CLASS DESCRIPTIONS—MATHCAMP 2018 Classes
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Game Theory Lecture Notes
Game Theory: Penn State Math 486 Lecture Notes Version 2.1.1 Christopher Griffin « 2010-2021 Licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License With Major Contributions By: James Fan George Kesidis and Other Contributions By: Arlan Stutler Sarthak Shah Contents List of Figuresv Preface xi 1. Using These Notes xi 2. An Overview of Game Theory xi Chapter 1. Probability Theory and Games Against the House1 1. Probability1 2. Random Variables and Expected Values6 3. Conditional Probability8 4. The Monty Hall Problem 11 Chapter 2. Game Trees and Extensive Form 15 1. Graphs and Trees 15 2. Game Trees with Complete Information and No Chance 18 3. Game Trees with Incomplete Information 22 4. Games of Chance 24 5. Pay-off Functions and Equilibria 26 Chapter 3. Normal and Strategic Form Games and Matrices 37 1. Normal and Strategic Form 37 2. Strategic Form Games 38 3. Review of Basic Matrix Properties 40 4. Special Matrices and Vectors 42 5. Strategy Vectors and Matrix Games 43 Chapter 4. Saddle Points, Mixed Strategies and the Minimax Theorem 45 1. Saddle Points 45 2. Zero-Sum Games without Saddle Points 48 3. Mixed Strategies 50 4. Mixed Strategies in Matrix Games 53 5. Dominated Strategies and Nash Equilibria 54 6. The Minimax Theorem 59 7. Finding Nash Equilibria in Simple Games 64 8. A Note on Nash Equilibria in General 66 Chapter 5. An Introduction to Optimization and the Karush-Kuhn-Tucker Conditions 69 1. A General Maximization Formulation 70 2. Some Geometry for Optimization 72 3. -
Frequently Asked Questions in Mathematics
Frequently Asked Questions in Mathematics The Sci.Math FAQ Team. Editor: Alex L´opez-Ortiz e-mail: [email protected] Contents 1 Introduction 4 1.1 Why a list of Frequently Asked Questions? . 4 1.2 Frequently Asked Questions in Mathematics? . 4 2 Fundamentals 5 2.1 Algebraic structures . 5 2.1.1 Monoids and Groups . 6 2.1.2 Rings . 7 2.1.3 Fields . 7 2.1.4 Ordering . 8 2.2 What are numbers? . 9 2.2.1 Introduction . 9 2.2.2 Construction of the Number System . 9 2.2.3 Construction of N ............................... 10 2.2.4 Construction of Z ................................ 10 2.2.5 Construction of Q ............................... 11 2.2.6 Construction of R ............................... 11 2.2.7 Construction of C ............................... 12 2.2.8 Rounding things up . 12 2.2.9 What’s next? . 12 3 Number Theory 14 3.1 Fermat’s Last Theorem . 14 3.1.1 History of Fermat’s Last Theorem . 14 3.1.2 What is the current status of FLT? . 14 3.1.3 Related Conjectures . 15 3.1.4 Did Fermat prove this theorem? . 16 3.2 Prime Numbers . 17 3.2.1 Largest known Mersenne prime . 17 3.2.2 Largest known prime . 17 3.2.3 Largest known twin primes . 18 3.2.4 Largest Fermat number with known factorization . 18 3.2.5 Algorithms to factor integer numbers . 18 3.2.6 Primality Testing . 19 3.2.7 List of record numbers . 20 3.2.8 What is the current status on Mersenne primes? . -
On Likely Solutions of the Stable Matching Problem with Unequal
ON LIKELY SOLUTIONS OF THE STABLE MATCHING PROBLEM WITH UNEQUAL NUMBERS OF MEN AND WOMEN BORIS PITTEL Abstract. Following up a recent work by Ashlagi, Kanoria and Leshno, we study a stable matching problem with unequal numbers of men and women, and independent uniform preferences. The asymptotic formulas for the expected number of stable matchings, and for the probabilities of one point–concentration for the range of husbands’ total ranks and for the range of wives’ total ranks are obtained. 1. Introduction and main results Consider the set of n1 men and n2 women facing a problem of selecting a marriage partner. For n1 = n2 = n, a marriage M is a matching (bijection) between the two sets. It is assumed that each man and each woman has his/her preferences for a marriage partner, with no ties allowed. That is, there are given n permutations σj of the men set and n permutations ρj of the women set, each σj (ρj resp.) ordering the women set (the men set resp.) according to the desirability degree of a woman (a man) as a marriage partner for man j (woman j). A marriage is called stable if there is no unmarried pair (a man, a woman) who prefer each other to their respective partners in the marriage. A classic theorem, due to Gale and Shapley [4], asserts that, given any system of preferences {ρj,σj}j∈[n], there exists at least one stable marriage M. arXiv:1701.08900v2 [math.CO] 13 Feb 2017 The proof of this theorem is algorithmic. A bijection is constructed in steps such that at each step every man not currently on hold makes a pro- posal to his best choice among women who haven’t rejected him before, and the chosen woman either provisionally puts the man on hold or rejects him, based on comparison of him to her current suitor if she has one already. -
