Article

The Optimum Strategy in Blackjack

Taylor & Francis
Journal of the American Statistical Association
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Abstract

This article discusses the card game blackjack as played in the casinos of Las Vegas. The basic rules for the game are described in detail. The player's strategic problems are analyzed with the objective of finding the strategy maximizing his mathematical expectation.A mathematical expression is derived giving a general solution to the player's problem of standing pat with a given hand versus drawing additional cards. No general solutions are possible for the other major strategic problems, however, and a detailed examination of individual situations is required. The formulas and methods for the case analysis are stated, but computational details are omitted. Similarly, the formula for the player's mathematical expectation is stated, but its numerical evaluation is not described. Detailed discussion is given to the problems arising in the combinatorial type of computations required by blackjack.The “optimum strategy” determined by the above analysis differs substantially from the published strategies of card experts and the usual style of play in the casinos.

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... The game of blackjack has received a considerable attention from game theorists (and professional gamblers) during the recent decades. The original work which has initiated such an active interest in blackjack was published in [1], and [2,3]. In [1] it has been shown that following the prescribed set of rules the player can get almost even with the dealer, reducing the casino's edge to just about 0.62%. ...
... The original work which has initiated such an active interest in blackjack was published in [1], and [2,3]. In [1] it has been shown that following the prescribed set of rules the player can get almost even with the dealer, reducing the casino's edge to just about 0.62%. § Later [3] refined the edge estimate for the strategy of [1] to the player's advantage of 0.09%. ...
... In [1] it has been shown that following the prescribed set of rules the player can get almost even with the dealer, reducing the casino's edge to just about 0.62%. § Later [3] refined the edge estimate for the strategy of [1] to the player's advantage of 0.09%. Thorp also improved the prescription of [1], and claimed that, under certain specific game rules, player following the Thorp's optimized basic strategy has 0.12% edge over the casino. ...
Article
We apply the approach of evolutionary programming to the problem of optimization of the blackjack basic strategy. We demonstrate that the population of initially random blackjack strategies evolves and saturates to a profitable performance in about one hundred generations. The resulting strategy resembles the known blackjack basic strategies in the specifics of its prescriptions, and has a similar performance. We also study evolution of the population of strategies initialized to the Thorp's basic strategy.
... This way, the players can choose the most appropriate move according to the cards on the table. However, BJ is still quite neglected in the scientific game literature and offers a relatively unexplored area for mathematical and statistical analysis [2]. ...
... FuzzyFCA is available for download at https://goo.gl/MAu8CY.2 FuzzyDT is available for download at https://goo.gl/Gneywo. ...
... The probability of losing while drawing a card [P(lose|hit)] over all low-risk trials was 0.34, and 0.56 over all high-risk trials. The trials were designed in a way that -according to the blackjack basis strategy [38] in all high-risk and low-risk situations a hit was more advantageous for the player than a stand ([P(lose|stand] = 0.77). Fifty fill-trials were composed of cards with pictures and numbers with no relation to the blackjack game, which potentially serve as low-level baseline condition in further analyses not reported here. ...
... However, slower RTs in high-risk compared to low-risk situations in both groups might be associated with heightened response conflict [44]. Moreover, both groups might have related their decisions to the same extent to the blackjack basis strategy [38]. ...
Article
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Background The temporo-spatial dynamics of risk assessment and reward processing in problem gamblers with a focus on an ecologically valid design has not been examined previously.Methods We investigated risk assessment and reward processing in 12 healthy male occasional gamblers (OG) and in 12 male problem gamblers (PG) with a combined EEG and fMRI approach to identify group-differences in successively activated brain regions during two stages within a quasi-realistic blackjack game.ResultsBoth groups did not differ in reaction times but event-related potentials in PG and OG produced significantly different amplitudes in middle and late time-windows during high-risk vs. low-risk decisions. Applying an fMRI-constrained regional source model during risk assessment resulted in larger source moments in PG in the high-risk vs. low-risk comparison in thalamic, orbitofrontal and superior frontal activations within the 600-800 ms time window. During reward processing, PG showed a trend to enhanced negativity in an early time window (100-150 ms) potentially related to higher rostral anterior cingulate activity and a trend to centro-parietal group-differences in a later time window (390-440 ms) accompanied by increased superior-frontal (i.e., premotor-related) source moments in PG vs. OG.Conclusions We suggest that problem gambling is characterized by stronger cue-related craving during risk assessment. Reward processing is associated with early affective modulation followed by increased action preparation for ongoing gambling in PG.
... This paper describes the development and programming of a robotic system that can perform the role of a dealer in the game of blackjack. The robot hands out and identifies cards, detects typical player gestures, and plays according to the rules of the game [1]. We use multimodal sensing to allow for natural and unobtrusive user interaction. ...
