Table 1 A summary of the opinion pooling process.

Description

Distribute an investment (colour) amongst actors through an interaction network across sites

Distribute an investment (colour) amongst sites through an interaction network across actors

Network

\(\textbf{P}=\textbf{X}\textbf{C}\)

\(\textbf{Q}=\textbf{C}\textbf{X}\)

Markov chain

\(\varvec{\alpha }^{(t)}={\alpha }^{(t-1)}\textbf{P}\)

\(\varvec{\beta }^{(t)}={\beta }^{(t-1)}\textbf{Q}\)

Steady state (definition)

\(\displaystyle \varvec{\alpha }:=\lim _{t \rightarrow \infty }\varvec{\alpha }^{(t)}\)

\(\varvec{\alpha }\textbf{P}=\varvec{\alpha }\)

\(\varvec{\alpha }\textbf{1}=1\)

\(\displaystyle \varvec{\beta }:=\lim _{t \rightarrow \infty }\varvec{\beta }^{(t)}\)

\(\varvec{\beta }\textbf{Q}=\varvec{\beta }\)

\(\varvec{\beta }\textbf{1}=1\)

Steady State (solution³³,  pp.250–252)

\(\displaystyle \varvec{\alpha }\left[ \left( \textbf{I}_{m\times m}-\textbf{P}\right) |\textbf{1}_{m\times 1}\right] =\left[ \textbf{0}_{1\times m}|1 \right] \)

\(\displaystyle \underbrace{\left[ \left( \textbf{I}_{m\times m}-\textbf{P}\right) |\textbf{1}_{m\times 1}\right] ^T)}_{\textbf{M}} \underbrace{\varvec{\alpha }^T}_{\textbf{x}}=\underbrace{\left[ \textbf{0}_{1\times m}|1 \right] ^T}_{\textbf{a}}\)

\(\displaystyle \varvec{\alpha }^T=\arg \min _{\textbf{x}}{\Vert \textbf{M} \textbf{x}-\textbf{a}\Vert ^2_2}\)

\(\displaystyle \varvec{\beta }\left[ \left( \textbf{I}_{n\times n}-\textbf{Q}\right) |\textbf{1}_{n\times 1}\right] =\left[ \textbf{0}_{1\times n}|1 \right] \)

\(\displaystyle \underbrace{\left[ \left( \textbf{I}_{n\times n}-\textbf{Q}\right) |\textbf{1}_{n\times 1}\right] ^T)}_{\textbf{N}} \underbrace{\varvec{\beta }^T}_{\textbf{y}}=\underbrace{\left[ \textbf{0}_{1\times n}|1 \right] ^T}_{\textbf{b}}\)

\(\displaystyle \varvec{\beta }^T=\arg \min _{\textbf{y}}{\Vert \textbf{N} \textbf{y}-\textbf{b}\Vert ^2_2}\)

Duality

\(\varvec{\alpha }=\varvec{\beta }\textbf{C}\)

\(\varvec{\beta }=\varvec{\alpha }\textbf{X}\)