Fermionic operatorial model of a system with competitive and cooperative interactions

TL;DR

This paper introduces a fermionic operatorial model for a system of agents with cooperative and competitive interactions, using a Hamiltonian framework and a novel dynamics rule to analyze wealth evolution among subgroups.

Contribution

It presents a new fermionic operatorial model with a $( ho, ext{Heisenberg})$-induced dynamics approach for systems with mixed interactions, highlighting the efficiency of cooperative behavior.

Findings

Cooperative subgroup consistently outperforms others in wealth improvement.

The $( ho, ext{Heisenberg})$-induced dynamics effectively models wealth evolution.

Numerical simulations confirm the dominance of cooperation in the system.

Abstract

An operatorial model of a system made by $N$ agents interacting each other with mechanisms that can be thought of as cooperative or competitive is presented. We associate to each agent an annihilation, creation and number fermionic operator, and interpret the mean values of the number operators over an initial condition as measures of the agents' wealth status. The dynamics of the system is assumed to be ruled by a Hermitian Hamiltonian operator $\mathcal{H}$, and the classical Heisenberg view is used. The dynamical outcome is then enriched by using the recently introduced variant of $(\mathcal{H},\rho)$--induced dynamics, where $\rho$ denotes a rule that periodically modifies some of the parameters involved in $\mathcal{H}$. The agents are partitioned in three subgroups, one interacting each other only with a competitive mechanism, one interacting each other only with a cooperative mechanism, and one opportunist subgroup able to compete and cooperate. Some numerical simulations show that the $(\mathcal{H},\rho)$--induced dynamics approach makes, in all the cases, the cooperative subgroup definitely to be more efficient in improving its wealth status than the other subgroups.