Independent Learning in Mean-Field Games: Satisficing Paths and Convergence to Subjective Equilibria

Independent learners are learning agents that naively employ single-agent learning algorithms in multi-agent systems, intentionally ignoring the effect of other strategic agents present in their environment. This paper studies $N$-player mean-field games from a decentralized learning perspective with two primary objectives: (i) to study the convergence properties of independent learners, and (ii) to identify structural properties of $N$-player mean-field games that can guide algorithm design. Toward the first objective, we study the learning iterates obtained by independent learners, and we use recent results from POMDP theory to show that these iterates converge under mild conditions. In particular, we consider four information structures corresponding to information at each agent: (1) global state + local action; (2) local state, mean-field state + local action; (3) local state, compressed mean-field state + local action; (4) local state with local action. We present a notion of subjective equilibrium suitable for the analysis of independent learners. Toward the second objective, we study a family of dynamical systems on the set of joint policies. The dynamical systems under consideration are subject to a so-called $\epsilon$-satisficing condition: agents who are subjectively $\epsilon$-best-responding at a given joint policy do not change their policy. We establish a useful structural property relating to such dynamical systems. Finally, we develop an independent learning algorithm for $N$-player mean-field games that drives play to subjective $\epsilon$-equilibrium under self-play, exploiting the aforementioned structural properties to guarantee convergence of policies. Notably, we avoid requiring agents to follow the same policy (via a representative agent) during the learning process, which has been the typical approach in the existing literature on learning for mean-field games.