$) are, respectively, the ordering-consistent regions for the given variances  which both satisfy~\eqref{e3.2}, thus are both contained inside the improvement regions. Moreover, in \cref{f2-d}, triangle $DCE$ (excluding the point $D$) is also contained in the improvement region since there is more than one permutation of $\sigma^{2}=(2, 1, 16)^{\top}$ satisfying~\eqref{e3.1}.

\section{The role of social power in undermining the wisdom}\label{S4}
In this section, we investigate the question of when social influence asymptotically undermines the wisdom of crowds. For a permutation $\tau$ of $(1,\dots,n)$, denote by $\tau^{-1}=(\tau_{n},\dots,\tau_{1})$ its inverse. Given $\sigma^{2}\in\mathbb{R}^{n}$, let $\mathcal{\bar{A}}_{\sigma^{2}}=\setdef{z\in\Delta_{n}}{\mathcal{E}_{\sigma^{2}}(z)>\mathcal{E}_{\sigma^{2}}(\frac{\mathbf{1}_{n}}{n})}$ be the undermining region. The following theorem shows that if the ordering of the influence system's social power allocation is in reverse to the ordering of individuals' accuracy, social influence undermines the wisdom.
\begin{theorem}[Ordering-reverse influence system undermines the wisdom]\label{T5}
For social influence process \eqref{eq1} with social power allocation $x\in\Delta_{n}$ and individuals' variances $\sigma^{2}\in\mathbb{R}^{n}$, let $\tau$ be a permutation such that $\sigma_{\tau_{i}}^{2}\leq\sigma_{\tau_{i+1}}^{2}$ for all $i\in\until{n-1}$, $\Delta^{\tau}_{n}=\setdef{z\in\Delta_{n}\setminus\{ \frac{\mathbf{1}_{n}}{n}\}}{z_{\tau_{i}}\geq z_{\tau_{i+1}}, i\in\until{n-1}}$. System~\eqref{eq1} with all social power allocation $x\in\Delta_{n}^{\tau^{-1}}$ asymptotically undermines the wisdom.
\end{theorem}
\begin{proof}
Since $x\in\Delta_{n}^{\tau^{-1}}$, we obtain $x_{\tau_{i}}\leq x_{\tau_{i+1}}$ for all $i\in\until{n-1}$. Without loss of generality, assume that $x_{\tau_{m}}<\frac{1}{n}$ and $x_{\tau_{m+1}}\geq\frac{1}{n}$ with $m<n$. Then, we have
\begin{align*}
\mathcal{E}_{\sigma_{\tau}^{2}}(x_{\tau})-\mathcal{E}_{\sigma_{\tau}^{2}}(\frac{\mathbf{1}_{n}}{n})
=&\sum_{j=1}^{n}x_{\tau_{j}}^{2}\sigma_{\tau_{j}}^{2}-\frac{1}{n^2}\sum_{j=1}^{n}\sigma_{\tau_{j}}^{2}\\
=&\sum_{j=1}^{m}(x_{\tau_{j}}+\frac{1}{n})(x_{\tau_{j}}-\frac{1}{n})\sigma_{\tau_{j}}^{2}
+\sum_{j=m+1}^{n}(x_{\tau_{j}}+\frac{1}{n})(x_{\tau_{j}}-\frac{1}{n})\sigma_{\tau_{j}}^{2}\\
>&\frac{2\sigma_{\tau_{m+1}}^{2}}{n}\sum_{j=m+1}^{n}(x_{\tau_{j}}-\frac{1}{n})-\frac{2\sigma_{\tau_{m}}^{2}}{n}\sum_{j=1}^{m}(\frac{1}{n}-x_{\tau_{j}})
\geq 0,
\end{align*}
where the strictly inequality holds since $x\neq\frac{\mathbf{1}_{n}}{n}$.
Therefore, we obtain $x_{\tau}\in\mathcal{\bar{A}}_{\sigma_{\tau}^{2}}$, which is equivalent to $x\in\mathcal{\bar{A}}_{\sigma^{2}}$. In conclusion, system~\eqref{eq1} with any social power allocation $x\in\Delta_{n}^{\tau^{-1}}$ asymptotically undermines the wisdom.
\end{proof}

\Cref{T5} suggests that if more accurate individuals are allocated less social power, wisdom of crowds is undermined. In other words, for any given $\sigma^{2}$, there exists an ordering of social power allocation such that social influence with social power allocation in that ordering undermines the wisdom, see the orange areas in \cref{f2-a,f2-b}. Comparing \cref{T5} with \cref{T3,T4}, to undermine the wisdom just needs to allocate more social power to less accurate individuals, while generally allocating more social power to more accurate individuals, i.e., ordering-consistency, is not sufficient to improve the wisdom.