Conference Paper
We describe a fully integrated blackjack dealing robot system utilizing multimodal input to interact with human players. It can deal cards to players and visually detect which cards have been played. Furthermore, it detects gestures commonly used in blackjack, such as knocking and swiping performed by human players to indicate whether they would like to receive another card. Both, visual and auditory input of the players is processed to achieve robust detection. We demonstrate robust and unobtrusive gameplay. A video showing the system interacting with two players can be seen at https://youtu.be/0XWj5tyZmnY.
... Probability formula of a natural for an n-deck shoe with numerical examples, various combinations and expectations in HA results, and special regulations, particularly, Las Vegas Strip Rules, are considered. Basic strategy by Baldwin et al. (1956) is described, as well as doubling down and insurance, wild jokers, naturals and splitting, and more. Card counting, including the high-low count, surrender option, side-counting and hi-opt I, and others are discussed. ...
... If player's total points are greater or equal to M(D), player should stand. And if total points are less than M(D), player should draw [5]. ...
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Blackjack is a popular casino poker game in which players make drawing decisions based on rules other than the size of their points. We found that despite the randomness of the card draw, blackjack is actually very mathematical and logical. By calculating probabilities and using game theory, we can allow players to make relatively "rational" decisions that maximize their expectations of winning, rather than relying solely on guesswork and vague feelings. Therefore, in this article, we built several models to simulate a blackjack game with different numbers of people. For players in each model, we give them a code that outputs the optimal decision, which helps them figure out the winning percentage (more specifically, the expectation, since once stakes are involved in multiplayer games, the winning percentage doesn't necessarily represent the expectation of winning money) for each decision, thus helping the player make a better decision.
... The strategies of play become different if a player's hand is "hard" or "soft". For this project, we will not need to make this distinction; we consider all totals to be "hard"totals [25]. Now, in order to be able to implement this strategy, we need our computer to be able to identify the cards. ...
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The field of computer vision is rapidly evolving, with a focus on analyzing, manipulating, and understanding images at a sophisticated level. The primary objective of this discipline is to interpret the visual input from cameras and utilize this knowledge to manage computer or robotic systems or to generate more informative and visually appealing images. The potential applications of computer vision are wide-ranging and include video surveillance, biometrics, automotive, photography, movie production, web search, medicine, augmented reality gaming, novel user interfaces, and many others. This paper outlines how computer vision technology will be utilized to achieve a winning outcome in the game of Blackjack. The game of Blackjack has long captivated the attention of enthusiasts and players worldwide. One area of particular interest is the development of a winning strategy that maximizes the player's chances of success. With the advent of sophisticated computer algorithms and machine learning techniques, there is enormous potential for research in this area. This paper explores the game-winning strategies for Blackjack, with a particular focus on utilizing advanced analytical methods to identify optimal plays. By analyzing large data sets and leveraging the power of predictive modeling, we aim to create a robust and reliable framework for achieving consistent success in this popular casino game. We believe that this research avenue holds enormous promise for unlocking new insights into the game of Blackjack and developing a more comprehensive understanding of its intricacies.
... In [17] the optimal strategy for Blackjack is discussed, where a mathematical expression is derived, providing a general solution to the player's problem of drawing additional cards, with a given hand, or not. The game is also used in [18] as a baseline for examining the advantages that quantum strategies allow in communication-limited games. ...
Preprint
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p>Cuckoo is a popular card game, which originated in France during the 15th century and then spread throughout Europe, where it is currently well-known under distinct names and with different variants. Cuckoo is an imperfect information game-of-chance, which makes the research regarding its optimal strategies determination interesting. The rules are simple: each player receives a covered card from the dealer; starting from the player at the dealer's left, each player looks at its own card and decides whether to exchange it with the player to their left, or keep it; the dealer plays at last and, if it decides to exchange card, it draws a random one from the remaining deck; the player(s) with the lowest valued card lose(s) the round. We formulate the gameplay mathematically and provide an analysis of the optimal decision policies. Different card decks can be used for this game, e.g., the standard 52-card deck or the Italian 40-card deck. We generalize the decision model for an arbitrary number decks' cards, suites, and players. Lastly, through numerical simulations, we compare the determined optimal decision strategy against different benchmarks, showing that the strategy outperforms the random and naive policies and approaches the performance of the ideal oracle. This preprint has been accepted for publication in IEEE Transactions on Games . How to cite : N. Mignoni, R. Carli and M. Dotoli, "Optimal Decision Strategies for the Generalized Cuckoo Card Game," in IEEE Transactions on Games. © 2023 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. </p
... Further examples of well-known CGs for which optimal strategies have been derived are Blackjack and Baccarat [16], both sharing a strong gambling component. In [17] the optimal strategy for Blackjack is discussed, where a mathematical expression is derived, providing a general solution to the player's problem of drawing additional cards, with a given hand, or not. The game is also used in [18] as a baseline for examining the advantages that quantum strategies allow in communicationlimited games. ...
