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. This limitation causes a critical breakdown in the positive feedback loop essential for self-play RL, preventing effective learning beyond rote memorisation of frequently observed states. These results align with broader concerns regarding AlphaZero-style algorithms’ vulnerability to adversarial attacks, highlighting their inability to truly master all legal game states. Our work underscores that simple hyperparameter adjustments are insufficient to overcome these challenges, establishing a crucial foundation for the development of fundamentally novel algorithmic approaches, potentially involving neuro-symbolic or meta-learning paradigms, to bridge the gap towards true expert-level AI in combinatorial games.

In this paper, we study Condorcet domains, sets of linear orders from which majority ranking produces a linear order. We introduce a new class of Condorcet domains, called coherent domains, which is natural from both a voting theoretic and combinatorial perspective. After studying the properties of these domains we introduce set-alternating schemes. This is a method for constructing well-behaved coherent domains. Using this we show that, for sufficiently large numbers of alternatives n , there are coherent domains of size more than 2 . 197 3 n . This improves the best existing asymptotic lower bounds for the size of the largest general Condorcet domains.