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


 Cody Little
 7 years ago
 Views:
Transcription
1 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 MicroeconomicsCharles W Upton Zero Sum Games Define a zerosum game, in which one firm s profits are another firm s losses Flipping coins or other betting games are straightforward examples of zerosum games Positive sum games such as buying a product are more common in economics Why Zero Sum Games? Zero sum games are easier to analyze They show us an important extension of game theory Since this is a zerosum game, we only display A s gains, for B s losses are exactly the opposite of A s gains
2 How should B play the game? There is not a dominant strategy here If B always follows strategy B, A will always follow A If B always follows strategy B, A will always follow A That would suggest that A can only win $ In fact A can do better A mixed strategy Suppose A follows strategy A sometimes; and other times, strategy A A will always win $ and sometimes $ or $3, depending on what B does Thus, it does better B s Response When B follows B, it l loses $ part of the time and $3 part of the time When it follows B, it loses $ part of the time and $ part of the time
3 B must mix strategies to minimize A s winnings Suppose B p percent of the time B (p ) percent of the time From Strategy A p () + (p )() From Strategy A p (3) + (p ) () Remember, B is following strategy p percent of the time If B is following the two strategies randomly, these are A s optimal decisions
4 A will follow his best strategy B must respond by minimizing his maximum winnings That means setting p = /3 This is the best B can do It is following a strategy to minimize A s maximum gain This is the minimax strategy There is an obvious analogy to playing poker If you always fold a poor hand and raise a good hand, you will not make much money You must, on occasion, bet on a poor hand and fold on a good hand If not, your opponent can read your bets and adjust his accordingly
5 3 From Strategy A p () + (p )() From Strategy A p (3) + (p )() 0 /3 /3 3 0 /3 /3 A s payoffs from following strategy A as a function of B s probability of following B A s payoffs from following strategy A as a function of B s probability of following B 3 If p = 0 (B never plays strategy B ), A maximizes his winnings by playing A 0 /3 /3 3 Given A s ability to choose strategies, B does best (or loses the least) by setting p =/3 0 /3 /3 Any attempt to carry this further will lead us into advanced mathematics This quick introduction illustrates what can be one to set up strategy problems in a game theoretic framework End 003 Charles W Upton
Understanding Options: Calls and Puts
2 Understanding Options: Calls and Puts Important: in their simplest forms, options trades sound like, and are, very high risk investments. If reading about options makes you think they are too risky for
More informationLab 11. Simulations. The Concept
Lab 11 Simulations In this lab you ll learn how to create simulations to provide approximate answers to probability questions. We ll make use of a particular kind of structure, called a box model, that
More informationExpected Value. 24 February 2014. Expected Value 24 February 2014 1/19
Expected Value 24 February 2014 Expected Value 24 February 2014 1/19 This week we discuss the notion of expected value and how it applies to probability situations, including the various New Mexico Lottery
More informationBayesian Nash Equilibrium
. Bayesian Nash Equilibrium . In the final two weeks: Goals Understand what a game of incomplete information (Bayesian game) is Understand how to model static Bayesian games Be able to apply Bayes Nash
More informationGame Theory and Algorithms Lecture 10: Extensive Games: Critiques and Extensions
Game Theory and Algorithms Lecture 0: Extensive Games: Critiques and Extensions March 3, 0 Summary: We discuss a game called the centipede game, a simple extensive game where the prediction made by backwards
More informationCoin Flip Questions. Suppose you flip a coin five times and write down the sequence of results, like HHHHH or HTTHT.
Coin Flip Questions Suppose you flip a coin five times and write down the sequence of results, like HHHHH or HTTHT. 1 How many ways can you get exactly 1 head? 2 How many ways can you get exactly 2 heads?
