Fake news detection increasingly relies on LLMs to extract semantic and pragmatic signals. We study a narrower problem than factual hallucination: when an LLM scores multiple intent dimensions, the resulting vector may contain internally inconsistent labels. We present IGTF, an intent-guided, game-inspired structured intent calibration framework for LLM-assisted fake news detection. IGTF defines an expert-validated nine-dimensional intent taxonomy, extracts intent-aware text representations with a Mixture-of-Experts encoder, and calibrates LLM-produced intent vectors through a fixed graph of cooperative and conflicting intent relations. The refinement is motivated by a potential-game formulation, whereas the deployed algorithm is a differentiable fixed-point update rather than an exact Nash-equilibrium solver. We prove convergence to a unique fixed point under a contraction condition and evaluate the refinement empirically. Across four benchmark datasets, IGTF improves over supervised GPT-4+BERT fusion baselines by 4.07 F1-macro points on average, with the largest gains when raw intent vectors show stronger relation-level inconsistency. Human evaluation and graph-perturbation controls suggest that these gains come from structured intent calibration rather than LLM features alone.