Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Supporting Online Material: Adaptive Dynamics of Continuous Games Michael Doebeli 1, Christoph Hauert 1,2 & Timothy Killingback 3 1 Departments of Mathematics and Zoology, University of British Columbia, Vancouver BC V6T 1Z4, Canada. 2 Present address: Program for Evolutionary Dynamics, Harvard University, One Brattle Square, Cambridge MA 02138, USA. 3 Ecology and Evolution, ETH Zürich, 8092 Zürich, Switzerland. 1 Adaptive Dynamics The evolution of a continuous trait x under mutation and selection can be analyzed using the mathematical framework of adaptive dynamics (S1, S2, S3). The central concept is that of invasion fitness f x (y), which denotes the growth rate of a rare mutant y in a resident population that is monomorphic for trait x. The adaptive dynamics of the trait x is then governed by the selection gradient D(x) = f x / y y=x, so that ẋ = D(x). For a detailed discussion of the underlying assumptions of this dynamic equation we refer to (S1, S2, S3). Note that in general, ẋ = md(x), where m depends on population size and reflects the mutational process providing the raw material for evolutionary change. For constant population sizes, m is simply a parameter that scales time, and one can set m = 1 without loss of generality. Singular points of the adaptive dynamics are given by solutions of D(x ) = 0. If there is no such solution, the trait x either always increases or decreases evolutionarily, depending on the sign of D(x). If a singular trait x exists, it is convergent stable and hence an attractor for the adaptive dynamics if dd/dx x=x < 0. If this inequality is reversed, x is a repellor, i.e. the trait x evolves to ever lower values if the initial trait of the population x 0 is x 0 < x, and to ever higher values if x 0 > x. Very interesting evolutionary dynamics can occur because convergence stability does not imply evolutionary stability. Generically, a convergent stable singular point x is either a 1

2 maximum or a minimum of the invasion fitness f x (y) (S2, S3). If x is a maximum, i.e., if 2 f x / y 2 y=x < 0, then x is evolutionarily stable, i.e., it cannot be invaded by any mutant. If, however, x is a minimum, i.e., if 2 f x / y 2 y=x > 0, then a population that is monomorphic for x can be invaded by mutants with trait values on either side of x. In this case, the population first converges evolutonarily towards x, but subsequently splits into two distinct and diverging phenotypic clusters. This phenomenon is called evolutionary branching, and the singular point is called an evolutionary branching point. 1.1 Continuous Snowdrift Game In the Continuous Snowdrift game, the quantitative trait x represents the level of cooperative investments. The growth rate of a rare mutant strategy y in a monomorphic resident x is determined by replicator dynamics (S4), and the invasion fitness of the rare mutant y is given by f x (y) = P (y, x) P (x, x), where P (y, x) is the payoff of y playing against x, and P (x, x) is the payoff of x playing against itself. In the Continuous Snowdrift game the payoff is given by P (x, y) = B(x + y) C(x) where B(x) and C(x) are the benefit and cost functions. The adaptive dynamics of the continuous trait x is then given by ẋ = D(x) = B (2x) C (x), and singular strategies are solutions of B (2x ) = C (x ). If a singular strategy x exists, it is convergent stable if dd/dx x=x = 2B (2x ) C (x ) < 0, and it is evolutionarily unstable if 2 f x / y 2 y=x = B (2x ) C (x ) > 0. Consequently, x is a branching point if 2B (2x ) < C (x ) < B (2x ) < 0. Note that adaptive dynamics never maximizes the monomorphic population payoff given by B(2x) C(x), because maximizing this payoff would yield the gradient dynamics 2B (2x) C (x), which is different form the adaptive dynamics given above Quadratic Cost and Benefit Functions If the cost and benefit functions B(x) and C(x) are linear, the gradient D(x) is constant and the evolution of the trait x is always directional, either leading to ever higher or ever lower investments (except in the degenerate case where x is a neutral trait). However, richer dynamics are already observed for quadratic cost and benefit functions C(x) = c 2 x 2 + c 1 x and B(x) = 2

