Large language models (LLMs) 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 LLMs each advise a population of clients who participate in instances of an underlying game, creating strategic interaction at the level of the LLMs themselves. This induces a meta-game among the LLMs, mediated through clients. We first analyze the one-shot setting, where shared instructions can change equilibrium behavior only when an LLM may influence 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. Our main result concerns the repeated setting. We prove a folk theorem for LLMs: 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. The result does not follow from the standard folk theorem and requires new proof techniques. Together, these results show that shared LLM guidance can sustain cooperation among populations of agents even when the underlying incentives are misaligned.

翻译：大语言模型（LLMs）日益被用于为相互交互的多个智能体提供指令。这种共享依赖关系使得表面上独立行动的智能体产生耦合：它们实际上可能受同一模型引导。这种耦合可能改变激励不一致的智能体之间的合作前景。我们研究了多个LLM各自为参与基础游戏实例的客户群体提供建议的场景，从而在LLM层面形成战略互动，由此催生LLM之间通过客户中介的元博弈。首先分析一次性博弈场景：只有当同一LLM可能影响同一交互中的多个角色时，共享指令才能改变均衡行为；在此类情形下，合作可能涌现，而客户份额的影响可能是有益的、有害的或非单调的，这取决于基础博弈。我们的主要结果针对重复博弈场景。我们为LLM证明了民间定理：尽管存在间接观察且客户无法识别对手所接受的LLM建议，所有可行且个体理性的结果均可作为ε-均衡维系。该结论无法通过标准民间定理推导，需要全新的证明技术。这些结果共同表明，即使底层激励不一致，共享LLM引导也能维系智能体群体中的合作。