Sequential Decomposition of Mean-Field Games

We consider both finite and infinite horizon discounted mean-field games where there is a large population of homogeneous players sequentially making strategic decisions and each player is affected by other players through an aggregate population state. Each player has a private type that only she observes and all players commonly observe a mean-field population state which represents the empirical distribution of other players' types. Mean-field equilibrium (MFE) in such games is defined as solution of coupled Bellman dynamic programming backward equation and Mckean-Vlasov forward equation, where a player's strategy in an MFE depend on both, her private type and current population state. In this paper, we present a novel backward recursive algorithm to compute all MFEs of the game. Each step in this algorithm consists of solving a fixed-point equation. We provide sufficient conditions that guarantee existence of this fixed-point equation for each time t. Using this algorithm, we study versions of security problem in cyber-physical system where infected nodes put negative externality on the system, and each node makes a decision to get vaccinated. We numerically compute MFE of the game.