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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Introduces the fundamentals of probability, statistics, decision theory, and game theory, and features interesting examples of games of chance and strategy to motivate and illustrate abstract mathematical concepts Covering both random and strategic games, Probability, Decisions and Games features a variety of gaming and gambling examples to build a better understanding of basic concepts of probability, statistics, decision theory, and game theory. The authors present fundamental concepts such as random variables, rational choice theory, mathematical expectation and variance, fair games, combinatorial calculus, conditional probability, Bayes Theorem, Bernoulli trials, zero-sum games and Nash equilibria, as well as their application in games such as Roulette, Craps, Lotto, Blackjack, Poker, Rock-Paper-Scissors, the Game of Chicken and Tic-Tac-Toe. Computer simulations, implemented using the popular R computing environment, are used to provide intuition on key concepts and verify complex calculations. The book starts by introducing simple concepts that are carefully motivated by the same historical examples that drove their original development of the field of probability, and then applies those concepts to popular contemporary games. The first two chapters of Probability, Decisions and Games: A Gentle Introduction using R feature an introductory discussion of probability and rational choice theory in finite and discrete spaces that builds upon the simple games discussed in the famous correspondence between Blaise Pascal and Pierre de Fermat. Subsequent chapters utilize popular casino games such as Roulette and Blackjack to expand on these concepts illustrate modern applications of these methodologies. Finally, the book concludes with discussions on game theory using a number of strategic games. This book: * Features introductory coverage of probability, statistics, decision theory and game theory, and has been class-tested at University of California, Santa Cruz for the past six years * Illustrates basic concepts in probability through interesting and fun examples using a number of popular casino games: roulette, lotto, craps, blackjack, and poker * Introduces key ideas in game theory using classic games such as Rock-Paper-Scissors, Chess, and Tic-Tac-Toe. * Features computer simulations using R throughout in order to illustrate complex concepts and help readers verify complex calculations * Contains exercises and approaches games and gambling at a level that is accessible for readers with minimal experience * Adopts a unique approach by motivating complex concepts using first simple games and then moving on to more complex, well-known games that illustrate how these concepts work together Probability, Decisions and Games: A Gentle Introduction using R is a unique and helpful textbook for undergraduate courses on statistical reasoning, introduction to probability, statistical literacy, and quantitative reasoning for students from a variety of disciplines.
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Table of Contents
Cover
Title Page
Copyright
Dedication
Preface
Why Gambling and Gaming?
