Nonlinear Markov Games on a Finite State Space (Mean-field and Binary Interactions)

Abstract

Managing large complex stochastic systems, including competitive interests, when one or several players can control the behavior of a large number of particles (agents, mechanisms, vehicles, subsidiaries, species, police units, etc), say $N_k$ for a player $k$, the complexity of the game-theoretical (or Markov decision) analysis can become immense as $N_k\to \infty$. However, under rather general assumptions, the limiting problem as all $N_k\to \infty$ can be described by a well manageable deterministic evolution. In this paper we analyze some simple situations of this kind proving the convergence of Nash-equilibria for finite games to equilibria of a limiting deterministic differential game.