Opinion on Opinions

Note

The following is an exceedingly opinionated write-up on how I feel opinions are formed, or at least how I tend to form opinions. Venture forth at your own peril.

We generally try not to have specific preformed opinions that are dug-in, and as a result attempt to form an opinion based on some overarching consensus. Of course, this would be an entirely ideal scenario which is almost always not the case. Every person living or dead will have some internal inclinations about almost everything. This is the result of what could be a combination of their uniquely individual upbringing and innate nature to an extent. These things are difficult to quantize in any legitimate sense. So we can generally treat these ‘inclinations’ as the initial-beliefs vector $\Lambda$ used to preserve the person’s identity when reshaped through their changing opinions. This may be a large vector in some ${\mathbb{R}}^n$ space that contains specific axes along which we measure some $n$ metrics, each telling us something about the person. Any random initial belief vector can be represented in the following manner:

$$\Lambda =\left(\begin{array}{c}{\Lambda }_1\\ {\Lambda }_2\\ {\Lambda }_3\\ ⋮\\ {\Lambda }_n\end{array}\right)=\left(\begin{array}{c}0.01\\ -2.83\\ 6.00\\ ⋮\\ 1.39\end{array}\right)$$

Where each ${\Lambda }_i\in \mathbb{R}$ is a factor that contributes to forming the initial belief when going into an opinion-formation process. For a more concrete example, we can take an initial belief of some person as relating to cats:

$${\Lambda }_{\text{cats}}=\left(\begin{array}{c}4.00\\ 1.01\\ -2.93\\ ⋮\\ 0.01\end{array}\right)\begin{array}{c}\begin{array}{rlrl}& & \to & \text{ Are they cute?}\\ & & \to & \text{ Would you adopt one?}\\ & & \to & \text{ Like animals in general?}\\ & & ⋮& \\ & & \to & \text{ Any relatives with cats?}\end{array}\end{array}$$

Each element ${\Lambda }_i$ of the initial belief vector represents a contributing factor that is measured over $\mathbb{R}$ and the overall opinion is a combination of these values corresponding to different axes that can be quantified appropriately. The initial-beliefs vector contains a stacking of multiple such metrics, some combination of which is used to express an opinion. While updating and refining an opinion or stance on some specific topic, involvement of all metrics might not be necessary; therefore, we may construct a smaller ${\mathbb{R}}^m(m\ll n)$ vector that will participate in the opinion-formation process to update specific metrics in the larger vector.

In doing so, we represent a subset of their initial-beliefs as a ${\mathbb{R}}^m$ vector that is to be updated. This vector (say ${\Lambda }^{\prime }$ ) is updated over multiple iterations in order to simulate evolving opinions and beliefs over time. The vector ${\Lambda }^{\prime }$ can, therefore, represent the beliefs that any person may hold when going into an iteration of opinion-formation. In cases where someone may not have any initial-beliefs regarding some matter, we can have ${\Lambda }^{\prime }=\vec{0}$.

Generally for our use case we can represent other opinions also as belonging to ${\mathbb{R}}^m$ with positive/negative inclinations expressed along multiple axes. Opinions in this context would influence a person’s belief by accumulating on the basis of some consensus. For a population size of $k$ this can be represented in the following manner:

$${\vec{O}}_1+{\vec{O}}_2+\cdots +{\vec{O}}_i+\cdots +{\vec{O}}_k+\Lambda$$

Where each ${\vec{O}}_i\in {\mathbb{R}}^m$ is a measure of the positiveness, negativeness or neutrality of their opinion on a particular topic along some predefined axes. We can expand these opinion vectors to the ${\mathbb{R}}^n$ space by converting it into a sparse vector:

$${\vec{O}}_i={\left(\begin{array}{c}{O}_{i1}\\ O_{i2}\\ 0\\ O_{i3}\\ ⋮\\ O_{ij}\\ ⋮\\ O_{im}\end{array}\right)}_{m\times 1}\stackrel{f:{\mathbb{R}}^m\to {\mathbb{R}}^n}{\to }{\left(\begin{array}{c}0\\ O_{i1}\\ O_{i2}\\ O_{i3}\\ ⋮\\ O_{ij}\\ ⋮\\ O_{im}\\ 0\\ 0\\ ⋮\end{array}\right)}_{n\times 1}$$

This implies that it is sufficient to involve the large initial-beliefs vector itself in the consensus accumulation process by setting the corresponding unrelated metrics in all ${\vec{O}}_i$ to $0$ during so that they do not contribute anything to the process.

