Do LLMs Know When to Flip a Coin? Strategic Randomization through Reasoning and Experience

Strategic randomization is a key principle in game theory, yet it remains underexplored in large language models (LLMs). Prior work often conflates the cognitive decision to randomize with the mechanical generation of randomness, leading to incomplete evaluations. To address this, we propose a novel zero-sum game inspired by the Tian Ji Horse Race, where the Nash equilibrium corresponds to a maximal entropy strategy. The game's complexity masks this property from untrained humans and underdeveloped LLMs. We evaluate five LLMs across prompt styles -- framed, neutral, and hinted -- using competitive multi-tournament gameplay with system-provided random choices, isolating the decision to randomize. Results show that weaker models remain deterministic regardless of prompts, while stronger models exhibit increased randomization under explicit hints. When facing weaker models, strong LLMs adopt deterministic strategies to exploit biases, but converge toward equilibrium play when facing peers. Through win/loss outcomes and Bayes factor analysis, we demonstrate meaningful variation in LLMs' strategic reasoning capabilities, highlighting opportunities for improvement in abstract reasoning and adaptive learning. We make our implementation publicly available at https://github.com/ocelopus/llm-when-to-throw-coin to ensure full reproducibility.