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italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Cuckoo is a popular card game, which originated in France during the 15th century and then spread throughout Europe, where it is currently well-known under distinct names and with different variants. Cuckoo is an imperfect information game-of-chance, which makes the research regarding its optimal strategies determination interesting. The rules are simple: each player receives a covered card from the dealer; starting from the player at the dealer's left, each player looks at its own card and decides whether to exchange it with the player to their left, or keep it; the dealer plays at last and, if it decides to exchange card, it draws a random one from the remaining deck; the player(s) with the lowest valued card lose(s) the round. We formulate the gameplay mathematically and provide an analysis of the optimal decision policies. Different card decks can be used for this game, e.g., the standard 52-card deck or the Italian 40-card deck. We generalize the decision model for an arbitrary number decks' cards, suites, and players. Lastly, through numerical simulations, we compare the determined optimal decision strategy against different benchmarks, showing that the strategy outperforms the random and naive policies and approaches the performance of the ideal oracle.
... In [17] the optimal strategy for Blackjack is discussed, where a mathematical expression is derived, providing a general solution to the player's problem of drawing additional cards, with a given hand, or not. The game is also used in [18] as a baseline for examining the advantages that quantum strategies allow in communication-limited games. ...
Preprint
Full-text available
p>Cuckoo is a popular card game, which originated in France during the 15th century and then spread throughout Europe, where it is currently well-known under distinct names and with different variants. Cuckoo is an imperfect information game-of-chance, which makes the research regarding its optimal strategies determination interesting. The rules are simple: each player receives a covered card from the dealer; starting from the player at the dealer's left, each player looks at its own card and decides whether to exchange it with the player to their left, or keep it; the dealer plays at last and, if it decides to exchange card, it draws a random one from the remaining deck; the player(s) with the lowest valued card lose(s) the round. We formulate the gameplay mathematically and provide an analysis of the optimal decision policies. Different card decks can be used for this game, e.g., the standard 52-card deck or the Italian 40-card deck. We generalize the decision model for an arbitrary number decks' cards, suites, and players. Lastly, through numerical simulations, we compare the determined optimal decision strategy against different benchmarks, showing that the strategy outperforms the random and naive policies and approaches the performance of the ideal oracle. This work has been submitted to IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible. </p
... In [17] the optimal strategy for Blackjack is discussed, where a mathematical expression is derived, providing a general solution to the player's problem of drawing additional cards, with a given hand, or not. The game is also used in [18] as a baseline for examining the advantages that quantum strategies allow in communication-limited games. ...
Preprint
Full-text available
p>Cuckoo is a popular card game, which originated in France during the 15th century and then spread throughout Europe, where it is currently well-known under distinct names and with different variants. Cuckoo is an imperfect information game-of-chance, which makes the research regarding its optimal strategies determination interesting. The rules are simple: each player receives a covered card from the dealer; starting from the player at the dealer's left, each player looks at its own card and decides whether to exchange it with the player to their left, or keep it; the dealer plays at last and, if it decides to exchange card, it draws a random one from the remaining deck; the player(s) with the lowest valued card lose(s) the round. We formulate the gameplay mathematically and provide an analysis of the optimal decision policies. Different card decks can be used for this game, e.g., the standard 52-card deck or the Italian 40-card deck. We generalize the decision model for an arbitrary number decks' cards, suites, and players. Lastly, through numerical simulations, we compare the determined optimal decision strategy against different benchmarks, showing that the strategy outperforms the random and naive policies and approaches the performance of the ideal oracle. This work has been submitted to IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible. </p
... The strategies of play become different if a player's hand is "hard" or "soft". For this project, we will not need to make this distinction, we consider all totals to be "hard" totals [4]. Now, in order to be able to implement this strategy we need our computer to be able to identify the cards. ...
Preprint
Full-text available
Computer vision is a fast-expanding discipline focusing on analyzing, altering, and comprehending images at a high level. Its goal is to figure out what's going on in front of a camera and use that knowledge to manage a computer or robotic system or to show people new visuals that are more instructive or attractive than the original camera photos. Video surveillance, biometrics, automotive, photography, movie production, Web search, medicine, augmented reality gaming, new user interfaces, and many other applications are all possible with computer vision technologies. This paper aims to describe how computer vision will be used to play a winning game of blackjack.
... As mentioned earlier, all factors of influence reduce Q-learning's overestimation bias, as summarized in Table 1, while also increasing the obtained reward significantly. Blackjack In Blackjack the optimal policy is known as basic strategy (Baldwin et al., 1956) and results in an average reward of −0.045. We determine the degree of overestimation of the different algorithms by setting γ = 1 and by subtracting the average of the starting state Q values weighted by the occurrence, by the optimal performance. ...