\begin{theorem}[Connection between improving and undermining of the wisdom]\label{T6}
For social influence process~\eqref{eq1} with social power allocation $x\in\Delta_{n}$ and individuals' variances $\sigma^{2}\in\mathbb{R}^{n}$, let $\tau$ be a permutation of $(1,\dots,n)$, $\Delta^{\tau}_{n}=\setdef{z\in\Delta_{n}\setminus\{ \frac{\mathbf{1}_{n}}{n}\}}{z_{\tau_{i}}\geq z_{\tau_{i+1}}, i\in\until{n-1}}$. If system~\eqref{eq1} with any social power allocation $x\in\Delta_{n}^{\tau}$ asymptotically improves the wisdom, then system~\eqref{eq1} with any social power allocation $x\in\Delta_{n}^{\tau^{-1}}$ asymptotically undermines the wisdom.
\end{theorem}
\begin{proof}
Since system~\eqref{eq1} with any social power allocation $x\in\Delta_{n}^{\tau}$ asymptotically improves the wisdom, we have $\Delta_{n}^{\tau}\subset\mathcal{A}_{\sigma^{2}}$, which, by \cref{L4}~\ref{L4-1}, suggests
\begin{equation*}
\frac{1}{i^2}\sum_{r=1}^{i}\sigma_{\tau_{r}}^{2}<\frac{1}{n^2}\sum_{j=1}^{n}\sigma_{j}^{2}
\end{equation*}
for all $i\in\until{n-1}$. Therefore,
\begin{equation*}
\sum_{j=1}^{n}\sigma_{j}^{2}=\sum_{r=1}^{i-1}\sigma_{\tau_{r}}^{2}+\sum_{r=i}^{n}\sigma_{\tau_{r}}^{2}<\frac{(i-1)^2}{n^2}\sum_{j=1}^{n}\sigma_{j}^{2}+\sum_{r=i}^{n}\sigma_{\tau_{r}}^{2},
\end{equation*}
which implies
\begin{equation}\label{e7.1}
\sum_{r=i}^{n}\sigma_{\tau_{r}}^{2}>(1-\frac{(i-1)^2}{n^2})\sum_{j=1}^{n}\sigma_{j}^{2} \quad \text{for all} \quad i\in\{2,\dots,n\}.
\end{equation}
For $x\in\Delta_{n}^{\tau^{-1}}$, i.e., $x_{\tau_{i}}\leq x_{\tau_{i+1}}$, $i\in\until{n-1}$, denote $q_{1}=x_{\tau_{1}}$ and $q_{i}=x_{\tau_{i}}-x_{\tau_{i-1}}$ for $i\in\{2,\dots,n\}$. Then, we have $q_{i}\geq 0$ and $x_{\tau_{i}}=\sum_{r=1}^{i}q_{r}$. Moreover,
\begin{align*}
\sum_{i=1}^{n}x_{\tau_{i}}^{2}\sigma_{\tau_{i}}^{2}
=\sum_{i=1}^{n}(\sum_{r=1}^{i}q_{r})^{2}\sigma_{\tau_{i}}^{2}
=&\sum_{i=1}^{n}\sigma_{\tau_{i}}^{2}\sum_{r=1}^{i}q_{r}^{2}+2\sum_{i=2}^{n}\sigma_{\tau_{i}}^{2}\sum_{r=1}^{i}q_{r}\sum_{j=r+1}^{i}q_{j}\\
=&q_{1}^{2}\sum_{r=1}^{n}\sigma_{\tau_{r}}^{2}+\sum_{i=2}^{n}q_{i}^{2}\sum_{r=i}^{n}\sigma_{\tau_{r}}^{2}+2\sum_{i=2}^{n}q_{i}\sum_{j=1}^{i-1}q_{j}\sum_{r=i}^{n}\sigma_{\tau_{r}}^{2}.
\end{align*}
By~\eqref{e7.1}, we obtain
\begin{align*}
\sum_{i=1}^{n}x_{\tau_{i}}^{2}\sigma_{\tau_{i}}^{2}>&(q_{1}^{2}+\sum_{i=2}^{n}(1-\frac{(i-1)^2}{n^2})(q_{i}^{2}+2q_{i}\sum_{j=1}^{i-1}q_{j}))\sum_{r=1}^{n}\sigma_{\tau_{r}}^{2},
\end{align*}
where
\begin{align*}
q_{1}^{2}+\sum_{i=2}^{n}(1-\frac{(i-1)^2}{n^2})(q_{i}^{2}+2q_{i}\sum_{j=1}^{i-1}q_{j})=&(\sum_{i=1}^{n}q_{i})^{2}-\frac{1}{n^2}\sum_{i=2}^{n}(i-1)^{2}(q_{i}^{2}+2q_{i}\sum_{j=1}^{i-1}q_{j})\\
=&x_{\tau_{n}}^{2}\!-\!\frac{1}{n^2}\sum_{i=2}^{n}(i\!-\!1)^{2}(x_{\tau_{i}}^{2}\!-\!x_{\tau_{i-1}}^{2}\!)\!\\
=&\frac{1}{n^2}\sum_{i=1}^{n}(2i-1)x_{\tau_{i}}^{2}
\end{align*}
due to $\sum_{i=2}^{n}(i\!-\!1)^{2}(x_{\tau_{i}}^{2}\!-\!x_{\tau_{i-1}}^{2}\!)=(n-1)^{2}x_{\tau_{n}}^{2}-\sum_{i=1}^{n-1}(2i-1)x_{\tau_{i}}^{2}$. Note that
\begin{align*}
\sum_{i=1}^{n}(2i-1)x_{\tau_{i}}^{2}-1=&\sum_{i=1}^{n}(2i-1)x_{\tau_{i}}^{2}-(\sum_{i=1}^{n}x_{\tau_{i}})^{2}\\
=&2\sum_{i=2}^{n}(i-1)x_{\tau_{i}}^{2}-2\sum_{i=2}^{n}\sum_{j=1}^{i-1}x_{\tau_{i}}x_{\tau_{j}}\\
=&2\sum_{i=2}^{n}x_{\tau_{i}}((i-1)x_{\tau_{i}}-\sum_{j=1}^{i-1}x_{\tau_{j}})\geq 0
\end{align*}
because $x_{\tau_{i}}\geq x_{\tau_{j}}$ for all $j\in\until{i-1}$. Hence, we obtain
\begin{equation*}
\sum_{i=1}^{n}x_{\tau_{i}}^{2}\sigma_{\tau_{i}}^{2}>\frac{1}{n^2}\sum_{i=1}^{n}(2i-1)x_{\tau_{i}}^{2}\sum_{r=1}^{n}\sigma_{r}^{2}\geq\frac{1}{n^2}\sum_{r=1}^{n}\sigma_{r}^{2},
\end{equation*}
which implies that social influence asymptotically undermines the wisdom.
\end{proof}

\Cref{T6} implies that if there exists a permutation $\tau$ of $(1,\dots,n)$ such that system~\eqref{eq1} with any social power allocation $x\in\Delta_{n}^{\tau}$ asymptotically improves the wisdom, then system~\eqref{eq1} with any social power allocation $x\in\Delta_{n}^{\tau^{-1}}$ asymptotically undermines the wisdom, see \cref{f2-c,f2-d}. Moreover, there may be more than one such permutation, as shown in \cref{f2-d}, and \cref{Alg1} can find all these permutations for given $\sigma^{2}$. However, the converse of \cref{T6} does not necessarily hold, i.e., all social power allocation $x\in\Delta_{n}^{\tau}$ asymptotically undermines the wisdom does not imply all social power $x\in\Delta_{n}^{\tau^{-1}}$ asymptotically improves the wisdom generally, see \cref{f2-a,f2-b} for counter examples.

