Game Theory Basics - Department of Mathematics

More documents

Recommendations

Info

viiiCONTENTS3.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.4 Expected-utility payoffs . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.5 Example: Compliance inspections . . . . . . . . . . . . . . . . . . . . . 623.6 Bimatrix games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.7 Matrix notation for expected payoffs . . . . . . . . . . . . . . . . . . . . 663.8 Convex combinations and mixed strategy sets . . . . . . . . . . . . . . . 683.9 The best response condition . . . . . . . . . . . . . . . . . . . . . . . . . 693.10 Existence of mixed equilibria . . . . . . . . . . . . . . . . . . . . . . . . 713.11 Finding mixed equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . 733.12 The upper envelope method . . . . . . . . . . . . . . . . . . . . . . . . . 773.13 Degenerate games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803.14 Zero-sum games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853.15 Exercises for chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 914 Game trees with imperfect information 954.1 Learning objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.2 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.4 Information sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.5 Extensive games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024.6 Strategies for extensive games and the strategic form . . . . . . . . . . . 1044.7 Reduced strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.8 Perfect recall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1064.9 Perfect recall and order of moves ∗ . . . . . . . . . . . . . . . . . . . . . 1104.10 Behaviour strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1124.11 Kuhn’s theorem: behaviour strategies suffice . . . . . . . . . . . . . . . . 1164.12 Subgames and subgame perfect equilibria . . . . . . . . . . . . . . . . . 1204.13 Exercises for chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1225 Bargaining 1255.1 Learning objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1255.2 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1255.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1255.4 Bargaining sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1265.5 Bargaining axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1285.6 The Nash bargaining solution . . . . . . . . . . . . . . . . . . . . . . . . 1305.7 Splitting a unit pie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1345.8 The ultimatum game . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1365.9 Alternating offers over two rounds . . . . . . . . . . . . . . . . . . . . . 1395.10 Alternating offers over several rounds . . . . . . . . . . . . . . . . . . . 1435.11 Stationary strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1465.12 The Nash bargaining solution via alternating offers . . . . . . . . . . . . 1505.13 Exercises for chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
CONTENTSix6 Geometric representation of equilibria 1556.1 Learning objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1556.2 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1556.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1566.4 Best response regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1566.5 Identifying equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1626.6 The Lemke–Howson algorithm . . . . . . . . . . . . . . . . . . . . . . 1636.7 Odd number of Nash equilibria . . . . . . . . . . . . . . . . . . . . . . . 1666.8 Exercises for chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1687 Linear programming and zero-sum games 1717.1 Learning objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1717.2 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1727.3 Linear programs and duality . . . . . . . . . . . . . . . . . . . . . . . . 1727.4 The minimax theorem via LP duality . . . . . . . . . . . . . . . . . . . . 1757.5 General LP duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1777.6 The Lemma of Farkas and proof of strong LP duality ∗ . . . . . . . . . . . 1817.7 Exercises for chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