On the Effi Ciency of Stable Matchings in Large Markets Sangmok Lee
On the Effi ciency of Stable Matchings in Large Markets SangMok Lee Leeat Yarivyz August 22, 2018 Abstract. Stability is often the goal for matching clearinghouses, such as those matching residents to hospitals, students to schools, etc. We study the wedge between stability and utilitarian effi ciency in large one-to-one matching markets. We distinguish between stable matchings’average effi ciency (or, effi ciency per-person), which is maximal asymptotically for a rich preference class, and their aggregate effi ciency, which is not. The speed at which average effi ciency of stable matchings converges to its optimum depends on the underlying preferences. Furthermore, for severely imbalanced markets governed by idiosyncratic preferences, or when preferences are sub-modular, stable outcomes may be average ineffi cient asymptotically. Our results can guide market designers who care about effi ciency as to when standard stable mechanisms are desirable and when new mechanisms, or the availability of transfers, might be useful. Keywords: Matching, Stability, Effi ciency, Market Design. Department of Economics, Washington University in St. Louis, http://www.sangmoklee.com yDepartment of Economics, Princeton University, http://www.princeton.edu/yariv zWe thank Federico Echenique, Matthew Elliott, Aytek Erdil, Mallesh Pai, Rakesh Vohra, and Dan Wal- ton for very useful comments and suggestions. Euncheol Shin provided us with superb research assistance. Financial support from the National Science Foundation (SES 0963583) and the Gordon and Betty Moore Foundation (through grant 1158) is gratefully acknowledged. On the Efficiency of Stable Matchings in Large Markets 1 1. Introduction 1.1. Overview. The design of most matching markets has focused predominantly on mechanisms in which only ordinal preferences are specified: the National Resident Match- ing Program (NRMP), clearinghouses for matching schools and students in New York City and Boston, and many others utilize algorithms that implement a stable matching correspond- ing to reported rank preferences. -
The Monty Hall Problem in the Game Theory Class
The Monty Hall Problem in the Game Theory Class Sasha Gnedin∗ October 29, 2018 1 Introduction Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice? With these famous words the Parade Magazine columnist vos Savant opened an exciting chapter in mathematical didactics. The puzzle, which has be- come known as the Monty Hall Problem (MHP), has broken the records of popularity among the probability paradoxes. The book by Rosenhouse [9] and the Wikipedia entry on the MHP present the history and variations of the problem. arXiv:1107.0326v1 [math.HO] 1 Jul 2011 In the basic version of the MHP the rules of the game specify that the host must always reveal one of the unchosen doors to show that there is no prize there. Two remaining unrevealed doors may hide the prize, creating the illusion of symmetry and suggesting that the action does not matter. How- ever, the symmetry is fallacious, and switching is a better action, doubling the probability of winning. There are two main explanations of the paradox. One of them, simplistic, amounts to just counting the mutually exclusive cases: either you win with ∗[email protected] 1 switching or with holding the first choice. -
Integers, Rational Numbers, and Algebraic Numbers
LECTURE 9 Integers, Rational Numbers, and Algebraic Numbers In the set N of natural numbers only the operations of addition and multiplication can be defined. For allowing the operations of subtraction and division quickly take us out of the set N; 2 ∈ N and3 ∈ N but2 − 3=−1 ∈/ N 1 1 ∈ N and2 ∈ N but1 ÷ 2= = N 2 The set Z of integers is formed by expanding N to obtain a set that is closed under subtraction as well as addition. Z = {0, −1, +1, −2, +2, −3, +3,...} . The new set Z is not closed under division, however. One therefore expands Z to include fractions as well and arrives at the number field Q, the rational numbers. More formally, the set Q of rational numbers is the set of all ratios of integers: p Q = | p, q ∈ Z ,q=0 q The rational numbers appear to be a very satisfactory algebraic system until one begins to tries to solve equations like x2 =2 . It turns out that there is no rational number that satisfies this equation. To see this, suppose there exists integers p, q such that p 2 2= . q p We can without loss of generality assume that p and q have no common divisors (i.e., that the fraction q is reduced as far as possible). We have 2q2 = p2 so p2 is even. Hence p is even. Therefore, p is of the form p =2k for some k ∈ Z.Butthen 2q2 =4k2 or q2 =2k2 so q is even, so p and q have a common divisor - a contradiction since p and q are can be assumed to be relatively prime. -