\section{Improving and undermining the wisdom with the French-DeGroot opinion dynamics}\label{S5}

As reported in~\cite{JB-DB-DC:17} and~\cite{GM-GGdP:15}, one of the most significant empirical findings on wisdom of crowds is the PAP hypothesis, which attributes the improvement of wisdom to that more accurate individuals are more resistant to social influence, while less accurate individuals are more susceptible to social influence. In this section, we apply our theoretical results to the FD influence process to verify the effects of the PAP hypothesis.
\subsection{The French-DeGroot opinion dynamics and the PAP hypothesis}

Suppose that individuals interact their estimates according to the FD model:
\begin{equation}\label{e1}
y(k+1)=Wy(k),
\end{equation}
where $W$ is the row-stochastic influence matrix. More specifically, individual $i$ updates its estimate to a convex combination of estimates of others and itself, that is, $y_{i}(k+1)=\sum_{j=1}^{n}W_{ij}y_{j}(k)$, where $W_{ij}$ is the influence weight individual $i$ assigns to individual $j$ and $W_{ii}$ is its self-weight. Sociologically, $W_{ii}$ is interpreted as
individual $i$'s self-appraisal and indicates its resistance to social influence, while $1-W_{ii}$ represents its susceptibility to social influence. Let $\gamma_{i}=1-W_{ii}$, then there exists row-stochastic, zero-diagonal matrix $C\in\mathbb{R}^{n\times n}$, called the relative interaction matrix, such that $C_{ij}=0$ for all $i=j$ and $W_{ij}=\gamma_{i}C_{ij}$ otherwise. Denote by $\gamma\in\mathbb{R}^{n}$ the susceptibility vector,~\eqref{e1} can be written as
\begin{equation}\label{e2}
y(k+1)=[\gamma]Cy(k)+(I_{n}-[\gamma])y(k)
\end{equation}
with $W=[\gamma]C+I_{n}-[\gamma]$. Note that $\gamma_{i}=0$ means that individual $i$ does not take into account others' estimates, and $\gamma_{i}=1$ indicates that individual $i$ does not consider its own estimates. We assume that every individual is open-minded about others' estimates and there exists at least one individual who is confident in its own estimates, i.e., $\gamma\in\Gamma=\setdef{z\in\mathbb{R}^{n}\setminus\{\mathbf{1}_{n}\}}{1\geq z_{i}>0, i\in\until{n}}$. With a slight abuse of terminology, we regard system~\eqref{e2} as a population $\mathcal{P}_{\sigma^{2},\gamma}$ in a relative interaction network $\mathcal{G}(C)$, where a population is a group of individuals with some characteristics, which mainly concerned here are individual's susceptibility and variance.
\begin{remark}[Model advantages]
It is reported in~\cite{JB-DB-DC:17} that over $80\%$ people in the experiments displayed behavior of opinion updating consistent with system~\eqref{e2}. Moreover,~\eqref{e2} can model not only influence networks in which the relative interaction matrix $C$ is determined or accessible by individuals, such as, face to face discussion or social media networks, but also the influence networks in which $C$ is unknown to individuals, such as the Delphi model. Recall that the four basic features of the Delphi method are anonymity, iteration, controlled feedback and statistical aggregation. If we assume that $C$ is controlled by a moderator and is unknown to individuals, then system~\eqref{e2} is a well-posed influence network formulation for the Delphi method, where anonymity and controlled feedback are guaranteed by the assumption that $C_{ij}$ is unknown to individuals; iteration is achieved by the dependence of $y(k)$ upon time scale $k$; and the statistical aggregation is arithmetic average.
\end{remark}

Motivated by the PAP hypothesis, we have the following definitions for populations, which are presented analogously to \cref{D2}.
\begin{definition}[PAP populations]\label{D3}
For a population $\mathcal{P}_{\sigma^{2},\gamma}$ consisting of individuals $\until{n}$ with variances $\sigma^{2}\in\mathbb{R}^{n}$ and susceptibilities $\gamma\in\setdef{z\in\mathbb{R}^{n}\setminus\{\mathbf{1}_{n}\}}{1\geq z_{i}>0, i\in\until{n}}$, we say $\mathcal{P}_{\sigma^{2},\gamma}$ is a
\begin{enumerate}
\item PAP population if $\gamma_{i}\geq\gamma_{j}$ for all $\sigma^{2}_{i}\geq\sigma^{2}_{j}$, and $\gamma\in\Span{\mathbf{1}_{n}}$ only if $\sigma^{2}\in\Span{\mathbf{1}_{n}}$;

\item strong PAP population if $\mathcal{P}_{\sigma^{2},\gamma}$ is a PAP population, and $\gamma_{j}\sigma^{2}_{i}\geq\gamma_{i}\sigma^{2}_{j}$ for all $\sigma^{2}_{i}\geq\sigma^{2}_{j}$;

\item maximal PAP population if $\frac{\gamma_{i}}{\gamma_{j}}=\frac{\sigma^{2}_{i}}{\sigma^{2}_{j}}$ for all $i,j\in\until{n}$.
\end{enumerate}
\end{definition}

In a PAP population, less accurate individuals are more susceptible to social influence, while more accurate individuals are more resistant to social influence. Strong PAP population requires that more accurate individuals are more resistant to social influence compared with less accurate individuals, but can not be too much more resistant. Maximal PAP population requires that the proportions of individuals' susceptibilities are exactly the same as the proportions of their variances.
\subsection{Improving and optimizing the wisdom in irreducible influence networks}
It is well known that for system~\eqref{e2}, $\lim_{k\to\infty}y(k)$ exists if and only if every sink SCC of $\mathcal{G}(W)$ is aperiodic; if $\mathcal{G}(W)$ is strongly connected and aperiodic (or equivalently, $W$ is primitive), $y(k)$ reaches consensus asymptotically, that is, $\lim_{k\to\infty}y(k)=\omega^{\top}y(0)\mathbf{1}_{n}$, where $\omega\in\operatorname{int}{\Delta_{n}}$ is the left dominant eigenvector of $W$ satisfying $\omega^{\top}W=\omega^{\top}$~\cite{FB:22}.
If the relative interaction matrix $C$ is reducible, then $W$ is reducible since $\gamma>0$. Consequently, individuals in sink SCCs have positive social power and all others' social power is $0$. In this case, whether system~\eqref{e2} improves the wisdom or not is decided by the social power of individuals in sink SCCs, which corresponds to row-stochastic and irreducible submatrices of $C$. For simplicity, we assume $C$ is irreducible, and $c\in\Delta_{n}$ is its left dominant eigenvector. Since $\gamma\in\Gamma=\setdef{z\in\mathbb{R}^{n}\setminus\{\mathbf{1}_{n}\}}{1\geq z_{i}>0, i\in\until{n}}$, $W=[\gamma]C+I_{n}-[\gamma]$ is irreducible and aperiodic~\cite[Corollary 8.4.7]{RAH-CRJ:12}. Therefore, we have $x^{\top}W=x^{\top}[\gamma]C+x^{\top}(I_{n}-[\gamma])=x^{\top}$,
by which there holds $x^{\top}[\gamma]C=x^{\top}[\gamma]$. That is, $x^{\top}[\gamma]>0$ is a left eigenvector of $C$ associated with eigenvalue $1$. By Perron-Frobenius theorem~\cite[Theorem 2.12]{FB:22}, $[\gamma]x=\eta c$ with $\eta=\sum_{i=1}^{n}\gamma_{i}x_{i}$, which implies
\begin{equation}\label{e8}
\frac{x_{i}}{x_{j}}=\frac{c_{i}}{c_{j}}\frac{\gamma_{j}}{\gamma_{i}} \quad \text{for all} \quad i,j\in\until{n}.
\end{equation}
\begin{corollary}[Improving/optimizing wisdom with PAP populations in irreducible influence networks]\label{C1}
For system~\eqref{e2} with population $\mathcal{P}_{\sigma^{2},\gamma}$ and relative interaction network $\mathcal{G}(C)$, where $\gamma\in\Gamma$ with $\Gamma=\setdef{z\in\mathbb{R}^{n}\setminus\{\mathbf{1}_{n}\}}{1\geq z_{i}>0, i\in\until{n}}$, $C$ is irreducible with dominant left eigenvector $c\in\operatorname{int}\Delta_{n}$, the following statements hold:
\begin{enumerate}
\item assume that $\sigma^{2}\notin\Span{\mathbf{1}_{n}}$, then system~\eqref{e2} asymptotically improves the wisdom if $\frac{\sigma_{j}^{2}}{\sigma_{i}^{2}}\frac{\gamma_{i}}{\gamma_{j}}\leq\frac{c_{i}}{c_{j}}\leq\frac{\gamma_{i}}{\gamma_{j}}$ for all $\sigma_{i}^{2}\geq\sigma_{j}^{2}$ and $\frac{c_{i}}{c_{j}}<\frac{\gamma_{i}}{\gamma_{j}}$ for at least one $\sigma_{i}^{2}\geq\sigma_{j}^{2}$;\label{C1-1}

