  Regret analysis is challenging in Multi-Agent Reinforcement Learning (MARL)
primarily due to the dynamical environments and the decentralized information
among agents. We attempt to solve this challenge in the context of
decentralized learning in multi-agent linear-quadratic (LQ) dynamical systems.
We begin with a simple setup consisting of two agents and two dynamically
decoupled stochastic linear systems, each system controlled by an agent. The
systems are coupled through a quadratic cost function. When both systems'
dynamics are unknown and there is no communication among the agents, we show
that no learning policy can generate sub-linear in $T$ regret, where $T$ is the
time horizon. When only one system's dynamics are unknown and there is
one-directional communication from the agent controlling the unknown system to
the other agent, we propose a MARL algorithm based on the construction of an
auxiliary single-agent LQ problem. The auxiliary single-agent problem in the
proposed MARL algorithm serves as an implicit coordination mechanism among the
two learning agents. This allows the agents to achieve a regret within
$O(\sqrt{T})$ of the regret of the auxiliary single-agent problem.
Consequently, using existing results for single-agent LQ regret, our algorithm
provides a $\tilde{O}(\sqrt{T})$ regret bound. (Here $\tilde{O}(\cdot)$ hides
constants and logarithmic factors). Our numerical experiments indicate that
this bound is matched in practice. From the two-agent problem, we extend our
results to multi-agent LQ systems with certain communication patterns.
