I am working on a simple individual based model that aggregates information. I am not a mathematician, but I would like to be as precise as possible with the terminology used to describe the system and the aggregation rules, so I thought this forum might be the appropriate place for advice (if you think there is another forum more suitable, please let me know). I try to define things as follows:
Let $X=\{x_{1}, x_{2},\ldots, x_{O}\}$ be a vector of $O$ opinions, and let $A=\{a_{i}, a_{j},\ldots, a_{N}\}$ be the set of agents in a population. In the initial state, each agent $a_{n} \in A$ is randomly assigned an opinion $x_{o} \in X$ selected from $X$ without replacement. We use $x_{nr}$ to denote the opinion of $a_{n}$ at round $r$.
At each round:
Each agent expresses one opinion from $X$ (according to the probabilistic model MI). Thus, the vector of opinions $X_{r}$ in the population is a vector of $N$ (one per agent) opinions $(x_{ir}, x_{jr},\ldots, x_{Nr})$, where $x_{ir}$ is the opinion produced by agent $a_{i}$ at round $r$, and so on.
Each agent is defined by a vector of preferences $S$, which is a probability distribution on $X$: that is, at each round, an agent $a_{n}$ has a vector of preferences $S_{n}$ = $(s_{n1r}, s_{n2r},\ldots, s_{nOr})$. $S$ assigns a number between $0$ and $1$ to each opinion in $X$, where $s_{n1r}$ is the value assigned by an agent $a_{n}$ to opinion $x_{1}$ at round $r$, and so on. Thus, $s_{nor}$ is the value assigned by a random agent $a_{n}$ to an opinion $x_{o}$ at round $r$. S evolves according to model MII.
I use two aggregation rules to create an emergent vector called $G$ = $(g_{1r}, g_{2r},\ldots, g_{Or})$, which contains one value for each possible opinion in X.
First rule: At each round $r$, the value $g_{or}$ assigned to a given opinion $x_{o}$ is calculated as an arithmetic mean of the $N$ values $s_{ior}, s_{jor},\ldots, s_{Nor}$, where $s_{ior}$ corresponds to the value assigned by agent $a_{i}$ to opinion $x_{o}$, and $s_{jor}$ corresponds to the value assigned by agent $a_{j}$ to opinion $x_{o}$, and so on. That is, the value of $g_{or}$ at round $r$ is:
\begin{equation} \label{Eq2} g_{or}= \frac{1}{N} \sum_{nor=1}^{N} s_{nor} = \frac{s_{ior}+s_{jor}+\ldots+s_{Nor}}{N} \end{equation}
Second rule: At each round $r$, the value $g_{or}$ assigned to a given opinion $x_{o}$ is calculated as the relative frequency $f$ of opinion $x_{o}$ in $X_{r}$.
\begin{equation} g_{or}= f_{x_{o}} = \frac{n_{x_{o}}}{O} = \frac{n_{x_{o}}}{\sum_{x_{o}}n_{x_{o}}} \end{equation}
where n stands for the number of the specific opinions $x_{o}$ found in $X_{r}$.
Probabilistic model MI:
At each round, and for each agent, the model yields a probability distribution of opinions for a given history ($h$) (We use the apostrophe ($\prime$) to denote the probabilistic complement): \begin{equation} \label{eq:3} \Pr(x_{nor}\mid h_{nr})= C f(x_{o}\mid h_{nr}) + C's_{nor} \end{equation}
where $\Pr(x_{nor}\mid h_{nr})$ corresponds to the probability that an agent $a_{n}$ produces opinion $x_{o}$ at round $r$ given the specific history of agent $a_{n}$ by round $r$. The term $f(x_{o}\mid h_{nr})$ stands for the relative frequency of opinion $x_{o}$ in agent's history. $C$ is a constant that takes values form 0 to 1.
Model MII:
At each round, for a given agent $a_{n}$, each value of its vector of preferences $S$ on $X$ is updated as follows:
\begin{equation} \label{eqvalues} s_{nor+1}(s_{nor},x_{nor},g_{or},\epsilon)= \begin{cases} \epsilon \cdot g_{or} + (\epsilon'){ |x_{o}|} & \text{if } x_{o} \in h_{nr} \\ s_{nor}, & \text{otherwise} \end{cases} \end{equation}
where $s_{nor+1}$ stands for the value that agent $a_{n}$ gives to opinion $x_{o}$ at round $r+1$, $\epsilon$ is a constant that takes values form 0 to 1. At each round, $|x_{o}|$ is 1 if the specific opinion $x_{o}$ is produced by the agent in the current round $r$, 0 otherwise.
Questions:
- Is this description mathematically acceptable and clear?
- Is there a better way to describe this system more precisely mathematically? If so, suggestions and rewriting of the system are greatly appreciated.
- Regarding the rules of opinion aggregation, is anyone aware of classical literature where these methods (simple arithmetic mean) and (relative frequency) have been used?
Thank you very much.