\item assume that $\mathcal{P}_{\sigma^{2},\gamma}$ is maximal PAP population, then system~\eqref{e2} asymptotically optimizes the wisdom if and only if $\mathcal{G}(C)$ is democratic; \label{C1-2}

\item assume there exists permutation $\tau$ of $(1,\dots,n)$ such that $\sigma_{\tau}^{2}$ satisfies~\eqref{e3.1}, then system~\eqref{e2} asymptotically improves the wisdom if $\frac{c_{\tau_{i}}}{c_{\tau_{i+1}}}\geq\frac{\gamma_{\tau_{i}}}{\gamma_{\tau_{i+1}}}$ for all $i\in\until{n-1}$ with at least one strict inequality holding. \label{C1-3}

\end{enumerate}
\end{corollary}

In \cref{C1}~\ref{C1-1}, it requires $\frac{c_{i}}{c_{j}}\leq\frac{\gamma_{i}}{\gamma_{j}}$ for $\sigma_{i}^{2}\geq\sigma_{j}^{2}$, which holds if $\gamma_{i}>\gamma_{j}$ and $c_{i}<c_{j}$. That is, more accurate individual is more resistant to social influence and has larger network centrality. If the condition of \cref{C1}~\ref{C1-1} holds, then system~\eqref{e2} is gap-consistent, thus it improves the wisdom. However, $\mathcal{P}_{\sigma^{2},\gamma}$ is not necessarily a PAP population under the condition of \cref{C1}~\ref{C1-1}. Generally, the PAP hypothesis is not enough for improving the wisdom in the FD opinion dynamics. \Cref{C1} shows that even the extreme case, namely, maximal PAP population, is not sufficient to improve the wisdom.
\begin{remark}
In Delphi experiments, the relative interaction matrix $C$ is unknown to individuals and is designed by the inquiry moderator. A widely-adopted algorithm for designing relative influence network $\mathcal{G}(C)$ is the equal-neighbor model~\cite[Chapter~5]{FB:22}. Let $M$ be a binary adjacent matrix associated with connected undirected graph $\mathcal{G}$, i.e., $M_{ij}=M_{ji}=1$ if $(i,j)$ is an edge of $\mathcal{G}$, and $M_{ij}=M_{ji}=0$ otherwise. The corresponding equal-neighbor relative interaction matrix is $C=[M\mathbf{1}_{n}]^{-1}M$. Let $d=M\mathbf{1}_{n}$ be the degree vector of $\mathcal{G}$. It is well known that the left dominant eigenvector of $C$ is $c=\frac{1}{d^{\top}\mathbf{1}_{n}}(d_{1},\dots,d_{n})^{\top}$, where $M$ and $d$ is designed by the moderator. By~\eqref{e2},
\begin{equation*}\label{e2.6}
\gamma_{i}=\frac{y_{i}(k+1)-y_{i}(k)}{\sum_{j=1}^{n}C_{ij}y_{j}(k)-y_{i}(k)},
\end{equation*}
which means that the moderator can approximate an individual's susceptibility using its estimates at successive time step. Since $\frac{c_{i}}{c_{j}}=\frac{d_{i}}{d_{j}}$, \cref{C1} suggests that the moderator may dynamically assign node degrees to individuals according to their observed susceptibilities to improve the collective estimates.
\end{remark}

In the investigation of influence networks, democratic and autocratic networks are studied frequently. It is of interest that if $\mathcal{G}(C)$ is democratic, the left dominant eigenvector of $C$ (also known as eigenvector centrality scores) is $\mathbf{1}_{n}/n$; if $\mathcal{G}(C)$ is autocratic with center node $i$, the center node has the largest centrality score $1/2$, which is strictly larger than that of every non-center node~\cite{PJ-AM-NEF-FB:13d}.
\begin{corollary}[Improving/optimizing wisdom in democratic and autocratic influence networks]\label{C2}
For system~\eqref{e2} with population $\mathcal{P}_{\sigma^{2},\gamma}$ and relative interaction network $\mathcal{G}(C)$, where $\gamma\in\Gamma=\setdef{z\in\mathbb{R}^{n}\setminus\{\mathbf{1}_{n}\}}{1\geq z_{i}>0, i\in\until{n}}$,
\begin{enumerate}
\item suppose that $\mathcal{G}(C)$ is democratic, then\label{C2-1}
\begin{enumerate}
\item system~\eqref{e2} asymptotically optimizes the wisdom if and only if $\mathcal{P}_{\sigma^{2},\gamma}$ is maximal PAP population;

\item if $\sigma^{2}\notin\Span{\mathbf{1}_{n}}$ and $\mathcal{P}_{\sigma^{2},\gamma}$ is strong PAP population, system~\eqref{e2} asymptotically improves the wisdom;

\item assume $\sigma_{i}^{2}\leq\sigma_{i+1}^{2}$ for all $i\in\until{n-1}$. System~\eqref{e2} with any PAP population asymptotically improves the wisdom if and only if~\eqref{e3.2} holds for all $i\in\until{n-1}$.

