The emergence of mathematical probability has something to do with dice games: all the early discussions (Cardano, Galileo, Pascal) suggest as much. Although this has long been recognized, the problem is that gambling at dice has been a popular pastime since antiquity. Why, then, did gamblers wait until the sixteenth century ce to calculate the math of dicing? Many theories have been offerred, but there may be a simple solution: early-modern gamblers played different sorts of dice games than in (...) antiquity. While ancients diced at communal risk, early-moderns diced at individualized risk. These individual wager-games, for example, the game “hazard”, incentivized solving questions like “how likely is it to roll a triple-six?”. Before the early modern period there was no gambling game to which working out mathematical probability would have brought an advantage. (shrink)

Gambling addiction is special type of addiction not only through the object of the addiction, the psychobiological constitution of the individual, and the pattern of the development of a pathological behaviour and condition, but also through the methods available for preventing and fighting against it and their effectiveness. This latter specificity has not been clearly established and thus has not been exploited much in problem-gambling research; but specificity should be reflected in warning messages and counselling content.

With the previous technical reports, we presented the main insights from the qualitative analysis of the reflection of the mathematical dimension of gambling in gambling-related sites, in its structural, linguistic, epistemic, and informative aspects. In this report we will focus on the factors responsible for the patterns that the content of the gambling-related sites exhibit and the way that the mathematical dimension of gambling is reflected in this content, namely the factors that influence the content policies (...) of these sites. The report will also point to further directions of research-for both this project and for other investigators-for finding the appropriate balance and relation between the commercial policies on the one hand, and the adequacy and ethics of the content on the other hand. (shrink)

In light of the observations and research design presented in the previous reports, the current technical report is focused on the relationship between the quality and specificity of the content of the gambling sites and the site’s SEO and marketing policy. This relationship is dependent upon the category of the gambling site and the difference in content quality, and the degree to which the mathematical dimension of gambling is reflected in this content is explained by this dependence.

Randomness, a core concept of gambling, is seen in problem gambling as responsible for the formation of the math-related cognitive distortions, especially the Gambler’s Fallacy. In problem-gambling research, the concept of randomness was traditionally referred to as having a mathematical nature and categorized and approached as such. Randomness is not a mathematical concept, and I argue that its weak mathematical dimension is not decisive at all for the randomness-related issues in gambling and problem gambling, including (...) the correction of the misconceptions and fallacies about probability and statistical concepts applied in gambling. I distinguish between mathematical and nonmathematical dimensions of randomness (the epistemic, the theoretical-methodological, the functional, and the ethical) falling within the general concept, and I argue that both the studies having as object the math-related cognitive distortions among gamblers and the educational programs aiming at correcting them should employ this distinction in their design and content. (shrink)

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. (shrink)

The current technical report presents the partial results of the quantitative analysis of the research project, after the review of 247 gambling websites. It is focused on and discusses the usage of the math terms specific to gambling in the reviewed sample. In particular, the fifth technical report discusses the usage of math terms associated with the game of slots, as found in the reviewed sample.

The current technical report of the research project investigating how the mathematical dimension of gambling is reflected in the communication and texts associated with the gambling industry raises the problem of the adequacy of sampling and proposes a new approach in this respect. The qualitative analysis of the reviewed websites is extended to a deeper analysis of language and also to the organization and structure of websites’ content. Although not stated as a goal of the initial project, the (...) research will also try to identify relationships between the specificity of content relative to the mathematical dimension of gambling, and the management and marketing of the website. An overview of the results of the statistical analysis of the current sample is also presented. (shrink)

The current report presents results of the qualitative analysis of the usage of the key math terms in the content of the reviewed websites in the current sample, with emphasis on the necessary distinctions for the word and concept of ‘odds’. Patterns, motivations, conceptual and contextual aspects of the usage are discussed. The theoretical norms of adequacy of this usage in the context of gambling are briefly presented. -/- Technical report of the research project 'Reflection of the mathematical dimension (...) of gambling in iGaming online content - A qualitative analysis'. (shrink)

The current study evaluates qualitatively how the mathematical dimension of gambling is reflected in the content of gambling websites. A number of gambling websites have been reviewed for their content in that respect. A statistical analysis recorded the presence of the mathematical dimension of gambling and its forms in the content of the participating websites, and a qualitative research study analyzed and assessed the quality of the content with respect to that dimension. The technical reports associated (...) with this study describe the methods used in the processes of review and content analysis, details of the processes, and the specific challenges and difficulties that they posed. (shrink)

The author proposes in this practical guide for both problem and non-problem gamblers a new pragmatic, conceptual approach of gambling mathematics. The primary aim of this guide is the adequate understanding of the essence and complexity of gambling through its mathematical dimension. The author starts from the premise that formal gambling mathematics, which is hardly even digestible for the non-math-inclined gamblers, is ineffective alone in correcting the specific cognitive distortions associated with gambling. By applying (...) the latest research results in this field, the author blends the gambling-mathematics concepts with the epistemology of applied mathematics and cognitive psychology for providing gamblers the knowledge required for rational and safe gambling. (shrink)

