To lower the barriers to game design development, automated game design, which generates game designs through computational processes, has been explored. In automated game design, machine learning-based techniques such as evolutionary algorithms have achieved success. Benefiting from the remarkable advancements in deep learning, applications in computer vision and natural language processing have progressed in level generation. However, due to the limited amount of data in game design, the application of deep learning has been insufficient for tasks such as game description generation. To pioneer a new approach for handling limited data in automated game design, we focus on the in-context learning of large language models (LLMs). LLMs can capture the features of a task from a few demonstration examples and apply the capabilities acquired during pre-training. We introduce the grammar of game descriptions, which effectively structures the game design space, into the LLMs' reasoning process. Grammar helps LLMs capture the characteristics of the complex task of game description generation. Furthermore, we propose a decoding method that iteratively improves the generated output by leveraging the grammar. Our experiments demonstrate that this approach performs well in generating game descriptions.