Game, Set, and Match Carl W
Game, Set, and Match Carl W. Lee September 2016 Note: Some of the text below comes from Martin Gardner's articles in Scientific American and some from Mathematical Circles by Fomin, Genkin, and Itenberg. 1 1 Fifteen This is a two player game. Take a set of nine cards, numbered one (ace) through nine. They are spread out on the table, face up. Players alternately select a card to add their hands, which they keep face up in front of them. The goal is to achieve a subset of three cards in your hand so that the values of these three cards sum to exactly fifteen. (The ace counts as 1.) This is an example of a game that is isomorphic to (the \same as") a well-known game: Tic-Tac-Toe. Number the cells of a tic-tac-toe board with the integers from 1 to 9 arranged in the form of a magic square. 8 1 6 3 5 7 4 9 2 Then winning combinations correspond exactly to triples of numbers that sum to 15. This is a combinatorial two person game (but one that permits ties). There is no hidden information (e.g., cards that one player has that the other cannot see) and no random elements (e.g., rolling of dice). 2 2 Ones and Twos Ten 1's and ten 2's are written on a blackboard. In one turn, a player may erase (or cross out) any two numbers. If the two numbers erased are identical, they are replaced with a single 2. -
Introduction to Probability 1 Monty Hall
Massachusetts Institute of Technology Course Notes, Week 12 6.042J/18.062J, Fall ’05: Mathematics for Computer Science November 21 Prof. Albert R. Meyer and Prof. Ronitt Rubinfeld revised November 29, 2005, 1285 minutes Introduction to Probability Probability is the last topic in this course and perhaps the most important. Many algo rithms rely on randomization. Investigating their correctness and performance requires probability theory. Moreover, many aspects of computer systems, such as memory man agement, branch prediction, packet routing, and load balancing are designed around probabilistic assumptions and analyses. Probability also comes up in information theory, cryptography, artificial intelligence, and game theory. Beyond these engineering applica tions, an understanding of probability gives insight into many everyday issues, such as polling, DNA testing, risk assessment, investing, and gambling. So probability is good stuff. 1 Monty Hall In the September 9, 1990 issue of Parade magazine, the columnist Marilyn vos Savant responded to this letter: Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say number 1, and the host, who knows what’s behind the doors, opens another door, say number 3, which has a goat. He says to you, ”Do you want to pick door number 2?” Is it to your advantage to switch your choice of doors? Craig. F. Whitaker Columbia, MD The letter roughly describes a situation faced by contestants on the 1970’s game show Let’s Make a Deal, hosted by Monty Hall and Carol Merrill. -
Algebraic Number Theory
Algebraic Number Theory William B. Hart Warwick Mathematics Institute Abstract. We give a short introduction to algebraic number theory. Algebraic number theory is the study of extension fields Q(α1; α2; : : : ; αn) of the rational numbers, known as algebraic number fields (sometimes number fields for short), in which each of the adjoined complex numbers αi is algebraic, i.e. the root of a polynomial with rational coefficients. Throughout this set of notes we use the notation Z[α1; α2; : : : ; αn] to denote the ring generated by the values αi. It is the smallest ring containing the integers Z and each of the αi. It can be described as the ring of all polynomial expressions in the αi with integer coefficients, i.e. the ring of all expressions built up from elements of Z and the complex numbers αi by finitely many applications of the arithmetic operations of addition and multiplication. The notation Q(α1; α2; : : : ; αn) denotes the field of all quotients of elements of Z[α1; α2; : : : ; αn] with nonzero denominator, i.e. the field of rational functions in the αi, with rational coefficients. It is the smallest field containing the rational numbers Q and all of the αi. It can be thought of as the field of all expressions built up from elements of Z and the numbers αi by finitely many applications of the arithmetic operations of addition, multiplication and division (excepting of course, divide by zero). 1 Algebraic numbers and integers A number α 2 C is called algebraic if it is the root of a monic polynomial n n−1 n−2 f(x) = x + an−1x + an−2x + ::: + a1x + a0 = 0 with rational coefficients ai. -