On the question of whether gambling behavior can be changed as result of teaching gamblers the mathematics of gambling, past studies have yielded contradictory results, and a clear conclusion has not yet been drawn. In this paper, I bring some criticisms to the empirical studies that tended to answer no to this hypothesis, regarding the sampling and laboratory testing, and I argue that an optimal mathematical scholastic intervention with the objective of preventing problem gambling is possible, (...) by providing the principles that would optimize the structure and content of the teaching module. Given the ethical aspects of the exposure of mathematical facts behind games of chance, and starting from the slots case – where the parametric design is missing, we have to draw a line between ethical and optional information with respect to the mathematical content provided by a scholastic intervention. Arguing for the role of mathematics in problem-gambling prevention and treatment, interdisciplinary research directions are drawn toward implementing an optimal mathematical module in cognitive therapies. (shrink)

Over the past two decades, gamblers have begun taking mathematics into account more seriously than ever before. While probability theory is the only rigorous theory modeling the uncertainty, even though in idealized conditions, numerical probabilities are viewed not only as mere mathematical information, but also as a decision-making criterion, especially in gambling. This book presents the mathematics underlying the major games of chance and provides a precise account of the odds associated with all gaming events. It begins (...) by explaining in simple terms the meaning of the concept of probability for the layman and goes on to become an enlightening journey through the mathematics of chance, randomness and risk. It then continues with the basics of discrete probability, combinatorics and counting arguments for those interested in the supporting mathematics. These mathematic sections may be skipped by readers who do not have a minimal background in mathematics; these readers can skip directly to the Guide to Numerical Results to pick the odds and recommendations they need for the desired gaming situation. Doing so is possible due to the organization of that chapter, in which the results are listed at the end of each section, mostly in the form of tables. The chapter titled The Mathematics of Games of Chance presents these games not only as a good application field for probability theory, but also in terms of human actions where probability-based strategies can be tried to achieve favorable results. Through suggestive examples, the reader can see what are the experiments, events and probability fields in games of chance and how probability calculus works there. The main portion of this work is a collection of probability results for each type of game. Each game s section is packed with formulas and tables. Each section also contains a description of the game, a classification of the gaming events and the applicable probability calculations. The primary goal of this work is to allow the reader to quickly find the odds for a specific gaming situation, in order to improve his or her betting/gaming decisions. Every type of gaming event is tabulated in a logical, consistent and comprehensive manner. The complete methodology and complete or partial calculations are shown to teach players how to calculate probability for any situation, for every stage of the game for any game. Here, readers can find the real odds, returned by precise mathematical formulas and not by partial simulations that most software uses. Collections of odds are presented, as well as strategic recommendations based on those odds, where necessary, for each type of gaming situation. The book contains much new and original material that has not been published previously and provides great coverage of probabilities for the following games of chance: Dice, Slots, Roulette, Baccarat, Blackjack, Texas Hold em Poker, Lottery and Sport Bets. Most of games of chance are predisposed to probability-based decisions. This is why the approach is not an exclusively statistical one, but analytical: every gaming event is taken as an individual applied probability problem to solve. A special chapter defines the probability-based strategy and mathematically shows why such strategy is theoretically optimal.". (shrink)

This second technical report shows some partial results for the variables of the proposed statistical analysis and a discussion about some changes in sampling. In what concerns the qualitative analysis of content, the report presents the general predominant tendencies that get contoured with the first two samples.

This paper explores the moral status of various gambling maxims, particularly as they relate to the bettor’s interest in profit and the mathematical expectation of the game being played. Certain difficulties with the prevailing interpretations of the Formula of Universalizability will also be discussed, particularly in relation to games for which the bettor can have a positive expectation.

Peirce writes "It is an indubitable result of the theory of probabilities that every gambler, if he continues long enough, must ultimately be ruined... if he plays long enough he will be sure some time to have such a run against him as to exhaust his entire fortune.... The same thing is true of an insurance company. Let the directors take the utmost pains... according to the doctrine of chances, the time must come, at last, when their losses will bring (...) them to a stop." This note shows that under the most plausible interpretation of the quoted passage Peirce is committed to a mathematical untruth. (shrink)

Understanding the interconnectedness of the gambling harms is essential for developing comprehensive strategies to mitigate the damage caused by problem gambling. Central to this effort is the concept of playing informed. The current review synthesizes current research on the cascading harms in relation to the critical role of informed gambling in the reduction of these harms, while pointing out interdisciplinary connections that might contribute to this research for crystallizing an adequate conceptual and theoretical framework.