However, this relation assumes that we allocate equal amount of importance to every opinion that we come across, which is simply not true. The weightage of someone’s opinion whom you just met is going to be far less than a longtime friend, family member, etc. Thus, we update the relation to reflect the same:

$${\vec{O}}_1{H}_1(\Lambda )+{\vec{O}}_2{H}_1(\Lambda )+\cdots +{\vec{O}}_i{H}_i(\Lambda )+\cdots +{\vec{O}}_k{H}_k(\Lambda )+\Lambda$$

Where each $H_i(\Lambda )$ is a measure of closeness between you and the $i$-th person providing the opinion. Similarly, we also tend to listen more to people when their opinions generally represent our own. In which case the closeness represented by $H_i(\Lambda )$ can also contain a metric derived from the closeness of opinion between you and the $i$-th person. The way to think about this particular metric is having two distinct contributing factors:

$$H_i(\Lambda )=({\lambda }_i\cdot \Lambda {)}^{c_i}$$

Where ${\lambda }_i\cdot \Lambda$ is the similarly score between the initial-beliefs of $i$-th and you for that particular iteration, and $c\in (0,1)$ is the scaling factor determining how closely related you and the $i$-th person are. Both ${\lambda }_i$ and $\Lambda$ vectors are inhabitants of the same vector space ${\mathbb{R}}^n$, which is the “beliefs space” for our purposes. The only chief difference between them is the assumption that $\Lambda$ is not yet a crystallized opinion and needs to be refined iteratively, while we assume that ${\lambda }_i$ is already an existing preformed opinion that contributes to the formation of $\Lambda$ for that iteration. In this case we have initial-beliefs influencing how much effect it lets the individual opinions have during the consensus accumulation process. When the $i$-th person is distantly related or not related to you at all we will have $c\to 0$, but not exactly $c=0$ as that would invalidate their opinion, disqualifying them from the consensus accumulation process (and preventing you from making any new friends). If the internet is any indication, people take strangers’ opinions to heart everyday, so disallowing people from the opinion-formation process seems completely unnecessary.

We can try restricting the the lower bound of $c_i$ to a very small number $c_{\text{min}}$ which acts as the maximum factor by which two people can be separated (six degrees of separation?). The new bounds thus determined is: $c\in [c_{\text{min}},1)$

Another case to consider is when $c_i\to 1$. In this case as $c_i\to 1$, the closeness keeps increasing until you and the $i$-th person are exactly the same person, i.e., $c_i=1$. It would be useful to make additional amends to the bound for $c_i$ so that this cannot be the case. Since your original opinion is already accounted for by the $\Lambda$ factor. So, an adjusted factor might look like:

$$H_i(\Lambda )=({\lambda }_i\cdot \Lambda {)}^{c_i+c_{\text{min}}}$$

Here since we are already accounting for $c_{\text{min}}$, we have to readjust the bounds of $c_i$. The new resulting bounds of $c_i$ are $c_i\in (0,1-c_{\text{min}})$, but for a simple model with an undecided $c_{\text{min}}$ we can just go ahead with $c_{\text{min}}=0$. Thus, reverting back to our original bounds of $(0,1)$ and $H_i(\Lambda )=({\lambda }_i\cdot \Lambda {)}^{c_i}$, which is sufficiently expressive of what we wish to accomplish.

Note

There is one specific scenario that we should bring our attention to - what were to happen if both the similarity of the opinions is and the closeness is exceptionally high.

In cases where these two factors are exactly $1$; ${\vec{O}}_i{H}_i(\Lambda )$ becomes $O_i$ and is able to have the highest contribution to the overall consensus other than $\Lambda$ itself. This has the effect of one person reinforcing his own beliefs over and over again to be more and more extreme. This is an unstable state, as this leads to some opinions having more outsized impact in proportion to their actual contributions w.r.t. population size. This is not to say that it does not happen, but that impact should not be exactly identical to whatever impact the original opinion has. Interestingly, in cases with ${\vec{O}}_i{H}_i(\Lambda )={\vec{O}}_i$, the resulting states after many iterations has the appearance of symptoms similar to some delusional disorder or paranoid schizophrenia.

In contrast, for scenarios where both similarly score and closeness score tends to $1$ we have:

$${\lambda }_i\cdot \Lambda \to 1\text{ and }{c}_i\to 1⟹{\left({\lambda }_i\cdot \Lambda \right)}^{c_i}\to 1⟹H_i(\Lambda )\to 1\text{ and }{\vec{O}}_i{H}_i(\Lambda )\to {\vec{O}}_i$$

The resulting $i$-th opinion has ${\vec{O}}_i{H}_i(\Lambda )\to \vec{O}i{H}_i(\lambda )\to {\vec{O}}_i$, almost an equal amount of weightage as your own starting opinion $\Lambda$ but still not quite equal to it. Thus, the affects can be similar to the case when ${\vec{O}}_i{H}_i(\lambda )={\vec{O}}_i$, only a little less extreme as we have $\forall i$, ${\lambda }_i\cdot \Lambda <1⟹{\vec{O}}_i{H}_i(\Lambda )<{\vec{O}}_i$.