Preprint
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We study whether the learning rate α\alpha, the discount factor γ\gamma and the reward signal r have an influence on the overestimation bias of the Q-Learning algorithm. Our preliminary results in environments which are stochastic and that require the use of neural networks as function approximators, show that all three parameters influence overestimation significantly. By carefully tuning α\alpha and γ\gamma, and by using an exponential moving average of r in Q-Learning's temporal difference target, we show that the algorithm can learn value estimates that are more accurate than the ones of several other popular model-free methods that have addressed its overestimation bias in the past.
... While such an experiment would sacrifice the long-range entanglements of, for example, a quantum optics experiment, we nevertheless hope these results provide an avenue for experimental followups in the near term. As mentioned in Ref. [26], most casinos play blackjack with the same, high level rules, but differ in the specifics. The authors of that paper set out a standardized ruleset, from which we deviate slightly for clarity and ease of analysis. ...
Article
We examine the advantages that quantum strategies afford in communication-limited games. Inspired by the card game blackjack, we focus on cooperative, two-party sequential games in which a single classical bit of communication is allowed from the player who moves first to the player who moves second. Within this setting, we explore the usage of quantum entanglement between the players and find analytic and numerical conditions for quantum advantage over classical strategies. Using these conditions, we study a family of blackjack-type games with varying numbers of card types, and find a range of parameters where quantum advantage is achieved.
... As mentioned in Reference [12], most casinos play blackjack with the same, high level rules, but differ in the specifics. The authors of that paper set out a standardized ruleset, which we deviate slightly from for clarity and ease of analysis. ...
Article
We examine the advantages that quantum strategies afford in communication-limited games. Inspired by the card game blackjack, we focus on cooperative, two-party sequential games in which a single classical bit of communication is allowed from the player who moves first to the player who moves second. Within this setting, we explore the usage of quantum entanglement between the players and find analytic and numerical conditions for quantum advantage over classical strategies. Using these conditions, we study a family of blackjack-type games with varying numbers of card types, and find a range of parameters where quantum advantage is achieved. Furthermore, we give an explicit quantum circuit for the strategy achieving quantum advantage.
... Many books and articles describe calculation of expected values in blackjack by computer methods. The first accurate derivation of correct strategy was done by Baldwin et al. [1]. Due to limited computer capabilities at that time, they used several approximations. ...
Preprint
Computer calculations for most exact expected values in blackjack have been available since the 1960's, but exact results for pair splitting and resplitting have previously been too computer intensive. This paper describes a new algorithm for exact pair-splitting. By using dealer probability caching methods and revising the method for recursively generating possible player hands, the estimated calculation time compared to standard methods was reduced by five orders of magnitude. The resulting algorithm was used to calculate the first exact and complete pair splitting results for a single deck game. The exact results were compared to prior approximate theories for resplitting. The prior theories are accurate for many calculations, but inaccurate for resplitting tens. A new approximation method was developed that is accurate for all resplitting calculations.
... As mentioned in Reference [12], most casinos play blackjack with the same, high level rules, but differ in the specifics. The authors of that paper set out a standardized ruleset, which we deviate slightly from for clarity and ease of analysis. ...
Preprint
We examine the advantages that quantum strategies afford in communication-limited games. Inspired by the card game blackjack, we focus on cooperative, two-party sequential games in which a single classical bit of communication is allowed from the player who moves first to the player who moves second. Within this setting, we explore the usage of quantum entanglement between the players and find analytic and numerical conditions for quantum advantage over classical strategies. Using these conditions, we study a family of blackjack-type games with varying numbers of card types, and find a range of parameters where quantum advantage is achieved. Furthermore, we give an explicit quantum circuit for the strategy achieving quantum advantage.
... This way, the players can choose the most appropriate move according to the cards on the table. However, BJ is still quite neglected in the scientific game literature and offers a relatively unexplored area for mathematical and statistical analysis [2]. ...
Conference Paper
Black Jack is a card game played by two or more players who aim at reaching 21 points. It is one of the most played card games as it presents an advantage for the player against the Casino: the player decides what to do before the dealer and he/she may stand (stop getting cards) when reaching a sum close to and smaller than 21. In the literature, it is possible to find some techniques proposed to support the prediction of the decision to be taken in accordance to the player's hand. Such techniques focus on finding a way to minimize the casino's advantage and turn the odds in favour of the player. Such strategies currently used were defined in the 1960's by mathematicians based on probability with hundreds of hands. However, these strategies are complex, making it difficult to memorize all possible actions to be taken. In this sense, this study aims to analyse data sets containing information on Black Jack hands in order to obtain rules to favour the player. This is an initial work, as there is a lack of proposals in the literature, based on the hypothesis that we could obtain sets of rules that could be used to support the player's decisions using machine learning algorithms. We used two fuzzy rule-based algorithms, namely FuzzyDT and FuzzyFCA, as well as the classic C4.5, PART, and Ripper algorithms to extract rules. The rules obtained are described and the results discussed.