The near-miss has been considered an important factor of reinforcement in gambling behavior, and previous research has focused more on its industry-related causes and effects and less on the gaming phenomenon itself. The near-miss has usually been associated with the games of slots and scratch cards, due to the special characteristics of these games, which include the possibility of pre-manipulation of award symbols in order to increase the frequency of these “engineered” near-misses. In this paper, we argue that starting (...) from an elementary mathematical description of the classical (by pure chance) near-miss, generalizable to any game, and focusing equally on the epistemology of its constitutive concepts and their mathematical description, we can identify more precisely the fallacious elements of the near-miss cognitive effects and the inadequate perception and representation of the observational-intentional “I was that close.” This approach further suggests a strategy of using non-standard mathematical knowledge of an epistemological type in problem-gambling prevention and cognitive therapies. (shrink)

The present theory leads to a set of subjective weights such that the utility of an uncertain alternative (gamble) is partitioned into three terms involving those weights—a conventional subjectively weighted utility function over pure consequences, a subjectively weighted value function over events, and a subjectively weighted function of the subjective weights. Under several assumptions, this becomes one of several standard utility representations, plus a weighted value function over events, plus an entropy term of the weights. In the finitely additive case, (...) the latter is the Shannon entropy; in all other cases it is entropy of degree not 1. The primary mathematical tool is the theory of inset entropy. (shrink)

This paper outlines a formal recursive wager resolution calculus (WRC) that provides a novel conceptual framework for sentential logic via bridge rules that link wager resolution with truth values. When paired with a traditional truth-centric criterion of logical soundness WRC generates a sentential logic that is broadly truth-conditional but not truth-functional, supports the rules of proof employed in standard mathematics, and is immune to the most vexing features of their traditional implementation. WRC also supports a novel probabilistic criterion of (...) logical soundness, the fair betting probability criterion (FBP). It guarantees that the conclusion of an FBP-valid argument is at least as credible as a conjunction of premises, and also that the conclusion is true if the premises are. In addition, WRC provides a platform for a novel non-probabilistic, computationally simpler criterion of logical soundness – the criterion of Super-validity - that issues the same logical appraisals as FBP, and hence the same guarantees. (shrink)

This eighth book of the author on gambling math presents in accessible terms the cold mathematics behind the sparkling slot machines, either physical or virtual. It contains all the mathematical facts grounding the configuration, functionality, outcome, and profits of the slot games. Therefore, it is not a so-called how-to-win book, but a complete, rigorous mathematical guide for the slot player and also for game producers, being unique in this respect. As it is primarily addressed to the slot player, (...) its goal is to present practical applications of the mathematical models of slot games, in order to provide numerical results that a player can use as criteria for gaming decisions or just as information for any slot game and any predicted winning event. These results are focused on probability and expected value, these being the most important parameters for decisional criteria in slots. The book is packed with plenty of figures, tables, and formulas. The content is organized so that readers can skip the theoretical parts and go picking the practical results (numerical, in tables of values where possible, or ready-to-compute formulas) for the desired situation. The practical results are gathered in the last chapter, titled "Practical Applications and Numerical Results," the largest part of the book, for the most popular categories of slot machines, namely with 3, 5, 9, and 16 reels. Any other category of slot games is covered in the theoretical part of the book, where the general formulas apply not only to existing slot games, but also to possible future slot games of any design and configuration. The author does not just throw the slot mathematics to the audience and run away, but offers an ultimate practical contribution with the chapter "How to estimate the number of stops and the symbol distribution on a reel", a surprise for both players and producers, where one can see that mathematics provides players with some statistical methods as well as methods based on physical measurements for retrieving these missing data. Having these data along with the mathematical results of this book, anyone can generate the PAR sheet of any slot machine. In the last decade, mathematics has been taken more and more seriously into account in gaming, as being the essence that governs the games of chance and the only rigorous tool providing information on optimal play, where possible. For the popular game of slots, mathematics already fulfilled its duty by providing all the data that it can provide and that cannot be found on the display of the slot machines - it is all here in this book. (shrink)

The recently celebrated discovery of the Higgs boson has captivated the public's imagination with the promise that it can explain the origins of everything in the universe. It's no wonder that the media refers to it grandly as the "God particle." Yet behind closed doors, physicists are admitting that there is much more to this story, and even years of gunning the Large Hadron Collider and herculean number crunching may still not lead to a deep understanding of the laws of (...) nature. In this fascinating and eye-opening account, theoretical physicist Alexander Unzicker and science writer Sheilla Jones offer a polemic. They question whether the large-scale, multinational enterprises actually lead us to the promised land of understanding the universe. The two scientists take us on a tour of contemporary physics and show how a series of highly publicized theories met a dead end. Unzicker and Jones systematically unpack the recent hot theories such as "parallel universes," "string theory," and "inflationary cosmology," and provide an accessible explanation of each. They argue that physics has abandoned its evidence-based roots and shifted to untestable mathematical theories, and they issue a clarion call for the science to return to its experimental foundation. (shrink)

Randomness, uncertainty, and lack of regularity had concerned savants for a long time. As early as the seventeenth century, Blaise Pascal conceived the arithmetic of chance for gambling. At the height of positional astronomy, mathematicians developed a theory of errors to cope with random deviations in astronomical observations. In the nineteenth century, pioneers of statistics employed probabilistic calculus to define “normal” and “pathological” in the distribution of social characters, while physicists devised the statistical-mechanical interpretation of thermodynamic effects. By the (...) end of the century, a probabilistic worldview—as historians Lorrain Daston, Theodore Porter, and Stephen Stigler. (shrink)