Using this Book
Acknowledgments
About the Companion Website
Chapter 1: An Introduction to Probability
1.1 What is Probability?
1.2 Odds and Probabilities
1.3 Equiprobable Outcome Spaces and De Méré's Problem
1.4 Probabilities for Compound Events
1.5 Exercises
Chapter 2: Expectations and Fair Values
2.1 Random Variables
2.2 Expected Values
2.3 Fair Value of a Bet
2.4 Comparing Wagers
2.5 Utility Functions and Rational Choice Theory
2.6 Limitations of Rational Choice Theory
2.7 Exercises
Chapter 3: Roulette
3.1 Rules and Bets
3.2 Combining Bets
3.3 Biased Wheels
3.4 Exercises
Chapter 4: Lotto and Combinatorial Numbers
4.1 Rules and Bets
4.2 Sharing Profits: De Méré's Second Problem
4.3 Exercises
Chapter 5: The Monty Hall Paradox and Conditional Probabilities
5.1 The Monty Hall Paradox
5.2 Conditional Probabilities
5.3 Independent Events
5.4 Bayes Theorem
5.5 Exercises
Chapter 6: Craps
6.1 Rules and Bets
6.2 Exercises
Chapter 7: Roulette Revisited
7.1 Gambling Systems
7.2 You are a Big Winner!
7.3 How Long will My Money Last?
7.4 Is This Wheel Biased?
7.5 Bernoulli Trials
7.6 Exercises
Chapter 8: Blackjack
8.1 Rules and Bets
8.2 Basic Strategy in Blackjack
8.3 A Gambling System that Works: Card Counting
8.4 Exercises
Chapter 9: Poker
9.1 Basic Rules
9.2 Variants of Poker
9.3 Additional Rules
9.4 Probabilities of Hands in Draw Poker
9.5 Probabilities of Hands in Texas Hold'em
9.6 Exercises
Chapter 10: Strategic Zero-Sum Games with Perfect Information
10.1 Games with Dominant Strategies
10.2 Solving Games with Dominant and Dominated Strategies
10.3 General Solutions for Two Person Zero-Sum Games
10.4 Exercises
Chapter 11: Rock–Paper–Scissors: Mixed Strategies in Zero-Sum Games
11.1 Finding Mixed-Strategy Equilibria
11.2 Mixed Strategy Equilibria in Sports
11.3 Bluffing as a Strategic Game with a Mixed-Strategy Equilibrium
11.4 Exercises
Chapter 12: The Prisoner's Dilemma and Other Strategic Non-zero-sum Games
12.1 The Prisoner's Dilemma
12.2 The Impact of Communication and Agreements
12.3 Which Equilibrium?
12.4 Asymmetric Games
12.5 Exercises
Chapter 13: Tic-Tac-Toe and Other Sequential Games of Perfect Information
13.1 The Centipede Game
13.2 Tic-Tac-Toe
13.3 The Game of Nim and the First- and Second-Mover Advantages
13.4 Can Sequential Games be Fun?
13.5 The Diplomacy Game
13.6 Exercises
Appendix A: A Brief Introduction to R
A.1 Installing
R
A.2 Simple Arithmetic
A.3 Variables
A.4 Vectors
A.5 Matrices
A.6 Logical Objects and Operations
A.7 Character Objects
A.8 Plots
A.9 Iterators
A.10 Selection and Forking
A.11 Other Things to Keep in Mind
Index
End User License Agreement
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Guide
Cover
Table of Contents
Preface
Begin Reading
List of Illustrations
Chapter 1: An Introduction to Probability
Figure 1.1 Cumulative empirical frequency of heads (black line) in 5000 simulated flips of a fair coin. The gray horizontal line corresponds to the true probability .
Figure 1.2 Venn diagram for the (a) union and (b) intersection of two events.
Figure 1.3 Venn diagram for the addition rule.
Chapter 2: Expectations and Fair Values
Figure 2.1 Running profits from a wager that costs $1 to join and pays nothing if a coin comes up tails and $1.50 if the coin comes up tails (solid line). The gray horizontal line corresponds to the expected profit.
Figure 2.2 Running profits from Wagers 1 (continuous line) and 2 (dashed line).
Figure 2.3 Running profits from Wagers 3 (continuous line) and 4 (dashed line).
Chapter 3: Roulette
Figure 3.1 The wheel in the French/European (left) and American (right) roulette and respective areas of the roulette table where bets are placed.
Figure 3.2 Running profits from a color (solid line) and straight-up (dashed line) bet.
Figure 3.3 Empirical frequency of each pocket in 5000 spins of a biased wheel.
Figure 3.4 Cumulative empirical frequency for a single pocket in an unbiased wheel.
Chapter 5: The Monty Hall Paradox and Conditional Probabilities
Figure 5.1 Each branch in this tree represents a different decision and the s represent the probability of each door being selected to contain the prize.
Figure 5.2 The tree structure now represents an extra level, representing the contestant decisions and the probability for each decision to be the one chosen.
Figure 5.3 Decision tree for the point when Monty decides which door to open assuming the prize is behind door 3.
Figure 5.4 Partitioning the event space.
Figure 5.5 Tree representation of the outcomes of the
game of urns
under the optimal strategy that calls yellow balls as coming from Urn 3 and blue and red balls as coming from urn 2.
Chapter 6: Craps
Figure 6.1 The layout of a craps table.
Figure 6.2 Tree representation for the possible results of the game of craps. Outcomes that lead to the pass line bet winning are marked with W, while those that lead to a lose are marked L.
Figure 6.3 Tree representation for the possible results of the game of craps with the probabilities for each of the come-out roll.
Figure 6.4 Tree representation for the possible results of the game of craps with the probabilities for all scenarios.
