Optimal Blackjack Betting Strategies Through Dynamic Programming and Expected Utility Theory

Abstract: This study presents a rigorous mathematical approach to the optimization of round and betting policies in Blackjack, using Markov Decision Processes (MDP) and Expected Utility Theory. The analysis considers a direct confrontation between a player and the dealer, simplifying the dynamics of the game. The objective is to develop optimal strategies that maximize expected utility for risk profiles defined by constant (CRRA) and absolute (CARA) aversion utility functions. Dynamic programming algorithms are implemented to estimate optimal gambling and betting policies with different levels of complexity. The evaluation is performed through simulations, analyzing histograms of final returns. The results indicate that the advantage of applying optimized round policies over the "basic strategy" is slight, highlighting the efficiency of the last one. In addition, betting strategies based on the exact composition of the deck slightly outperform the Hi-Lo counting system, showing its effectiveness. The optimized strategies include versions suitable for mental use in physical environments and more complex ones requiring computational processing. Although the computed strategies approximate the theoretical optimal performance, this study is limited to a specific configuration of rules. As a future challenge, it is proposed to explore strategies under other game configurations, considering additional players or deeper penetration of the deck, which could pose new technical challenges.