Matchings and Games on Networks
Matchings and games on networks by Linda Farczadi A thesis presented to the University of Waterloo in fulfilment of the thesis requirement for the degree of Doctor of Philosophy in Combinatorics and Optimization Waterloo, Ontario, Canada, 2015 c Linda Farczadi 2015 Author's Declaration I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Abstract We investigate computational aspects of popular solution concepts for different models of network games. In chapter 3 we study balanced solutions for network bargaining games with general capacities, where agents can participate in a fixed but arbitrary number of contracts. We fully characterize the existence of balanced solutions and provide the first polynomial time algorithm for their computation. Our methods use a new idea of reducing an instance with general capacities to an instance with unit capacities defined on an auxiliary graph. This chapter is an extended version of the conference paper [32]. In chapter 4 we propose a generalization of the classical stable marriage problem. In our model the preferences on one side of the partition are given in terms of arbitrary bi- nary relations, that need not be transitive nor acyclic. This generalization is practically well-motivated, and as we show, encompasses the well studied hard variant of stable mar- riage where preferences are allowed to have ties and to be incomplete. Our main result shows that deciding the existence of a stable matching in our model is NP-complete. -
Many More Names of (7, 3, 1)
VOL. 88, NO. 2, APRIL 2015 103 Many More Names of (7, 3, 1) EZRA BROWN Virginia Polytechnic Institute and State University Blacksburg, VA 24061-0123 [email protected] Combinatorial designs are collections of subsets of a finite set that satisfy specified conditions, usually involving regularity or symmetry. As the scope of the 984-page Handbook of Combinatorial Designs [7] suggests, this field of study is vast and far reaching. Here is a picture of the very first design to appear in “Opening the Door,” the first of the Handbook’s 109 chapters: Figure 1 The design that opens the door This design, which we call the (7, 3, 1) design, makes appearances in many areas of mathematics. It seems to turn up again and again in unexpected places. An earlier paper in this MAGAZINE [4] described (7, 3, 1)’s appearance in a number of different areas, including finite projective planes, as the Fano plane (FIGURE 1); graph theory, as the Heawood graph and the doubly regular round-robin tournament of order 7; topology, as an arrangement of six mutually adjacent hexagons on the torus; (−1, 1) matrices, as a skew-Hadamard matrix of order 8; and algebraic number theory, as the splitting field of the polynomial (x 2 − 2)(x 2 − 3)(x 2 − 5). In this paper, we show how (7, 3, 1) makes appearances in three areas, namely (1) Hamming’s error-correcting codes, (2) Singer designs and difference sets based on n-dimensional finite projective geometries, and (3) normed algebras. We begin with an overview of block designs, including two descriptions of (7, 3, 1). -
Algebraic Number Theory Summary of Notes
Algebraic Number Theory summary of notes Robin Chapman 3 May 2000, revised 28 March 2004, corrected 4 January 2005 This is a summary of the 1999–2000 course on algebraic number the- ory. Proofs will generally be sketched rather than presented in detail. Also, examples will be very thin on the ground. I first learnt algebraic number theory from Stewart and Tall’s textbook Algebraic Number Theory (Chapman & Hall, 1979) (latest edition retitled Algebraic Number Theory and Fermat’s Last Theorem (A. K. Peters, 2002)) and these notes owe much to this book. I am indebted to Artur Costa Steiner for pointing out an error in an earlier version. 1 Algebraic numbers and integers We say that α ∈ C is an algebraic number if f(α) = 0 for some monic polynomial f ∈ Q[X]. We say that β ∈ C is an algebraic integer if g(α) = 0 for some monic polynomial g ∈ Z[X]. We let A and B denote the sets of algebraic numbers and algebraic integers respectively. Clearly B ⊆ A, Z ⊆ B and Q ⊆ A. Lemma 1.1 Let α ∈ A. Then there is β ∈ B and a nonzero m ∈ Z with α = β/m. Proof There is a monic polynomial f ∈ Q[X] with f(α) = 0. Let m be the product of the denominators of the coefficients of f. Then g = mf ∈ Z[X]. Pn j Write g = j=0 ajX . Then an = m. Now n n−1 X n−1+j j h(X) = m g(X/m) = m ajX j=0 1 is monic with integer coefficients (the only slightly problematical coefficient n −1 n−1 is that of X which equals m Am = 1).