3 b 2 x 2 + b 1 x, for which the adaptive dynamics are given by ẋ = 4b 2 x + b 1 2c 2 x c 1. The singular strategy (if it exists) is given by x = c 1 b 1 2(2b 2 c 2 ) and is convergent stable if 2b 2 c 2 < 0 and evolutionarily stable if b 2 c 2 < 0. The existence of a positive x requires either (i) 4b 2 2c 2 > c 1 b 1 > 0 or (ii) 4b 2 2c 2 < c 1 b 1 < 0. In the first case, x is always a repellor. Whether the repellor is evolutionarily stable or not is irrelevant as this state is never reached from generic intitial conditions. In the second case x is always convergent stable. If, in addition, b 2 c 2 < 0, then x is evolutionarily stable strategy (ESS). In this case, the population evolves towards the evolutionary end state x in which all individuals make intermediate cooperative investments. If, however, b 2 c 2 > 0, then x is an evolutionary branching point, and after converging towards x, the population splits into two distinct and diverging phenotypic clusters Adaptive Dynamics after Branching After evolutionary branching, the adaptive dynamics can be calculated based on the equilibrium frequencies of the two co-existing strategies x > x > y (where x is the branching point). According to traditional replicator dynamics (S4) the equilibrium frequency p of strategy x is the solution of p P (x, x) + (1 p )P (x, y) = p P (y, x) + (1 p )P (y, y). With quadratic cost and benefit functions, this yields p = c 1 b 1 +x(c 2 b 2 )+y(c 2 3b 2 ) 2b 2. The invasion fitness of a mutant (x y) v with respect to the two resident branches x and y is then given by f x,y (v) = p P (v, x) + (1 p )P (v, y) P (x, y) where P (x, y) = p P (x, x)+(1 p )P (x, y) = p P (y, x)+(1 p )P (y, y) denotes the average population payoff. Adaptive dynamics in the two branches is then given by ẋ = m 1 (x, y) fx,y(v) and ẏ = m v 2 (x, y) fx,y(v), respectively, where m v=x v 1 (x, y) p and v=y m 1 (x, y) 1 p are positive quantities describing the mutational process in the two branches (S2, S3). For the quadratic cost and benefit functions one finds ẋ = m 1 (x, y)(b 2 c 2 )(x y) and ẏ = m 2 (x, y)(b 2 c 2 )(x y). This implies that after branching, evolution is always directional and drives the trait values in the two branches to the boundaries of the strategy range. With more complicated cost and benefit functions, such as those used for Fig. 3 (see main text), analytical solutions for p, and hence for the 2-dimensional adaptive dynamics, can in general not be obtained. 3

4 1.1.3 Groups of N Interacting Individuals The Continuous Snowdrift can be generalized to groups of N > 2 interacting individuals, in which the N individuals make cooperative investments x 1,..., x N towards a common good. There are a number of ways in which this can be done, but perhaps the most straightforward approach is to assume that the payoff to the i-th player is 1 B( N x N j ) C(x i ), reflecting the fact that the benefit accrued from the sum of the individual investments is equally shared among the N interacting players. In principle, the factor 1/N can be incorporated into the benefit function, but explicitly retaining it facilitates comparison of results for different group sizes N (see below). The payoff of a mutant y in a monomorphic resident population with strategy x is P (y, x) = 1 B((N 1)x + y) C(y). Using similar arguments as above, this yields the N adaptive dynamics ẋ = 1 N B (Nx) C (x). Singular strategies x are now given as solutions of 1 N B (Nx ) C (x ) = 0. A singular strategy is convergent stable if B (Nx ) C (x ) < 0, and it is evolutionarily stable if 1 N B (Nx ) C (x ) < 0. The condition for evolutionary branching becomes B (Nx ) < C (x ) < 1 N B (Nx ) < 0. With quadratic cost and benefit functions, the singular strategy for N-player games is thus given by x = c 1 1 N b 1 2(b 2 c 2 ). In complete analogy to pairwise interactions, the existence of x requires either (i) 2(b 2 c 2 ) > c 1 1 N b 1 > 0 or (ii) 2(b 2 c 2 ) < c 1 1 N b 1 < 0. In the first case, x is again a repellor. In the second case it is an ESS if b 2 c 2 < 0, while x is an evolutionary branching point if b 2 < c 2 < 1 N b 2 < 0. It follows that the range of parameters generating evolutionary branching increases with group size N. 2 Individual-Based Simulations The analytical predictions derived from adaptive dynamics can be illustrated and verified through individual-based simulations of the Continuous Snowdrift game in finite populations of fixed size N pop (S5). These simulations emulate replicator dynamics in populations in which individuals are characterized by their investment strategy x. The population is updated asynchronously by sequentially choosing a random focal individual x to be replaced by an offspring as follows. The payoff of the focal individual, P x = P (x, z), is determined through a single interaction j=1 4

