A Markov Chain Analysis of Blackjack Strategy
Markov chains; blackjack; gambling; simulation
Blackjack receives considerable attention from mathematicians and entrepreneurs alike, due to its simple rules, its inherent random nature, and the abundance of "prior" information available to an observant player. Many attempts have been made to propose card-counting systems that exploit such information to the player's advantage. Because blackjack is a complicated game, attempts to actually calculate the expected gain from a particular system often rely on simulation techniques. While such techniques may yield correct results, they may also fail to explore the interesting mathematical properties of the game. Despite the apparent complexity, there is a great deal of structure inherent in both the blackjack rules and the card-counting systems. Exploiting this structure and elementary results from the theory of Markov chains, we present a novel framework for analyzing the expected advantage of a card-counting system entirely without simulation. The method presented here requires only a few, mild simplifying assumptions, can account for many rule variations, and is applicable to a large class of counting systems. As a specific example, we verify the reported advantage provided by one of the earliest systems, the Complete Point-Count System, discussed in Edward Thorp's famous book, <i>Beat the Dealer</i>. While verifying this analysis is satisfying, in our opinion the primary value of this work lies in the exposition of an interesting mathematical framework for analyzing a complicated "real-world" problem.