Slot machines gained a high popularity despite a specific element that could limit their appeal: non-transparency with respect to mathematical parameters. The PAR sheets, exposing the parameters of the design of slot machines and probabilities associated with the winning combinations are kept secret by game producers, and the lack of data regarding the configuration of a machine prevents people from computing probabilities and other mathematical indicators. In this article, I argue that there is no rational justification for this secrecy by (...) giving two reasons, one psychological and the other mathematical. For the latter, I show that mathematics provides us with some statistical methods of retrieving the missing data, which are essential for the numerical probability computations in slots. The slots case raises the problem of the exposure of the parametric configuration and mathematical facts of any game of chance as an ethical obligation. (shrink)

Continuing his series of books on the mathematics of gambling, the author shows how a simple-rule game such as roulette is suited to a complex mathematical model whose applications generate improved betting systems that take into account a player's personal playing criteria. The book is both practical and theoretical, but is mainly devoted to the application of theory. About two-thirds of the content is lists of categories and sub-categories of improved betting systems, along with all the parameters that (...) might stand as the main objective criteria in a personal strategy - odds, profits and losses. The work contains new and original material not published before. The mathematical chapter describes complex bets, the profit function, the equivalence between bets and all their properties. All theoretical results are accompanied by suggestive concrete examples and can be followed by anyone with a minimal mathematical background because they involve only basic algebraic skills and set theory basics. The reader may also choose to skip the math and go directly to the sections containing applications, where he or she can pick desired numerical results from tables. The book offers no new so-called winning strategies, although it discusses them from a mathematical point of view. It does, however, offer improved betting systems and helps to organize a player's choices in roulette betting, according to mathematical facts and personal strategies. It is a must-have roulette handbook to be studied before placing your bets on the turn of either a European or American roulette wheel. (shrink)

Book Review Luck, logic, and white lies: the mathematics of games, second edition by Jörg Bewersdorff, New York, Taylor & Francis, CRC Press, 2021, 568 pp., GBP 42.99 (paperback), ISBN 9780367548414, Number of chapters 51.

The Empire of Chance tells how quantitative ideas of chance transformed the natural and social sciences, as well as daily life over the last three centuries. A continuous narrative connects the earliest application of probability and statistics in gambling and insurance to the most recent forays into law, medicine, polling and baseball. Separate chapters explore the theoretical and methodological impact in biology, physics and psychology. Themes recur - determinism, inference, causality, free will, evidence, the shifting meaning of probability - (...) but in dramatically different disciplinary and historical contexts. In contrast to the literature on the mathematical development of probability and statistics, this book centres on how these technical innovations remade our conceptions of nature, mind and society. Written by an interdisciplinary team of historians and philosophers, this readable, lucid account keeps technical material to an absolute minimum. It is aimed not only at specialists in the history and philosophy of science, but also at the general reader and scholars in other disciplines. (shrink)

Before examining de Moivre's contributions to the science of mathematics, this article reviews the source materials, consisting of the printed works and the correspondence of de Moivre, and constructs his biography from them. The analytical part examines de Moivre's contributions and achievements in the study of equations, series, and the calculus of probability. De Moivre contributed to the continuing development from Viète to Abel and Galois of the theory of solving equations by means of constructing particular equations, the roots (...) of which can be written in the form $$\sqrt[n]{{a + \sqrt b }} + \sqrt[n]{{a - \sqrt b }}$$. He also discovered the reciprocal equations. In the course of this work de Moivre discovered an expression equivalent to (cos α+i sin α) n =cos n α+i sin n α and, following Cotes, he succeeded in expressing the nth roots of unity in trigonometric form. In the theory of series, de Moivre developed a polynomial theorem encom-passing Newton's binomial theorem and, in particular, a theorem of recurrent series useful in the calculus of probability. The demands of the calculus of probability led de Moivre to an approximation for the binomial coefficients $$\left( {_m^n } \right)$$ for large values of n. The interaction between de Moivre and James Stirling, particularly in regard to the asymptotic series for log (n!), is treated at length. This work supplied the foundation for de Moivre's limit theorem for the binomial distribution. The calculus of probability, which occupied him from 1708 onward, became in time ever more the center of de Moivre's inquiries. Proceeding from contemporary collections of gambling exercises, de Moivre, by introducing an explicit measure of probability for the so-called Laplace experiments, found the beginnings of a theory of probability. De Moivre expanded the classic application of probability calculus to games of chance by addressing himself to the problem of annuities and by adopting Halley's work with its conception of “Probability of life”. De Moivre was the first to publish a mathematically formulated law for the decrements of life derived from mortality tables. (shrink)

Chance-based gambling has been a recurrent cultural activity throughout history and across many diverse human societies. In this paper, I combine quantitative and qualitative data and present a cultural evolutionary framework to explain why the odds in games of chance in premodern China appeared “designed” to ensure a moderate yet favorable house advantage. This is especially intriguing since extensive research in the history of probability has shown that, prior to the development of probability theory, people had very limited understanding (...) of the nature of random events and were generally disinclined to think mathematically about the frequency of their occurrence. I argue that games of chance in the context of gambling may have culturally evolved into their documented forms via a process of selective imitation and retention, and neither the customers nor the gambling houses understood the probability calculus involved in these games. (shrink)