Page 2: iiThe material in this book has bee
Page 5 and 6: vthe game that is not too complicat
Page 7: Contents1 Nim and combinatorial gam
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
Page 57 and 58: 2.13. SUBGAME PERFECT NASH EQUILIBR
Page 59 and 60:
2.14. COMMITMENT GAMES 49of moves d
Page 61 and 62:
2.14. COMMITMENT GAMES 51This game
Page 63 and 64:
2.15. EXERCISES FOR CHAPTER 2 53010
Page 65 and 66:
2.15. EXERCISES FOR CHAPTER 2 55(a)
Page 67 and 68:
2.15. EXERCISES FOR CHAPTER 2 57(b)
Page 69 and 70:
Chapter 3Mixed strategy equilibriaT
Page 71 and 72:
3.4. EXPECTED-UTILITY PAYOFFS 61Sec
Page 73 and 74:
3.5. EXAMPLE: COMPLIANCE INSPECTION
Page 75 and 76:
3.6. BIMATRIX GAMES 65Secondly, mix
Page 77 and 78:
3.7. MATRIX NOTATION FOR EXPECTED P
Page 79 and 80:
3.9. THE BEST RESPONSE CONDITION 69
Page 81 and 82:
3.10. EXISTENCE OF MIXED EQUILIBRIA
Page 83 and 84:
3.11. FINDING MIXED EQUILIBRIA 73is
Page 85 and 86:
3.11. FINDING MIXED EQUILIBRIA 75
Page 87 and 88:
3.12. THE UPPER ENVELOPE METHOD 773
Page 89 and 90:
3.12. THE UPPER ENVELOPE METHOD 79G
Page 91 and 92:
3.13. DEGENERATE GAMES 81Figure 3.9
Page 93 and 94:
3.13. DEGENERATE GAMES 83equilibriu
Page 95 and 96:
3.14. ZERO-SUM GAMES 85Proof. Let x
Page 97 and 98:
3.14. ZERO-SUM GAMES 87numbers. (Ot
Page 99 and 100:
3.14. ZERO-SUM GAMES 89of payoffs t
Page 101 and 102:
3.15. EXERCISES FOR CHAPTER 3 91A f
Page 103 and 104:
3.15. EXERCISES FOR CHAPTER 3 93❅
Page 105 and 106:
Chapter 4Game trees with imperfect
Page 107 and 108:
4.4. INFORMATION SETS 97large compa
Page 109 and 110:
4.4. INFORMATION SETS 99full circle
Page 111 and 112:
4.4. INFORMATION SETS 101We can say
Page 113 and 114:
4.5. EXTENSIVE GAMES 103information
Page 115 and 116:
4.7. REDUCED STRATEGIES 105c 1II❅
Page 117 and 118:
4.8. PERFECT RECALL 107❅❅ITBII
Page 119 and 120:
4.8. PERFECT RECALL 109chance1/21/2
Page 121 and 122:
4.9. PERFECT RECALL AND ORDER OF MO
Page 123 and 124:
4.10. BEHAVIOUR STRATEGIES 113payof
Page 125 and 126:
4.10. BEHAVIOUR STRATEGIES 115The c
Page 127 and 128:
4.11. KUHN’S THEOREM: BEHAVIOUR S
Page 129 and 130:
4.11. KUHN’S THEOREM: BEHAVIOUR S
Page 131 and 132:
4.12. SUBGAMES AND SUBGAME PERFECT
Page 133 and 134:
4.13. EXERCISES FOR CHAPTER 4 123sh
Page 135 and 136:
Chapter 5BargainingThis chapter pre
Page 137 and 138:
5.4. BARGAINING SETS 127and vertica
Page 139 and 140:
5.5. BARGAINING AXIOMS 129game. In
Page 141 and 142:
5.6. THE NASH BARGAINING SOLUTION 1
Page 143 and 144:
5.6. THE NASH BARGAINING SOLUTION 1
Page 145 and 146:
5.7. SPLITTING A UNIT PIE 1351vS( )
Page 147 and 148:
5.8. THE ULTIMATUM GAME 137III0II.0
Page 149 and 150:
5.9. ALTERNATING OFFERS OVER TWO RO
Page 151 and 152:
5.9. ALTERNATING OFFERS OVER TWO RO
Page 153 and 154:
5.10. ALTERNATING OFFERS OVER SEVER
Page 155 and 156:
5.10. ALTERNATING OFFERS OVER SEVER
Page 157 and 158:
5.11. STATIONARY STRATEGIES 147I0x1
Page 159 and 160:
5.11. STATIONARY STRATEGIES 149s <
Page 161 and 162:
5.13. EXERCISES FOR CHAPTER 5 151to
Page 163 and 164:
5.13. EXERCISES FOR CHAPTER 5 153(e
Page 165 and 166:
Chapter 6Geometric representation o
Page 167 and 168:
6.4. BEST RESPONSE REGIONS 157and t
Page 169 and 170:
6.4. BEST RESPONSE REGIONS 159and c
Page 171 and 172:
6.4. BEST RESPONSE REGIONS 161label
Page 173 and 174:
6.6. THE LEMKE-HOWSON ALGORITHM 163
Page 175 and 176:
6.6. THE LEMKE-HOWSON ALGORITHM 165
Page 177 and 178:
6.7. ODD NUMBER OF NASH EQUILIBRIA
Page 179 and 180:
6.8. EXERCISES FOR CHAPTER 6 169(a)
Page 181 and 182:
Chapter 7Linear programming and zer
Page 183 and 184:
7.3. LINEAR PROGRAMS AND DUALITY 17
Page 185 and 186:
7.4. THE MINIMAX THEOREM VIA LP DUA
Page 187 and 188:
7.5. GENERAL LP DUALITY 177x ≥ 0y
Page 189 and 190:
7.5. GENERAL LP DUALITY 179These tw
Page 191 and 192:
7.6. THE LEMMA OF FARKAS AND PROOF
Page 193 and 194:
7.6. THE LEMMA OF FARKAS AND PROOF
Page 195 and 196:
7.7. EXERCISES FOR CHAPTER 7 185but
show all

Game Theory Basics - Department of Mathematics

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?