A Necessary and Sufficient Condition Beyond Monotonicity for Convergence of the Gradient Play in Continuous Games

In this article, we aim to answer the following question: What kind of games can guarantee convergence of the (full-information or partial-information, continuous-time or discrete-time) gradient play? To the best of our knowledge, current works on Nash equilibrium seeking are mainly established on the monotonicity condition. We introduce a concept called stability condition to continuous games, which includes the monotonicity condition as a special case. We prove that the stability condition is necessary and sufficient for convergence of gradient play. In detail, we show that, if the step size is fixed and within a given bound, the full-information and partial-information gradient play is guaranteed to converge to the Nash equilibrium in strongly stable games. If the step size is diminishing, then convergence of the gradient play can be obtained for strictly stable games. We present a game that is stable but not monotone to illustrate our theoretical developments.