I introduce a mathematical account of expectation based on a qualitative criterion of coherence for qualitative comparisons between gambles (or random quantities). The qualitative comparisons may be interpreted as an agent’s comparative preference judgments over options or more directly as an agent’s comparative expectation judgments over random quantities. The criterion of coherence is reminiscent of de Finetti’s quantitative criterion of coherence for betting, yet it does not impose an Archimedean condition on an agent’s comparative judgments, it does not require the (...) binary relation reflecting an agent’s comparative judgments to be reflexive, complete or even transitive, and it applies to an absolutely arbitrary collection of gambles, free of structural conditions (e.g., closure, measurability, etc.). Moreover, unlike de Finetti’s criterion of coherence, the qualitative criterion respects the principle of weak dominance, a standard of rational decision making that obliges an agent to reject a gamble that is possibly worse and certainly no better than another gamble available for choice. Despite these weak assumptions, I establish a qualitative analogue of de Finetti’s Fundamental Theorem of Prevision, from which it follows that any coherent system of comparative expectations can be extended to a weakly ordered coherent system of comparative expectations over any collection of gambles containing the initial set of gambles of interest. The extended weakly ordered coherent system of comparative expectations satisfies familiar additivity and scale invariance postulates (i.e., independence) when the extended collection forms a linear space. In the course of these developments, I recast de Finetti’s quantitative account of coherent prevision in the qualitative framework adopted in this article. I show that comparative previsions satisfy qualitative analogues of de Finetti’s famous bookmaking theorem and his Fundamental Theorem of Prevision.The results of this article complement those of another article (Pedersen, Strictly coherent preferences, no holds barred, Manuscript, 2013). I explain how those results entail that any coherent weakly ordered system of comparative expectations over a unital linear space can be represented by an expectation function taking values in a (possibly non-Archimedean) totally ordered field extension of the system of real numbers. The ordered field extension consists of formal power series in a single infinitesimal, a natural and economical representation that provides a relief map tracing numerical non-Archimedean features to qualitative non-Archimedean features. (shrink)

This paper evaluates a recent method proposed by Jeremy Gwiazda for calculating the value of gambles that fail to have expected values in the standard sense. I show that Gwiazda’s method fails to give answers for many gambles that do have standardly defined expected values. However, a slight modification of his method (based on the mathematical notion of the ‘Cauchy principal value’ of an integral), is in fact a proper extension of both his method and the method of ‘weak expectations’. (...) I show that this method gives an appropriate value when the ‘tails’ of the gambles that are eliminated in the truncation are ‘stable’, but that the value is not appropriate when the tails are not stable. I do not attempt to give an argument for the use of this method, but just note that it is more general than Gwiazda’s method, and is mathematically quite natural. (shrink)

Humanity has thrown everything we have at implacable luck—novel theologies, entire philosophical movements, fresh branches of mathematics—and yet we seem to have gained only the smallest edge on the power of fortune. The Myth of Luck tells us why we have been fighting an unconquerable foe. Taking us on a guided tour of one of our oldest concepts, we begin in ancient Greece and Rome, considering how Plato, Plutarch, and the Stoics understood luck, before entering the theoretical world of (...) probability and exploring how luck relates to theology, sports, ethics, gambling, knowledge, and present-day psychology. As we travel across traditions, times and cultures, we come to realize that it’s not that as soon as we solve one philosophical problem with luck that two more appear, like heads on a hydra, but rather that the monster is altogether mythological. We cannot master luck because there is nothing to defeat: luck is no more than a persistent and troubling illusion. By introducing us to compelling arguments and convincing reasons that explain why there is no such thing as luck, we finally see why in a very real sense we make our own luck, that luck is our own doing. The Myth of Luck helps us to regain our own agency in the world - telling the entertaining story of the philosophy and history of luck along the way. (shrink)

This article explores some open questions related to the problem of verification of theories in the context of empirical sciences by contrasting three epistemological frameworks. Each of these epistemological frameworks is based on a corresponding central metaphor, namely: (a) Neo-empiricism and the gambling metaphor; (b) Popperian falsificationism and the scientific tribunal metaphor; (c) Cognitive constructivism and the object as eigen-solution metaphor. Each of one of these epistemological frameworks has also historically co-evolved with a certain statistical theory and method for (...) testing scientific hypotheses, respectively: (a) Decision theoretic Bayesian statistics and Bayes factors; (b) Frequentist statistics and p-values; (c) Constructive Bayesian statistics and e-values. This article examines with special care the Zero Probability Paradox (ZPP), related to the verification of sharp or precise hypotheses. Finally, this article makes some remarks on Lakatos’ view of mathematics as a quasi-empirical science. (shrink)

