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Game Theory Through Examples Game through EXAMPLES ERICH PRISNER CLASSROOM RESOURCE MATERIALS i i “Alles” — 2014/5/8 — 11:19 — page i — #1 i i Game Theory Through Examples i i i i i i “Alles” — 2014/5/8 — 11:36 — page ii — #2 i i c 2014 by the Mathematical Association of America, Inc. Electronic edition ISBN 978-1-61444-115-1 i i i i i i “Alles” — 2014/5/8 — 11:19 — page iii — #3 i i Game Theory Through Examples Erich Prisner Franklin University Switzerland Published and Distributed by The Mathematical Association of America i i i i i i “Alles” — 2014/5/8 — 11:19 — page iv — #4 i i Council on Publications and Communications Frank Farris, Chair Committee on Books Fernando Gouvˆea, Chair Classroom Resource Materials Editorial Board Susan G Staples, Editor Michael Bardzell Jennifer Bergner Caren L Diefenderfer Christopher Hallstrom Cynthia J Huffman Paul R Klingsberg Brian Lins Mary Morley Philip P Mummert Barbara E Reynolds Darryl Yong i i i i i i “Alles” — 2014/5/8 — 11:19 — page v — #5 i i CLASSROOM RESOURCE MATERIALS Classroom Resource Materials is intended to provide supplementary classroom material for students— laboratory exercises, projects, historical information, textbooks with unusual approaches for presenting math- ematical ideas, career information, etc. 101 Careers in Mathematics, 3rd edition edited by Andrew Sterrett Archimedes: What Did He Do Besides Cry Eureka?, Sherman Stein Calculus: An Active Approach with Projects, Stephen Hilbert, Diane Driscoll Schwartz, Stan Seltzer, John Maceli, and Eric Robinson Calculus Mysteries and Thrillers, R. Grant Woods Conjecture and Proof, Mikl´os Laczkovich Counterexamples in Calculus, Sergiy Klymchuk Creative Mathematics, H. S. Wall Environmental Mathematics in the Classroom, edited by B. A. Fusaro and P. C. Kenschaft Excursions in Classical Analysis: Pathways to Advanced Problem Solving and Undergraduate Research, by Hongwei Chen Explorations in Complex Analysis, Michael A. Brilleslyper, Michael J. Dorff, Jane M. McDougall, James S. Rolf, Lisbeth E. Schaubroeck, Richard L. Stankewitz, and Kenneth Stephenson Exploratory Examples for Real Analysis, Joanne E. Snow and Kirk E. Weller Exploring Advanced Euclidean Geometry with GeoGebra, Gerard A. Venema Game Theory Through Examples, Erich Prisner Geometry From Africa: Mathematical and Educational Explorations, Paulus Gerdes Historical Modules for the Teaching and Learning of Mathematics (CD), edited by Victor Katz and Karen Dee Michalowicz Identification Numbers and Check Digit Schemes, Joseph Kirtland Interdisciplinary Lively Application Projects, edited by Chris Arney Inverse Problems: Activities for Undergraduates, Charles W. Groetsch Keeping it R.E.A.L.: Research Experiences for All Learners, Carla D. Martin and Anthony Tongen Laboratory Experiences in Group Theory, Ellen Maycock Parker Learn from the Masters, Frank Swetz, John Fauvel, Otto Bekken, Bengt Johansson, and Victor Katz Math Made Visual: Creating Images for Understanding Mathematics, Claudi Alsina and Roger B. Nelsen Mathematics Galore!: The First Five Years of the St. Marks Institute of Mathematics, James Tanton Methods for Euclidean Geometry, Owen Byer, Felix Lazebnik, and Deirdre L. Smeltzer Ordinary Differential Equations: A Brief Eclectic Tour, David A. S´anchez Oval Track and Other Permutation Puzzles, John O. Kiltinen Paradoxes and Sophisms in Calculus, Sergiy Klymchuk and Susan Staples A Primer of Abstract Mathematics, Robert B. Ash Proofs Without Words, Roger B. Nelsen Proofs Without Words II, Roger B. Nelsen Rediscovering Mathematics: You Do the Math, Shai Simonson i i i i i i “Alles” — 2014/5/8 — 11:19 — page vi — #6 i i She Does Math!, edited by Marla Parker Solve This: Math Activities for Students and Clubs, James S. Tanton Student Manual for Mathematics for Business Decisions Part 1: Probability and Simulation, David Williamson, Marilou Mendel, Julie Tarr, and Deborah Yoklic Student Manual for Mathematics for Business Decisions Part 2: Calculus and Optimization, David Williamson, Marilou Mendel, Julie Tarr, and Deborah Yoklic Teaching Statistics Using Baseball, Jim Albert Visual Group Theory, Nathan C. Carter Which Numbers are Real?, Michael Henle Writing Projects for Mathematics Courses: Crushed Clowns, Cars, and Coffee to Go, Annalisa Crannell, Gavin LaRose, Thomas Ratliff, and Elyn Rykken MAA Service Center P.O. Box 91112 Washington, DC 20090-1112 1-800-331-1MAA FAX: 1-301-206-9789 i i i i i i “Alles” — 2014/5/8 — 11:19 — page vii — #7 i i Contents Preface xvi 1 Theory 1: Introduction 1 1.1 What’s a Game? .......................................... 1 1.2 Game, Play, Move: Some Definitions .............................. 1 1.3 Classification of Games ...................................... 2 Exercises ................................................ 