Cooperation and Decision-Making of LLM Agents in Bayesian-Informed Infinitely Repeated Games

The rapid development of large language models (LLMs) has significantly increased their engagement frequency in our daily lives. Due to the large amount usage of LLM agents, users are beginning to consider whether agents can serve as reliable and cooperative assistants for their benefit. To this end, game theory, as an effective tool in the study of strategic interactions, has gathered attention and has been employed in the research field of LLMs, particularly in exploring their interactions with users. Most previous studies focused on the performance of LLMs in static games or finitely repeated games, but these studies are relatively simplistic and fail to fully capture the characteristics of User-LLM interactions. To address the issues, our research proposes a novel approach using infinitely repeated games to analyze LLMs’ behavioral traits. Bayesian inference is incorporated by providing probabilistic information to assist LLMs in their decision-making process, using the beta distribution as both the prior and posterior for consistency. We conduct a case study over the trending LLMs: GPT3, GPT4 and Llama3. Experimental results demonstrate that LLMs outperform the mixed Nash equilibrium in infinitely repeated games, indicating their capability for cooperation in repeated interactions. The integration of Bayesian inference shows that LLMs can effectively process probabilistic information and improve their performance by an average of 5.26%. Our findings suggest that LLM agents have the preference to consider future payoffs rather than only caring about single-stage rewards, as well as the ability to build and maintain long-term cooperative relationships with users.