In his classic book “the Foundations of Statistics” Savage developed a formal system of rational decision making. The system is based on (i) a set of possible states of the world, (ii) a set of consequences, (iii) a set of acts, which are functions from states to consequences, and (iv) a preference relation over the acts, which represents the preferences of an idealized rational agent. The goal and the culmination of the enterprise is a representation theorem: Any preference relation that (...) satisfies certain arguably acceptable postulates determines a (finitely additive) probability distribution over the states and a utility assignment to the consequences, such that the preferences among acts are determined by their expected utilities. Additional problematic assumptions are however required in Savage's proofs. First, there is a Boolean algebra of events (sets of states) which determines the richness of the set of acts. The probabilities are assigned to members of this algebra. Savage's proof requires that this be a σ-algebra (i.e., closed under infinite countable unions and intersections), which makes for an extremely rich preference relation. On Savage's view we should not require subjective probabilities to be σ-additive. He therefore finds the insistence on a σ-algebra peculiar and is unhappy with it. But he sees no way of avoiding it. Second, the assignment of utilities requires the constant act assumption: for every consequence there is a constant act, which produces that consequence in every state. This assumption is known to be highly counterintuitive. The present work contains two mathematical results. The first, and the more difficult one, shows that the σ-algebra assumption can be dropped. The second states that, as long as utilities are assigned to finite gambles only, the constant act assumption can be replaced by the more plausible and much weaker assumption that there are at least two non-equivalent constant acts. The second result also employs a novel way of deriving utilities in Savage-style systems -- without appealing to von Neumann-Morgenstern lotteries. The paper discusses the notion of “idealized agent" that underlies Savage's approach, and argues that the simplified system, which is adequate for all the actual purposes for which the system is designed, involves a more realistic notion of an idealized agent. (shrink)

The article addresses the necessity of increasing the role of mathematics in the psychological intervention in problem gambling, including cognitive therapies. It also calls for interdisciplinary research with the direct contribution of mathematics. The current contributions and limitations of the role of mathematics are analysed with an eye toward the professional profiles of the researchers. An enhanced collaboration between these two disciplines is suggested and predicted.

A predictive model applicable in both neurophysiological and decision-making studies is proposed, bridging the gap between psychological/behavioral and neurophysiological studies. Supposing the electromagnetic waves are carriers of decision-making, and electromagnetic waves with the same frequency, individual amplitude and constant phase triggered by conditions interfere with each other and the resultant intensity determines the probability of the decision. Accordingly, brainwave-interference decision-making model is built mathematically and empirically test with neurophysiological and behavioral data. Event-related potential data confirmed the stability of the phase (...) differences in a given decision context. Behavioral data analysis shows that phase stability exists across categorization-decision, two-stage gambling, and prisoner’s dilemma decisions. Irrational decisions occurring in those experiments are actually rational as their phases could be quantitatively derived from the phases of the riskiest and safest choices. Model fitting result reveals that the root-mean-square deviations between the fitted and actual phases of irrational decisions are less than 10°, and the mean absolute percentage errors of the fitted probabilities are less than 0.06. The proposed model is similar in mathematical form compared with the quantum modeling approach, but endowed with physiological/psychological connection and predictive ability, and promising in the integration of neurophysiological and behavioral research to explore the origin of the decision. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Martha Nussbaum on Dickens's Hard TimesPaulette KidderAt the heart of Martha Nussbaum's work in capability ethics is a rejection of utilitarianism. Nussbaum has repeatedly recounted a pivotal moment in Dickens's Hard Times (1854), in which the young Sissy Jupe delivers an innocent but devastating critique of the utilitarian system.1 Nussbaum's most extended and compelling reading of Hard Times appears in Poetic Justice: The Literary Imagination and Public Life.2 Nussbaum (...) convincingly shows that Dickens's novel is a "deep attack" on utilitarianism, going to the heart of the assumption that the human good consists in maximizing the satisfaction of preferences of the greatest aggregate of persons.In what follows, I will review Nussbaum's philosophical reading of the novel and its critique of utilitarianism, and then return to Hard Times to show that Nussbaum has systematically omitted references to the Biblical themes and references that pervade the novel. Arguing that Dickens's attack on utilitarianism draws much of its power from the resonance of these historically rooted symbols of transcendence, I will conclude with a reflection on the reasons for Nussbaum's avoidance of these themes in her own reading of the book.Hard Times, which focuses on a set of characters living in the polluted and class-divided manufacturing city of "Coketown," is known as Dickens's most polemical novel. The novel announces its central theme immediately in the opening speech by Thomas Gradgrind to the young students of the Gradgrind School: "Now, what I want is, Facts. Teach [End Page 417] these boys and girls nothing but Facts. Facts alone are wanted in life. Plant nothing else, and root out everything else...."3From the youngest age, the children in Gradgrind's school are taught to avoid wonder and imagination and only to focus on science and mathematics—the study of what is observable and measurable. This education leaves his two eldest children, Tom and Louisa, miserably unequipped for the choices they face as adults. Young Tom Gradgrind grows up to be an undisciplined wastrel, incurring gambling debts that lead him to embezzle money and then to frame a factory worker, Stephen Blackpool, for the crime. Louisa Gradgrind endures a miserable marriage to her father's wealthy friend, Josiah Bounderby.At the moral center of the novel is Sissy Jupe, a girl who proves impervious to Gradgrind's educational system. Sissy, the daughter of a horse trainer who works for Sleary's Circus, is abandoned by her father and taken into the Gradgrind home. Sharing the moral center with Sissy is Stephen Blackpool, a weaver whose honesty and conviction cause him to run afoul of both his employer Bounderby and the demagogic Slackbridge, organizer for the trade union.In her reading of Hard Times, Nussbaum identifies four elements of utilitarianism: commensurability, aggregation, maximizing, and exogenous preferences, all of which are both dramatized in the novel and critiqued by it (PJ, p. 14). Commensurability means that "all the valuable things under consideration [are regarded as] valuable because they contain some one thing that itself varies only in quantity" (PJ, p. 14). Characters that Dickens presents as memorable individuals are seen by Gradgrind's system as "parcels of human nature." Students in Gradgrind's School are called by number instead of name, and the workers of Coke-town are seen as "hands and stomachs," and are likened to "ants and beetles." Individuality is erased by what Nussbaum terms Gradgrind's "abstracting mathematical mind" (PJ, p. 21).Aggregation, the second utilitarian element, means that data about individual lives is pooled, "without regarding the boundaries between lives as especially salient for the purposes of choice" (p. 14). Louisa Gradgrind is taught, for example, to view the working classes as "Something to be worked so much and paid so much, and there ended; something to be infallibly settled by laws of supply and demand; something that blundered against those laws, and floundered into difficulty; something that was a little pinched when wheat was dear, and overate itself when wheat was cheap... But, she had scarcely thought more of separating [End Page 418] [the people of Coketown] into units, than of separating the sea itself into its... (shrink)

