Higher-Order Uncoupled Dynamics Do Not Lead to Nash Equilibrium - Except When They Do

The framework of multi-agent learning explores the dynamics of how an agent's strategies evolve in response to the evolving strategies of other agents. Of particular interest is whether or not agent strategies converge to well known solution concepts such as Nash Equilibrium (NE). In "higher order'' learning, agent dynamics include auxiliary states that can capture phenomena such as path dependencies. We introduce higher-order gradient play dynamics that resemble projected gradient ascent with auxiliary states. The dynamics are "payoff based'' and "uncoupled'' in that each agent's dynamics depend on its own evolving payoff and has no explicit dependence on the utilities of other agents. We first show that for any specific game with an isolated completely mixed-strategy NE, there exist higher-order gradient play dynamics that lead (locally) to that NE, both for the specific game and nearby games with perturbed utility functions. Conversely, we show that for any higher-order gradient play dynamics, there exists a game with a unique isolated completely mixed-strategy NE for which the dynamics do not lead to NE. Finally, we show that convergence to the mixed-strategy equilibrium in coordination games, comes at the expense of the dynamics being inherently internally unstable.