Sequential decomposition of discrete-time partially observed mean-field games

We consider both finite and infinite horizon discounted mean-field games with imperfect observations 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 state that she doesn’t perfectly observe but rather makes noisy observations about it. We first define a private belief stateof a player that is her belief on her own type given her information till time t and a joint common state on an archetypical player’s type and her private belief. Mean-field equilibrium (MFE) is then defined as a solution of coupled Bellman dynamic programming backward equation and Fokker Planck forward equation, where a player’s strategy in an MFE depends on both, her private belief state and current joint common 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 the existence of this fixed-point equation for each time t.