5 with a random member z of the population. P x is then compared to the payoff of another randomly chosen individual y, whose payoff P y is also obtained through a random interaction. With a probability w proportional to the payoff difference, w = (P y P x )/α (where α = max x,y,u,v P (x, y) P (u, v) ensures w 1), the offspring replacing the focal individual has y as parent; otherwise the parent is the focal individual itself (which is also the case if P y < P x ). The offspring inherits the parental strategy, except if a mutation occurs (which happens with probability µ), in which case the offspring strategy is drawn from a Gaussian distribution with the parental strategy as mean and a small standard deviation σ. This numerical scheme implements the deterministic replicator dynamics in the limit of large population sizes N pop (S6). Our individual-based models of the Continuous Snowdrift game can be explored interactively at References and Notes S1. U. Dieckmann, R. Law, J. Math. Biol. 34, 579 (1996). S2. S. A. H. Geritz, E. Kisdi, G. Meszena, J. A. J. Metz, Evol. Ecol. 12, 35 (1998). S3. J. A. J. Metz, S. A. H. Geritz, G. Meszena, F. J. A. Jacobs, J. S. van Heerwaarden, Stochastic and Spatial Structures of Dynamical Systems, S. J. van Strien, S. M. Verduyn Lunel, eds. (North Holland, Amsterdam, 1996), pp S4. J. Hofbauer, K. Sigmund, Evolutionary Games and Population Dynamics (Cambridge University Press, Cambridge, 1998). S5. C. Hauert, Virtuallabs: Interactive tutorials on evolutionary game theory, (2004). S6. C. Hauert, M. Doebeli, Nature 428, 643 (2004). 5

### The Evolutionary Origin of Cooperators and Defectors

The Evolutionary Origin of Cooperators and Defectors Michael Doebeli, 1 * Christoph Hauert, 1. Timothy Killingback 2 Coexistence of cooperators and defectors is common in nature, yet the evolutionary origin

### 10 Evolutionarily Stable Strategies

10 Evolutionarily Stable Strategies There is but a step between the sublime and the ridiculous. Leo Tolstoy In 1973 the biologist John Maynard Smith and the mathematician G. R. Price wrote an article in

### Online Appendix to Stochastic Imitative Game Dynamics with Committed Agents

Online Appendix to Stochastic Imitative Game Dynamics with Committed Agents William H. Sandholm January 6, 22 O.. Imitative protocols, mean dynamics, and equilibrium selection In this section, we consider

### Stochastic dynamics of invasion and fixation

Stochastic dynamics of invasion and fixation Arne Traulsen, 1 Martin A. Nowak, 1,2 and Jorge M. Pacheco 1,3 1 Program for Evolutionary Dynamics, Harvard University, Cambridge, Massachusetts 02138, USA

### 1.5. Factorisation. Introduction. Prerequisites. Learning Outcomes. Learning Style

Factorisation 1.5 Introduction In Block 4 we showed the way in which brackets were removed from algebraic expressions. Factorisation, which can be considered as the reverse of this process, is dealt with

### European Options Pricing Using Monte Carlo Simulation

European Options Pricing Using Monte Carlo Simulation Alexandros Kyrtsos Division of Materials Science and Engineering, Boston University akyrtsos@bu.edu European options can be priced using the analytical

### A stochastic individual-based model for immunotherapy of cancer

A stochastic individual-based model for immunotherapy of cancer Loren Coquille - Joint work with Martina Baar, Anton Bovier, Hannah Mayer (IAM Bonn) Michael Hölzel, Meri Rogava, Thomas Tüting (UniKlinik

### Brännström, Ake; Johansson, Jacob; von Festenberg, Niels. Article The Hitchhiker's Guide to adaptive dynamics

econstor www.econstor.eu Der OpenAccessPublikationsserver der ZBW LeibnizInformationszentrum Wirtschaft The Open Access Publication Server of the ZBW Leibniz Information Centre for Economics Brännström,