More informationVideo Poker in South Carolina: A Mathematical Study
Video Poker in South Carolina: A Mathematical Study by Joel V. Brawley and Todd D. Mateer Since its debut in South Carolina in 1986, video poker has become a game of great popularity as well as a game
More informationPoker with a Three Card Deck 1
Poker with a Three Card Deck We start with a three card deck containing one Ace, one King and one Queen. Alice and Bob are each dealt one card at random. There is a pot of $ P (and we assume P 0). Alice
More informationProbabilistic Strategies: Solutions
Probability Victor Xu Probabilistic Strategies: Solutions Western PA ARML Practice April 3, 2016 1 Problems 1. You roll two 6sided dice. What s the probability of rolling at least one 6? There is a 1
More informationEconomics 1011a: Intermediate Microeconomics
Lecture 12: More Uncertainty Economics 1011a: Intermediate Microeconomics Lecture 12: More on Uncertainty Thursday, October 23, 2008 Last class we introduced choice under uncertainty. Today we will explore
More informationQueuing Theory. Long Term Averages. Assumptions. Interesting Values. Queuing Model
Queuing Theory Queuing Theory Queuing theory is the mathematics of waiting lines. It is extremely useful in predicting and evaluating system performance. Queuing theory has been used for operations research.
More informationOptimization in ICT and Physical Systems
27. OKTOBER 2010 in ICT and Physical Systems @ Aarhus University, Course outline, formal stuff Prerequisite Lectures Homework Textbook, Homepage and CampusNet, http://kurser.iha.dk/eeictmaster/tiopti/
More informationSource. http://en.wikipedia.org/wiki/poker
AI of poker game 1 C H U N F U N G L E E 1 0 5 4 3 0 4 6 1 C S E 3 5 2 A R T I F I C I A L I N T E L L I G E N C E P R O. A N I T A W A S I L E W S K A Source CAWSEY, ALISON. THE ESSENCE OF ARTIFICAL INTELLIGENCE.
More informationNewPokerSoft. Texas Holdem Poker Game Simulator
NewPokerSoft poker for life Texas Holdem Poker Game Simulator www.newpokersoft.com Poker training simulator for Texas Holdem Here, we present the simulator of the Texas Holdem PokerGame. It can be used
More informationTexas Hold em. From highest to lowest, the possible five card hands in poker are ranked as follows:
Texas Hold em Poker is one of the most popular card games, especially among betting games. While poker is played in a multitude of variations, Texas Hold em is the version played most often at casinos
More informationDecision Making Under Uncertainty. Professor Peter Cramton Economics 300
Decision Making Under Uncertainty Professor Peter Cramton Economics 300 Uncertainty Consumers and firms are usually uncertain about the payoffs from their choices Example 1: A farmer chooses to cultivate
More informationECON 459 Game Theory. Lecture Notes Auctions. Luca Anderlini Spring 2015
ECON 459 Game Theory Lecture Notes Auctions Luca Anderlini Spring 2015 These notes have been used before. If you can still spot any errors or have any suggestions for improvement, please let me know. 1
More informationEconomics 1011a: Intermediate Microeconomics
Lecture 11: Choice Under Uncertainty Economics 1011a: Intermediate Microeconomics Lecture 11: Choice Under Uncertainty Tuesday, October 21, 2008 Last class we wrapped up consumption over time. Today we
More informationPlaying with Numbers
PLAYING WITH NUMBERS 249 Playing with Numbers CHAPTER 16 16.1 Introduction You have studied various types of numbers such as natural numbers, whole numbers, integers and rational numbers. You have also
More informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice,
More informationMath 728 Lesson Plan
Math 728 Lesson Plan Tatsiana Maskalevich January 27, 2011 Topic: Probability involving sampling without replacement and dependent trials. Grade Level: 812 Objective: Compute the probability of winning
More informationECE316 Tutorial for the week of June 15
ECE316 Tutorial for the week of June 15 Problem 35 Page 176: refer to lecture notes part 2, slides 8, 15 A box contains 5 red and 5 blue marbles. Two marbles are withdrawn randomly. If they are the same
More informationRemedies to the Lemons Problem. Warranties. Warranties. Remedies to the Lemons Problem. Warranties and Moral Hazard
Lectures in MicroeconomicsCharles W. Upton Warranties You are selling used cars People worry about lemons Warranties You are selling used cars People worry about lemons Provide a warranty! Warranties
More information6.207/14.15: Networks Lecture 15: Repeated Games and Cooperation
6.207/14.15: Networks Lecture 15: Repeated Games and Cooperation Daron Acemoglu and Asu Ozdaglar MIT November 2, 2009 1 Introduction Outline The problem of cooperation Finitelyrepeated prisoner s dilemma
More informationMicroeconomic Theory Jamison / Kohlberg / Avery Problem Set 4 Solutions Spring 2012. (a) LEFT CENTER RIGHT TOP 8, 5 0, 0 6, 3 BOTTOM 0, 0 7, 6 6, 3
Microeconomic Theory Jamison / Kohlberg / Avery Problem Set 4 Solutions Spring 2012 1. Subgame Perfect Equilibrium and Dominance (a) LEFT CENTER RIGHT TOP 8, 5 0, 0 6, 3 BOTTOM 0, 0 7, 6 6, 3 Highlighting
More informationFeb 7 Homework Solutions Math 151, Winter 2012. Chapter 4 Problems (pages 172179)
Feb 7 Homework Solutions Math 151, Winter 2012 Chapter Problems (pages 172179) Problem 3 Three dice are rolled. By assuming that each of the 6 3 216 possible outcomes is equally likely, find the probabilities
More informationCVA: Default Probability ain t matter?