Chapter 7: Roulette Revisited
Figure 7.1 The solid line represents the running profits from a martingale doubling system with $1 initial wagers for an even bet in roulette. The dashed horizontal line indicates the zero-profit level.
Figure 7.2 Running profits from a Labouchère system with an initial list of $50 entries of $10 for an even bet in roulette. Note that the simulation stops when the cumulative profit is ; the number of spins necessary to reach this number will vary from simulation to simulation.
Figure 7.3 Running profits over 10,000 spins from a D'Alembert system with an initial bet of $5, change in bets of $1, minimum bet of $1 and maximum bet of $20 to an even roulette bet.
Chapter 8: Blackjack
Figure 8.1 A 52-card French-style deck.
Chapter 9: Poker
Figure 9.1 Examples of poker hands.
Chapter 11: Rock–Paper–Scissors: Mixed Strategies in Zero-Sum Games
Figure 11.1 Graphical representation of decisions in a simplified version of poker.
Chapter 12: The Prisoner's Dilemma and Other Strategic Non-zero-sum Games
Figure 12.1 Expected utilities for Ileena (solid line) and Hans (dashed line) in the game of chicken as function of the probability that Hans will swerve with probability if we assume that Ileena swerves with probability .
Figure 12.2 Expected utilities for Ileena (solid line) and Hans (dashed line) in the game of chicken as function of the probability that Ileena will swerve with probability if we assume that Hans swerves with probability .
Figure 12.3 Expected utilities for Ileena (solid line) and Hans (dashed line) in the game of chicken as function of the probability that Hans will swerve with probability if we assume that Ileena always swerves.
Chapter 13: Tic-Tac-Toe and Other Sequential Games of Perfect Information
Figure 13.1 Extensive-form representation of the centipede game.
Figure 13.2 Reduced extensive-form representation of the centipede game after solving for Carissa's optimal decision during the third round of play.
Figure 13.3 Reduced extensive-form representation of the centipede game after solving for Sahar's optimal decision during the second round of play and Carissa's optimal decision during the third round of play.
Figure 13.4 A game of tic-tac-toe where the player represented by X plays first and the player represented by O wins the game. The boards should be read left to right and then top to bottom.
Figure 13.5 A small subsection of the extensive-form representation of tic-tac-toe.
Figure 13.6 Examples of boards in which the player using the X mark created a fork for themselves, a situation that should be avoided by their opponent. In the left figure, player 1 (who is using the X) claimed the top left corner in their first move, then player 2 claimed the top right corner, player 1 responded by claiming the bottom right corner, which forces player 2 to claim the center square (in order to block a win), and player 1 claims the bottom left corner too. At this point, player 1 has created a fork since they can win by placing a mark on either of the cells marked with an F. Similarly, in the right Figure player 1 claimed the top left corner, player 2 responded by claiming the bottom edge square, then player 1 took the center square, which forced player 2 to take the bottom right corner to block a win. After that, if player 1 places their mark on the bottom left corner they would have created a fork.
Figure 13.7 Extended-form representation of the game of Nim with four initial pieces.
Figure 13.8 Pruned tree for a game of Nim with four initial pieces after the optimal strategy at the third round has been elucidated.
Figure 13.9 Pruned tree for a game of Nim with four initial pieces after the optimal strategy at the second round has been elucidated.
Figure 13.10 The diplomacy game in extensive form.
Figure 13.11 First branches pruned in the diplomacy game.
Figure 13.12 Pruned tree associated with the diplomacy game.
Appendix A: A Brief Introduction to R
Figure A.1 The R interactive command console in a Mac OS X computer. The symbol > is a prompt for users to provide instructions; these will be executed immediately after the user presses the RETURN key.
Figure A.2 A representation of a vector x of length 6 as a series of containers, each one of them corresponding to a different number.
Figure A.3 An example of a scatterplot in R]An example of a scatterplot in R.
Figure A.4 An example of a line plot in R.
Figure A.5 Adding multiple plots and reference lines to a single graph.
Figure A.6 Example of a barplot in R.
