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We introduce $\alpha$-Rank, a principled evolutionary dynamics methodologyfor the evaluation and ranking of agents in large-scale multi-agentinteractions, grounded in a novel dynamical game-theoretic solution conceptcalled Markov-Conley chains (MCCs). The approach leverages continuous- anddiscrete-time evolutionary dynamical systems applied to empirical games, andscales tractably in the number of agents, the type of interactions, and thetype of empirical games (symmetric and asymmetric). Current models arefundamentally limited in one or more of these dimensions and are not guaranteedto converge to the desired game-theoretic solution concept (typically the Nashequilibrium). $\alpha$-Rank provides a ranking over the set of agents underevaluation and provides insights into their strengths, weaknesses, andlong-term dynamics. This is a consequence of the links we establish to the MCCsolution concept when the underlying evolutionary model's ranking-intensityparameter, $\alpha$, is chosen to be large, which exactly forms the basis of$\alpha$-Rank. In contrast to the Nash equilibrium, which is a static conceptbased on fixed points, MCCs are a dynamical solution concept based on theMarkov chain formalism, Conley's Fundamental Theorem of Dynamical Systems, andthe core ingredients of dynamical systems: fixed points, recurrent sets,periodic orbits, and limit cycles. $\alpha$-Rank runs in polynomial time withrespect to the total number of pure strategy profiles, whereas computing a Nashequilibrium for a general-sum game is known to be intractable. We introduceproofs that not only provide a unifying perspective of existing continuous- anddiscrete-time evolutionary evaluation models, but also reveal the formalunderpinnings of the $\alpha$-Rank methodology. We empirically validate themethod in several domains including AlphaGo, AlphaZero, MuJoCo Soccer, andPoker.

$\alpha$-Rank: Multi-Agent Evaluation by Evolution