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

Pigtail: A Pig Addendum

The object of the jeopardy dice game Pig is to be the first player to reach 100 points. Each turn, a player repeatedly rolls a die until either a 1 is rolled or the player holds and scores the sum of the rolls (i.e., the turn total). At any time during a player’s turn, the player is faced with two choices: roll or hold. If the player rolls a 1, the player scores nothing and it becomes the opponent’s turn. If the player rolls a number other than 1, the number is added to the player’s turn total and the player’s turn continues. If the player instead chooses to hold, the turn total is added to the player’s score and it becomes the opponent’s turn. In our original article [Neller and Presser 2004], we described a means to compute optimal play for Pig. Since that time, we have also solved a number of Pig variants. In this addendum, we review the optimality equations for Pig, show how these equations change for several Pig variants, and show how the resulting optimal policies change accordingly. [excerpt

Practical 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 turn, a player repeatedly rolls a die until either a 1 is rolled or the player holds and scores the sum of the rolls (i.e., the turn total). At any time during a player’s turn, the player is faced with two choices: roll or hold. If the player rolls a 1, the player scores nothing and it becomes the opponent’s turn. If the player rolls a number other than 1, the number is added to the player’s turn total and the player’s turn continues. If the player instead chooses to hold, the turn total is added to the player’s score and it becomes the opponent’s turn. In our original article [Neller and Presser 2004], we described a means to compute optimal play for Pig. However, optimal play is surprisingly complex and beyond human potential to memorize and apply. In this paper, we mathematically explore a more subjective question: What is the simplest human-playable policy that most closely approximates optimal play? While one cannot enumerate and search the space of all possible simple policies for Pig play, our exploration will present interesting insights and yield a surprisingly good policy that one can play by memorizing only three integers and using simple mental arithmetic. [excerpt

U.S. Attitudes toward Terrorism and Counterterrorism: Report to the Resilient Systems Division, Science and Technology Directorate, U.S Department of Homeland Security

Existing survey data do not provide comprehensive baseline information about U.S. beliefs and attitudes on terrorism and counterterrorism. Improved understanding of public attitudes can inform programs and tools related to managing public risk perception, increasing effectiveness of pre- and post-event communication by Federal, state, and local officials, and building and supporting more resilient social networks within and across communities.In this project, systematic survey data was collected from a sample of Americans in response to a range of newly developed survey questions. The survey was developed by two leading survey methodologists, following consultations with a research team of experts who study the dynamics of terrorism, counterterrorism, and community resilience, as well as with practitioners and officials from throughout the homeland security community. The questions were administered to members of a web panel by the on-line survey firm Knowledge Networks, and a second wave of the survey will be issued approximately six months after the first wave to allow for analysis of attitudes over time.The first wave of the questionnaire was completed, from September 28, 2012 to October 12, 2012, by 1,576 individuals 18 years of age and older. The first section of the questionnaire assessed the salience of terrorism by asking respondents whether they had thought about terrorism in the preceding week, how likely they thought a terrorist attack in the United States was in the next year, and whether they had done anything differently in the past year because of the possibility of such an attack. The second section of the questionnaire posed questions about how likely respondents would be to call the police in response to various actions potentially related to terrorism and how concerned respondents felt the government should be about these actions. Respondents who said they had thought about a terrorist attack in the last week were more likely than other respondents to say they were likely to call the police in response to the various situations described to them. The survey then assessed respondents' awareness and evaluation of government efforts related to terrorism in the United States. A large majority of the respondents said that the U.S. government has been very effective (33 percent) or somewhat effective (54 percent) at preventing terrorism; less than 13 percent characterized the government as not too effective or not effective at all.In a final section of the survey, we asked respondents about two specific programs focused on increasing communication between members of the public and the government on topics related to terrorism

The Smug Satisfaction of the Media Mentioned. Book Review Of: Public Intellectuals: A Study of Decline. by Richard A. Posner

Book review of: Public intellectuals: a study of decline. By Richard A. Posner. Harvard University Press. 2001 Pp. 408. Reviewed by: Stephen Presse

Carbons and electrolytes for advanced supercapacitors

Electrical energy storage (EES) is one of the most critical areas of technological research around the world. Storing and efficiently using electricity generated by intermittent sources and the transition of our transportation fleet to electric drive depend fundamentally on the development of EES systems with high energy and power densities. Supercapacitors are promising devices for highly efficient energy storage and power management, yet they still suffer from moderate energy densities compared to batteries. To establish a detailed understanding of the science and technology of carbon/carbon supercapacitors, this review discusses the basic principles of the electrical double-layer (EDL), especially regarding the correlation between ion size/ion solvation and the pore size of porous carbon electrodes. We summarize the key aspects of various carbon materials synthesized for use in supercapacitors. With the objective of improving the energy density, the last two sections are dedicated to strategies to increase the capacitance by either introducing pseudocapacitive materials or by using novel electrolytes that allow to increasing the cell voltage. In particular, advances in ionic liquids, but also in the field of organic electrolytes, are discussed and electrode mass balancing is expanded because of its importance to create higher performance asymmetric electrochemical capacitors

Samuel Chase: In Defense of the Rule of Law and Against the Jeffersonians

Samuel Chase is not exactly unknown. Indeed, as the only U.S. Supreme Court Justice to be impeached, he achieved a sort of instant fame, or instant infamy. He is, I think, fairly characterized as a neglected Justice, however, because, in our exclusive focus on his impeachment, we tend to forget that he did possess considerable intelligence, virtue, legal ability, and energy that make him worth our study. His life is also something of an object lesson in how a judge\u27s self-destructive tendencies can harm his reputation. As Richard Peters, his colleague on the Pennsylvania Circuit Court remarked, Chase had a singular instinct for tumult and appeared to have sought controversy whenever he could. I never sat with him without pain, Peters remembered, as he was forever getting into some intemperate and unnecessary squabble. \u27 Chase is remembered as a rabid Federalist partisan and a vehemently anti-Jeffersonian judge. I am not aware of any other Supreme Court Justice who apparently deprived his Court of a full complement of required personnel because he went out on the political hustings to give speeches in support of a presidential candidate he favored (John Adams) and against one he feared (Thomas Jefferson). During the 1800 election campaign, Chase made himself an easy target for Jeffersonian newspapers when he appeared to sympathize zealously with the prosecution of Jeffersonian editors and writers. Indeed, more than one historian has suggested that in the trial of one of these writers, the notorious John Thompson Callender, Chase actively sought to prevent all creatures called democrats from serving on the jury. This assertion, however, is dubious

Legal History or the History of Law: A Primer on Bringing the Law\u27s Past into the Present

The increasing opportunities to teach legal history in law schools and the lamentable decline of positions available to historians in undergraduate institutions have resulted in more historians either teaching in law schools or combining graduate training in history with graduate training in law. As a result, several methodologies or approaches to legal history have emerged. Although legal history has generated a great deal of comment, few have written about how this spate of scholarship and criticism might affect law school teaching. This Article attempts to categorize and to review,therefore, the kinds of insights that American legal history currently offers both to law students and to law professors. Parts II and III of the Article sketch the parameters of the legal history discipline and describe the variant historiographical approaches that scholars recently have adopted. Part IV then offers some suggestions for integrating legal history into selected law school courses