Large Language Models (LLMs) have demonstrated remarkable general capabilities, yet they falter in domains requiring deep strategic reasoning. A primary obstacle is the need to navigate a game tree that grows exponentially with search depth, a task for which their generative nature is ill-suited. To address this, we introduce Generative Gamer (GenGamer), a framework that trains LLMs to reason like an expert player. Instead of attempting an exhaustive search, GenGamer learns to generate a compact, pruned reasoning trajectory termed as a Dynamic Deduction. This is achieved by integrating three key strategies: action pruning based on policy confidence, state pruning via value estimation, and branch pruning inspired by alpha-beta principles. Furthermore, to train the model effectively, we propose the Deduction Tree Reward (DTR), a process-oriented mechanism that provides step-by-step feedback on the quality of the reasoning process, rather than relying solely on the final game outcome. Experiments on complex games such as Tic-Tac-Toe and Leduc Poker demonstrate that GenGamer significantly enhances the strategic capabilities of LLMs, enabling them to achieve performance that surpasses current state-of-the-art language models.