... Although the game has an element of randomness in the card drawing, Blackjack has some skill [31]. Strategies for playing the game to reduce the dealer's advantage go from what is called Basic Strategy [9] or Optimum Strategy [32], which closely relates to the work we do in this paper, to more complex such as card counting, which we do not address as it is not for novice play. ...
... But because face-down cards are (obviously) unobservable, and the cards coming out of the deck (in the absence of card-counting techniques) are essentially random, no amount of data, analytic complexity, or machine learning can increase a good blackjack player's longterm win (ie, predictive) percentage above ≈40%. 10 The same 2 issues of unobservability and randomness likewise bedevil prediction in health care. Not infrequently, important predictors of health outcomes may be unobservable-defined as neither measured directly nor measured by some combination of proxy measures, analogous to the facedown card in blackjack. ...
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Big data analytics are widely touted as a key innovation to improve health care. Although the term big data is variably defined, it generally implies the application of advanced statistical analyses, with names such as machine learning, artificial intelligence, or cognitive computing, to data sources that greatly exceed the size and complexity of databases traditionally used for healthcare analyses.1 Somewhat counterintuitively, data volume alone does not qualify an analysis as big data analytics. The term more appropriately refers to analytic and computational techniques, combined with information technology innovations, that were specifically developed to yield insights from the large quantities of data that are increasingly common in the digital economy. Big data analytics offer the promise of turning large amounts of data into superior predictive models that can be used to improve healthcare quality and outcomes. Since its inception, Circulation: Cardiovascular Quality and Outcomes has focused on using scientifically rigorous data analyses to improve health care,2 thus the journal is a natural home for scientific investigations using big data analytic techniques. The current Special Issue highlights several high-quality studies that reveal the depth and breadth of these new methods and points toward a future where big data analytics are incorporated into the daily practice of all clinicians, much as big data analytics routinely impact the everyday online experiences of millions of users of Google, Amazon, and other internet-based companies. Although the term big data is a relatively new invention,1 many of the techniques of big data analytics have been in existence for decades, and the field of biomedical informatics has been critical to their development.3 Several recent phenomena have converged to move big data analytics to the forefront of health care, including the widespread adoption of electronic medical records and subsequent digitization of large volumes of healthcare …
... There are many books describing the counting strategies previously presented in this paper. However, BlackJack is still quite neglected in the game scientific literature and offers a relatively unexplored area for the mathematical and statistical analysis [2]. The reasons for that may include the fact that the computational representation of the game can be challenging. ...
... De matematiska och statistiska egenskaperna hos Black Jack studerades först av Baldwin et al. (1956) och Thorp (1961). Utvecklingen av dessa strategier och förståelse tvingade kasinon att modifiera sina regler på 1970-talet (Manson et al., 1975 ...
... The experimental results of the proposed algorithm in the blackjack test bed (Baldwin, Cantey, Maisel, & McDermott, 1956) are discussed in this section. A series of computersimulated experiments consisting of 20,000 consecutive hands of blackjack were carried out and the reported results are achieved by averaging over 1000 separate instances of 20,000 hands. ...
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Communication and coordination are the main cores for reaching a constructive agreement among multi-agent systems (MASs). Dividing the overall performance of MAS to individual agents may lead to group learning as opposed to individual learning, which is one of the weak points of MASs. This paper proposes a recursive genetic framework for solving problems with high dynamism. In this framework, a combination of genetic algorithm and multi-agent capabilities is utilised to accelerate team learning and accurate credit assignment. The argumentation feature is used to accomplish agent learning and the negotiation features of MASs are used to achieve a credit assignment. The proposed framework is quite general and its recursive hierarchical structure could be extended. We have dedicated one special controlling module for increasing convergence time. Due to the complexity of blackjack, we have applied it as a possible test bed to evaluate the system’s performance. The learning rate of agents is measured as well as their credit assignment. The analysis of the obtained results led us to believe that our robust framework with the proposed negotiation operator is a promising methodology to solve similar problems in other areas with high dynamism.
... Many books and articles have been written about the calculation of expected values in blackjack by computer methods. The first accurate derivation of correct strategy was done by Baldwin et al. [1]. Due to limited computer capabilities at that time, they used several approximations. ...
Article
Computer calculations for most exact expected values in blackjack have been available since the 1960's, but the exact results for pair splitting and resplitting have previously been too computer intensive. This paper describes a new algorithm for exact pair-splitting. By using dealer probability caching methods and revising the method for recursively generating possi-ble player hands, the estimated calculation time compared to standard methods was reduced by five orders of magnitude. The resulting algorithm was used to calculate the first exact and complete pair splitting results for the single deck game. The exact results were compared to prior approximate theories for resplitting. The prior theories are accurate for many calcula-tions, but inaccurate for resplitting tens. A new approximation method was developed that is accurate for all resplitting calculations.
