Ergodic properties of multi-agent iterated function systems with subjective probabilities

We propose a novel framework for analysing multi-agent systems (MASs) that integrates behavioural subjectivity into the stochastic dynamics of agent interactions. Specifically, we extend classical Iterated Function Systems (IFS) by introducing subjective probability distortions into agent decision-making, inspired by Prospect Theory. Each agent updates its state via contractive mappings influenced by peers, weighted by both structural interaction strengths and cognitively biased probability weights. We derive sufficient conditions for ergodicity–namely, contraction of influence maps, irreducibility of the interaction network, and full support of subjective weights–by establishing contraction in the Wasserstein-1 metric. Numerical experiments validate our theoretical findings, demonstrating convergence to a unique invariant measure across varied initial conditions. This is the first work to rigorously establish ergodicity in a multi-agent IFS framework incorporating behavioural distortions, opening new directions in the control of complex systems with human-like agents.