CVA: Default Probability ain t matter? Ignacio Ruiz July 211 Version 1.1 Abstract CVA can be priced using market implied riskneutral or historical realworld parameters. There is no consensus in the market
More informationSection 7C: The Law of Large Numbers
Section 7C: The Law of Large Numbers Example. You flip a coin 00 times. Suppose the coin is fair. How many times would you expect to get heads? tails? One would expect a fair coin to come up heads half
More informationAMS 5 CHANCE VARIABILITY
AMS 5 CHANCE VARIABILITY The Law of Averages When tossing a fair coin the chances of tails and heads are the same: 50% and 50%. So if the coin is tossed a large number of times, the number of heads and
More informationBayesian Tutorial (Sheet Updated 20 March)
Bayesian Tutorial (Sheet Updated 20 March) Practice Questions (for discussing in Class) Week starting 21 March 2016 1. What is the probability that the total of two dice will be greater than 8, given that
More informationQuestion: What is the probability that a fivecard poker hand contains a flush, that is, five cards of the same suit?
ECS20 Discrete Mathematics Quarter: Spring 2007 Instructor: John Steinberger Assistant: Sophie Engle (prepared by Sophie Engle) Homework 8 Hints Due Wednesday June 6 th 2007 Section 6.1 #16 What is the
More informationEconomics 335, Spring 1999 Problem Set #7
Economics 335, Spring 1999 Problem Set #7 Name: 1. A monopolist has two sets of customers, group 1 and group 2. The inverse demand for group 1 may be described by P 1 = 200? Q 1, where P 1 is the price
More informationHoover High School Math League. Counting and Probability
Hoover High School Math League Counting and Probability Problems. At a sandwich shop there are 2 kinds of bread, 5 kinds of cold cuts, 3 kinds of cheese, and 2 kinds of dressing. How many different sandwiches
More informationBetting systems: how not to lose your money gambling
Betting systems: how not to lose your money gambling G. Berkolaiko Department of Mathematics Texas A&M University 28 April 2007 / Mini Fair, Math Awareness Month 2007 Gambling and Games of Chance Simple
More informationPERPETUITIES NARRATIVE SCRIPT 2004 SOUTHWESTERN, A THOMSON BUSINESS
NARRATIVE SCRIPT 2004 SOUTHWESTERN, A THOMSON BUSINESS NARRATIVE SCRIPT: SLIDE 2 A good understanding of the time value of money is crucial for anybody who wants to deal in financial markets. It does
More informationMTH6120 Further Topics in Mathematical Finance Lesson 2
MTH6120 Further Topics in Mathematical Finance Lesson 2 Contents 1.2.3 Nonconstant interest rates....................... 15 1.3 Arbitrage and BlackScholes Theory....................... 16 1.3.1 Informal
More informationAP Stats  Probability Review
AP Stats  Probability Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. I toss a penny and observe whether it lands heads up or tails up. Suppose
More informationMathematical goals. Starting points. Materials required. Time needed
Level S2 of challenge: B/C S2 Mathematical goals Starting points Materials required Time needed Evaluating probability statements To help learners to: discuss and clarify some common misconceptions about
More informationTHREE CARD BRAG (FLASH)
THREE CARD BRAG (FLASH) Introduction Three Card Brag is a gambling game played with a standard 52 card pack without jokers. The cards in each suit rank in the usual order from high to low: AKQJ1098765432.
