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Recent advances in large language model (LLM) reasoning, led by reinforcement learning with verifiable rewards (RLVR), have inspired self-play post-training, where models improve by generating and solving their own problems. While selfplay has shown strong in-domain and out-of-domain gains, the mechanisms behind these improvements remain poorly understood. In this work, we analyze the training dynamics of self-play through the lens of the Absolute Zero Reasoner, comparing it against RLVR and supervised fine-tuning (SFT). Our study examines parameter update sparsity, entropy dynamics of token distributions, and alternative proposer reward functions. We further connect these dynamics to reasoning performance using pass@k evaluations. Together, our findings clarify how self-play differs from other post-training strategies, highlight its inherent limitations, and point toward future directions for improving LLM math reasoning through self-play.