### Evolution leads to Kantian morality

Evolution leads to Kantian morality Ingela Alger (Toulouse School of Economics and IAST) Jörgen Weibull (Stockholm School of Economics and IAST) Oslo, 6 October 6, 2015 Introduction For decades economics

### A Note on Best Response Dynamics

Games and Economic Behavior 29, 138 150 (1999) Article ID game.1997.0636, available online at http://www.idealibrary.com on A Note on Best Response Dynamics Ed Hopkins Department of Economics, University

### Additive decompositions for Fisher, Törnqvist and geometric mean indexes

Journal of Economic and Social Measurement 28 (2002) 51 61 51 IOS Press Additive decompositions for Fisher, Törnqvist and geometric mean indexes Marshall B. Reinsdorf a, W. Erwin Diewert b and Christian

### Introduction to nonparametric regression: Least squares vs. Nearest neighbors

Introduction to nonparametric regression: Least squares vs. Nearest neighbors Patrick Breheny October 30 Patrick Breheny STA 621: Nonparametric Statistics 1/16 Introduction For the remainder of the course,

### Linear and quadratic Taylor polynomials for functions of several variables.

ams/econ 11b supplementary notes ucsc Linear quadratic Taylor polynomials for functions of several variables. c 010, Yonatan Katznelson Finding the extreme (minimum or maximum) values of a function, is

### On the Evolution of Behavioral Heterogeneity in Individuals and Populations

Biology and Philosophy 13: 205 231, 1998. c 1998 Kluwer Academic Publishers. Printed in the Netherlands. On the Evolution of Behavioral Heterogeneity in Individuals and Populations CARL T. BERGSTROM Department

### HOMEWORK 4 SOLUTIONS. All questions are from Vector Calculus, by Marsden and Tromba

HOMEWORK SOLUTIONS All questions are from Vector Calculus, by Marsden and Tromba Question :..6 Let w = f(x, y) be a function of two variables, and let x = u + v, y = u v. Show that Solution. By the chain

### Evolutionary Branching and Sympatric Speciation in Diploid Populations

International Institute for Applied Systems Analysis Schlossplatz 1 A-361 Laxenburg Austria Telephone: (+43 36) 807 34 Fax: (+43 36) 71313 E-mail: publications@iiasa.ac.at Internet: www.iiasa.ac.at Interim

### Sensitivity analysis of utility based prices and risk-tolerance wealth processes

Sensitivity analysis of utility based prices and risk-tolerance wealth processes Dmitry Kramkov, Carnegie Mellon University Based on a paper with Mihai Sirbu from Columbia University Math Finance Seminar,

### Tetris: Experiments with the LP Approach to Approximate DP

Tetris: Experiments with the LP Approach to Approximate DP Vivek F. Farias Electrical Engineering Stanford University Stanford, CA 94403 vivekf@stanford.edu Benjamin Van Roy Management Science and Engineering

### ELEC-E8104 Stochastics models and estimation, Lecture 3b: Linear Estimation in Static Systems

Stochastics models and estimation, Lecture 3b: Linear Estimation in Static Systems Minimum Mean Square Error (MMSE) MMSE estimation of Gaussian random vectors Linear MMSE estimator for arbitrarily distributed

### Mistakes Allow Evolutionary Stability in the Repeated Prisoner's Dilemma Game. I. Introduction

J. theor. Biol. (1989) 136, 47-56 Mistakes Allow Evolutionary Stability in the Repeated Prisoner's Dilemma Game ROBERT BOYD Department of Anthropology, University of California, Los Angeles, California

### Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model

Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model 1 September 004 A. Introduction and assumptions The classical normal linear regression model can be written

### How do populations evolve?... Are there any trends?...

How do populations evolve?... Are there any trends?... Gene pool: all of the genes of a population Allele frequency: the percentage of any particular allele in a gene pool A population in which an allele

### Sealed Bid Second Price Auctions with Discrete Bidding

Sealed Bid Second Price Auctions with Discrete Bidding Timothy Mathews and Abhijit Sengupta August 16, 2006 Abstract A single item is sold to two bidders by way of a sealed bid second price auction in