\end{enumerate}
\item Suppose that $\mathcal{G}(C)$ is autocratic with center node $l$, $C_{li}$ is the influence weight the center node accords to $i$, then \label{C2-2}
\begin{enumerate}
\item if $\sigma^{2}\notin\Span{\mathbf{1}_{n}}$, $\frac{\gamma_{i}}{\gamma_{j}}\frac{\sigma_{j}^{2}}{\sigma^{2}_{i}}\leq\frac{C_{li}}{C_{lj}}\leq\frac{\gamma_{i}}{\gamma_{j}}$ for all $\sigma_{i}^{2}\geq\sigma_{j}^{2}$ and $\frac{C_{li}}{C_{lj}}<\frac{\gamma_{i}}{\gamma_{j}}$ at least for one $\sigma_{i}^{2}\geq\sigma_{j}^{2}$, system~\eqref{e2} asymptotically improves the wisdom;

\item system~\eqref{e2} asymptotically optimizes the wisdom asymptotically if and only if $\frac{\gamma_{i}}{\gamma_{l}}=C_{li}\frac{\sigma^{2}_{i}}{\sigma^{2}_{l}}$ for all $i\neq l$.
\end{enumerate}
\end{enumerate}
\end{corollary}

\Cref{C2}~\ref{C2-1} shows that the PAP hypothesis is significant in democratic influence networks. If more accurate individuals are more resistant to social influence in some extent, the wisdom is improved or optimized in democratic influence networks. \Cref{C2}~\ref{C2-2} shows that the FD opinion dynamics can improve or optimize wisdom even in autocratic influence networks.
\subsection{Undermining the wisdom in irreducible influence networks}
We end this section by summarizing results on undermining the wisdom with the FD opinion dynamics in irreducible influence networks.
\begin{corollary}[Undermining wisdom in irreducible influence networks]\label{C3}
For system~\eqref{e2} with population $\mathcal{P}_{\sigma^{2},\gamma}$ and relative interaction network $\mathcal{G}(C)$, where $\gamma\in\Gamma=\setdef{z\in\mathbb{R}^{n}\setminus\{\mathbf{1}_{n}\}}{1\geq z_{i}>0, i\in\until{n}}$, $C$ is irreducible with dominant left eigenvector $c\in\operatorname{int}\Delta_{n}$,
\begin{enumerate}
\item if $\frac{c_{i}}{c_{j}}\geq\frac{\gamma_{i}}{\gamma_{j}}$ for all $\sigma_{i}^{2}\geq\sigma_{j}^{2}$ with at least one inequality holding strictly, system~\eqref{e2} asymptotically undermines the wisdom;

\item assume there exists permutation $\tau$ of $(1,\dots,n)$ such that $\sigma_{\tau}^{2}$ satisfies~\eqref{e3.1}, then system~\eqref{e2} asymptotically undermines the wisdom if $\frac{c_{\tau_{i}}}{c_{\tau_{i+1}}}\leq\frac{\gamma_{\tau_{i}}}{\gamma_{\tau_{i+1}}}$ for all $i\in\until{n-1}$ with at least one strict inequality holding.

\end{enumerate}

\end{corollary}

Comparing \cref{C3} with \cref{C1}, if more accurate individuals have less network centralities and are more susceptible to social influence, the wisdom is undermined; on the contrary, if more accurate individuals have larger network centralities and are more resistant to social influence, the wisdom is improved.

\section{Conclusions}\label{S6}
This paper has investigated the problem of improving, optimizing and
undermining the collective wisdom in influence networks. Mathematical
formulations and rigorous analysis are provided. Various results for
improving, optimizing and undermining of the wisdom are derived from the
perspective of influence systems theory. Our theoretical results contribute
to the study of influence networks and the wisdom of crowds effect in the
following aspects: first, we show that wisdom in influence networks can be
both improved and undermined by social influence. Second, we provide
necessary and/or sufficient conditions for improving, optimizing and
undermining the wisdom. Finally, we provide theoretical explanations for
empirical evidence reported in the literature, such as the PAP
hypothesis. Future work shall focus on the effect of reconsideration and
the plasticity of populations.


text:
Hardy County, West Virginia
Hardy County in the mountains of eastern West Virginia was having an identity crisis. Their newly formed CVB and board knew instinctively that they had much to offer visitors. They just had to figure out how to sell their county as a destination. Hardy County consists of two incorporated towns and several tiny, unincorporated ones. Moorefield is the county seat and center of commerce. Wardensville is a small but hip town that happens to be located on a heavily traveled route between DC and other West Virginia destinations like Seneca Rocks and Canaan Valley. There is also a region of the county known as The Lost River Valley. Though it is breathtakingly beautiful and home to several award-winning inns that draw many repeat customers, The Lost River Valley does not appear on an official state map.
The Mikula-Harris team brought an unbiased perspective to this challenge. To develop a brand that was both true to Hardy County's strengths and also differentiated from nearby destinations, we relied heavily on good research. We also spent some time exploring the county from end to end. We conducted surveys of stakeholders and visitors. We found that an overwhelming number of visitors referred to their trip as a visit to the Lost River Valley. The research also showed that people visited to relax and enjoy time in the peaceful mountain surroundings. While nearby destinations promote non-stop adventure, we plan to position Hardy County as the most relaxing getaway in West Virginia. In addition, The Lost River Valley will play a large role in the county's identity. Wardensville, which is already home to restaurants and shops featuring the Lost River name, will serve as a gateway town. The abundant outdoor recreation opportunities mesh nicely with the brand strategy. Hardy County has paddling, hiking, biking, fishing and other activities, but no adrenaline rush opportunities like rock climbing or whitewater rafting.
The Hardy County CVB staff and board understood from the beginning the importance of basing strategic decisions on research. They eagerly digested all of the data and ultimately adopted the branding recommendations. We then developed a logo and visual identity for the county.
With the foundation of a good brand now in place, we moved on to developing an ad campaign. Advertising is a creative interpretation of a brand and must contribute to the public's understanding of it. Our team offered several ideas. Hardy County CVB chose to proceed with a versatile ad concept that supports the "most relaxing getaway in West Virginia" claim while also touting one or more available activities. For example, an ad in the series promoting family getaways features a photo of kids in a canoe along with the message: "A family trip to Hardy County changes everything. Instead of going separate ways, the kids learn to paddle in the same direction. Turn off the phones so your family can truly reconnect. Experience the Hardy Effect." Each ad in the series makes a bold statement about the benefit of spending time in a mountain paradise like Hardy County.

redpajama_set_name: RedPajamaCommonCrawl


Specializes in pizza, manicotti and other Italian dishes, as well as salads.
Dine in or order online for takeout from China Buffet, featuring Chinese food made with the freshest ingredients.
A varied menu ranges from steaks to Carolina barbecue to jambalaya to salads and sandwiches.
Specialities include smoked brisket, meatball marinara, fried pickles, Thai chicken sandwiches and veggie wraps.