More informationThe Binomial Distribution
The Binomial Distribution James H. Steiger November 10, 00 1 Topics for this Module 1. The Binomial Process. The Binomial Random Variable. The Binomial Distribution (a) Computing the Binomial pdf (b) Computing
More informationIs it possible to beat the lottery system?
Is it possible to beat the lottery system? Michael Lydeamore The University of Adelaide Postgraduate Seminar, 2014 The story One day, while sitting at home (working hard)... The story Michael Lydeamore
More informationMath Games For Skills and Concepts
Math Games p.1 Math Games For Skills and Concepts Original material 20012006, John Golden, GVSU permission granted for educational use Other material copyright: Investigations in Number, Data and Space,
More informationLecture V: Mixed Strategies
Lecture V: Mixed Strategies Markus M. Möbius February 26, 2008 Osborne, chapter 4 Gibbons, sections 1.31.3.A 1 The Advantage of Mixed Strategies Consider the following RockPaperScissors game: Note that
More informationAN ANALYSIS OF A WARLIKE CARD GAME. Introduction
AN ANALYSIS OF A WARLIKE CARD GAME BORIS ALEXEEV AND JACOB TSIMERMAN Abstract. In his book Mathematical MindBenders, Peter Winkler poses the following open problem, originally due to the first author:
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Practice Test Chapter 9 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the odds. ) Two dice are rolled. What are the odds against a sum
More informationProfit Maximization. 2. product homogeneity
Perfectly Competitive Markets It is essentially a market in which there is enough competition that it doesn t make sense to identify your rivals. There are so many competitors that you cannot single out
More informationThere are a number of superb online resources as well that provide excellent blackjack information as well. We recommend the following web sites:
3. Once you have mastered basic strategy, you are ready to begin learning to count cards. By counting cards and using this information to properly vary your bets and plays, you can get a statistical edge
More informationInClass Game. The Transformation Game (Lesson 69)
EACHING SUGGESIONS he ransformation Game (Lesson 69) Separate the class into groups of four. he ransformation Game master, p. 22 he ransformation Game Board master, p. 23 he ransformation Game Pieces
More informationTwo against the Bean Mafia By Uwe Rosenberg
Two against the Bean Mafia By Uwe Rosenberg Players: 1 2 persons Age: 12 + Duration: 30 60 minutes Contents: 20 Blue Beans 19 Kidney Beans 18 Fire Beans 16 Puff Beans 16 Broad Beans 14 French Beans 13
More informationGame Theory and Poker
Game Theory and Poker Jason Swanson April, 2005 Abstract An extremely simplified version of poker is completely solved from a game theoretic standpoint. The actual properties of the optimal solution are
More information2. How many ways can the letters in PHOENIX be rearranged? 7! = 5,040 ways.
Math 142 September 27, 2011 1. How many ways can 9 people be arranged in order? 9! = 362,880 ways 2. How many ways can the letters in PHOENIX be rearranged? 7! = 5,040 ways. 3. The letters in MATH are
More informationVon Neumann and Newman poker with a flip of hand values
Von Neumann and Newman poker with a flip of hand values Nicla Bernasconi Julian Lorenz Reto Spöhel Institute of Theoretical Computer Science ETH Zürich, 8092 Zürich, Switzerland {nicla jlorenz rspoehel}@inf.ethz.ch
More informationECO 199 B GAMES OF STRATEGY Spring Term 2004 PROBLEM SET 4 B DRAFT ANSWER KEY 1003 9099 21 8089 14 7079 4 069 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 1003 9099 21 8089 14 7079 4 069 11 Question 1: 30 points Games
More informationGames of Incomplete Information
Games of Incomplete Information Jonathan Levin February 00 Introduction We now start to explore models of incomplete information. Informally, a game of incomplete information is a game where the players
More informationCh. 13.2: Mathematical Expectation
Ch. 13.2: Mathematical Expectation Random Variables Very often, we are interested in sample spaces in which the outcomes are distinct real numbers. For example, in the experiment of rolling two dice, we
More informationLesson 4: Efficiently Adding Integers and Other Rational Numbers
Classwork Example 1: Rule for Adding Integers with Same Signs a. Represent the sum of 3 + 5 using arrows on the number line. i. How long is the arrow that represents 3? ii. iii. How long is the arrow that
More informationMITOCW watch?v=kn92wxckr0m
MITOCW watch?v=kn92wxckr0m The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To
More informationNash Equilibria and. Related Observations in OneStage Poker
Nash Equilibria and Related Observations in OneStage Poker Zach Puller MMSS Thesis Advisor: Todd Sarver Northwestern University June 4, 2013 Contents 1 Abstract 2 2 Acknowledgements 3 3 Literature Review
More informationMA 1125 Lecture 14  Expected Values. Friday, February 28, 2014. Objectives: Introduce expected values.
