Deployable Large Language Models (LLMs) must conform to the criterion of helpfulness and harmlessness, thereby achieving consistency between LLMs outputs and human values. Red-teaming techniques constitute a critical way towards this criterion. Existing work rely solely on manual red team designs and heuristic adversarial prompts for vulnerability detection and optimization. These approaches lack rigorous mathematical formulation, thus limiting the exploration of diverse attack strategy within quantifiable measure and optimization of LLMs under convergence guarantees. In this paper, we present Red-teaming Game (RTG), a general game-theoretic framework without manual annotation. RTG is designed for analyzing the multi-turn attack and defense interactions between Red-team language Models (RLMs) and Blue-team Language Model (BLM). Within the RTG, we propose Gamified Red-teaming Solver (GRTS) with diversity measure of the semantic space. GRTS is an automated red teaming technique to solve RTG towards Nash equilibrium through meta-game analysis, which corresponds to the theoretically guaranteed optimization direction of both RLMs and BLM. Empirical results in multi-turn attacks with RLMs show that GRTS autonomously discovered diverse attack strategies and effectively improved security of LLMs, outperforming existing heuristic red-team designs. Overall, RTG has established a foundational framework for red teaming tasks and constructed a new scalable oversight technique for alignment.

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