text:
Education and Inspiration for Your Photography Business
Home Business The Power of a Hand-Written Note
Skip Cohen
No Comments on The Power of a Hand-Written Note
The Power of a Hand-Written Note
I was reading one of my old favorites recently, The Book of Awesome. Written by Neil Pasricha, it's a series of simple awesome events, products and stuff we too often don't experience anymore. Many of the things are events we've all forgotten about, like one of my favorites, "Getting something with actual handwriting on it in the mail."
We live in the digital world. We're surrounded by emails, links, posts, tweets and voicemails. We live a life of default prompts. We rarely personalize anything. I'm not talking about your website or blog, but how you communicate with clients and associates, and where appropriate making them more personal.
Neil wrote:
"Checking the mail can be a bit depressing. Sometimes there isn't anything in there. Nope, nothing at all. Just one big empty mailbox telling the world that everyone forgot about you today. Then again, the alternative is typically a fistful of bills and flyers…just kind of boring, kind of bland, kind of blah.
But that's what makes it so great when something with actual handwriting on it turns up…those little, endangered parcels have something very special about them…"
Now, let's apply this concept to a thank you note the next time somebody does something nice for you. It's such an easy thing to do. Think about a hand-written note to a rep you met in a booth at a convention who took the time to help you understand a new feature or a new product. Maybe you want to thank a client for a referral or being a lot of fun to work with. How about an associate who makes your life easier?
We all understand and agree it's a great thing to do, but there's a second important aspect to sending hand-written notes; You need to send them on your personal, customized stationery! You need to use your own images, and on the back of a thank-you card for example, just like the Hallmark logo, should be your logo and contact information.
Designing your personal stationery is one of the easiest ways to help build more brand awareness. No professional photographer should ever be using store-bought stationery or holiday cards. Use your images and then take the time to help save those "endangered" parcels that Neil Pasricha talked about.
Making it even easier is Marathon's ability to help you design everything you need! Take the time NOW to design your own notecards! Stop using store-bought stationery and take the time to create something AWESOME!
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http://www.SkipCohenUniversity.com/
Skip Cohen is an industry executive recognized for his diversity. He has served as past president of Hasselblad USA, Rangefinder/WPPI and in 2009 founded his own educational consulting company. In 2013 he launched Skip Cohen University dedicated to helping artists build a stronger business. He's a regular speaker at a variety of conventions and writes for several different magazines, as well as having two business classes at Lynda.com. Click above to visit the SCU blog.
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Chamira Young (16)
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Chamira started her journey into the arts with a goal to be an artist. From the University of Michigan School of Art & Design (Go Blue!) to working as an excited graphic designer and illustrator at a book publishing company, to serving as an eager photographer, print designer, and web designer at an international motorcycle magazine. She admits to being a tech nerd and today is a successful photographer, podcaster and writer. She co-hosts two popular podcasts with Skip Cohen, writes for Photofocus.com, her own blog, ProPhotographerJourney, and has a never-ending love for the craft!
Six Common Denominators Professional Photographers Hate
Justin & Mary Marantz (18)
marantz@noemail.com
Justin & Mary are an international wedding photography duo, who travel the globe shooting weddings, teaching, & inspiring other small business owners. Their "Walk Through a Wedding" Workshops have been presented worldwide including London, Australia & The Netherlands. They've authored two books- The Guide to Marketing and the J&M Lighting Guide, presented at international conferences including WPPI and The Dutch Wedding Congress and have been named as one of Profoto's Legends of Lighting.
Are You In a Booking Rut?
Lori Nordstrom (11)
lorinordstrom@noemail.com
Lori Nordstrom {M.Photog., Cr., CPP, ABI} owns a boutique studio specializing in children and family portraits located in the small, picturesque town of Winterset, Iowa. Lori has been a photographer for over 20 years and is known not only for her simple and sweet portraits of children but as a leader in the photography industry in the areas of business, marketing and sales. Lori is a PPA approved business instructor, and sharing with other photographers and small business owners is one of the things she most enjoys. http://www.nordstromphoto.com
New Photographer or Seasoned Pro – Five Points for Success
Mark Weber (20)
markw@marathonpress.net
Mark Weber is a veteran professional photographer and marketing consultant with Marathon Press. His photography has won numerous awards for recognition of a variety of unique and innovative styles. In addition, his educational programs on marketing have helped studio owners increase the profitability of their business. He spearheads the MAP at Marathon Press which helps photographers gain control of their business and execute a customized, strategic marketing plan to maximize profits and take their business to a new level with ongoing marketing support.
Weber is a former past President of the Professional Photographers of Greater Omaha and has served on the Professional Photographers of Nebraska board. He holds the B.F.A. degree from the University of Nebraska at Omaha where he completed his thesis on abstract painting.
Top 5 Reasons You Need to Send Direct Mail to High School Seniors
Michele Celentano (8)
Celentano@noemail.com
Michele Celentano is one of America's leading family photographers, Canon Explorer of Light and a published author. She's a sought-after educator and artist who strongly believes in the importance of photographs. "The most important part of this journey through life are the people we love. For me, the second most important thing we have are the photographs that help us to remember those we love when they are no longer with us...Photographs of the people we love the most and the times we spend with them will outlast any material thing we have."
Sometimes You Just Need to Rant – Part Two
Nicole Begley (19)
NicoleBegley@nomail.com
Nicole Begley, M.Photog.Cr, CPP, is an animal trainer turned pet
photographer based in Pittsburgh, PA who also travels the world teaching
pet photographers from every corner of the globe. She is the creator of
Hair of the Dog, a site dedicated to helping pet photographers run a
profitable pet photography studio and has authored a book - Pet and Horse
Photography for Everybody. A member of PPA since 2010, Nicole has earned
her Master of Photography degree, Photographic Craftsman degree, as well as
her Certified Professional Photographer designation. Her work has won
several awards at local and district competition, as well as a four-time
medalist in the International Print Competition.
Are You Stuck in Analysis Paralysis?
Robert Nowell (8)
RobertNowell@nomail.com
Robert Nowell is a veteran full-time professional photographer. He has photographed over 650 weddings and has a sterling reputation as a portrait artist. Robert has taught thousands of photographers in the last three decades. He is currently an adjunct professor at Mohawk college In the Creative Photography program where he teaches business and marketing. Robert is co-founder and content creator at Sprouting Photographer and Vice President and co-founder at Sprout Studio.
Taking the Fear Out of Sales
Sarah Petty (58)
sarahpetty@noemail.com
Sarah is a New York Times best-selling author, highly-acclaimed speaker, author, MBA and coach who started her own boutique photography studio after working for Coca Cola for 20 years and then meeting the marketing goals of a top regional advertising agency's clients. She attributes the rapid growth of her boutique photography studio, which was named one of the most profitable in the country within just five years in business by PPA, to the creation of her own strong brand. Click on Sarah's photograph to visit her blog.
Great Tips for Studios on the Move
Skip Cohen (70)
SkipCohen@nomail.com
Building Blocks for the New Year!
Suzette Allen (24)
SuzetteAllen@nomail.com
Suzette Allen, Lumix Ambassador and Adobe Influencer, is a long- time Photoshop instructor and Hybrid pioneer who continues to reinvent herself to stay current with the changing landscape of photography. Pursuing creativity by embracing technology has become a hallmark of her style and a continual driving force to share her creativity with the world. She passionately creates and shares her vision as a Photographer and Educator, being a lover of people AND photographic art!
Getting the Most Out of Window Light
Tamara Lackey (6)
tlackey@nomail.com
Tamara Lackey is a photographer, writer and educator. While her photographic specialty is primarily children and families, there's probably nothing she hasn't photographed. She's also an outstanding talk-show host, mother, wife and friend to so many in the industry. She's one of the few artists who has helped photographers establish their priorities between being a business-style and lifestyle.
Mother's Day – But Skip it's September!
Ann Monteith
Bruce Hudson
Chris Wunder
Mr. Wunder is a 30 year veteran in the photo marketing business. His expertise extends from Portrait Studios to Special Events including School Photography, Sports, Yearbooks and Directory publishing. An industry insider, he has worked with smaller, entrepreneurial firms as well as large corporate clients. He is in demand as a consultant, trainer and presenter for conventions and workshops at the regional, national and international level.
Gavin Phillips
Jen Basford
Jerry Weiner
Jessica Drossin
John Hartman
Josh Hudson
Judy Grann
Kay Eskridge
Linda McCarthy
Marilyn Sholin
Mary Fisk-Taylor
Melissa Gallo
Copyright © 2019 Marathon Press. All rights reserved.


