Game Theory Basics - Department of Mathematics

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viThe book is relatively swift in treating games in strategic form and game trees in asingle chapter (chapter 3), which are often considered separately. I find these conceptssimple enough to allow for that approach.Mixed strategies and the best response condition are very useful for finding equilibriain games, so they are given detailed treatment. Similarly, game trees with imperfect information,and the concepts of perfect recall and behaviour strategies, and Kuhn’s theorem,are also given an in-depth treatment, because this not found in many textbooks at thislevel.The book does not try to be comprehensive in presenting the most relevant economicapplications of game theory. Corresponding “stories” are only told for illustration of theconcepts.One book served as a starting point for the selection of the material, namely• K. Binmore, Fun and Games, D. C. Heath, 1991. Its recent, significantly revisedversion is Playing for Real, Oxford University Press, 2007.This book describes nim as a game that one can learn to play perfectly. In turn, thisled me to the question of why the binary system has such an important role, and towardsthe study of combinatorial games. The topic of bargaining is also motivated by Binmore’sbook. At the same time, the present text is very different from Binmore’s book, becauseit tries to treat as few side topics as possible.Two short sections can be considered as slightly non-standard side topics, namely thesections 2.9, Symmetries involving strategies, and 4.9 Perfect recall and order of moves,both marked with a star as “can be skipped at first reading”. I included them as topicsthat I have not found elsewhere. Section 4.9 demonstrates how to reason carefully aboutmoves in extensive games, and should help understand the perfect recall condition.Methods, not philosophyThe general emphasis of the book is to teach methods, not philosophy. Game theoriststend to question and to justify the approaches they take, for example the concept of Nashequilibrium, and the assumed common knowledge of all players about the rules of thegame. These questions are of course very important. In fact, in practice these are probablythe very issues that make a game-theoretic analysis questionable. However, this problemis not remedied by a lengthy discussion of why one should play Nash equilibrium.I think a student who learns about game theory should first become fluent in knowingthe modelling tools (such as game trees and the strategic form) and in analysing games(finding their Nash equilibria). That toolbox will then be useful when comparing differentgame-theoretic models and looking at their implications. In many disciplines that arefurther away from mathematics, these possible implications are typically more interestingthan the mathematical model itself (the model is necessarily imperfect, so there is no pointin being dogmatic about the analysis).
Contents1 Nim and combinatorial games 11.1 Aims of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Learning objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 What is combinatorial game theory? . . . . . . . . . . . . . . . . . . . . 21.5 Nim – rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.6 Combinatorial games, in particular impartial games . . . . . . . . . . . . 51.7 Simpler games and notation for nim heaps . . . . . . . . . . . . . . . . . 61.8 Sums of games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.9 Equivalent games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.10 Sums of nim heaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.11 Poker nim and the mex rule . . . . . . . . . . . . . . . . . . . . . . . . . 161.12 Finding nim values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.13 Exercises for chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 Games as trees and in strategic form 272.1 Learning objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.4 Definition of game trees . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.5 Backward induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.6 Strategies and strategy profiles . . . . . . . . . . . . . . . . . . . . . . . 322.7 Games in strategic form . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.8 Symmetric games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.9 Symmetries involving strategies ∗ . . . . . . . . . . . . . . . . . . . . . . 372.10 Dominance and elimination of dominated strategies . . . . . . . . . . . . 382.11 Nash equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.12 Reduced strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.13 Subgame perfect Nash equilibrium (SPNE) . . . . . . . . . . . . . . . . 452.14 Commitment games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.15 Exercises for chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 523 Mixed strategy equilibria 593.1 Learning objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.2 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59vii
Page 2: iiThe material in this book has bee
Page 5: vthe game that is not too complicat
Page 9 and 10: CONTENTSix6 Geometric representatio
Page 11 and 12: Chapter 1Nim and combinatorial game
Page 13 and 14: 1.4. WHAT IS COMBINATORIAL GAME THE
Page 15 and 16: 1.6. COMBINATORIAL GAMES, IN PARTIC
Page 17 and 18: 1.8. SUMS OF GAMES 7principle of in
Page 19 and 20: 1.9. EQUIVALENT GAMES 9The games th
Page 21 and 22: 1.9. EQUIVALENT GAMES 11It is an ea
Page 23 and 24: 1.10. SUMS OF NIM HEAPS 13Theorem 1
Page 25 and 26: 1.10. SUMS OF NIM HEAPS 15How does
Page 27 and 28: 1.12. FINDING NIM VALUES 17K of G s
Page 29 and 30: 1.12. FINDING NIM VALUES 19by movin
Page 31 and 32: 1.13. EXERCISES FOR CHAPTER 1 21(b)
Page 33 and 34: 1.13. EXERCISES FOR CHAPTER 1 23end
Page 35 and 36: 1.13. EXERCISES FOR CHAPTER 1 25, ,
Page 37 and 38: Chapter 2Games as trees and in stra
Page 39 and 40: 2.4. DEFINITION OF GAME TREES 29IIa
Page 41 and 42: 2.5. BACKWARD INDUCTION 31of the ga
Page 43 and 44: 2.7. GAMES IN STRATEGIC FORM 332.7
Page 45 and 46: 2.8. SYMMETRIC GAMES 35case letters
Page 47 and 48: 2.9. SYMMETRIES INVOLVING STRATEGIE
Page 49 and 50: 2.10. DOMINANCE AND ELIMINATION OF
Page 51 and 52: 2.11. NASH EQUILIBRIUM 41⇒ Exerci
Page 53 and 54: 2.12. REDUCED STRATEGIES 43⇒ In o