... In order to assess the performance of any evolved strategy, a set of benchmarks must be obtained for comparison purposes. While there have been many attempts to calculate the performance of blackjack strategies using simulation and probabilistic techniques [24,25,21], the values produced tend to vary by a rather large margin. For instance, the success of a player employing the standard dealer strategy is reported at between 39% and 44% wins. ...
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This paper examines a new approach to the evolution of blackjack strategies, that of cultural learning. Many traditional machine learning approaches have concentrated on reinforcement learning ap- proaches and report satisfactory results. Populations of neural network agents evolve using genetic algorithms (population learning) and at each generation the best performing agents are selected as teachers. Cultural learning is implemented through a hidden layer in each teacher's neural network that is used to produce utterances which are imitated by its pupils during many games of blackjack. Results show that the cultural learning approach outperforms previous work as well as the best known non-card counting human approaches. The game of blackjack has been the subject of much research, particularly in the reinforcement learning domain. This paper introduces a cultural learning approach for the evolution of high quality blackjack playing agents. Using a combination of genetic algorithms and neural networks, we evolve a population of neural network agents which play games of blackjack against an automated dealer. Cultural learning is introduced by taking a percentage of the population and allowing it to teach the following generation through specialised verbal out- put nodes. Three experiments are performed, each one increasing the information available to each agent. We compare the evolved strategies with bench-marks obtained from a blackjack simulator. The remainder of this paper is arranged as follows: Section 2 discusses re- lated work, including a summary of the learning models employed in this set of experiments and evolutionary computation approaches to the game of black- jack. Section 3 presents the results of bench-marking several popular blackjack strategies, Section 4 introduces the artificial life simulator employed in the ex- periments, Section 5 presents the experimental results and Section 6 concludes and suggests future work.
... This exploration is much inspired by Thorp's (1966) book, Beat the Deafer. Whereas Baldwin et al. (1956) proposed a statistical approach which supplied groundwork, Thorp's study is credited with providing the first published strategy to give a positive expected return to the player. The point of our experiment into machine learning for blackjack is to see whether Thorp's 'ten-csunt' standing number table can be approximated by a generalpurpose learning algorithm in conjunction with a blackjack simulator. ...
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Search for optimal counting strategies for blackjack, like many other tasks in artificial intelligence, can be transcribed into the task of finding a global minimum of a multi-modal discontinuous objective function on the basis of noisy measurements. Herein we relate an algorithm for such ‘stochastic minimization’ problems, and derive some general properties. The general framework is extended to be applicable to finding the ‘ten-count’ standing-number strategy for blackjack. Computational experiments explore this learning application. The feature here, as opposed to blackjack analyses by Thorp, Braum, and others, who rely on combinatorics in their constructions, is that our methodology requires minimal modelling or understanding of the problem structure. This is the first reasonably detailed account of a procedure for finding ten-count standing numbers. But it is general enough to be applicable to other blackjack problems and, in fact, to a wide variety of sequential decision tasks.
... In order to assess the performance of any evolved strategy, a set of benchmarks must be obtained for comparison purposes. While there have been many attempts to calculate the performance of blackjack strategies using simulation and probabilistic techniques (Thorp 1984, Baldwin, Cantey, Maisel & McDermott 1956, Thorp 1963, the values produced tend to vary by a rather large margin. For instance, the success of a player employing the standard dealer strategy is reported at between 39% and 44% wins. ...
... There are many books describing the counting strategies previously presented in this paper. However, BlackJack is still quite neglected in the game scientific literature and offers a relatively unexplored area for the mathematical and statistical analysis [2]. The reasons for that may include the fact that the computational representation of the game can be challenging. ...
Chapter
In this first chapter probability theory is placed on a firm foundation. Then a variety of interesting and entertaining examples are introduced and investigated. These include French and American roulette; dice games such as craps, hazard, chuck-a-luck, crown and anchor, and sic-bo; and card games such as blackjack, macao, baccarat, and solitaire. Beyond the realms of card and casino games, topics ranging from the Martingale betting system and the Fibonacci betting system, setting winnings and losses bounds, the “optimum” strategy for blackjack, the gamblers’ fallacy, inherited diseases, monopoly, weather forecasting, chaos theory, horseracing odds, elections, your intuition, and paradoxes are examined. While we should expect that in each and every casino game the house has a statistical advantage, in this chapter we shed light on which games offer players a good chance.
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This paper uses computational and stochastic measures to demonstrate the negative impacts on the casino business of advantage players that legally rely on their knowledge to continuously make profits at licensed casinos worldwide. We present the stochastic process utilized by the advantage players to make profits from premium offers. We show in detail how players estimate the probability of reaching the win goal, the net-expected-win value, and the total expected return. In addition, a simulation is conducted to determine the amount of the starting capital that is needed before participating in games. The findings provide significant implications that advantage players are capable of exploiting premium offers and profiting from the casinos.