List of Tables
Chapter 1: An Introduction to Probability
Table 1.1 Two different ways to think about the outcome space associated with rolling two dice
Chapter 2: Expectations and Fair Values
Table 2.1 Winnings for the different lotteries in Allais paradox
Table 2.2 Winnings for 11% of the time for the different lotteries in Allais paradox
Chapter 3: Roulette
Table 3.1 Inside bets for the American wheel
Table 3.2 Outside bets for the American wheel
Table 3.3 Outcomes of a combined bet of $2 on red and $1 on the second dozen
Chapter 4: Lotto and Combinatorial Numbers
Table 4.1 List of possible groups of 3 out of 6 numbers, if the order of the numbers is not important
Chapter 5: The Monty Hall Paradox and Conditional Probabilities
Table 5.1 Probabilities of winning if the contestant in the Monty problem switches doors
Table 5.2 Studying the relationship between smoking and lung cancer
Chapter 6: Craps
Table 6.1 Names associated with different combinations of dice in craps
Table 6.2 All possible equiprobable outcomes associated with two dice being rolled
Table 6.3 Sum of points associated with the roll of two dice
Chapter 7: Roulette Revisited
Table 7.1 Accumulated losses from playing a martingale doubling system with an initial bet of $1 and an initial bankroll of $1000
Table 7.2 Probability that you play exactly rounds before you lose your first dollar for between 1 and 6
Chapter 8: Blackjack
Table 8.1 Probability of different hands assuming that the house stays on all 17s and that the game is being played with a large number of decks
Table 8.2 Probability of different hands assuming that the house stays on all 17s, conditional on the face-up card
Table 8.3 Optimal splitting strategy
Table 8.4 Probability of different hands assuming that the house stays on all 17s and that the game is being played with a single deck where all Aces, 2s, 3s, 4s, 5s, and 6s have been removed
Table 8.5 Probability of different hands assuming that the house stays on all 17s, conditional on the face-up card
Chapter 9: Poker
Table 9.1 List of poker hands
Table 9.2 List of opponent's poker hands that can beat our two-pair
Chapter 10: Strategic Zero-Sum Games with Perfect Information
Table 10.1 Profits in the game between Pevier and Errian
Table 10.2 Poll results for Matt versus Ling (first scenario)
Table 10.3 Best responses for Matt (first scenario)
Table 10.4 Best responses for Ling (first scenario)
Table 10.5 Poll results for Matt versus Ling (second scenario)
Table 10.6 Best responses for Matt (second scenario)
Table 10.7 Best responses for Ling (second scenario)
Table 10.8 Poll results for Matt versus Ling (third scenario)
Table 10.9 Best responses for Ling (third scenario)
Table 10.10 Best responses for Matt (third scenario)
Table 10.11 Reduced Table for poll results for Matt versus Ling
Table 10.12 A game without dominant or dominated strategies
Table 10.13 Best responses for Player 1 in our game without dominant or dominated strategies
Table 10.14 Best responses for Player 2 in our game without dominant or dominated strategies
Table 10.15 Example of a game with multiple equilibria
Chapter 11: Rock–Paper–Scissors: Mixed Strategies in Zero-Sum Games
Table 11.1 Player's profit in rock–paper–scissors
Table 11.2 Best responses for Jiahao in the game of rock–paper–scissors
Table 11.3 Utility associated with different actions that Jiahao can take if he assumes that Antonio selects rock with probability , paper with probability and scissors with probability
Table 11.4 Utilities associated with different penalty kick decisions
Table 11.5 Utility associated with different actions taken by the kicker if he assumes that goal keeper selects left with probability , center with probability , and right with probability
Table 11.6 Expected profits in the simplified poker
Table 11.7 Best responses for you in the simplified poker game
Table 11.8 Best responses for Alya in the simplified poker game
Table 11.9 Expected profits in the simplified poker game after eliminating dominated strategies
Table 11.10 Expected profits associated with different actions you take if you assume that Alya will select with probability and with probability
Chapter 12: The Prisoner's Dilemma and Other Strategic Non-zero-sum Games
Table 12.1 Payoffs for the prisoner's dilemma
Table 12.2 Best responses for Prisoner 2 in the prisoner's dilemma game
Table 12.3 Communication game in normal form
Table 12.4 Best responses for Anastasiya in the communication game
Table 12.5 Best responses for Anil in the communication game
Table 12.6 Expected utility for Anil in the communication game
Table 12.7 The game of chicken
Table 12.8 Best responses for Ileena in the game of chicken
Table 12.9 Expected utility for Ileena in the game of chicken
Table 12.10 A fictional game of swords in Star Wars
Table 12.11 Best responses for Ki-Adi in the sword game
Table 12.12 Best responses for Asajj in the sword game
Table 12.13 Expected utility for Ki-Adi in the asymmetric sword game
Table 12.14 Expected utility for Asajj in the asymmetric sword game
Probability, Decisions and Games
A Gentle Introduction using R
Abel RodríguezBruno Mendes
This edition first published 2018
© 2018 John Wiley & Sons, Inc.
