  We study distributed planning for multi-robot systems to provide optimal
service to cooperative tasks that are distributed over space and time. Each
task requires service by sufficiently many robots at the specified location
within the specified time window. Tasks arrive over episodes and the robots try
to maximize the total value of service in each episode by planning their own
trajectories based on the specifications of incoming tasks. Robots are required
to start and end each episode at their assigned stations in the environment. We
present a game theoretic solution to this problem by mapping it to a game,
where the action of each robot is its trajectory in an episode, and using a
suitable learning algorithm to obtain optimal joint plans in a distributed
manner. We present a systematic way to design minimal action sets (subsets of
feasible trajectories) for robots based on the specifications of incoming tasks
to facilitate fast learning. We then provide the performance guarantees for the
cases where all the robots follow a best response or noisy best response
algorithm to iteratively plan their trajectories. While the best response
algorithm leads to a Nash equilibrium, the noisy best response algorithm leads
to globally optimal joint plans with high probability. We show that the
proposed game can in general have arbitrarily poor Nash equilibria, which makes
the noisy best response algorithm preferable unless the task specifications are
known to have some special structure. We also describe a family of special
cases where all the equilibria are guaranteed to have bounded suboptimality.
Simulations and experimental results are provided to demonstrate the proposed
approach.