text:
Berlusconi wants younger AC Milan team
by David on July 29, 2009
AC Milan president Silvio Berlusconi has called on coach Leonardo to only target younger players in the transfer market.
Milan has been criticized in recent years for relying heavily on players older than 30, including Paolo Maldini, Clarence Seedorf and David Beckham.
Berlusconi told the ANSA news agency on Tuesday that the team "will look for players who are under 23."
He did not specify whether that would affect Beckham's chances of rejoining the team following the end of the MLS season. Beckham reluctantly returned to the Los Angeles Galaxy following a loan stint with Milan, and has said he wants go back to Italy.


text:
START_ATF_NAMESPACE
    namespace Detail
    {

        Info::CNationSettingDataKRctor_CNationSettingDataKR2_ptr CNationSettingDataKRctor_CNationSettingDataKR2_next(nullptr);
        Info::CNationSettingDataKRctor_CNationSettingDataKR2_clbk CNationSettingDataKRctor_CNationSettingDataKR2_user(nullptr);

        Info::CNationSettingDataKRCreateBilling4_ptr CNationSettingDataKRCreateBilling4_next(nullptr);
        Info::CNationSettingDataKRCreateBilling4_clbk CNationSettingDataKRCreateBilling4_user(nullptr);

        Info::CNationSettingDataKRCreateWorker6_ptr CNationSettingDataKRCreateWorker6_next(nullptr);
        Info::CNationSettingDataKRCreateWorker6_clbk CNationSettingDataKRCreateWorker6_user(nullptr);

        Info::CNationSettingDataKRGetCashItemPrice8_ptr CNationSettingDataKRGetCashItemPrice8_next(nullptr);
        Info::CNationSettingDataKRGetCashItemPrice8_clbk CNationSettingDataKRGetCashItemPrice8_user(nullptr);

        Info::CNationSettingDataKRGetItemName10_ptr CNationSettingDataKRGetItemName10_next(nullptr);
        Info::CNationSettingDataKRGetItemName10_clbk CNationSettingDataKRGetItemName10_user(nullptr);

        Info::CNationSettingDataKRInit12_ptr CNationSettingDataKRInit12_next(nullptr);
        Info::CNationSettingDataKRInit12_clbk CNationSettingDataKRInit12_user(nullptr);

        Info::CNationSettingDataKRIsNormalChar14_ptr CNationSettingDataKRIsNormalChar14_next(nullptr);
        Info::CNationSettingDataKRIsNormalChar14_clbk CNationSettingDataKRIsNormalChar14_user(nullptr);

        Info::CNationSettingDataKRReadSystemPass16_ptr CNationSettingDataKRReadSystemPass16_next(nullptr);
        Info::CNationSettingDataKRReadSystemPass16_clbk CNationSettingDataKRReadSystemPass16_user(nullptr);

        Info::CNationSettingDataKRSendCashDBDSNRequest18_ptr CNationSettingDataKRSendCashDBDSNRequest18_next(nullptr);
        Info::CNationSettingDataKRSendCashDBDSNRequest18_clbk CNationSettingDataKRSendCashDBDSNRequest18_user(nullptr);

        void CNationSettingDataKRctor_CNationSettingDataKR2_wrapper(struct CNationSettingDataKR* _this)
        {
           CNationSettingDataKRctor_CNationSettingDataKR2_user(_this, CNationSettingDataKRctor_CNationSettingDataKR2_next);
        };
        struct CBilling* CNationSettingDataKRCreateBilling4_wrapper(struct CNationSettingDataKR* _this)
        {
           return CNationSettingDataKRCreateBilling4_user(_this, CNationSettingDataKRCreateBilling4_next);
        };
        struct CashDbWorker* CNationSettingDataKRCreateWorker6_wrapper(struct CNationSettingDataKR* _this)
        {
           return CNationSettingDataKRCreateWorker6_user(_this, CNationSettingDataKRCreateWorker6_next);
        };
        int CNationSettingDataKRGetCashItemPrice8_wrapper(struct CNationSettingDataKR* _this, struct _CashShop_str_fld* pFld)
        {
           return CNationSettingDataKRGetCashItemPrice8_user(_this, pFld, CNationSettingDataKRGetCashItemPrice8_next);
        };
        char* CNationSettingDataKRGetItemName10_wrapper(struct CNationSettingDataKR* _this, struct _NameTxt_fld* pFld)
        {
           return CNationSettingDataKRGetItemName10_user(_this, pFld, CNationSettingDataKRGetItemName10_next);
        };
        int CNationSettingDataKRInit12_wrapper(struct CNationSettingDataKR* _this)
        {
           return CNationSettingDataKRInit12_user(_this, CNationSettingDataKRInit12_next);
        };
        bool CNationSettingDataKRIsNormalChar14_wrapper(struct CNationSettingDataKR* _this, wchar_t wcChar)
        {
           return CNationSettingDataKRIsNormalChar14_user(_this, wcChar, CNationSettingDataKRIsNormalChar14_next);
        };
        bool CNationSettingDataKRReadSystemPass16_wrapper(struct CNationSettingDataKR* _this)
        {
           return CNationSettingDataKRReadSystemPass16_user(_this, CNationSettingDataKRReadSystemPass16_next);
        };
        void CNationSettingDataKRSendCashDBDSNRequest18_wrapper(struct CNationSettingDataKR* _this)
        {
           CNationSettingDataKRSendCashDBDSNRequest18_user(_this, CNationSettingDataKRSendCashDBDSNRequest18_next);
        };

