Stationary Equilibria of Mean Field Games with Finite State and Action Space

Abstract

Mean field games formalize dynamic games with a continuum of players and explicit interaction where the players can have heterogeneous states. As they additionally yield approximate equilibria of corresponding N -player games, they are of great interest for socio-economic applications. However, most techniques used for mean field games rely on assumptions that imply that for each population distribution there is a unique optimizer of the Hamiltonian. For finite action spaces, this will only hold for trivial models. Thus, the techniques used so far are not applicable. We propose a model with finite state and action space, where the dynamics are given by a time-inhomogeneous Markov chain that might depend on the current population distribution. We show existence of stationary mean field equilibria in mixed strategies under mild assumptions and propose techniques to compute all these equilibria. More precisely, our results allow -given that the generators are irreducible -to characterize the set of stationary mean field equilibria as the set of all fixed points of a map completely characterized by the transition rates and rewards for deterministic strategies. Additionally, we propose several partial results for the case of non-irreducible generators and we demonstrate the presented techniques on two examples.

Continuous-Time Mean Field Games with Finite State Space and Common Noise

“…There is also a strand of literature on mean field games with finite state spaces, including [2,15,18,24,30,31,34,49] as well as [9, §7.2]. In a recent article, [22] provide an extension of [31] to mean field interactions that occur not only through the agents' states, but also through their controls.…”

Competition versus Cooperation: A class of solvable mean field impulse control problems

“…Only in the case of linear dynamics and quadratic costs, explicit solutions have been obtained. More recently, several other types of mean field games have been introduced, such as finite state mean field games [16,24,34] or mean field games of stopping [35,36,15], in which case it is sometimes possible to obtain other explicitly solvable models which might yield a deeper understanding of the nature of mean field equilibria.…”

Continuous-Time Mean Field Games with Finite State Space and Common Noise

“…There is also a strand of literature on mean field games with finite state spaces, including [GMS10], [GMS13], [Gué15], [BC18], [CP19a], [DGG19], [CF20], [Neu20] as well as [CD18a, §7.2]. In a recent article, [CW18] provide an extension of [GMS13] to mean field interactions that occur not only through the agents' states, but also through their controls.…”