### 1.4. Removing Brackets. Introduction. Prerequisites. Learning Outcomes. Learning Style

Removing Brackets 1. Introduction In order to simplify an expression which contains brackets it is often necessary to rewrite the expression in an equivalent form but without any brackets. This process

### Partial Fractions. (x 1)(x 2 + 1)

Partial Fractions Adding rational functions involves finding a common denominator, rewriting each fraction so that it has that denominator, then adding. For example, 3x x 1 3x(x 1) (x + 1)(x 1) + 1(x +

### Estimation and Inference in Cointegration Models Economics 582

Estimation and Inference in Cointegration Models Economics 582 Eric Zivot May 17, 2012 Tests for Cointegration Let the ( 1) vector Y be (1). Recall, Y is cointegrated with 0 cointegrating vectors if there

### Lecture L18 - Exploring the Neighborhood: the Restricted Three-Body Problem

S. Widnall 16.07 Dynamics Fall 008 Version 1.0 Lecture L18 - Exploring the Neighborhood: the Restricted Three-Body Problem The Three-Body Problem In Lecture 15-17, we presented the solution to the two-body

### L10: Probability, statistics, and estimation theory

L10: Probability, statistics, and estimation theory Review of probability theory Bayes theorem Statistics and the Normal distribution Least Squares Error estimation Maximum Likelihood estimation Bayesian

### LECTURES ON REAL OPTIONS: PART II TECHNICAL ANALYSIS

LECTURES ON REAL OPTIONS: PART II TECHNICAL ANALYSIS Robert S. Pindyck Massachusetts Institute of Technology Cambridge, MA 02142 Robert Pindyck (MIT) LECTURES ON REAL OPTIONS PART II August, 2008 1 / 50

### A Programme Implementation of Several Inventory Control Algorithms

BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume, No Sofia 20 A Programme Implementation of Several Inventory Control Algorithms Vladimir Monov, Tasho Tashev Institute of Information

### PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.

PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include

### 3. INNER PRODUCT SPACES

. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.

### Vector and Matrix Norms

Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a non-empty

### Joint Probability Distributions and Random Samples (Devore Chapter Five)

Joint Probability Distributions and Random Samples (Devore Chapter Five) 1016-345-01 Probability and Statistics for Engineers Winter 2010-2011 Contents 1 Joint Probability Distributions 1 1.1 Two Discrete

### P (A) = lim P (A) = N(A)/N,

1.1 Probability, Relative Frequency and Classical Definition. Probability is the study of random or non-deterministic experiments. Suppose an experiment can be repeated any number of times, so that we

### 1 Mathematical Models of Cost, Revenue and Profit

Section 1.: Mathematical Modeling Math 14 Business Mathematics II Minh Kha Goals: to understand what a mathematical model is, and some of its examples in business. Definition 0.1. Mathematical Modeling

### CHAPTER 23 THE EVOLUTIONS OF POPULATIONS. Section B: Causes of Microevolution

CHAPTER 23 THE EVOLUTIONS OF POPULATIONS Section B: Causes of Microevolution 1. Microevolution is generation-to-generation change in a population s allele frequencies 2. The two main causes of microevolution

### Regression Using Support Vector Machines: Basic Foundations

Regression Using Support Vector Machines: Basic Foundations Technical Report December 2004 Aly Farag and Refaat M Mohamed Computer Vision and Image Processing Laboratory Electrical and Computer Engineering

### Review of Fundamental Mathematics

Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools

### Numerical Methods for Option Pricing

Chapter 9 Numerical Methods for Option Pricing Equation (8.26) provides a way to evaluate option prices. For some simple options, such as the European call and put options, one can integrate (8.26) directly

### Equations, Inequalities & Partial Fractions

Contents Equations, Inequalities & Partial Fractions.1 Solving Linear Equations 2.2 Solving Quadratic Equations 1. Solving Polynomial Equations 1.4 Solving Simultaneous Linear Equations 42.5 Solving Inequalities

### Covariance and Correlation

Covariance and Correlation ( c Robert J. Serfling Not for reproduction or distribution) We have seen how to summarize a data-based relative frequency distribution by measures of location and spread, such