MA 5 Lecture 4  Expected Values Friday, February 2, 24. Objectives: Introduce expected values.. Means, Variances, and Standard Deviations of Probability Distributions Two classes ago, we computed the
More informationExpected Value and the Game of Craps
Expected Value and the Game of Craps Blake Thornton Craps is a gambling game found in most casinos based on rolling two six sided dice. Most players who walk into a casino and try to play craps for the
More informationGame Theory for Humans. Matt Hawrilenko MIT: Poker Theory and Analytics
Game Theory for Humans Matt Hawrilenko MIT: Poker Theory and Analytics 1 Play Good Poker ReadBased Approach Game Theoretic Approach Why Game Theory? Computer Scientists Humans Audience Theory Practice
More informationSequential lmove Games. Using Backward Induction (Rollback) to Find Equilibrium
Sequential lmove Games Using Backward Induction (Rollback) to Find Equilibrium Sequential Move Class Game: Century Mark Played by fixed pairs of players taking turns. At each turn, each player chooses
More informationDecision & Risk Analysis Lecture 6. Risk and Utility
Risk and Utility Risk  Introduction Payoff Game 1 $14.50 0.5 0.5 $30  $1 EMV 30*0.5+(1)*0.5= 14.5 Game 2 Which game will you play? Which game is risky? $50.00 Figure 13.1 0.5 0.5 $2,000  $1,900 EMV
More information6.042/18.062J Mathematics for Computer Science. Expected Value I
6.42/8.62J Mathematics for Computer Science Srini Devadas and Eric Lehman May 3, 25 Lecture otes Expected Value I The expectation or expected value of a random variable is a single number that tells you
More informationRentSeeking and Corruption
RentSeeking and Corruption Holger Sieg University of Pennsylvania October 1, 2015 Corruption Inefficiencies in the provision of public goods and services arise due to corruption and rentseeking of public
More informationCommon sense, and the model that we have used, suggest that an increase in p means a decrease in demand, but this is not the only possibility.
Lecture 6: Income and Substitution E ects c 2009 Je rey A. Miron Outline 1. Introduction 2. The Substitution E ect 3. The Income E ect 4. The Sign of the Substitution E ect 5. The Total Change in Demand
More informationElementary Statistics and Inference. Elementary Statistics and Inference. 17 Expected Value and Standard Error. 22S:025 or 7P:025.
Elementary Statistics and Inference S:05 or 7P:05 Lecture Elementary Statistics and Inference S:05 or 7P:05 Chapter 7 A. The Expected Value In a chance process (probability experiment) the outcomes of
More informationRisk and Uncertainty. Vani K Borooah University of Ulster
Risk and Uncertainty Vani K Borooah University of Ulster Basic Concepts Gamble: An action with more than one possible outcome, such that with each outcome there is an associated probability of that outcome
More informationBonus Maths 2: Variable Bet Sizing in the Simplest Possible Game of Poker (JB)
Bonus Maths 2: Variable Bet Sizing in the Simplest Possible Game of Poker (JB) I recently decided to read Part Three of The Mathematics of Poker (TMOP) more carefully than I did the first time around.