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Library of Congress Cataloging-in-Publication Data:
Names: Rodríguez, Abel, 1975- author. | Mendes, Bruno, 1970- author.
Title: Probability, decisions, and games : a gentle introduction using R / by Abel Rodríguez, Bruno Mendes.
Description: Hoboken, NJ : Wiley, 2018. | Includes index. |
Identifiers: LCCN 2017047636 (print) | LCCN 2017059013 (ebook) | ISBN 9781119302612 (pdf) | ISBN 9781119302629 (epub) | ISBN 9781119302605 (pbk.)
Subjects: LCSH: Game theory-Textbooks. | Game theory-Data processing. | Statistical decision-Textbooks. | Statistical decision-Data processing. | Probabilities--Textbooks. | Probabilities-Data processing. | R (Computer program language)
Classification: LCC QA269 (ebook) | LCC QA269 .R63 2018 (print) | DDC 519.30285/5133-dc23
LC record available at https://lccn.loc.gov/2017047636
Cover design: Wiley
Cover image: © Jupiterimages/Getty Images
To SabrinaAbel To my familyBruno
Preface
Why Gambling and Gaming?
Games are a universal part of human experience and are present in almost every culture; the earliest games known (such as senet in Egypt or the Royal Game of Ur in Iraq) date back to at least 2600 B.C. Games are characterized by a set of rules regulating the behavior of players and by a set of challenges faced by those players, which might involve a monetary or nonmonetary wager. Indeed, the history of gaming is inextricably linked to the history of gambling, and both have played an important role in the development of modern society.
Games have also played a very important role in the development of modern mathematical methods, and they provide a natural framework to introduce simple concepts that have wide applicability in real-life problems. From the point of view of the mathematical tools used for their analysis, games can be broadly divided between random games and strategic games. Random games pit one or more players against “nature” that is, an unintelligent opponent whose acts cannot be predicted with certainty. Roulette is the quintessential example of a random game. On the other hand, strategic games pit two or more intelligent players against each other; the challenge is for one player to outwit their opponents. Strategic games are often subdivided into simultaneous (e.g., rock–paper–scissors) and sequential (e.g., chess, tic-tac-toe) games, depending on the order in which the players take their actions. However, these categories are not mutually exclusive; most modern games involve aspects of both strategic and random games. For example, poker incorporates elements of random games (cards are dealt at random) with those of a sequential strategic game (betting is made in rounds and “bluffing” can win you a game even if your cards are worse than those of your opponent).
One of the key ideas behind the mathematical analysis of games is the rationality assumption, that is, that players are indeed interested in winning the game and that they will take “optimal” (i.e., rational) steps to achieve this. Under these assumptions, we can postulate a theory of how decisions are made, which relies on the maximization of a utility function (often, but certainly not always, related to the amount of money that is made by playing the game). Players attempt to maximize their own utility given the information available to them at any given moment. In the case of random games, this involves making decisions under uncertainty, which naturally leads to the study of probability. In fact, the formal study of probability was born in the seventeenth century from a series of questions posed by an inveterate gambler (Antoine Gambaud, known as the Chevalier de Méré). De Méré, suffered severe financial losses for assessing incorrectly his chances of winning in certain games of dice. Contrary to the ordinary gambler of the time, he pursued the cause of his error with the help of Blaise Pascal, which in turn led to an exchange of letters with Pierre de Fermat and the development of probability theory.