### 3.6. Partial Fractions. Introduction. Prerequisites. Learning Outcomes

Partial Fractions 3.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. For 4x + 7 example it can be shown that x 2 + 3x + 2 has the same

### Scientific Computing: An Introductory Survey

Scientific Computing: An Introductory Survey Chapter 10 Boundary Value Problems for Ordinary Differential Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign

### An Introduction to Superprocesses

Olav Kallenberg, March 2007: An Introduction to Superprocesses The Mathematics of Life 1. Branching Brownian motion 2. Diffusion approximation 3. Persistence extinction 4. Fractal properties 5. Family

### Chapter 23. (Mendelian) Population. Gene Pool. Genetic Variation. Population Genetics

30 25 Chapter 23 Population Genetics Frequency 20 15 10 5 0 A B C D F Grade = 57 Avg = 79.5 % (Mendelian) Population A group of interbreeding, sexually reproducing organisms that share a common set of

### Joint Probability Distributions and Random Samples. Week 5, 2011 Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage

5 Joint Probability Distributions and Random Samples Week 5, 2011 Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage Two Discrete Random Variables The probability mass function (pmf) of a single

### Week 2 Quiz: Equations and Graphs, Functions, and Systems of Equations

Week Quiz: Equations and Graphs, Functions, and Systems of Equations SGPE Summer School 014 June 4, 014 Lines: Slopes and Intercepts Question 1: Find the slope, y-intercept, and x-intercept of the following

### Computational Learning Theory Spring Semester, 2003/4. Lecture 1: March 2

Computational Learning Theory Spring Semester, 2003/4 Lecture 1: March 2 Lecturer: Yishay Mansour Scribe: Gur Yaari, Idan Szpektor 1.1 Introduction Several fields in computer science and economics are

### A Game Theoretical Framework for Adversarial Learning

A Game Theoretical Framework for Adversarial Learning Murat Kantarcioglu University of Texas at Dallas Richardson, TX 75083, USA muratk@utdallas Chris Clifton Purdue University West Lafayette, IN 47907,

### is identically equal to x 2 +3x +2

Partial fractions 3.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. 4x+7 For example it can be shown that has the same value as 1 + 3

### 15.062 Data Mining: Algorithms and Applications Matrix Math Review

.6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop

### Don t forget the degrees of freedom: evaluating uncertainty from small numbers of repeated measurements

Don t forget the degrees of freedom: evaluating uncertainty from small numbers of repeated measurements Blair Hall b.hall@irl.cri.nz Talk given via internet to the 35 th ANAMET Meeting, October 20, 2011.

19 LINEAR QUADRATIC REGULATOR 19.1 Introduction The simple form of loopshaping in scalar systems does not extend directly to multivariable (MIMO) plants, which are characterized by transfer matrices instead

### FE670 Algorithmic Trading Strategies. Stevens Institute of Technology

FE670 Algorithmic Trading Strategies Lecture 6. Portfolio Optimization: Basic Theory and Practice Steve Yang Stevens Institute of Technology 10/03/2013 Outline 1 Mean-Variance Analysis: Overview 2 Classical

### Introduction to the Finite Element Method (FEM)

Introduction to the Finite Element Method (FEM) ecture First and Second Order One Dimensional Shape Functions Dr. J. Dean Discretisation Consider the temperature distribution along the one-dimensional

### JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson

JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials

### 1 Lecture: Integration of rational functions by decomposition

Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.

### 4.5 Linear Dependence and Linear Independence

4.5 Linear Dependence and Linear Independence 267 32. {v 1, v 2 }, where v 1, v 2 are collinear vectors in R 3. 33. Prove that if S and S are subsets of a vector space V such that S is a subset of S, then

### ECO 199 B GAMES OF STRATEGY Spring Term 2004 PROBLEM SET 4 B DRAFT ANSWER KEY 100-3 90-99 21 80-89 14 70-79 4 0-69 11

The distribution of grades was as follows. ECO 199 B GAMES OF STRATEGY Spring Term 2004 PROBLEM SET 4 B DRAFT ANSWER KEY Range Numbers 100-3 90-99 21 80-89 14 70-79 4 0-69 11 Question 1: 30 points Games

### Iterated Mutual Observation with Genetic Programming

Iterated Mutual Observation with Genetic Programming Peter Dittrich (), Thomas Kron (2), Christian Kuck (), Wolfgang Banzhaf () () University of Dortmund (2) University of Hagen Department of Computer

