Game theory with translucent players

A traditional assumption in game theory is that players are opaque to one another—if a player changes strategies, then this change in strategies does not affect the choice of other players’ strategies. In many situations this is an unrealistic assumption. We develop a framework for reasoning about games where the players may be translucent to one another; in particular, a player may believe that if she were to change strategies, then the other player would also change strategies. Translucent players may achieve significantly more efficient outcomes than opaque ones. Our main result is a characterization of strategies consistent with appropriate analogues of common belief of rationality. Common Counterfactual Belief of Rationality (CCBR) holds if (1) everyone is rational, (2) everyone counterfactually believes that everyone else is rational (i.e., all players i believe that everyone else would still be rational even if i were to switch strategies), (3) everyone counterfactually believes that everyone else is rational, and counterfactually believes that everyone else is rational, and so on. CCBR characterizes the set of strategies surviving iterated removal of minimax-dominated strategies, where a strategy \(\sigma \) for player i is minimax dominated by \(\sigma '\) if the worst-case payoff for i using \(\sigma '\) is better than the best possible payoff using \(\sigma \).     

As American as Apple Pi

An interesting history of Pi can be found in Howard Eve's An Introduction to the History of Mathematics.One of the more exciting chapters in this story came about in 1897 with the introduction of House Bill No. 246 in the Indiana State Legislature: ’Be it enacted by the General Assembly of the State of Indiana: It has been found that a circular area is to the square on a line equal to the quadrant of the circumference, as the area of an equilateral rectangle is to the square on one side’.Essentially this little gem of elocution says that circles and squares of equal perimeters have the same area which in turn implies that Pi is equal to four. The bill passed the House but because of newspaper ridicule it was shelved by the Senate. At the time everyone from mathematical quacks to religious zealots got into the act. Here is a typical incident that just might have happened during the confusion.     

血液分组化验问题二次分组化验法最佳分组方式的算法

针对血液分组化验问题,进行了严密的数学分析和数学证明,给出了一次分组化验法最佳k_1值的简单精确计算方法;特别是通过对引理及定理3证明推导出二次分组化验法中每人平均化验次数计算公式;通过对定理4-5证明,总结出二次分组化验法最佳分组时k1和k2的计算方法.同时给出了二次分组化验法最佳分组时p与k_1和k_2对照表,以及一次分组化验法最佳分组时p与k_1对照表.     

Toward optimal gossiping schemes with conference calls

n people have distinct bits of information which they can communicate by k-person conference calls. The object of gossip is to inform everyone of everyone else's information. Which networks admit a minimum call gossip scheme? For k = 2 a necessary and sufficient condition for such a network is that it is connected and contains a cycle of length 4. We address the same question for k>2. The values of n are partitioned into 2 classes: trivial and nontrivial. A necessary and sufficient condition is given for networks of size n for trivial n. For nontrivial n a sufficient condition is given and its necessity is conjectured.     

Floats, Integers, and Single Source Shortest Paths

Floats are ugly, but to everyone but theoretical computer scientists, they are the real thing. A linear time algorithm is presented for the undirected single-source shortest paths problem with nonnegative floating point weights.     

Improving the Fundamental Theorem of Algebra

Theorems 2 and 3 establish the minimum algebraic conditions necessary for a field to be algebraically closed, and they can therefore be said to “optimize” the Fundamental Theorem of Algebra. But each specific“degree implication” is a first-order consequence of the axioms for fields, and could have been discovered two centuries ago; the existence of these finitary relationships appears to have been unsuspected by practically everyone, with one important exception.     

What you as a manager can do to overcome verbal abuse of staff

Employees and their work performance determine the success of any business. Permitting verbal abuse and other disruptive behaviors is bad management. Well-written policies, physician champions, strong executives, and the willingness of OR managers to try new approaches to assist the staff to develop new skill sets will help to create healthier workplaces where everyone knows words don't hurt.     

