Discrete mean field games: Existence of equilibria and convergence

Abstract

We consider mean field games with discrete state spaces (called discrete mean field games in the following) and we analyze these games in continuous and discrete time, over finite as well as infinite time horizons. We prove the existence of a mean field equilibrium assuming continuity of the cost and of the drift. These conditions are more general than the existing papers studying finite state space mean field games. Besides, we also study the convergence of the equilibria of N -player games to mean field equilibria in our four settings. On the one hand, we define a class of strategies in which any sequence of equilibria of the finite games converges weakly to a mean field equilibrium when the number of players goes to infinity. On the other hand, we exhibit equilibria outside this class that do not converge to mean field equilibria and for which the value of the game does not converge. In discrete time this nonconvergence phenomenon implies that the Folk theorem does not scale to the mean field limit.

SIR Dynamics with Vaccination in a Large Configuration Model

“…In Theorem 2 of [17], the authors show that if the dynamics and the costs are continuous functions of the involved variables there always exists a mean-field equilibrium in such games; thus the existence of a solution for our problem follows from their result, since the cost is linear in I , the function g is analytic, and the rates of transition for the new individual depend linearly on p I or are constant for a fixed π .…”

“…An infected individual recovers at rate γ , and a susceptible can choose its own vaccination rate π , going to recovery state. The optimal strategy π played for all the players is called a mean-field equilibrium, defined as a fixed point of the best response functional, this is, π ∈ B R(π ), which minimizes a properly defined cost functional [17]. A mean eld equilibrium consists in a strategy where no player has an incentive to deviate from the common strategy for their own benefit.…”

Vaccination in a Large Population: Mean Field Equilibrium Versus Social Optimum

“…SO always exist in our SIRV model (The proof uses the compacity of the strategy space for the weak topology and is not detailed here). For more details on these two proofs of existence, see [4].…”

“…) behaves as a fluid, whose evolution follows the Kolmogorov equations of the individual Markov chain (see for example [4,5]). Under a given vaccination strategy π.…”

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.…”