  We initiate the study of a quantity that we call coordination complexity. In
a distributed optimization problem, the information defining a problem instance
is distributed among $n$ parties, who need to each choose an action, which
jointly will form a solution to the optimization problem. The coordination
complexity represents the minimal amount of information that a centralized
coordinator, who has full knowledge of the problem instance, needs to broadcast
in order to coordinate the $n$ parties to play a nearly optimal solution.
  We show that upper bounds on the coordination complexity of a problem imply
the existence of good jointly differentially private algorithms for solving
that problem, which in turn are known to upper bound the price of anarchy in
certain games with dynamically changing populations.
  We show several results. We fully characterize the coordination complexity
for the problem of computing a many-to-one matching in a bipartite graph by
giving almost matching lower and upper bounds.Our upper bound in fact extends
much more generally, to the problem of solving a linearly separable convex
program. We also give a different upper bound technique, which we use to bound
the coordination complexity of coordinating a Nash equilibrium in a routing
game, and of computing a stable matching.
