A Generalized Theory of Card Counting

Existing card counting systems have largely been developed through heuristic design, without a unified mathematical definition of what constitutes an optimal count. This paper introduces a framework for defining and deriving optimal card-counting systems by formulating card counting as a long-term growth-rate maximization problem. The central idea is to view a count as a low-dimensional representation of the high-dimensional information contained in the exposed-card composition: the representation should preserve the information most relevant to future returns while remaining sufficiently compact for practical use. Based on this formulation, an alternating optimization procedure jointly adapts the count representation and the playing policy, combining a growth-optimal count update with reinforcement learning for count-dependent strategy deviations. The framework establishes connections between card counting, Kelly criterion, information-theoretic value of side information, and representation learning, and can be generalized beyond classical blackjack to variants with different rules and payoff structures. Experiments improve both long-term growth and fixed-spread return, increasing growth by up to 30.92% over Hi-Lo with the Illustrious 18 under 3:2 blackjack and by 76.98% under 6:5; the framework also demonstrates its value on new variants, achieving positive growth and spread in the evaluated Spanish 21 game where baseline methods loses.