Who Is Really Playing? Strategic Interaction in AI-Guided Populations

AI systems in general, and Large language models (LLMs), in particular, are increasingly used to provide instructions to many agents who interact with one another. Such shared reliance couples agents who appear to act independently: they may in fact be guided by a common model. This coupling can change the prospects for cooperation among agents with misaligned incentives. We study settings in which multiple \emph{guidance providers} each advise a population of clients who participate in instances of an underlying game, creating strategic interaction at the level of the providers themselves. This induces a meta-game among the providers, mediated through clients. We first analyze the one-shot setting, where we show that shared instructions can change equilibrium behavior only when some provider influences more than one role in the same interaction. In such cases, cooperation may emerge, and the effect of client share can be beneficial, harmful, or non-monotone, depending on the base game. For the repeated setting, we prove a folk theorem for guidance providers: despite indirect observation and the clients' inability to identify which LLM advised their opponents, all feasible and individually rational outcomes can be sustained as $\varepsilon$-equilibria.