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The framework of multi-agent learning explores the dynamics of how individualagent strategies evolve in response to the evolving strategies of other agents.Of particular interest is whether or not agent strategies converge to wellknown solution concepts such as Nash Equilibrium (NE). Most "fixed order"learning dynamics restrict an agent's underlying state to be its own strategy.In "higher order" learning, agent dynamics can include auxiliary states thatcan capture phenomena such as path dependencies. We introduce higher-ordergradient play dynamics that resemble projected gradient ascent with auxiliarystates. The dynamics are "payoff based" in that each agent's dynamics depend onits own evolving payoff. While these payoffs depend on the strategies of otheragents in a game setting, agent dynamics do not depend explicitly on the natureof the game or the strategies of other agents. In this sense, dynamics are"uncoupled" since an agent's dynamics do not depend explicitly on the utilityfunctions of other agents. We first show that for any specific game with anisolated completely mixed-strategy NE, there exist higher-order gradient playdynamics that lead (locally) to that NE, both for the specific game and nearbygames with perturbed utility functions. Conversely, we show that for anyhigher-order gradient play dynamics, there exists a game with a unique isolatedcompletely mixed-strategy NE for which the dynamics do not lead to NE. Theseresults build on prior work that showed that uncoupled fixed-order learningcannot lead to NE in certain instances, whereas higher-order variants can.Finally, we consider the mixed-strategy equilibrium associated withcoordination games. While higher-order gradient play can converge to suchequilibria, we show such dynamics must be inherently internally unstable.

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