Impartial Games: A Challenge for Reinforcement Learning

Abstract AlphaZero-style reinforcement learning (RL) algorithms have achieved superhuman performance in many complex board games such as Chess, Shogi, and Go. However, we showcase that these algorithms encounter significant and fundamental challenges when applied to impartial games, a class where players share game pieces and optimal strategy often relies on abstract mathematical principles. Specifically, we utilise the game of Nim as a concrete and illustrative case study to reveal critical limitations of AlphaZero-style and similar self-play RL algorithms. We introduce a novel conceptual framework distinguishing between champion and expert mastery to evaluate RL agent performance. Our findings reveal that while AlphaZero-style agents can achieve champion-level play on very small Nim boards, their learning progression severely degrades as the board size increases. This difficulty stems not merely from complex data distributions or noisy labels, but from a deeper representational bottleneck: the inherent struggle of generic neural networks to implicitly learn abstract, non-associative functions like parity, which are crucial for optimal play in impartial games. Th