  We study a multi-agent mean field type control problem in discrete time where
the agents aim to find a socially optimal strategy and where the state and
action spaces for the agents are assumed to be continuous. The agents are only
weakly coupled through the distribution of their state variables. The problem
in its original form can be formulated as a classical Markov decision process
(MDP), however, this formulation suffers from several practical difficulties.
In this work, we attempt to overcome the curse of dimensionality, coordination
complexity between the agents, and the necessity of perfect feedback collection
from all the agents (which might be hard to do for large populations.)
  We provide several approximations: we establish the near optimality of the
action and state space discretization of the agents under standard regularity
assumptions for the considered formulation by constructing and studying the
measure valued MDP counterpart for finite and infinite population settings. It
is a well known approach to consider the infinite population problem for
mean-field type models, since it provides symmetric policies for the agents
which simplifies the coordination between the agents. However, the optimality
analysis is harder as the state space of the measure valued infinite population
MDP is continuous (even after space discretization of the agents). Therefore,
as a final step, we provide further approximations for the infinite population
problem by focusing on smaller sized sub-population distributions.
