Homological Barriers to Stable Local Nash Dynamics in Quadratic Zero-Sum Games

We identify a topological obstruction to the standard stable-attractor template for learning local Nash equilibria, also known as first-order Nash equilibria (FONE), in constrained non-concave differentiable games. We consider continuous deterministic learning dynamics that leave each FONE point fixed and aim to attract every initial condition to the full FONE set. Our main message is that this familiar convergence template can fail for a topological reason. Targets with holes, such as loops, cannot be stable global attractors for such dynamics on contractible state spaces. We then exhibit this obstruction in a simple non-concave quadratic zero-sum game on a box. Its exact FONE set is a diagonal six-edge loop together with an isolated point, so the loop creates the required topological obstruction. As a result, no continuous pointwise-stationary learning rule covered by our framework can make this exact FONE set a stable global attractor. We show that degree two is the minimal polynomial degree at which this obstruction can arise. We further amplify the construction to arbitrary homological degree. Finally, by studying the topology in sufficiently small perturbed version of the game, we show that an open positive-volume family of quadratic games retain the same homological obstruction. Thus, even in quadratic zero-sum games, this phenomenon is not a pathology of a single construction; instead, the geometry of projected first-order stationarity can create an intrinsic barrier to stable global FONE dynamics.