  What does it mean to fully understand the behavior of a network of adaptive
agents? The golden standard typically is the behavior of learning dynamics in
potential games, where many evolutionary dynamics, e.g., replicator, are known
to converge to sets of equilibria. Even in such classic settings many critical
questions remain unanswered. We examine issues such as:
  Point-wise convergence: Does the system actually equilibrate even in the
presence of continuums of equilibria?
  Computing regions of attraction: Given point-wise convergence can we compute
the region of asymptotic stability of each equilibrium (e.g., estimate its
volume, geometry)?
  System invariants: Invariant functions remain constant along every system
trajectory. This notion is orthogonal to the game theoretic concept of a
potential function, which always strictly increases/decreases along system
trajectories. Do dynamics in potential games exhibit invariant functions? If
so, how many? How do these functions look like?
  Based on these geometric characterizations, we propose a novel quantitative
framework for analyzing the efficiency of potential games with many equilibria.
The predictions of different equilibria are weighted by their probability to
arise under evolutionary dynamics given uniformly random initial conditions.
This average case analysis is shown to offer novel insights in classic game
theoretic challenges, including quantifying the risk dominance in stag-hunt
games and allowing for more nuanced performance analysis in networked
coordination and congestion games with large gaps between price of stability
and price of anarchy.