More informationThe Concept of Present Value
The Concept of Present Value If you could have $100 today or $100 next week which would you choose? Of course you would choose the $100 today. Why? Hopefully you said because you could invest it and make
More informationReady, Set, Go! Math Games for Serious Minds
Math Games with Cards and Dice presented at NAGC November, 2013 Ready, Set, Go! Math Games for Serious Minds Rande McCreight Lincoln Public Schools Lincoln, Nebraska Math Games with Cards Close to 20 
More informationComparing Simple and Compound Interest
Comparing Simple and Compound Interest GRADE 11 In this lesson, students compare various savings and investment vehicles by calculating simple and compound interest. Prerequisite knowledge: Students should
More informationWorldwide Casino Consulting Inc.
Card Count Exercises George Joseph The first step in the study of card counting is the recognition of those groups of cards known as Plus, Minus & Zero. It is important to understand that the House has
More informationThe Basics of Game Theory
Sloan School of Management 15.010/15.011 Massachusetts Institute of Technology RECITATION NOTES #7 The Basics of Game Theory Friday  November 5, 2004 OUTLINE OF TODAY S RECITATION 1. Game theory definitions:
More informationEasy Casino Profits. Congratulations!!
Easy Casino Profits The Easy Way To Beat The Online Casinos Everytime! www.easycasinoprofits.com Disclaimer The authors of this ebook do not promote illegal, underage gambling or gambling to those living
More informationStat 20: Intro to Probability and Statistics
Stat 20: Intro to Probability and Statistics Lecture 16: More Box Models Tessa L. ChildersDay UC Berkeley 22 July 2014 By the end of this lecture... You will be able to: Determine what we expect the sum
More informationCh5: Discrete Probability Distributions Section 51: Probability Distribution
Recall: Ch5: Discrete Probability Distributions Section 51: Probability Distribution A variable is a characteristic or attribute that can assume different values. o Various letters of the alphabet (e.g.
More informationOne pile, two pile, three piles
CHAPTER 4 One pile, two pile, three piles 1. One pile Rules: One pile is a twoplayer game. Place a small handful of stones in the middle. At every turn, the player decided whether to take one, two, or
More informationMathematical Analysis Of Packs Poker. September 22, 2014. Prepared For John Feola New Vision Gaming 5 Samuel Phelps Way North Reading, MA 01864
Mathematical Analysis Of Packs Poker September 22, 2014 Prepared For John Feola New Vision Gaming 5 Samuel Phelps Way North Reading, MA 01864 Office: 978 6641515 Fax: 978664  5117 www.newvisiongaming.com
More informationGame Theory and Nash Equilibrium
Game Theory and Nash Equilibrium by Jenny Duffy A project submitted to the Department of Mathematical Sciences in conformity with the requirements for Math 4301 (Honours Seminar) Lakehead University Thunder
More informationMath 2020 Quizzes Winter 2009
Quiz : Basic Probability Ten Scrabble tiles are placed in a bag Four of the tiles have the letter printed on them, and there are two tiles each with the letters B, C and D on them (a) Suppose one tile
More informationJust Visiting. Visiting. Just. Sell. Sell $20 $20. Buy. Buy $20 $20. Sell. Sell. Just Visiting. Visiting. Just
Down South East North Up West Julian D. A. Wiseman 1986 to 1991, and 2015. Available at http://www.jdawiseman.com/hexagonal_thing.html North West Up Down East South ONE NORTH 1N 1N ONE NORTH ONE UP 1U
More informationUsing games to support. WinWin Math Games. by Marilyn Burns
4 WinWin Math Games by Marilyn Burns photos: bob adler Games can motivate students, capture their interest, and are a great way to get in that paperandpencil practice. Using games to support students
More informationGOD S BIG STORY Week 1: Creation God Saw That It Was Good 1. LEADER PREPARATION
This includes: 1. Leader Preparation 2. Lesson Guide GOD S BIG STORY Week 1: Creation God Saw That It Was Good 1. LEADER PREPARATION LESSON OVERVIEW Exploring the first two chapters of Genesis provides
More informationGame theory and AI: a unified approach to poker games
Game theory and AI: a unified approach to poker games Thesis for graduation as Master of Artificial Intelligence University of Amsterdam Frans Oliehoek 2 September 2005 ii Abstract This thesis focuses
More informationhp calculators HP 17bII+ Net Present Value and Internal Rate of Return Cash Flow Zero A Series of Cash Flows What Net Present Value Is
HP 17bII+ Net Present Value and Internal Rate of Return Cash Flow Zero A Series of Cash Flows What Net Present Value Is Present Value and Net Present Value Getting the Present Value And Now For the Internal
More informationChampion Poker Texas Hold em
Champion Poker Texas Hold em Procedures & Training For the State of Washington 4054 Dean Martin Drive, Las Vegas, Nevada 89103 1 Procedures & Training Guidelines for Champion Poker PLAYING THE GAME Champion
More informationPascal is here expressing a kind of skepticism about the ability of human reason to deliver an answer to this question.
