Evolvability in Rule-Making: A Self-Amendment Game Among LLM Agents

We introduce the Self-Amendment Game, a simulation framework where Large Language Model (LLM) agents iteratively propose, vote on, and revise the very rules that govern them. Inspired by the self-modifying game Nomic, this system enables agents to operate within a fully amendable environment in which institutional constraints are themselves subject to change. Using leading LLMs, we examine two agent strategies: "Add" agents that focus on introducing new rules, and "Modify" agents that refine existing ones. Add agents tend to generate denser but less stable rule systems, with rapid point accumulation and high leadership turnover. Modify agents foster greater stability, with fewer but more consistent leaders. Analyzing the semantic structure of proposals, we find that varied justification styles often lead to instability, while coherence in reasoning promotes rule adoption. Our findings highlight a crucial trade-off between innovation and governance stability, offering a foundation for studying adaptive, self-governing systems involving both artificial agents and hybrid human-AI institutions.