Stochastic Games for Fuel Follower Problem: $N$ versus Mean Field Game

Abstract

In this paper we formulate and analyze an N -player stochastic game of the classical fuel follower problem and its Mean Field Game (MFG) counterpart. For the N -player game, we obtain the Nash Equilibrium (NE) explicitly by deriving and analyzing a system of Hamilton-Jacobi-Bellman (HJB) equations, and by establishing the existence of a unique strong solution to the associated Skorokhod problem on an unbounded polyhedron with an oblique reflection. For the MFG, we derive a bang-bang type NE under some mild technical conditions and by the viscosity solution approach. We also show that this solution is an -NE to the N -player game, with = O( 1 √ N ). The N -player game and the MFG differ in that the NE for the former is state dependent while the NE for the latter is threshold-type bang-bang policy where the threshold is state independent . Our analysis shows that the NE for a stationary MFG may not be the NE for the corresponding MFG.

Pareto Optima for a Class of Singular Control Games

“…In the case where a i = 1 N , q i = q j , ν i = ν j and K ± i = K ± j for i = j, the payoff structure is symmetric under permutation of indices and this can be formulated as mean field game, which was studied in (Lasry & Lions, 2007;Guo & Xu, 2019). However we shall not need this assumption and will treat below the case of a more general, not necessarily symmetric, cost function h i (X X X t ).…”

“…The first two terms in (1.3) correspond to the 'baseline' (uncontrolled) diffusion dynamics, and the last two term correspond to the control ξ ξ ξ i = (ξ i,+ , ξ i,− ), modeled as a pair of non-decreasing càdlàg processes, leading to a singular control problem (Karatzas, 1982) for each player. Nash equilibria for such singular control games have been studied in (Guo & Xu, 2019). In the present work, we focus on Pareto-optimal outcomes.…”

“…Proof. Similar to the derivation in (Guo & Xu, 2019), we have the following Quasi-variational inequalities for the Nash equilibrium of game (5.1) with J 1 and J 2 and K 1 = K 2 = K,…”

Interbank lending with benchmark rates: Pareto optima for a class of singular control games

“…In the case where , , and for , the payoff structure is symmetric under permutation of indices and this can be formulated as mean field game (Lasry & Lions, 2007; Huang et al., 2006), which was studied under Nash equilibrium in (Guo & Xu, 2019). However we shall not need this assumption and will treat below the case of a more general, not necessarily symmetric, cost function .…”

Mean Field Games with Singular Controls

“…Mean field games (MFGs) with singular controls were introduced in [20], in which we consider general MFGs with singular controls and establish the existence of equilibria result by a probabilistic approach. Analytically, [24] and [25] characterized the equilibria of MFGs with singular controls of bounded velocity and mean field fuel follower games in infinite horizon, respectively. By now, all results on MFGs with singular controls are based on models with the interaction through states and there is no result on MFGs with singular controls under strategic interaction, to the best of our knowledge.…”