  A fundamental open problem in monotone game theory is the computation of a
specific generalized Nash equilibrium (GNE) among all the available ones, e.g.
the optimal equilibrium with respect to a system-level objective. The existing
GNE seeking algorithms have in fact convergence guarantees toward an arbitrary,
possibly inefficient, equilibrium. In this paper, we solve this open problem by
leveraging results from fixed-point selection theory and in turn derive
distributed algorithms for the computation of an optimal GNE in monotone games.
We then extend the technical results to the time-varying setting and propose an
algorithm that tracks the sequence of optimal equilibria up to an asymptotic
error, whose bound depends on the local computational capabilities of the
agents.
