An Agent-based Model for Competitive Agents
Mohammad Daneshvar, Mandana Delavari
TL;DR
This paper addresses competitive multi-agent wealth-exchange dynamics by building a stochastic agent-based model grounded in continuous-time Markov chains. It develops a microscopic, discrete-state description with state $X_i(t)$ and pairwise transitions, derives the Master Equation for the transition flux, and imposes wealth-conservation with exit upon depletion. A continuum limit is then obtained, yielding a Fokker-Planck equation with drift, diffusion, and cross-terms, and the authors solve this for linear consumption in two- and three-agent setups, including diffusion with elastic and sticky boundaries. The work provides a mathematically tractable framework to study transient densities in competitive environments, with potential applications in finance and social science, and offers explicit expressions and boundary-conditioned solutions for low-dimensional cases.
Abstract
In this paper, we analyze the behavior of a multi-agent system driven by the interactions of agents within a competitive environment. To achieve this, we describe the transition probabilities that underlie the system's stochastic nature. We also derive the Fokker-Planck equations for the density distribution of the number of agents in the system and solve these equations for specific cases.