This paper provides several theoretical results for empirical game theory. Specifically, we introduce bounds for empirical game theoretical analysis of complex multi-agent interactions. In doing so we provide insights in the empirical meta game showing that a Nash equilibrium of the estimated meta-game is an approximate Nash equilibrium of the true underlying metagame. We investigate and show how many data samples are required to obtain a close enough approximation of the underlying game. Additionally, we extend the evolutionary dynamics analysis of meta-games using heuristic payoff tables (HPTs) to asymmetric games. The stateof-the-art has only considered evolutionary dynamics of symmetric HPTs in which agents have access to the same strategy sets and the payoff structure is symmetric, implying that agents are interchangeable. Finally, we carry out an empirical illustration of the generalised method in several domains, illustrating the theory and evolutionary dynamics of several versions of the AlphaGo algorithm (symmetric), the dynamics of the Colonel Blotto game played by human players on Facebook (symmetric), the dynamics of several teams of players in the capture the flag game (symmetric), and an example of a meta-game in Leduc Poker (asymmetric), generated by the policy-space response oracle multi-agent learning algorithm. Keywords Empirical games • Asymmetric games • Replicator dynamics 1 Introduction Using game theory to examine multi-agent interactions in complex systems is a non-trivial task, especially when a payoff table or normal form representation is not directly available. Works by Walsh et al. [39,40], Wellman et al. [43,44], and Phelps et al. [23], have shown the great potential of using heuristic strategies and empirical game theory to examine such B Karl Tuyls

How do societies learn and maintain social norms? Here we use multiagent reinforcement learning to investigate the learning dynamics of enforcement and compliance behaviors. Artificial agents populate a foraging environment and need to learn to avoid a poisonous berry. Agents learn to avoid eating poisonous berries better when doing so is taboo, meaning the behavior is punished by other agents. The taboo helps overcome a credit assignment problem in discovering delayed health effects. Critically, introducing an additional taboo, which results in punishment for eating a harmless berry, further improves overall returns. This “silly rule” counterintuitively has a positive effect because it gives agents more practice in learning rule enforcement. By probing what individual agents have learned, we demonstrate that normative behavior relies on a sequence of learned skills. Learning rule compliance builds upon prior learning of rule enforcement by other agents. Our results highlight the benefit of employing a multiagent reinforcement learning computational model focused on learning to implement complex actions.

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Generalization is a major challenge for multi-agent reinforcement learning. How well does an agent perform when placed in novel environments and in interactions with new co-players? In this paper, we investigate and quantify the relationship between generalization and diversity in the multi-agent domain. Across the range of multi-agent environments considered here, procedurally generating training levels significantly improves agent performance on held-out levels. However, agent performance on the specific levels used in training sometimes declines as a result. To better understand the effects of co-player variation, our experiments introduce a new environment-agnostic measure of behavioral diversity. Results demonstrate that population size and intrinsic motivation are both effective methods of generating greater population diversity. In turn, training with a diverse set of co-players strengthens agent performance in some (but not all) cases.

In systems of multiple agents, identifying the cause of observed agent dynamics is challenging. Often, these agents operate in diverse, non-stationary environments, where models rely on handcrafted environment-specific features to infer influential regions in the system's surroundings. To overcome the limitations of these inflexible models, we present GP-LAPLACE, a technique for locating sources and sinks from trajectories in time-varying fields. Using Gaussian processes, we jointly infer a spatio-temporal vector field, as well as canonical vector calculus operations on that field. Notably, we do this from only agent trajectories without requiring knowledge of the environment, and also obtain a metric for denoting the significance of inferred causal features in the environment by exploiting our probabilistic method. To evaluate our approach, we apply it to both synthetic and real-world GPS data, demonstrating the applicability of our technique in the presence of multiple agents, as well as its superiority over existing methods.