Decision theory also plays an important role in strategic games. In this case, optimality often means evaluating the alternatives available to other players and finding a “best response” to them. This is often taken to mean minimizing losses, but the two concepts are not necessarily identical. Indeed, one important insight gleaned from game theory (the area of mathematics that studies strategic games) is that optimal strategies for zero-sum games (i.e., those games where a player can win only if another loses the same amount) and non zero-sum games can be very different. Also, it is important to highlight that randomness plays a role even in purely strategic games. An excellent example is the game of rock–paper–scissors. In principle, there is nothing inherently random in the rules of this game. However, the optimal strategy for any given player is to select his or her move uniformly at random among the three possible options that give the game its name.
The mathematical concepts underlying the analysis of games and gambles have practical applications in all realms of science. Take for example the game of blackjack. When you play blackjack, you need to sequentially decide whether to hit (i.e., get an extra card), stay (i.e., stop receiving cards) or, when appropriate, double down, split, or surrender. Optimally playing the game means that these decisions must be taken not only on the basis of the cards you have in your hand but also on the basis of the cards shown by the dealer and all other players. A similar problem arises in the diagnosis and treatment of medical conditions. A doctor has access to a series of diagnostic tests and treatment options; decisions on which one is to be used next needs to be taken sequentially based on the outcomes of previous tests or treatments for this as well as other patients. Poker provides another interesting example. As any experienced player can attest, bluffing is one of the most important parts of the game. The same rules that can be used to decide how to optimally bluff in poker can also be used to design optimal auctions that allow the auctioneer to extract the highest value assigned by the bidders to the object begin auctioned. These strategies are used by companies such as Google and Yahoo to allocate advertising spots.
Using this Book
The goal of this book is to introduce basic concepts of probability, statistics, decision theory, and game theory using games. The material should be suitable for a college-level general education course for undergraduate college students who have taken an algebra or pre-algebra class. In our experience, motivated high-school students who have taken an algebra course should also be capable of handling the material.
The book is organized into 13 chapters, with about half focusing on general concepts that are illustrated using a wide variety of games, and about half focusing specifically on well-known casino games. More specifically, the first two chapters of the book are dedicated to a basic discussion of utility and probability theory in finite, discrete spaces. Then we move to a discussion of five popular casino games: roulette, lotto, craps, blackjack, and poker. Roulette, which is one of the simplest casino games to play and analyze, is used to illustrate the basic concepts in probability such as expectations. Lotto is used to motivate counting rules and the notions of permutations and combinatorial numbers that allow us to compute probabilities in large equiprobable spaces. The games of craps and blackjack are used to illustrate and develop conditional probabilities. Finally, the discussion of poker is helpful to illustrate how many of the ideas from previous chapters fit in together. The last four chapters of the book are dedicated to game theory and strategic games. Since this book is meant to support a general education course, we restrict attention to simultaneous and sequential games of perfect information and avoid games of imperfect information.
The book uses computer simulations to illustrate complex concepts and convince students that the calculations presented in the book are correct. Computer simulations have become a key tool in many areas of scientific inquiry, and we believe that it is important for students to experience how easy access to computing power has changed science over the last 25 years. During the development of the book, we experimented with using spreadsheets but decided that they did not provide enough flexibility. In the end we settled for using R (https://www.r-project.org). R is an interactive environment that allows users to easily implement simple simulations even if they have limited experience with programming. To facilitate its use, we have included an overview and introduction to the R in Appendix A, as well as sidebars in each chapter that introduces features of the language that are relevant for the examples discussed in them. With a little extra work, this book could be used as the basis for a course that introduces students to both probability/statistics and programming. Alternatively, the book can also be read while ignoring the R commands and focusing only on the graphs and other output generated by it.
In the past, we have paired the content