### The Delta Method and Applications

Chapter 5 The Delta Method and Applications 5.1 Linear approximations of functions In the simplest form of the central limit theorem, Theorem 4.18, we consider a sequence X 1, X,... of independent and

### An Analysis on Behaviors of Real Estate Developers and Government in Sustainable Building Decision Making

Journal of Industrial Engineering and Management JIEM, 2014 7(2): 491-505 Online ISSN: 2013-0953 Print ISSN: 2013-8423 http://dx.doi.org/10.3926/jiem.1042 An Analysis on Behaviors of Real Estate Developers

### Marketing Mix Modelling and Big Data P. M Cain

1) Introduction Marketing Mix Modelling and Big Data P. M Cain Big data is generally defined in terms of the volume and variety of structured and unstructured information. Whereas structured data is stored

### Monte Carlo and Empirical Methods for Stochastic Inference (MASM11/FMS091)

Monte Carlo and Empirical Methods for Stochastic Inference (MASM11/FMS091) Magnus Wiktorsson Centre for Mathematical Sciences Lund University, Sweden Lecture 5 Sequential Monte Carlo methods I February

### Integrals of Rational Functions

Integrals of Rational Functions Scott R. Fulton Overview A rational function has the form where p and q are polynomials. For example, r(x) = p(x) q(x) f(x) = x2 3 x 4 + 3, g(t) = t6 + 4t 2 3, 7t 5 + 3t

### Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization

Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization 2.1. Introduction Suppose that an economic relationship can be described by a real-valued

### a 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)

ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x

### Mathematical Finance

Mathematical Finance Option Pricing under the Risk-Neutral Measure Cory Barnes Department of Mathematics University of Washington June 11, 2013 Outline 1 Probability Background 2 Black Scholes for European

### Mesh Discretization Error and Criteria for Accuracy of Finite Element Solutions

Mesh Discretization Error and Criteria for Accuracy of Finite Element Solutions Chandresh Shah Cummins, Inc. Abstract Any finite element analysis performed by an engineer is subject to several types of

### Age to Age Factor Selection under Changing Development Chris G. Gross, ACAS, MAAA

Age to Age Factor Selection under Changing Development Chris G. Gross, ACAS, MAAA Introduction A common question faced by many actuaries when selecting loss development factors is whether to base the selected

### ECE302 Spring 2006 HW7 Solutions March 11, 2006 1

ECE32 Spring 26 HW7 Solutions March, 26 Solutions to HW7 Note: Most of these solutions were generated by R. D. Yates and D. J. Goodman, the authors of our textbook. I have added comments in italics where

### Matrix Norms. Tom Lyche. September 28, Centre of Mathematics for Applications, Department of Informatics, University of Oslo

Matrix Norms Tom Lyche Centre of Mathematics for Applications, Department of Informatics, University of Oslo September 28, 2009 Matrix Norms We consider matrix norms on (C m,n, C). All results holds for

### The Heat Equation. Lectures INF2320 p. 1/88

The Heat Equation Lectures INF232 p. 1/88 Lectures INF232 p. 2/88 The Heat Equation We study the heat equation: u t = u xx for x (,1), t >, (1) u(,t) = u(1,t) = for t >, (2) u(x,) = f(x) for x (,1), (3)

### 6.254 : Game Theory with Engineering Applications Lecture 1: Introduction

6.254 : Game Theory with Engineering Applications Lecture 1: Introduction Asu Ozdaglar MIT February 2, 2010 1 Introduction Optimization Theory: Optimize a single objective over a decision variable x R

Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650

### Functions and Equations

Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c

### Data reduction and descriptive statistics

Data reduction and descriptive statistics dr. Reinout Heijungs Department of Econometrics and Operations Research VU University Amsterdam August 2014 1 Introduction Economics, marketing, finance, and most

### Høgskolen i Narvik Sivilingeniørutdanningen STE6237 ELEMENTMETODER. Oppgaver

Høgskolen i Narvik Sivilingeniørutdanningen STE637 ELEMENTMETODER Oppgaver Klasse: 4.ID, 4.IT Ekstern Professor: Gregory A. Chechkin e-mail: chechkin@mech.math.msu.su Narvik 6 PART I Task. Consider two-point