Grammatical Verification for Mathematical Formula Recognition Based on Context-Free Tree Grammar

This paper proposes the use of a formal grammar for the verification of mathematical formulae for a practical mathematical OCR system. Like a C compiler detecting syntax errors in a source file, we want to have a verification mechanism to find errors in the output of mathematical OCR. A linear monadic context-free tree grammar (LM-CFTG) is employed as a formal framework to define “well-formed” mathematical formulae. A cubic time parsing algorithm for LM-CFTGs is presented. For the purpose of practical evaluation, a verification system for mathematical OCR is developed, and the effectiveness of the system is demonstrated by using the ground-truthed mathematical document database InftyCDB-1 and a misrecognition database newly constructed for this study.     

摸彩决策的数学模型

生活中 ,摸彩的人们总以为越先摸 ,中彩的可能性越大 ,其实不然 !为此本文从概率角度建立了数学模型 ,证明了每个摸彩者中彩的机会均等 ,与先后摸彩顺序无关 .同时 ,通过分析在中彩时 ,已摸过的彩票数目的数学期望 ,向彩票发行者提出了几点建议 .     

Order of Operations

<正>In mathematics,A numerical expression like 4+3×5 or 2²+6÷2 is a combination of numbers and operations.The order of operations tells you which operation to perform first so that everyone gets the same final answer.What you will learn?Evaluate expressions using the order of operations.Today we will talk about"Order of Operations".     

Demystification of the geometric Fourier transforms and resulting convolution theorems

T. Qian As it will turn out in this paper, the recent hype about most of the Clifford–Fourier transforms is not thoroughly worth the pain. Almost everyone that has a real application is separable, and these transforms can be decomposed into a sum of real valued transforms with constant multivecor factors. This fact makes their interpretation, their analysis, and their implementation almost trivial. Copyright © 2015 John Wiley & Sons, Ltd.     

席位分配的平均公平度方法

定义个体相对于总体的公平程度,即个体公平度与总体绝对公平度的比值,当比值趋于1时,就说明分配方案使该个体满意.利用方差的概念定义平均公平度,使个体公平程度相对于总体的公平程度的差距最小,等价于每一个个体公平度都很接近,并且趋于1,每个个体的公平程度达到最大,此时座位分配最为公平.     

Mathematical Sociology: Some Educational and Organizational Problems of an Emergent Sub‐Discipline

Three major areas of mathematical sociology are critically reviewed: analysis of measurement, statistical analysis and model building. Next, some social problems, created by the introduction of mathematics into sociology, are discussed. These include the emergence of inflated expectations for mathematical sociology which are subsequently disappointed, and the potential status threat which mathematical sociology poses for non‐mathematical sociologists. Examples of mathematical applications in the construction of causal models, population projection and the analysis of stability in social groups are discussed. Following this, the role of mathematics in the education of undergraduate sociology majors is considered. Neither mathematics nor statistics should be required of such persons, but they should be encouraged to acquire a mathematical background if interested. Statistics should, however, be required of sociology graduate students. The graduate training of mathematical sociologists should emphasize research over course work. An apprenticeship relationship with a faculty member working in mathematical sociology is highly desirable for these students. A substantive specialty is also useful since it enables mathematical sociologists to stay in contact with mainstream sociology. Emphasis is placed on the function of present research in legitimating future expanded mathematical education of sociologists.     

The mathematical beliefs and behavior of high school students: Insights from a longitudinal study

There is a documented need for more research on the mathematical beliefs of students below college. In particular, there is a need for more studies on how the mathematical beliefs of these students impact their mathematical behavior in challenging mathematical tasks. This study examines the beliefs on mathematical learning of five high school students and the students’ mathematical behavior in a challenging probability task. The students were participants in an after-school, classroom-based, longitudinal study on students’ development of mathematical ideas funded by the United States National Science Foundation. The results show that particular educational experiences can alter results from previous studies on the mathematical beliefs and behavior of students below college, some of which have been used to justify non-reform pedagogical approaches in mathematics classrooms. Implications for classroom practice and ideas for future research are discussed.     