Decision theory comprises, first, a mathematical formalization of the relations among value, belief, and preference; and second, a set of prescriptions for rational preference. Both aspects of the theory are embodied in a single mathematical proof. The problem in the foundations of decision theory is to explain how elements of one and the same proof can serve both functions. I hope to solve this problem in a way that anchors the decision-theoretic norms of rational preference in fundamental intuitions about rationality (...) in general. I will thus depart from the tradition of anchoring those norms in intui-tions about gambling strategies or preference structures of the sort that are the special concern of the theory itself. Although my interpretation is meant to capture what is right about the decision-theoretic conception of rational preference, it will lead me to argue that there is also something fundamentally wrong about that conception. In my view, decision theory tells us how to be rational in our preferences because it tells us how to have preferences that make sense; but there are ways of making sense that outrun, and may in fact conflict with, the prescriptions of decision theory. (shrink)

“Dutch Book” arguments and references to gambling theorems are typical in the debate between Bayesians and scientists committed to “classical” statistical methods. These arguments have rarely convinced non-Bayesian scientists to abandon certain conventional practices, partially because many scientists feel that gambling theorems have little relevance to their research activities. In other words, scientists “don’t bet.” This article examines one attempt, by Schervish, Seidenfeld, and Kadane, to progress beyond such apparent stalemates by connecting “Dutch Book”–type mathematical results with principles (...) actually endorsed by practicing experimentalists. (shrink)

The field of of imprecise probability has matured, in no small part because of Teddy Seidenfeld’s decades of original scholarship and essential contributions to building and sustaining the ISIPTA community. Although the basic idea behind imprecise probability is (at least) 150 years old, a mature mathematical theory has only taken full form in the last 30 years. Interest in imprecise probability during this period has also grown, but many of the ideas that the mature theory serves can be difficult to (...) apprehend to those new to the subject. Although these fundamental ideas are common knowledge in the ISIPTA community, they are expressed, when they are expressed at all, obliquely, over the course of years with students and colleagues. -/- A single essay cannot convey the store of common knowledge from any research community, let alone the ISIPTA community. But, this essay nevertheless is an attempt to guide those familiar with the basic Bayesian framework to appreciate some of the elegant and powerful ideas that underpin the contemporary theory of lower previsions, which is the theory that most people associate with the term ‘imprecise probabilities’. (shrink)

This note addresses 3 issues that seem to pervade much of economic thought about individual decisions among uncertain alternatives: (1) Restricting primitives to just orderings of first-order gambles and not admitting, e.g., compound acts or joint receipt of consequences and gambles. (2) Great resistance to experimental findings that strongly suggest that most current theories fail descriptively. (3) Taking for granted the innocence of some assumptions when, in fact, they are not innocent, e.g., that constant acts are idempotent. My conclusion is (...) that these restrictions have greatly inhibited developments in utility theory. (shrink)