### Application of Game Theory in Inventory Management

Application of Game Theory in Inventory Management Rodrigo Tranamil-Vidal Universidad de Chile, Santiago de Chile, Chile Rodrigo.tranamil@ug.udechile.cl Abstract. Game theory has been successfully applied

### Multiple Linear Regression in Data Mining

Multiple Linear Regression in Data Mining Contents 2.1. A Review of Multiple Linear Regression 2.2. Illustration of the Regression Process 2.3. Subset Selection in Linear Regression 1 2 Chap. 2 Multiple

### WHERE DOES THE 10% CONDITION COME FROM?

1 WHERE DOES THE 10% CONDITION COME FROM? The text has mentioned The 10% Condition (at least) twice so far: p. 407 Bernoulli trials must be independent. If that assumption is violated, it is still okay

### Schooling, Political Participation, and the Economy. (Online Supplementary Appendix: Not for Publication)

Schooling, Political Participation, and the Economy Online Supplementary Appendix: Not for Publication) Filipe R. Campante Davin Chor July 200 Abstract In this online appendix, we present the proofs for

### Example 1: Competing Species

Local Linear Analysis of Nonlinear Autonomous DEs Local linear analysis is the process by which we analyze a nonlinear system of differential equations about its equilibrium solutions (also known as critical

### Fairfield Public Schools

Mathematics Fairfield Public Schools AP Statistics AP Statistics BOE Approved 04/08/2014 1 AP STATISTICS Critical Areas of Focus AP Statistics is a rigorous course that offers advanced students an opportunity

### Statistics - Written Examination MEC Students - BOVISA

Statistics - Written Examination MEC Students - BOVISA Prof.ssa A. Guglielmi 26.0.2 All rights reserved. Legal action will be taken against infringement. Reproduction is prohibited without prior consent.

### Lecture Notes to Accompany. Scientific Computing An Introductory Survey. by Michael T. Heath. Chapter 10

Lecture Notes to Accompany Scientific Computing An Introductory Survey Second Edition by Michael T. Heath Chapter 10 Boundary Value Problems for Ordinary Differential Equations Copyright c 2001. Reproduction

### The Bivariate Normal Distribution

The Bivariate Normal Distribution This is Section 4.7 of the st edition (2002) of the book Introduction to Probability, by D. P. Bertsekas and J. N. Tsitsiklis. The material in this section was not included

### Monte Carlo-based statistical methods (MASM11/FMS091)

Monte Carlo-based statistical methods (MASM11/FMS091) Jimmy Olsson Centre for Mathematical Sciences Lund University, Sweden Lecture 5 Sequential Monte Carlo methods I February 5, 2013 J. Olsson Monte Carlo-based

### Review of Basic Options Concepts and Terminology

Review of Basic Options Concepts and Terminology March 24, 2005 1 Introduction The purchase of an options contract gives the buyer the right to buy call options contract or sell put options contract some

### Correlation key concepts:

CORRELATION Correlation key concepts: Types of correlation Methods of studying correlation a) Scatter diagram b) Karl pearson s coefficient of correlation c) Spearman s Rank correlation coefficient d)

### Taylor Polynomials and Taylor Series Math 126

Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will

### Metric Spaces. Chapter 7. 7.1. Metrics

Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some

### A more robust unscented transform

A more robust unscented transform James R. Van Zandt a a MITRE Corporation, MS-M, Burlington Road, Bedford MA 7, USA ABSTRACT The unscented transformation is extended to use extra test points beyond the

Economics tutorial David G. Messerschmitt CS 294-6, EE 290X, BA 296.5 Copyright 1997, David G. Messerschmitt 3/5/97 1 Here we give a quick tutorial on the economics material that has been covered in class.

### Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test

Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important

### Minimax Strategies. Minimax Strategies. Zero Sum Games. Why Zero Sum Games? An Example. An Example

Everyone who has studied a game like poker knows the importance of mixing strategies With a bad hand, you often fold But you must bluff sometimes Lectures in Microeconomics-Charles W Upton Zero Sum Games

Factorising quadratics An essential skill in many applications is the ability to factorise quadratic expressions. In this unit you will see that this can be thought of as reversing the process used to