The framework of multi-agent learning explores the dynamics of how an agent's

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.

The framework of multi-agent learning explores the dynamics of how an agent's

We study learnability of mixed-strategy Nash Equilibrium (NE) in general finite games using higher-order replicator dynamics as well as classes of higher-order uncoupled heterogeneous dynamics. In higher-order uncoupled learning dynamics, players have no access to utilities of opponents (uncoupled) but are allowed to use auxiliary states to further process information (higher-order). We establish a link between uncoupled learning and feedback stabilization with decentralized control. Using this association, we show that for any finite game with an isolated completely mixed-strategy NE, there exist higher-order uncoupled learning dynamics that lead (locally) to that NE. We further establish the lack of universality of learning dynamics by linking learning to the control theoretic concept of simultaneous stabilization. We construct two games such that any higher-order dynamics that learn the completely mixed-strategy NE of one of these games can never learn the completely mixed-strategy NE of the other. Next, motivated by imposing natural restrictions on allowable learning dynamics, we introduce the Asymptotic Best Response (ABR) property. Dynamics with the ABR property asymptotically learn a best response in environments that are asymptotically stationary. We show that the ABR property relates to an internal stability condition on higher-order learning dynamics. We provide conditions under which NE are compatible with the ABR property. Finally, we address learnability of mixed-strategy NE in the bandit setting using a bandit version of higher-order replicator dynamics.

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.