Prior research indicates that although large language models (LLMs) can precisely articulate the theoretical probability distributions associated with optimal strategic choices, their actual decision-making systematically diverges from these prescriptions—a phenomenon we define as the cognition–behaviour gap in LLMs. For example, in a Rock–Paper–Scissors (RPS) game, LLMs correctly identify the strategy of Nash equilibrium as selecting each action (Rock, Paper, Scissors) with equal probability ¹⁄₃, but their observed choices systematically deviate from this uniform distribution. Through a comprehensive evaluation of 20 state-of-the-art LLMs, we identify two critical insights: (1) we demonstrate that intrinsic biases inherited from pre-training corpora alone are insufficient to explain the observed deviations; (2) we introduce a semantic-free paradigm that strips away intrinsic biases to isolate pure positional bias-LLMs exhibit distinct position preferences—for example, o1 favours the first option, DeepSeek-V3 peaks the middle and DeepSeek-R1 shows a bimodal bias toward first and last positions. Our findings advocate innovation to bridge the gap between strategic reasoning and decision-making in LLMs.