We evaluate Large Language Models (LLMs) in repeated game-theoretic settings to assess whether strategic performance reflects genuine reasoning or reliance on memorized patterns. We consider two canonical games, Prisoner's Dilemma (PD) and Rock-Paper-Scissors (RPS), upon which we introduce counterfactual variants that alter payoff structures and action labels, breaking familiar symmetries and dominance relations. Our multi-metric evaluation framework compares default and counterfactual instantiations, showcasing LLM limitations in incentive sensitivity, structural generalization and strategic reasoning within counterfactual environments.