Page 55 and 56: 2.13. SUBGAME PERFECT NASH EQUILIBR
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2.13. SUBGAME PERFECT NASH EQUILIBR
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2.14. COMMITMENT GAMES 49of moves d
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2.14. COMMITMENT GAMES 51This game
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2.15. EXERCISES FOR CHAPTER 2 53010
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2.15. EXERCISES FOR CHAPTER 2 55(a)
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2.15. EXERCISES FOR CHAPTER 2 57(b)
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Chapter 3Mixed strategy equilibriaT
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3.4. EXPECTED-UTILITY PAYOFFS 61Sec
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3.5. EXAMPLE: COMPLIANCE INSPECTION
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3.6. BIMATRIX GAMES 65Secondly, mix
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3.7. MATRIX NOTATION FOR EXPECTED P
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3.9. THE BEST RESPONSE CONDITION 69
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3.10. EXISTENCE OF MIXED EQUILIBRIA
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3.11. FINDING MIXED EQUILIBRIA 73is
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3.11. FINDING MIXED EQUILIBRIA 75
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3.12. THE UPPER ENVELOPE METHOD 773
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3.12. THE UPPER ENVELOPE METHOD 79G
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3.13. DEGENERATE GAMES 81Figure 3.9
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3.13. DEGENERATE GAMES 83equilibriu
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3.14. ZERO-SUM GAMES 85Proof. Let x
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3.14. ZERO-SUM GAMES 87numbers. (Ot
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3.14. ZERO-SUM GAMES 89of payoffs t
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3.15. EXERCISES FOR CHAPTER 3 91A f
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3.15. EXERCISES FOR CHAPTER 3 93❅
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Chapter 4Game trees with imperfect
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4.4. INFORMATION SETS 97large compa
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4.4. INFORMATION SETS 99full circle
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4.4. INFORMATION SETS 101We can say
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4.5. EXTENSIVE GAMES 103information
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4.7. REDUCED STRATEGIES 105c 1II❅
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4.8. PERFECT RECALL 107❅❅ITBII
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4.8. PERFECT RECALL 109chance1/21/2
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4.9. PERFECT RECALL AND ORDER OF MO
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4.10. BEHAVIOUR STRATEGIES 113payof
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4.10. BEHAVIOUR STRATEGIES 115The c
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4.11. KUHN’S THEOREM: BEHAVIOUR S
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4.11. KUHN’S THEOREM: BEHAVIOUR S
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4.12. SUBGAMES AND SUBGAME PERFECT
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4.13. EXERCISES FOR CHAPTER 4 123sh
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Chapter 5BargainingThis chapter pre
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5.4. BARGAINING SETS 127and vertica
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5.5. BARGAINING AXIOMS 129game. In
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5.6. THE NASH BARGAINING SOLUTION 1
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5.6. THE NASH BARGAINING SOLUTION 1
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5.7. SPLITTING A UNIT PIE 1351vS( )
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5.8. THE ULTIMATUM GAME 137III0II.0
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5.9. ALTERNATING OFFERS OVER TWO RO
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5.9. ALTERNATING OFFERS OVER TWO RO
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5.10. ALTERNATING OFFERS OVER SEVER
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5.10. ALTERNATING OFFERS OVER SEVER
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5.11. STATIONARY STRATEGIES 147I0x1
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5.11. STATIONARY STRATEGIES 149s <
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5.13. EXERCISES FOR CHAPTER 5 151to
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5.13. EXERCISES FOR CHAPTER 5 153(e
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Chapter 6Geometric representation o
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6.4. BEST RESPONSE REGIONS 157and t
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6.4. BEST RESPONSE REGIONS 159and c
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6.4. BEST RESPONSE REGIONS 161label
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6.6. THE LEMKE-HOWSON ALGORITHM 163
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6.6. THE LEMKE-HOWSON ALGORITHM 165
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6.7. ODD NUMBER OF NASH EQUILIBRIA
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6.8. EXERCISES FOR CHAPTER 6 169(a)
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Chapter 7Linear programming and zer
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7.3. LINEAR PROGRAMS AND DUALITY 17
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7.4. THE MINIMAX THEOREM VIA LP DUA
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7.5. GENERAL LP DUALITY 177x ≥ 0y
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7.5. GENERAL LP DUALITY 179These tw
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7.6. THE LEMMA OF FARKAS AND PROOF
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7.6. THE LEMMA OF FARKAS AND PROOF
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7.7. EXERCISES FOR CHAPTER 7 185but
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Game Theory Basics - Department of Mathematics

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