Chapter
Casino gambling has been booming steadily for decades, throughout the world. In the United States, from a time before the 1930s when casinos were illegal everywhere, they have grown in number and in popularity. Legalized initially in Nevada, they took off after World War II, particularly in Las Vegas and Reno, then around Lake Tahoe. Regulated casinos later mushroomed in Puerto Rico and, beginning in the 1970s, in Atlantic City, NJ. Casinos then sprang up on river boats in parts of the Midwest and South. More recently, American Indian tribes have been using their distinctive legal status to develop casinos on their reservations, especially in Connecticut (the largest anywhere in the world) but also in Illinois, California, Washington, and elsewhere. Older casinos in Las Vegas have even been bulldozed and replaced by much larger and grander ones. In almost every instance, each new casino has stimulated another increment in the public’s interest. Total casino gaming revenue has risen year after year, into the tens of billions of dollars in the U.S. alone, although evidence is growing that the market is saturating: some older venues are losing share to newer ones. The expansion has been worldwide as well; Macau and Singapore are particularly striking examples of the casino-building boom, and are drawing customers from their previous U.S. destinations.
Chapter
The crux of Chap. 3 is that the true count is a valuable clue to the expected return on the next round. But Player is concerned operationally only with how much to bet, not with the expected return: he really needs a rule for converting true count directly into bet size. Furthermore, the mental exertion of counting cards and adjusting bet size is only justified to the extent that it gives Player an overall advantage over the casino. An appropriate quantity by which to measure that advantage is the average, over all rounds between shuffles, of the product of the expected return on each round and the bet size on that round, weighted by its probability of occurrence; this is Player’s yield, or his average cash flow per round. The average over the shoe is necessary because the first (and probably several) rounds after a shuffle have a negative expected return (except for the single-deck game, as seen in Sect. 2.1). Only when enough of the shoe is dealt to permit sufficiently likely fluctuations in composition that result in a positive return, and the bet is increased correspondingly, can Player make up for his expected losses early in the shoe and achieve a positive yield.
Chapter
Textbook risk analysis examples generally are trivial and unrelated to real life. A comprehensive risk analysis of the casino game of blackjack is presented which illustrates practical, effective techniques for coping with problem complexity. A significant part of the effort has been devoted to developing new and better blackjack systems, a process much like the effort devoted to finding more attractive alternatives in a conventional comprehensive risk analysis.
Article
Casino pontoon is a convenient descriptive name for a variant of the American game of blackjack. Often under the name of blackjack it was played in clubs in the Midlands during the decade 1960-70, before the implementation of the Gaming Act of 1968. It was the subject of several successful prosecutions for unlawful gaming under the Acts of 1960 and 1963. The results of an investigation into its properties, which formed the basis of these prosecutions, are presented in this paper. This investigation yielded, inter alia, an optimum strategy for the player. Although the game is not lawful under present gaming regulations some of the results presented here are qualitatively relevant to the domestic game of pontoon.
Book
For decades, casino gaming has been steadily increasing in popularity worldwide. Blackjack is among the most popular of the casino table games, one where astute choices of playing strategy can create an advantage for the player. Risk and Reward analyzes the game in depth, pinpointing not just its optimal strategies but also its financial performance, in terms of both expected cash flow and associated risk. The book begins by describing the strategies and their performance in a clear, straightforward style. The presentation is self-contained, nonmathematical, and accessible to readers at all levels of playing skill, from the novice to the blackjack expert. Careful attention is also given to simplified, but still nearly optimal strategies that are easier to use in a casino. Unlike other books in the literature the author then derives each aspect of the strategy mathematically, to justify its claim to optimality. The derivations mostly use algebra and calculus, although some require more advanced analysis detailed in supporting appendices. For easy comprehension, formulae are translated into tables and graphs through extensive computation. This book will appeal to everyone interested in blackjack: those with mathematical training intrigued by its application to this popular game as well as all players seeking to improve their performance. N. Richard Werthamer is retired from a successful career as a scientist and executive, most recently as the Executive Officer of the American Physical Society. He graduated summa cum laude from Harvard College before receiving his PhD in Theoretical Physics from the University of California at Berkley. His original research has been published extensively in the world's leading journals. In this book, he applies his scientific background to the analysis of blackjack. © 2009 Springer Science+Business Media, LLC. All rights reserved.