The resulting relation thus far can be summed up as:

$$\begin{array}{c}\mathcal{O}(\Lambda )=\sum _{i=1}^k{\vec{O}}_i{H}_i(\Lambda )+\Lambda ={\vec{O}}_1{H}_1(\Lambda )+{\vec{O}}_2{H}_1(\Lambda )+\cdots +{\vec{O}}_i{H}_i(\Lambda )+\cdots +{\vec{O}}_k{H}_k(\Lambda )+\Lambda \\ \mathcal{O}(\Lambda )={\vec{O}}_1({\lambda }_1\cdot \Lambda {)}^{c_1}+{\vec{O}}_2({\lambda }_2\cdot \Lambda {)}^{c_2}+\cdots +{\vec{O}}_i({\lambda }_i\cdot \Lambda {)}^{c_i}+\cdots +{\vec{O}}_k({\lambda }_k\cdot \Lambda {)}^{c_k}+\Lambda \\ \\ \mathcal{O}(\Lambda )=\sum _{i=1}^k{\vec{O}}_i({\lambda }_i\cdot \Lambda {)}^{c_i}+\Lambda \end{array}$$

Another particular scenario that is worth looking into is how people react to specific opinions which is both a factor of novelty and the way it is received for each individual. Say there are two factors: reception ($r\in \{-1,1\}$) and novelty ($N\in {\mathbb{R}}^{+}$), we will general perceive an opinion that is coloured by these factors. Reception provides some context as to how the individual reacts to the stated opinion, it can either be positive or negative. This is influenced by how the opinion has been stated and the manner in which a person comes across the mentioned opinion. Novelty provides some additional context into how the opinion has been perceived by the target individual initially. Novelty is tempered by anticipation factor $\epsilon$, which has the effect of dampening the effect of novelty over time. The updated weightage factor $H_i(\Lambda )$ can be represented as:

$$H_i(\Lambda )=(1+rN{e}^{-\epsilon t})({\lambda }_i\cdot \Lambda {)}^{c_i}$$

Thus, the resulting relation can be summed up as:

$$\begin{array}{c}\mathcal{O}(\Lambda )=\sum _{i=1}^k{\vec{O}}_i{H}_i(\Lambda )+\Lambda ={\vec{O}}_1{H}_1(\Lambda )+{\vec{O}}_2{H}_1(\Lambda )+\cdots +{\vec{O}}_i{H}_i(\Lambda )+\cdots +{\vec{O}}_k{H}_k(\Lambda )+\Lambda \\ \mathcal{O}(\Lambda )=\sum _{i=1}^k{\vec{O}}_i(1+rN{e}^{-\epsilon t})({\lambda }_i\cdot \Lambda {)}^{c_i}+\Lambda \end{array}$$

The updation rule per iteration for a single person in a population size $k$ can be seen as:

$$\boxed{\begin{array}{rlrl}{\mathcal{O}}_t({\Lambda }_t)& :=& & \sum _{i=1}^k{\vec{O}}_i{H}_i({\Lambda }_t)+{\Lambda }_t\\ {\Lambda }_{t+1}& :=& & {\mathcal{O}}_t({\Lambda }_t)\end{array}}$$

Note

The formulation thus far, has the effect of showing how opinions are influencing the initial-beliefs and vice versa. It it not entirely clear whether there is a stable state that the population might reach, especially considering the dynamic factors that vary the influence of each vector for every iteration.

Similarly, every initial-beliefs vectors ${\lambda }_i$ will updated by a separate consensus accumulation process. For representing each member of the $k$-sized population we iterate over the a set ${\aleph }_k=\{{\Lambda }^{(1)},{\Lambda }^{(2)},\dots ,{\Lambda }^{(k)}\}$, for each ${\Lambda }^{(j)}$. For each iteration $t$, we take every ${\Lambda }^{(j)}$ from ${\aleph }_k$ and set it as the auxiliary variable ${\Lambda }_t$ over which consensus accumulation is done for that iteration, this is done for each member of the population. Thus, $\forall {\Lambda }^{(j)}\in {\aleph }_k$ we will have a set of vectors of cardinality $k$, where each element of the set is an opinion ${\vec{O}}_i\in {\mathbb{R}}^n$, the resulting set ${\text{O}}_{t,{\Lambda }^{(j)}}$ contains all the opinion vectors for a single ${\Lambda }^{(j)}$ at iteration $t$. Finally, we can have additional level of hierarchy that contains the set of ${\text{O}}_{t,{\Lambda }^{(j)}}$ for all ${\Lambda }^{(j)}$, for some specific iteration $t$ (resulting set is ${\text{O}}_t$) and finally the set of all opinions for all iterations in $\text{O}$.