3 2 Theory 2: Simultaneous Games 4 2.1 Normal Form—Bimatrix Description ............................... 4 2.1.1 Two Players ........................................ 4 2.1.2 Two Players, Zero-sum .................................. 5 2.1.3 Three or More Players .................................. 5 2.1.4 Symmetric Games .................................... 6 2.2 Which Option to Choose ..................................... 6 2.2.1 Maximin Move and Security Level ............................ 6 2.2.2 Dominated Moves .................................... 7 2.2.3 Best Response ...................................... 8 2.2.4 Nash Equilibria ...................................... 9 2.3 Additional Topics ......................................... 13 2.3.1 Best Response Digraphs ................................. 13 2.3.2 2-Player Zero-sum Symmetric Games .......................... 14 Exercises ................................................ 15 Project 1: Reacting fast or slow ..................................... 18 3 Example: Selecting a Class 19 3.1 Three Players, Two Classes .................................... 19 3.1.1 “I likeyouboth” ..................................... 19 3.1.2 Dislikingthe Rival .................................... 21 3.1.3 Outsider .......................................... 21 3.2 Larger Cases ........................................... 22 3.3 Assumptions ........................................... 23 Exercises ................................................ 23 Project 2 ................................................. 23 Project 3 ................................................. 24 Project 4 ................................................. 24 vii i i i i i i “Alles” — 2014/5/8 — 11:19 — page viii — #8 i i viii Contents 4 Example: Doctor Location Games 25 4.1 Doctor Location .......................................... 25 4.1.1 An Example Graph .................................... 26 4.1.2 No (Pure) Nash Equilibrium? .............................. 27 4.1.3 How Good are the Nash Equilibriafor the Public? ................... 28 4.2 Trees ................................................ 28 4.3 More than one Office (optional) ................................. 31 Exercises ................................................ 31 Project 5: Doctor location on MOPs .................................. 33 Project 6 ................................................. 33 Project 7 ................................................. 33 5 Example: Restaurant Location Games 34 5.1 A FirstGraph ........................................... 35 5.2 A Second Graph .......................................... 36 5.3 Existence of Pure Nash Equilibria ................................ 37 5.4 More than one Restaurant (optional) ............................... 38 Exercises ................................................ 39 6 Using Excel 42 6.1 Spreadsheet Programs like Excel ................................. 42 6.2 Two-Person Simultaneous Games ................................ 43 6.3 Three-Person Simultaneous Games ................................ 43 Exercises ................................................ 43 Project 8: Simultaneous Quatro-Uno .................................. 44 Project 9: Restaurant Location Games ................................. 44 Project 10: 5 Knights .......................................... 45 Project 11: 5 Cardinals ......................................... 45 7 Example: Election I 47 7.1 First Example ........................................... 47 7.2 Second Example ......................................... 48 7.3 The General Model ........................................ 50 7.4 Third Example .......................................... 50 7.5 The Eight Cases .......................................... 51 7.6 Voting Power Indices (optional) ................................. 51 Exercises ................................................ 52 8 Theory 3: Sequential Games I: Perfect Information and no Randomness 53 8.1 Extensive Form: Game Tree and Game Digraph ......................... 53 8.2 Analyzing the Game: Backward Induction ............................ 56 8.2.1 Finite Games ....................................... 56 8.2.2 The Procedure ...................................... 57 8.2.3 Zermelo’s Theorem .................................... 59 8.3 Additional Topics ......................................... 59 8.3.1 Reality Check ....................................... 59 8.3.2 Playing it Safe—Guaranteed Payoffs .......................... 61 8.3.3 Two-person Zero-sum Games .............................. 63 i i i i i i “Alles” — 2014/5/8 — 11:19 — page ix — #9 i i Contents ix 8.3.4 Breaking Ties ....................................... 64 8.3.5 Existing Games ...................................... 64 8.3.6 Greedy Strategies ....................................
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