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Cultural learning describes the process of information trans- fer between individuals in a population through non-genetic means. Typ- ically this is achieved through communication or the creation of artifacts available to all members of a population. Cultural learning has been simulated by combining genetic algorithms and neural networks using a teacher/pupil scenario where highly fit individuals are selected as teach- ers and instruct the next generation. This paper explores the effect of a cultural learning approach to the development of solutions for three test-case sequential decision tasks: connect-four, tic-tac-toe and blackjack. Experiments are conducted with populations employing population learning alone and populations com- bining population and cultural learning. A number of learning models may be readily observed from nature and have been the focus of much study in artificial intelligence research. Population learning (i.e. learning which occurs at a population level through genetic material) is typically simulated using genetic algorithms. Life-time learning (i.e. learning which takes place during an organisms's life time through reactions with its environment) can be simulated in a variety of ways, typically employing neural networks or reinforcement learning models. A relatively new field of study in artificial intelligence is synthetic ethology. The field is based on the premise that language and culture are too complex to be readily analysed in nature and that insight can be gained by simulating its emergence in populations of artificial organisms. While many studies have shown that lexical, syntactical and grammatical structures may spontaneously emerge from populations of artificial organisms, few discuss the impact such structures have on the relative fitness of individuals and of the entire population. The focus of this paper is to attempt to understand the effect of cultural learn- ing on a population of artificial organisms attempting to find solutions to three distinct sequential decision problems. The remainder of this paper is arranged as follows. Section 2 introduces background research, including descriptions of diversity measures and cultural learning techniques that have been employed for this study. Section 3 describes the experimental setup. Section 4 presents the a description of each experiment as well as their results and Section 5 provides a conclusion.
Article
Fifty-three Blackjack gamblers in four Nevada casinos were unobtrusively observed, and scored according to whether their playing decisions corresponded to the optimal “Zero-Memory” Basic decision strategy. Of the 940 hands recorded, at least 16% were played differently from the Basic prescription; presumably as a result of this “nonoptimal” play, average losses at Blackjack were quite high, on the order of ten times the theoretical loss rate for Basic play. Also, by wagering small bets in a subfair game, Blackjack gamblers practically guaranteed loss of their betting capital to the casinos.
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A great deal of research on the psychology of gambling has been conducted that has looked at non-experienced gamblers in laboratory or classroom settings. Yet there has been comparatively little research examining the practices and beliefs of actual gamblers within their natural gambling context. The current research contributes to the naturalistic study of casino gamblers. It reports the results of 10 weeks of ethnographic participant observation conducted in 1999 in two Indiana riverboat casinos located about ½ hour from Chicago. The research examines blackjack players' strategies for and beliefs about winning as explained and understood by the gamblers themselves. It uses blackjack's basic strategy and card counting as organizing principles around which to discuss and assess these strategies and beliefs.
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Man invented a concept that has since been variously viewed as a vice, a crime, a business, a pleasure, a type of magic, a disease, a folly, a weakness, a form of sexual substitution, an expression of the human instinct. He invented gambling. Recently, there has been a surge of interest in the statistical and mathematical theory behind gambling. Columbia pictures released a new movie 21 in July 2008 staring Jim Sturgess, Kevin Spacey, Kate Bosworth, Laurence Fishburne, Aaron Yoo, and Liza Lapira. Inspired by the true story of MIT students who mastered the art of card counting and took Vegas casinos for millions in winnings. Richard Epstein's classic book on gambling and its mathematical analysis covers the full range of games from penny matching, to blackjack and other casino games, to the stock market (including Black-Scholes analysis). He even considers what light statistical inference can shed on the study of paranormal phenomena. Epstein is witty and insightful, a pleasure to dip into and read and rewarding to study. The book is written at a fairly sophisticated mathematical level, this is not- Gambling for Dummies- or 'how to beat the odds' book, and a background in upper level undergraduate mathematics is essential to reading and understanding this book. o Comprehensive and exciting analysis of all major casino games and variants o Covers a wide range of interesting topics not covered in other books on the subject o Depth and breadth of its material is unique compared to other books of this nature.
Article
Numerous playing and betting strategies for the game of twenty-one have been computed assuming the deck or decks are randomly shuffled. In practice, dealers do not spend the time necessary (it takes too long) to completely randomly shuffle the decks used. Hence, there is information not only from the current round of play, but potentially from the previous round of play. We present a model for a non-random shuffle and assert ways in which this information can be used. Rules are derived using a normal approximation which updates the current strategies utilizing information from a non-random shuffle.
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This paper uses proprietary data from a blackjack table in Las Vegas to analyze how the expectation of regret affects peoples' decisions during gambles. Even among a group of people who choose to participate in a risk-taking activity, we find strong evidence of an economically significant omission bias: 80\% of the mistakes at the table are caused by playing too conservatively, resulting in substantial monetary losses. This behavior is equally prevalent among large-stakes gamblers and does not change in the face of more complicated strategic decisions.
Culbertson's Card Games Com-plete
  • E Culbertson
  • A H Morehead
  • G Mott-Smith
Culbertson, E., Morehead, A. H., and Mott-Smith, G., Culbertson's Card Games Com-plete. New York: The Greystone Press, 1952.
How to Be a Consistent Winner in the Most Popular Card Games. Garden City
  • J R Crawford
Crawford, J. R., How to Be a Consistent Winner in the Most Popular Card Games. Garden City, N. Y.: Doubleday and Company, Ine., 1953.
MacDougall on Dice and Cards
  • M Macdougall
MacDougall, M., MacDougall on Dice and Cards. New York: Coward-McCann, 1944.