An Operatorial View of Competition and Cooperation in a Network of Economic Agents

A network of agents interacting both with competitive and/or cooperative mechanisms is modeled by using fermionic ladder operators. The time evolution of the network is assumed to be governed by a Hermitian time-independent Hamiltonian operator, and the mean values of the number operators are interpreted as a measure of the wealth status of the agents. Besides the standard Heisenberg view, we use the recently introduced <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi mathvariant="script">H</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ρ</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> -induced dynamics approach to account for some actions able to provide a self-adjustment of the network according to its time evolution. Some numerical simulations are presented and discussed. Remarkably, we show that, in a network where cooperation may emerge, the average wealth of the agents is higher, and there is a very low level of inequality.