Therefore, the set $\text{O}$ has 3 degrees of hierarchy:

$$\begin{array}{rl}& \text{O}\{\begin{array}{l}{\text{O}}_1=\left\{{\text{O}}_{1,{\Lambda }^{(1)}},{\text{O}}_{1,{\Lambda }^{(2)}},\dots ,{\text{O}}_{1,{\Lambda }^{(j)}},{\text{O}}_{1,{\Lambda }^{(k)}}\right\}\\ {\text{O}}_2=\left\{{\text{O}}_{2,{\Lambda }^{(1)}},{\text{O}}_{2,{\Lambda }^{(2)}},\dots ,{\text{O}}_{2,{\Lambda }^{(j)}},{\text{O}}_{2,{\Lambda }^{(k)}}\right\}\\ ⋮\\ {\text{O}}_t=\left\{{\text{O}}_{t,{\Lambda }^{(1)}},{\text{O}}_{t,{\Lambda }^{(2)}},\dots ,{\text{O}}_{t,{\Lambda }^{(j)}},{\text{O}}_{t,{\Lambda }^{(k)}}\right\}\{\begin{array}{l}{\text{O}}_{t,{\Lambda }^{(1)}}={\left\{{\vec{O}}_1,{\vec{O}}_2,\dots ,{\vec{O}}_i,\dots ,{\vec{O}}_k\right\}}_1\\ {\text{O}}_{t,{\Lambda }^{(2)}}={\left\{{\vec{O}}_1,{\vec{O}}_2,\dots ,{\vec{O}}_i,\dots ,{\vec{O}}_k\right\}}_2\\ ⋮\\ {\text{O}}_{t,{\Lambda }^{(j)}}={\left\{{\vec{O}}_1,{\vec{O}}_2,\dots ,{\vec{O}}_i,\dots ,{\vec{O}}_k\right\}}_j\\ ⋮\\ {\text{O}}_{t,{\Lambda }^{(k)}}={\left\{{\vec{O}}_1,{\vec{O}}_2,\dots ,{\vec{O}}_i,\dots ,{\vec{O}}_k\right\}}_k\end{array}\\ ⋮\\ {\text{O}}_T=\left\{{\text{O}}_{T,{\Lambda }^{(1)}},{\text{O}}_{T,{\Lambda }^{(2)}},\dots ,{\text{O}}_{T,{\Lambda }^{(j)}},{\text{O}}_{T,{\Lambda }^{(k)}}\right\}\end{array}\end{array}$$

There may be some method to procedurally generate all opinion vectors for each iteration, for each member of the population instead of storing them in a hierarchical set-like structure. In any case, the collection of all such vectors at all levels will resemble the set $\text{O}$ in general. The overall algorithm for how a population evolves over several iteration has been mentioned below:

$$\begin{array}{rl}& \text{Algorithm }\text{Evolving Population Pattern}\\ & \text{Input: }\text{population }{\aleph }_k\text{, iterations }T\text{, opinions }\text{O}\\ & \text{Output: }\text{population }{\hat{\aleph }}_k\\ & t\leftarrow 0\\ & S\in {\mathbb{R}}^{k\times T}\\ & \text{while }t<T\text{ do}\\ & \text{for each }{\Lambda }^{(j)}\in {\aleph }_k\text{ do}\\ & {\Lambda }_t\leftarrow {\Lambda }^{(j)}\\ & \text{N }\leftarrow \{N_1,N_2,\dots ,N_k\}\\ & \epsilon \leftarrow \{{\epsilon }_1,{\epsilon }_2,\dots ,{\epsilon }_k\}\\ & \text{r }\leftarrow \{r_1,r_2,\dots ,r_k\}\\ & \lambda \leftarrow \{{\lambda }_i\mid \forall i\ne j,{\lambda }_i={\Lambda }^{(i)}\}\\ & \text{for each }{\vec{O}}_i\in {\text{O}}_{t,{\Lambda }^{(j)}}\text{ do}\\ & H_i({\Lambda }_t)\leftarrow (1+r_i{N}_i{e}^{-{\epsilon }_{i}t})({\lambda }_i\cdot \Lambda {)}^{c_i}\\ & S_{j,t}\leftarrow S_{j,t}+{\vec{O}}_i{H}_i({\Lambda }_t)\\ & S_{j,t+1}\leftarrow S_{j,t}+{\Lambda }_t\\ & \text{for each }{\Lambda }^{(j)}\in {\aleph }_k\text{ do}\\ & {\Lambda }^{(j)}\leftarrow S_{j,t+1}\\ & t\leftarrow t+1\end{array}$$

Info

This is a very subjective mathematical framework built on many shaky assumptions none of which might be true (or even accurate). However, I feel that this tries to adequately represent the role of consensus in the opinion-forming process. There will of course be factors that I may not have considered, and additional constraints I might have thought of.