Locally logical mathematics: An emerging teacher honoring both students and mathematics

This case study explores the mathematics engagement and teaching practice of a beginning secondary school teacher. The focus is on the mathematical opportunities available to her students (the classroom mathematics) and how they relate to the teacher's personal capacity and tendencies for mathematical engagement (her personal mathematics). We use a mathematical process-and-action approach to analyze mathematical engagement and then employ the teaching triad—mathematical challenge, sensitivity to students, and management of learning—to situate mathematical engagement within the larger context of teaching practice. The article develops the construct of locally logical mathematics to underscore the cogency of mathematical engagement in the classroom as part of a coherent mathematical system that is embedded within a teaching practice. Contributions of the study include the process-and-action approach, especially in tandem with the teaching triad, as a tool to understand nuances of mathematical engagement and differences in demand between written and implemented tasks.     

The effect of data and derivation

<正>The Spring Festival Gala is always an indispensable main course on the Chinese Spring Festival dinner over the years.But since I can remember,the show falls into the situation that controversy occurs every year.In junior high school,my explanation of this phenomenon is that the reaction nerve will be numb with repeating stimulus.The first Spring Festival Gala made everyone     

With a focus on ‘Grundvorstellungen’ Part 1: a theoretical integration into current concepts

Current comparative studies such as PISA assess individual achievement in an attempt to grasp the concept of competence. Working with mathematics is then put into concrete terms in the area of application. Thereby, mathematical work is understood as a process of modelling: At first, mathematical models are taken from a real problem; then the mathematical model is solved; finally the mathematical solution is interpreted with a view to reality and the original problem is validated by the solution. During this cycle the main focus is on the transition between reality and the mathematical level. Mental objects are necessary for this transition. These mental objects are described in the German didactic with the concept of Grundvorstellungen'. In the delimitation to related educational constructs, ‘Grundvorstellungen’ can be described as mental models of a mathematical concept.     

Mathematical creativity in generally gifted and mathematically excelling adolescents: what makes the difference?

Due to uncertainty regarding the relationship between mathematical creativity, mathematical expertise and general giftedness, we have conducted a large-scale study that explores the relationship between mathematical creativity and mathematical ability. We distinguish between relative and absolute creativity in order to address personal creativity as a characteristic that can be developed in schoolchildren. This paper presents part of a study that focuses on the power of multiple solution tasks (MSTs) as a tool for the evaluation of relative creativity. We discuss relationships between mathematical creativity, mathematical ability and general giftedness as reflected in the present empirical study of senior high school students in Israel which implemented the MST tool. The study demonstrates that between-group differences are task dependent and are a function of mathematical insight as it is integrated in the mathematical task. Thus, we conclude that different types of MSTs can be used for different research purposes, which we discuss at the end of this paper.     

Mathematics education: Procedures, rituals and man's search for meaning

An attempt is made to analyze mathematical behavior from more general psychological perspectives. The mathematical language is a special case of the human language, which is a form of expression. Many people use common language in a meaningless way. The same is true about the mathematical language. Rituals are other forms of expression. Many people identify rituals in many mathematical contexts (procedures, argumentation). Thus, quite often, they behave in a meaningless way as required by many rituals. On the other hand, the community of mathematics education struggles for meaningful learning. This can be regarded as a special case of man's search for meaning. The general claims will be illustrated by some examples from various mathematical contexts.     

Field-based hypotheses on advancing standards for mathematical practice

The Common Core State Standards in Mathematics (CCSSM, 2010) are organized around two types of standards: the standards for mathematical content and standards for mathematical practice. The central goal of this paper is to present cognitive and instructional analyses of standards for mathematical practice through a discussion of field-based activities with inservice secondary mathematics teachers and students. A potential value of the study is that it provides researchers with specific field-based hypotheses on advancing standards for mathematical practice.