Best-Response Dynamics in Continuous Potential Games: Non-Convergence to Saddle Points

The paper studies properties of best-response (BR) dynamics in potential games with continuous action sets. It is known that BR dynamics converge to the set of Nash equilibria (NE) in potential games. The set of NE in potential games is composed of local maximizers and saddle points of the potential function. The paper studies non-convergence of BR dynamics to saddle points of the potential function. Under relatively mild assumptions it is shown that BR dynamics may only converge to an interior saddle-point from a measure-zero set of initial conditions. This provides a weak stable manifold theorem in this context.