Optimal Play of the Dice Game Pig

The object of the jeopardy dice game Pig is to be the first player to reach 100 points. Each player\u27s turn consists of repeatedly rolling a die. After each roll, the player is faced with two choices: roll again, or hold (decline to roll again). If the player rolls a 1, the player scores nothing and it becomes the opponent\u27s turn. If the player rolls a number other than 1, the number is added to the player\u27s turn total and the player\u27s turn continues. If the player holds, the turn total, the sum of the rolls during the turn, is added to the player\u27s score, and it becomes the opponent\u27s turn. For such a simple dice game, one might expect a simple optimal strategy, such as in Blackjack (e.g., stand on 17 under certain circumstances, etc.). As we shall see, this simple dice game yields a much more complex and intriguing optimal policy, described here for the first time. The reader should be familiar with basic concepts and notation of probability and linear algebra

Teaching a University Course on the Mathematics of Gambling

Courses on the mathematics of gambling have been offered by a number of colleges and universities, and for a number of reasons. In the past 15 years, at least seven potential textbooks for such a course have been published. In this article we objectively compare these books for their probability content, their gambling content, and their mathematical level, to see which ones might be most suitable, depending on student interests and abilities. This is not a book review (e.g., none of the books is recommended over others) but rather an essay offering advice about which topics to include in a course on the mathematics of gambling

Mathematical models of games of chance: Epistemological taxonomy and potential in problem-gambling research

Games of chance are developed in their physical consumer-ready form on the basis of mathematical models, which stand as the premises of their existence and represent their physical processes. There is a prevalence of statistical and probabilistic models in the interest of all parties involved in the study of gambling – researchers, game producers and operators, and players – while functional models are of interest more to math-inclined players than problem-gambling researchers. In this paper I present a structural analysis of the knowledge attached to mathematical models of games of chance and the act of modeling, arguing that such knowledge holds potential in the prevention and cognitive treatment of excessive gambling, and I propose further research in this direction

Reinforcement learning strategies using Monte-Carlo to solve the blackjack problem

Blackjack is a classic casino game in which the player attempts to outsmart the dealer by drawing a combination of cards with face values that add up to just under or equal to 21 but are more incredible than the hand of the dealer he manages to come up with. This study considers a simplified variation of blackjack, which has a dealer and plays no active role after the first two draws. A different game regime will be modeled for everyone to ten multiples of the conventional 52-card deck. Irrespective of the number of standard decks utilized, the game is played as a randomized discrete-time process. For determining the optimum course of action in terms of policy, we teach an agent-a decision maker-to optimize across the decision space of the game, considering the procedure as a finite Markov decision chain. To choose the most effective course of action, we mainly research Monte Carlo-based reinforcement learning approaches and compare them with q-learning, dynamic programming, and temporal difference. The performance of the distinct model-free policy iteration techniques is presented in this study, framing the game as a reinforcement learning problem

A Dual-Process Account of Reactions to General and Specific Events: The Roles of Counterfactual Thinking and Pre-Event Expectations

Thesis (PhD) - Indiana University, Psychology, 2007On the basis of a dual-process account of reactions to general and specific cases, counterfactual thinking was hypothesized to occur more frequently in response to specific events than to general events. Cognitive reactions to general events were expected to be influenced by pre-event expectations, whereas reactions to specific events were expected to be influenced by counterfactual thinking. Such differences in processing may result in different comparison cases that ultimately influence reactions to the event as well as decisions regarding similar, future events. When people experience undesirable outcomes, counterfactual thoughts allow them to imagine more desirable possibilities, and thus greater confidence for future occurrences. Five experiments were designed to investigate these and other related hypotheses. In Experiment 1, participants were visually presented with general or specific outcomes of a golfing competition. Experiment 2 asked participants to complete a trivia test and provided them with global or specific performance feedback. Experiment 3 examined the impact of both upward and downward counterfactuals following the visual presentation of general or specific highlights of a tennis match. In Experiment 4, participants were asked to play several games of blackjack and were provided with global or specific performance feedback. Participants in Experiment 5 observed actual horse racing events, received general information about the events, or received thought-listing statements made by another participant in addition to general information about the events. In each experiment, some participants were also asked to complete a thought-listing task. Participants indicated their subjective confidence about the outcome of a similar, future event or placed bets on the outcome. Overall, the results supported the hypotheses. Counterfactual thoughts were observed more frequently in response to specific events than to general events. Counterfactual thought frequency also mediated the relationships between event type and decisions about similar, future events. A follow-up experiment to Experiment 5 provided evidence that counterfactual thought confidence also influences decisions regarding similar, future events. Although pre-event expectations were more clearly linked to reactions regarding general events than to specific events in Experiment 1, these links were inconsistent across experiments. Results are discussed in light of related dual-process theories and decision-making research

Campus Connection, April 20, 2000, Vol. 1 No. 9

CSUMB Professors Honored With Carnegie Scholarships -- Faculty/Staff Profile: Meet Sandy Hale -- WASC Accreditation Update -- Granite Construction Challenges CSUMB with $22 Million Science Center Donation -- Tech Notes Part I -- NEWS BRIEFS -- CALENDAR -- CalPERS Long-Term Care Plans -- Zmak and CSUMB Win Award for Viewbook -- Looking for Faculty/Staff Kudoshttps://digitalcommons.csumb.edu/campusconnection/1008/thumbnail.jp

A Type-2 signal detection analysis of gambling behaviour: cognitions, metacognitions, expertise and optimality

Cognitive gambling research has focused mainly on the irrational beliefs and cognitive biases that differentiate problem (PGs) and non-problem (RGs) gamblers. Whilst this research has been informative by highlighting that greater irrational beliefs are associated with gambling severity, the research has failed to determine cause and effect. This thesis proposes that metacognition is an area that may play a central role in the development and/or maintenance of problem gambling. Type-2 Signal Detection Theory (SDT) was used to analyse the data to measure three cognitive and metacognitive components of gambling performance: accuracy, resolution (metacognitive monitoring) and gambling criterion (metacognitive control). Optimality of gambling decisions was also explored. Experiment 1 used a simplified blackjack task, which demonstrated resolution differences between non-gamblers(NGs) and RGs. Experiments 2 to 5 examined the transference of gambling expertise of RGs and NGs in a novel dice gambling task. Experiment 6 demonstrated that the type of task can account for some cognitive and metacognitive variation observed between PGs and RGs, but impaired gambling criterion setting is a pertinent component of PGs’ gambling performance that is not dependent on gambling task. Finally, Experiment 7 showed that feedback enables participants to effectively shift gambling criteria to a more optimal position - and may have considerable implications for the treatment of problem gambling. The results are discussed in relation to four specific research questions and underscore the relative contribution of using a SDT approach in the study of gambling behaviour

Gambling in Mythical Temporality: Ontological Excess and Virtual Reality

This paper looks at certain features of games of chance to examine in greater detail the ways in which they thrive in a ‘mythical’ temporality. By examining the origins of modern gambling in the emergence of reason and its creation of an ontological excess that we refer to as ‘chance’, I argue that in order to understand the real significance of gambling in an age of virtual reality and virtual gaming, we must be able to grasp its existential dimensions and the ecstatic ‘disappearance’ it permits. I suggest that the ways in which chance has been manifested as the non-human or automatic in gambling draw us closer to the essence of its existential character. Games of chance are explored as encounters with the sacred or divine which, in their connection to a mythic temporality, are seen as precursors of virtual gaming, whose technologies are now driving the growth of gambling