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PRISNER

GameCLASSROOM RESOURCE MATERIALS

“Alles" — 2014/5/8 — 11:19 — page i — #1?

Game Theory

Through Examples

"Alles" - 2014/5/8 - 11:36 - page ii - #2? c?2014 by the Mathematical Association of America, Inc.

Electronic edition ISBN 978-1-61444-115-1

“Alles" — 2014/5/8 — 11:19 — page iii — #3?

Game Theory

Through Examples

Erich Prisner

Franklin University Switzerland

Published and Distributedby

The Mathematical Association of America

“Alles" — 2014/5/8 — 11:19 — page iv — #4?

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

ChristopherHallstrom

Cynthia J Huffman

Paul R Klingsberg

Brian Lins

Mary Morley

Philip P Mummert

Barbara E Reynolds

Darryl Yong

“Alles" — 2014/5/8 — 11:19 — page v — #5?

CLASSROOM RESOURCE MATERIALS

Classroom Resource Materials is intended to provide supplementary classroom material for students—

laboratoryexercises, projects, historical information,textbooks withunusual 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

IdentificationNumbers and Check Digit Schemes,Joseph Kirtland Interdisciplinary Lively ApplicationProjects,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, andVictor Katz Math Made Visual: Creating Images for UnderstandingMathematics,Claudi Alsina and Roger B. Nelsen MathematicsGalore!: 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 WithoutWords,Roger B. Nelsen

Proofs WithoutWords II,Roger B. Nelsen

Rediscovering Mathematics: You Do the Math,Shai Simonson “Alles" — 2014/5/8 — 11:19 — page vi — #6?

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 Part1: Probability and Simulation,David Williamson, Marilou Mendel, Julie Tarr, and Deborah Yoklic Student Manual for Mathematics for Business Decisions Part2: Calculus and Optimization,David Williamson, Marilou Mendel, Julie Tarr, and Deborah Yoklic

Teaching StatisticsUsing 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

“Alles" — 2014/5/8 — 11:19 — page vii — #7?

Contents

Prefacexvi

1 Theory 1: Introduction1

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 Games4

2.1 Normal Form—BimatrixDescription. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Class19

3.1 Three Players, Two Classes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1.1 “I like you both". . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 “Alles" — 2014/5/8 — 11:19 — page viii — #8? viiiContents

4 Example: Doctor Location Games25

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 Games34

5.1 A First Graph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Excel42

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 I47

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 Randomness53

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

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Contentsix

8.3.4 Breaking Ties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

8.3.5 Existing Games. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

8.3.6 Greedy Strategies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Project 12: TAKE SOME. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Project 13: WHO"s NEXT(n). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Project 14: LISA"s GAME. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Project 15: 2-AUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Project 16: 3-AUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

9 Example: Dividing A Few Items I70

9.1 Greedy Strategy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

9.2 Backward Induction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

9.2.1 Game Tree. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

9.2.2 Game Digraph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

9.2.3 Example: Game Digraph for ABBAB. . . . . . . . . . . . . . . . . . . . . . . . . . 72

9.3 An Abbreviated Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

9.3.1 Why it Matters: Complexity (optional). . . . . . . . . . . . . . . . . . . . . . . . . 74

9.4 Bottom-Up Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

9.5 Interdependencies between the Items (optional). . . . . . . . . . . . . . . . . . . . . . . . . 76

Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

10 Example: Shubik Auction I77

Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Project 17: SHUBIK AUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

11 Example: Sequential Doctor and Restaurant Location80

11.1 General Observations for Symmetric Games. . . . . . . . . . . . . . . . . . . . . . . . . . . 80

11.2 Doctor Location. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

11.3 Constant-Sum Games. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

11.4 Restaurant Location. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

11.5 Nash Equilibriaand First Mover Advantage for Symmetric Games. . . . . . . . . . . . . . . 84

Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Project 18. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Project 19: Hostile versus Friendly Play. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

12 Theory 4: Probability86

12.1 Terminology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

12.2 Computing Probabilities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

12.2.1 Equally Likely Simple Events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

12.2.2 Simple Events not Equally Likely. . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

12.3 Expected Value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

12.4 Multistep Experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

12.4.1 ProbabilityTrees. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

12.4.2 Conditional Probabilities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

12.4.3 ProbabilityDigraphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

12.5 Randomness in Simultaneous Games. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

12.6 Countingwithout Counting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

“Alles" — 2014/5/8 — 11:19 — page x — #10? xContents

Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Project 20: Tennis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Project 21: Final Exam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

13 France 165499

Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

14 Example: DMA Soccer I102

14.1 1-Round 2-Step Experiment for Given Player Distributions. . . . . . . . . . . . . . . . . . . 103

14.2 Expected Goal Difference for the One-Round Game. . . . . . . . . . . . . . . . . . . . . . . 104

14.3 3-Rounds Experiment for Given Player Distributions. . . . . . . . . . . . . . . . . . . . . . 105

14.4 Static Three-round Game. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

14.5 Static Nine-round DMA Soccer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Project 22: DMA6* Soccer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Project 23: DMA7* Soccer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

15 Example: Dividing A Few Items II110

15.1 Goals of Fairness and Efficiency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

15.1.1 Fairness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

15.1.2 Efficiency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

15.1.3 Three Additional Features. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

15.1.4 Mechanism Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

15.2 Some Games. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

15.2.1 Selecting one by one Games. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

15.2.2 Cut and Choose. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

15.2.3 Random and Exchange. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

15.3 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

15.4 Comparison of the Games for Seven Items and Complete Information. . . . . . . . . . . . . 115

15.4.1 Opposing or Similar Preferences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

15.5 Incomplete Information. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Project 24: Dividingfive items A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

Project 25: Dividingfive items B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

16 Theory 5: Sequential Games with Randomness121

16.1 Extensive Form Extended. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

16.2 Analyzing the Game: Backward Inductionagain. . . . . . . . . . . . . . . . . . . . . . . . . 121

16.3 Decision Theory: Alone against Nature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Project 26: Job Interviews. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

Project 27: 5 Envelopes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

Project 28: Oh-No or Oh-No6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Project 29:3?4version of PolyominoREC THE SQUARE with randomness. . . . . . . . . . . . 128

17 Example: Sequential Quiz Show I129

17.1 Candidates with Little Knowledge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

17.1.1 More May be Less. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

“Alles" — 2014/5/8 — 11:19 — page xi — #11?

Contentsxi

quotesdbs_dbs27.pdfusesText_33
[PDF] Beginner1. Lesson #27. Les invitations - Anciens Et Réunions

[PDF] Beginner1. Lesson #49. Au restaurant 3

[PDF] Beginner2. Lesson #17. Au bois de Boulogne - France

[PDF] Beginner2. Lesson #39. Week-end en amoureux - Anciens Et Réunions

[PDF] beginners` vinyasa

[PDF] Beginner`s Jig - Anciens Et Réunions

[PDF] Beginning a Life in Australia - English - Assurance

[PDF] Beginning algebra lial hornsby mcginnis 11th

[PDF] Beglaubigter Auszug aus dem Sitzungsbuch des Kreisausschusses

[PDF] Begleitbericht: Verordnung zum Bundesegsetz über den Konsumkredit

[PDF] Begleiten Sie - Wacker Burghausen

[PDF] Begleiten Sie uns auf dem Weg zur globalen Nr. 1

[PDF] Begleitende Folien zur Vorlesung

[PDF] Begleiterkrankungen bei Morbus Crohn

[PDF] Begleitete Gruppenreise Schottland Auf den Spuren der Highlander