The swift growth of artificial intelligence has led to the development of large language models, revolutionising various scientific domains and professional fields. This research explores collaborative, cooperative, and competitive game designs, that enhance knowledge elicitation using LLMs in games with a purpose. Through a systematic literature review using the PRISMA workflow, we identify that collaborative and cooperative game designs are more effective for knowledge elicitation than competitive designs. We also address the potential and limitations of incorporating LLMs into GWAPs, such as their use as Non-Player Characters to create engaging interactions. This study provides design principles for GWAPs leveraging LLMs, offering insights for researchers and developers, and discusses ethical considerations and future research directions.