        ::std::array<hook_record, 9> CNationSettingDataKR_functions =
        {
            _hook_record {
                (LPVOID)0x14022b160L,
                (LPVOID *)&CNationSettingDataKRctor_CNationSettingDataKR2_user,
                (LPVOID *)&CNationSettingDataKRctor_CNationSettingDataKR2_next,
                (LPVOID)cast_pointer_function(CNationSettingDataKRctor_CNationSettingDataKR2_wrapper),
                (LPVOID)cast_pointer_function((void(CNationSettingDataKR::*)())&CNationSettingDataKR::ctor_CNationSettingDataKR)
            },
            _hook_record {
                (LPVOID)0x14022b3f0L,
                (LPVOID *)&CNationSettingDataKRCreateBilling4_user,
                (LPVOID *)&CNationSettingDataKRCreateBilling4_next,
                (LPVOID)cast_pointer_function(CNationSettingDataKRCreateBilling4_wrapper),
                (LPVOID)cast_pointer_function((struct CBilling*(CNationSettingDataKR::*)())&CNationSettingDataKR::CreateBilling)
            },
            _hook_record {
                (LPVOID)0x14022b340L,
                (LPVOID *)&CNationSettingDataKRCreateWorker6_user,
                (LPVOID *)&CNationSettingDataKRCreateWorker6_next,
                (LPVOID)cast_pointer_function(CNationSettingDataKRCreateWorker6_wrapper),
                (LPVOID)cast_pointer_function((struct CashDbWorker*(CNationSettingDataKR::*)())&CNationSettingDataKR::CreateWorker)
            },
            _hook_record {
                (LPVOID)0x14022b4a0L,
                (LPVOID *)&CNationSettingDataKRGetCashItemPrice8_user,
                (LPVOID *)&CNationSettingDataKRGetCashItemPrice8_next,
                (LPVOID)cast_pointer_function(CNationSettingDataKRGetCashItemPrice8_wrapper),
                (LPVOID)cast_pointer_function((int(CNationSettingDataKR::*)(struct _CashShop_str_fld*))&CNationSettingDataKR::GetCashItemPrice)
            },
            _hook_record {
                (LPVOID)0x14022b320L,
                (LPVOID *)&CNationSettingDataKRGetItemName10_user,
                (LPVOID *)&CNationSettingDataKRGetItemName10_next,
                (LPVOID)cast_pointer_function(CNationSettingDataKRGetItemName10_wrapper),
                (LPVOID)cast_pointer_function((char*(CNationSettingDataKR::*)(struct _NameTxt_fld*))&CNationSettingDataKR::GetItemName)
            },
            _hook_record {
                (LPVOID)0x14022b230L,
                (LPVOID *)&CNationSettingDataKRInit12_user,
                (LPVOID *)&CNationSettingDataKRInit12_next,
                (LPVOID)cast_pointer_function(CNationSettingDataKRInit12_wrapper),
                (LPVOID)cast_pointer_function((int(CNationSettingDataKR::*)())&CNationSettingDataKR::Init)
            },
            _hook_record {
                (LPVOID)0x14022b1c0L,
                (LPVOID *)&CNationSettingDataKRIsNormalChar14_user,
                (LPVOID *)&CNationSettingDataKRIsNormalChar14_next,
                (LPVOID)cast_pointer_function(CNationSettingDataKRIsNormalChar14_wrapper),
                (LPVOID)cast_pointer_function((bool(CNationSettingDataKR::*)(wchar_t))&CNationSettingDataKR::IsNormalChar)
            },
            _hook_record {
                (LPVOID)0x14022b4d0L,
                (LPVOID *)&CNationSettingDataKRReadSystemPass16_user,
                (LPVOID *)&CNationSettingDataKRReadSystemPass16_next,
                (LPVOID)cast_pointer_function(CNationSettingDataKRReadSystemPass16_wrapper),
                (LPVOID)cast_pointer_function((bool(CNationSettingDataKR::*)())&CNationSettingDataKR::ReadSystemPass)
            },
            _hook_record {
                (LPVOID)0x14022b4c0L,
                (LPVOID *)&CNationSettingDataKRSendCashDBDSNRequest18_user,
                (LPVOID *)&CNationSettingDataKRSendCashDBDSNRequest18_next,
                (LPVOID)cast_pointer_function(CNationSettingDataKRSendCashDBDSNRequest18_wrapper),
                (LPVOID)cast_pointer_function((void(CNationSettingDataKR::*)())&CNationSettingDataKR::SendCashDBDSNRequest)
            },
        };
    }; // end namespace Detail
END_ATF_NAMESPACE

redpajama_set_name: RedPajamaGithub


Located in the lower end of Honey Dew Valley, Utah Vintage Estate Cellars grows a variety of wine grapes, and produces and bottles its own vintages. We specialize in Pinots, Merlots, Syrahs, and our own special Cava Rose Brut and Cava Sauvignon Brut. We also supply grapes to many of the Valley's finest wineries.
Our Master Wine Maker, Papa Bartelucci was born in Toscano, Italy and came to the US in 1973. He brought with him 10 generations of wine making expertise. Today he is Honey Dew Valley's most recognizable personality.
Mama came to the New World with Papa in 1973. She has been the chief Spaghetti chef ever since. Her garlic bread is also to die for, and her Cheesecske...malto benie.
and his beautiful wife, Betty. Their son, Tony, Jr. and his beautiful wife, Randi. They are the next generation of Wine Masters at Vintage Estate Cellars. Tony runs the business, and Tony, Jr. assists his grandfather in the Cellar.
Come for wine. Stay for breakfast.
Our Victorian Cottages are sweet and elegant, with all of the old world charm you would want for your weekend stay.
Our museum is the history of wine making in our charming Honey Dew Valley with enough Old World history to make it all fit together.
Take a picnic lunch and a couple of bottles on a bicycle picnic in the hills behind the vineyard. Come back for a hot dinner and another bottle.
We have been serving wine in the Honey Dew Valley since 1963. Our founder, Vino Bardalucci is a first generation Italian. He brings 10 generations of viticulture to Utah, and produces the finest grapes in all the Valley. Come by for a taste.
Come by for a glass of wine and go home with a case. Or stay the night in our B&B.
Visit our Museum of Wine and Wonder.