Relating maximum entropy, resilient behavior and game-theoretic equilibrium feedback operators in multi-channel systems

TL;DR

This paper explores the relationship between maximum entropy, equilibrium states, and game-theoretic feedback operators in multi-channel systems, revealing how these elements influence system resilience and dynamics.

Contribution

It introduces a novel connection between stationary density functions, game-theoretic equilibrium feedback, and maximum entropy behavior in multi-channel systems.

Findings

Existence of equilibrium feedback operators leading to maximum entropy states.

Stationary density functions characterize system dynamics and equilibrium.

Resilience of equilibrium feedback under small perturbations is analyzed.

Abstract

In this paper, we first draw a connection between the existence of a stationary density function (which corresponds to an equilibrium state in the sense of statistical mechanics) and a set of feedback operators in a multi-channel system that strategically interacts in a game-theoretic framework. In particular, we show that there exists a set of (game-theoretic) equilibrium feedback operators such that the composition of the multi-channel system with this set of equilibrium feedback operators, when described by density functions, will evolve towards an equilibrium state in such a way that the entropy of the whole system is maximized. As a result of this, we are led to study, by a means of a stationary density function (i.e., a common fixed-point) for a family of Frobenius-Perron operators, how the dynamics of the system together with the equilibrium feedback operators determine the evolution of the density functions, and how this information translates into the maximum entropy behavior of the system. Later, we use such results to examine the resilient behavior of this set of equilibrium feedback operators, when there is a small random perturbation in the system.