Pascal s wager So far we have discussed a number of arguments for or against the existence of God. In the reading for today, Pascal asks not Does God exist? but Should we believe in God? What is distinctive
More informationBetting on Volatility: A Delta Hedging Approach. Liang Zhong
Betting on Volatility: A Delta Hedging Approach Liang Zhong Department of Mathematics, KTH, Stockholm, Sweden April, 211 Abstract In the financial market, investors prefer to estimate the stock price
More informationThat s Not Fair! ASSESSMENT #HSMA20. Benchmark Grades: 912
That s Not Fair! ASSESSMENT # Benchmark Grades: 912 Summary: Students consider the difference between fair and unfair games, using probability to analyze games. The probability will be used to find ways
More informationWalking Through Some Examples of Futures and Options Contracts Speculation and Hedging
Walking Through Some Examples of Futures and Options Contracts Speculation and Hedging As Dr. Cogley said in class the other day, sometimes futures contracts and options are hard to wrap your head around
More informationREPEATED TRIALS. The probability of winning those k chosen times and losing the other times is then p k q n k.
REPEATED TRIALS Suppose you toss a fair coin one time. Let E be the event that the coin lands heads. We know from basic counting that p(e) = 1 since n(e) = 1 and 2 n(s) = 2. Now suppose we play a game
More information6.254 : Game Theory with Engineering Applications Lecture 2: Strategic Form Games
6.254 : Game Theory with Engineering Applications Lecture 2: Strategic Form Games Asu Ozdaglar MIT February 4, 2009 1 Introduction Outline Decisions, utility maximization Strategic form games Best responses
More informationTrader s Guide. Updated March 2014
Trader s Guide Updated March 2014 Welcome to this basic introduction to the Trigger Trade Report, FTMDaily s unique stock trading system. This simple guide will explain how to get started using our trading
More informationRadicals  Multiply and Divide Radicals
8. Radicals  Multiply and Divide Radicals Objective: Multiply and divide radicals using the product and quotient rules of radicals. Multiplying radicals is very simple if the index on all the radicals
More informationOptions on Beans For People Who Don t Know Beans About Options
Options on Beans For People Who Don t Know Beans About Options Remember when things were simple? When a call was something you got when you were in the bathtub? When premium was what you put in your car?
More informationChapter 15: Debt Policy
FIN 302 Class Notes Chapter 15: Debt Policy Two Cases: Case one: NO TAX All Equity Half Debt Number of shares 100,000 50,000 Price per share $10 $10 Equity Value $1,000,000 $500,000 Debt Value $0 $500,000
More informationContemporary Mathematics MAT 130. Probability. a) What is the probability of obtaining a number less than 4?
Contemporary Mathematics MAT 30 Solve the following problems:. A fair die is tossed. What is the probability of obtaining a number less than 4? What is the probability of obtaining a number less than
More informationThe Taxman Game. Robert K. Moniot September 5, 2003
The Taxman Game Robert K. Moniot September 5, 2003 1 Introduction Want to know how to beat the taxman? Legally, that is? Read on, and we will explore this cute little mathematical game. The taxman game
More informationPlaying around with Risks
Playing around with Risks Jurgen Cleuren April 19th 2012 2011 CTG, Inc. Introduction Projects are done in a probabilistic environment Incomplete information Parameters change over time What is true in
More information