# Modeling interactions between political parties and electors

## TL;DR

This paper develops and compares three models of political party interactions with electors and among themselves, analyzing how these interactions influence parties' decisions to form alliances through decision functions.

## Contribution

It introduces and contrasts three novel models of political dynamics, incorporating electors' influence and internal party interactions, with detailed analysis and numerical simulations.

## Key findings

- Electors' influence drives parties towards asymptotic decision values.
- Inclusion of rules modifies electors' status and decision functions at specific times.
- Without electors, parties' decisions oscillate near initial values.

## Abstract

In this paper we extend some recent results on an operatorial approach to the description of alliances between political parties interacting among themselves and with a basin of electors. In particular, we propose and compare three different models, deducing the dynamics of their related {\em decision functions}, i.e. the attitude of each party to form or not an alliance. In the first model the interactions between each party and their electors are considered. We show that these interactions drive the decision functions towards certain asymptotic values depending on the electors only: this is the {\em perfect party}, which behaves following the electors' suggestions. The second model is an extension of the first one in which we include a $rule$ which modifies the status of the electors, and of the decision functions as a consequence, at some specific time step. In the third model we neglect the interactions with the electors while we consider cubic and quartic interactions between the parties and we show that we get (slightly oscillating) asymptotic values for the decision functions, close to their initial values. This is the {\em real party}, which does not listen to the electors. Several explicit situations are considered in details and numerical results are also shown.