  Multiagent systems where agents interact among themselves and with a
stochastic environment can be formalized as stochastic games. We study a
subclass named Markov potential games (MPGs) that appear often in economic and
engineering applications when the agents share a common resource. We consider
MPGs with continuous state-action variables, coupled constraints and nonconvex
rewards. Previous analysis followed a variational approach that is only valid
for very simple cases (convex rewards, invertible dynamics, and no coupled
constraints); or considered deterministic dynamics and provided open-loop (OL)
analysis, studying strategies that consist in predefined action sequences,
which are not optimal for stochastic environments. We present a closed-loop
(CL) analysis for MPGs and consider parametric policies that depend on the
current state. We provide easily verifiable, sufficient and necessary
conditions for a stochastic game to be an MPG, even for complex parametric
functions (e.g., deep neural networks); and show that a closed-loop Nash
equilibrium (NE) can be found (or at least approximated) by solving a related
optimal control problem (OCP). This is useful since solving an OCP--which is a
single-objective problem--is usually much simpler than solving the original set
of coupled OCPs that form the game--which is a multiobjective control problem.
This is a considerable improvement over the previously standard approach for
the CL analysis of MPGs, which gives no approximate solution if no NE belongs
to the chosen parametric family, and which is practical only for simple
parametric forms. We illustrate the theoretical contributions with an example
by applying our approach to a noncooperative communications engineering game.
We then solve the game with a deep reinforcement learning algorithm that learns
policies that closely approximates an exact variational NE of the game.
