We introduce -Rank, a principled evolutionary dynamics methodology for the evaluation and ranking of agents in large-scale multi-agent interactions, grounded in a novel dynamical game-theoretic solution concept called Markov-Conley chains (MCCs). The approach leverages continuous- and discrete-time evolutionary dynamical systems applied to empirical games, and scales tractably in the number of agents, the type of interactions, and the type of empirical games (symmetric and asymmetric). Current models are fundamentally limited in one or more of these dimensions and are not guaranteed to converge to the desired game-theoretic solution concept (typically the Nash equilibrium). -Rank provides a ranking over the set of agents under evaluation and provides insights into their strengths, weaknesses, and long-term dynamics. This is a consequence of the links we establish to the MCC solution concept when the underlying evolutionary model's ranking-intensity parameter, , is chosen to be large, which exactly forms the basis of -Rank. In contrast to the Nash equilibrium, which is a static concept based on fixed points, MCCs are a dynamical solution concept based on the Markov chain formalism, Conley's Fundamental Theorem of Dynamical Systems, and the core ingredients of dynamical systems: fixed points, recurrent sets, periodic orbits, and limit cycles. -Rank runs in polynomial time with respect to the total number of pure strategy profiles, whereas computing a Nash equilibrium for a general-sum game is known to be intractable. We introduce proofs that not only provide a unifying perspective of existing continuous- and discrete-time evolutionary evaluation models, but also reveal the formal underpinnings of the -Rank methodology. We empirically validate the method in several domains including AlphaGo, AlphaZero, MuJoCo Soccer, and Poker.
Evolutionary dynamics have often been presented as a practical tool for analyzing interactions among meta-strategies found in EGTA @cite_53 @cite_14 @cite_41 , and for studying the change in policies of multiple learning agents @cite_41 , as the EGTA approach is largely based on the same assumptions as evolutionary game-theory, viz. repeated interactions among sub-groups sampled independently at random from an arbitrarily-large population of agents.
@cite_41 @cite_53 @cite_14
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The interaction of multiple autonomous agents gives rise to highly dynamic and nondeterministic environments, contributing to the complexity in applications such as automated financial markets, smart grids, or robotics. Due to the sheer number of situations that may arise, it is not possible to foresee and program the optimal behaviour for all agents beforehand. Consequently, it becomes essential for the success of the system that the agents can learn their optimal behaviour and adapt to new situations or circumstances. The past two decades have seen the emergence of reinforcement learning, both in single and multi-agent settings, as a strong, robust and adaptive learning paradigm. Progress has been substantial, and a wide range of algorithms are now available. An important challenge in the domain of multi-agent learning is to gain qualitative insights into the resulting system dynamics. In the past decade, tools and methods from evolutionary game theory have been successfully employed to study multi-agent learning dynamics formally in strategic interactions. This article surveys the dynamical models that have been derived for various multi-agent reinforcement learning algorithms, making it possible to study and compare them qualitatively. Furthermore, new learning algorithms that have been introduced using these evolutionary game theoretic tools are reviewed. The evolutionary models can be used to study complex strategic interactions. Examples of such analysis are given for the domains of automated trading in stock markets and collision avoidance in multi-robot systems. The paper provides a roadmap on the progress that has been achieved in analysing the evolutionary dynamics of multi-agent learning by highlighting the main results and accomplishments. We develop a model for analyzing complex games with repeated interactions, for which a full game-theoretic analysis is intractable. Our approach treats exogenously specified, heuristic strategies, rather than the atomic actions, as primitive, and computes a heuristic-payoff table specifying the expected payoffs of the joint heuristic strategy space. We analyze two games based on (i) automated dynamic pricing and (ii) continuous double auction. For each game we compute Nash equilibria of previously published heuristic strategies. To determine the most plausible equilibria, we study the replicator dynamics of a large population playing the strategies. In order to account for errors in estimation of payoffs or improvements in strategies, we also analyze the dynamics and equilibria based on perturbed payoffs.