In 1815, the clever, courted and cherished Annabella Milbanke married the notorious and brilliant Lord Byron. Just one year later, she fled, taking with her their baby daughter, the future Ada Lovelace. Byron himself escaped into exile and died as a revolutionary hero in 1824, aged 36. The one thing he had asked his wife to do was to make sure that their daughter never became a poet. Ada didn't. Brought up by a mother who became one of the most (...) progressive reformers of Victorian England, Byron's little girl was introduced to mathematics as a means of calming her wild spirits. Educated by some of the most learned minds in England, she combined that scholarly discipline with a rebellious heart and a visionary imagination. As a child invalid, Ada dreamed of building a steam-driven flying horse. As an exuberant and boldly unconventional young woman, she amplified her explanations of Charles Babbage's unbuilt calculating engine to predict, as nobody would do for another century, the dawn today of our modern computer age. When Ada died - like her father, she was only 36 - great things seemed still to lie ahead for her as a passionate astronomer. Even while mired in debt from gambling and crippled by cancer, she was frenetically employing Faraday's experiments with light refraction to explore the analysis of distant stars. Drawing on fascinating new material, Seymour reveals the ways in which Byron, long after his death, continued to shape the lives and reputations both of his wife and his daughter. During her life, Lady Byron was praised as a paragon of virtue; within ten years of her death, she was vilified as a disgrace to her sex. Well over a hundred years later, Annabella Milbanke is still perceived as a prudish wife and cruelly controlling mother. But her hidden devotion to Byron and her tender ambitions for his mercurial, brilliant daughter reveal a deeply complex but unsuspectedly sympathetic personality. Miranda Seymour has written a masterful portrait of two remarkable women, revealing how two turbulent lives were often governed and always haunted by the dangerously enchanting, quicksilver spirit of that extraordinary father whom Ada never knew. (shrink)

Incomplete information is a commonly cited barrier to the adoption of new innovations. We present a decision-making experiment, conducted with farmers in the field, that explores the extent to which information which reduces ambiguity may be provided as a public good. In the experiment, participants make a series of decisions between a risky gamble and an ambiguous gamble. An initial private decision is followed by second choice in which participants know that their chosen gambles and outcomes will be publicly but (...) anonymously revealed. Selection of the ambiguous option in this decision thus provide public good information which can be used to update beliefs about the ambiguous gamble. After the public revelation, participants make one final choice between the two options which we use to assess learning. We also test communication as a mechanism to increase provision of the public good. We find evidence that people with preference to avoid ambiguity contribute to public good information while risk averse people free ride. Communication does not affect the overall public good provision, either in a positive or negative direction. Finally, we find that people draw appropriate inference from the information that the public good provides. We relate our findings to the issue of new technology adoption in agriculture which is of particular importance for our participants. (shrink)

Third generation prospect theory is a theory of choices and of judgments of highest buying and lowest selling prices of risky prospects, i.e., of willingness to pay and willingness to accept. The gap between WTP and WTA is sometimes called the “endowment effect” and was previously called the “point of view” effect. Third generation prospect theory combines cumulative prospect theory for risky prospects with the theory that judged values are based on the integration of price paid or price received with (...) the consequences of gambles. In TGPT, the discrepancy between WTP and WTA is due to loss aversion-losses have greater absolute utility than gains of the same value. TGPT was developed independently of similar developments by Birnbaum and Zimmermann :145–187, 1998) and Luce. This paper reviews theoretical and empirical findings, to show that TGPT fails as a descriptive model of both choices and judgments. Evidence refutes three implications of TGPT, but they are consistent with configural weight models in which loss aversion is not needed to describe the results. In the configural weight models, buyers place greater weight on lower consequences, attributes or estimates of value compared to sellers, who place greater configural weight on higher aspects of an object or prospect. (shrink)

Hayek, one of the key thinkers of the twentieth century, has also been much misunderstood. His work has crossed disciplines—economics, philosophy, and political science—as well as national boundaries. He was an early critic of Keynes and became famous in the 1940s for his warnings that the advance of collectivism in Western democracies was the road to serfdom. He was a key figure in the post-war revival of free market liberalism and achieved renewed notoriety and some political influence in the 1970s (...) and 1980s as one of the chief intellectual inspirations for the New Right in Britain and the United States.This book traces Hayek’s intellectual formation in Austrian economics and English liberalism. It analyzes the main themes of his thought such as the idea of market order, the nature of knowledge, the limits of government, and his critiques of socialism and conservatism; assesses the originality and internal coherence of his account of liberalism and modernity; and explores as his interventions in policy debates. Gamble argues that Hayek the social scientist must be disentangled from Hayek the ideologue before you can appreciate the importance and implications of some of his insights into the nature of modern societies.An excellent critical guide to one of the most influential thinkers of our times, this book will be of interest to students in politics, economics, and philosophy, as well as to all those interested in a comprehensive introduction to one of the most controversial thinkers of the twentieth century. (shrink)

In mid-2019, the controversy regarding South African runner Caster Semenya’s eligibility to participate in competitions against other female runners culminated in a Court of Arbitration for Sport judgement. Semenya possessed high endogenous testosterone levels (arguably a performance advantage), secondary to a disorder of sexual development. In this commentary, Aristotelean teleology is used to defend the existence of ‘male’ and ‘female’ as discrete categories. It is argued that once the athlete’s sex is established, they should be allowed to compete in the (...) category of their sex without obligatory medical treatment. Indeed, other athletes who possess advantageous genetic or phenotypic traits that fall outside of the human norm have been allowed to compete as humans without restraint. In both cases, if an athlete possesses the essential attributes of being a human or being male or female they should be permitted to compete in those respective categories; athletes’ eligibilities should not be based upon accidental attributes. (shrink)

New materialism is one of the most important emerging trends in the humanities and social sciences, but it is also one of the least understood. This is because, as a term of ongoing contest...