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Lecturenotesongametheory Lecture Notes on Game Theory Theory and Examples Xiang Sun August 22, 2015 ii Contents Acknowledgement vii 1 Introduction 1 1.1 Timeline of the main evolution of game theory ................................ 1 1.2 Nobel prize laureates .............................................. 16 1.3 Potential Nobel prize winners ......................................... 19 1.4 Rational behavior ................................................ 20 1.5 Common knowledge .............................................. 20 2 Strategic games with complete information 23 2.1 Strategic games ................................................. 23 2.2 Nash equilibrium ................................................ 24 2.3 Examples .................................................... 25 2.4 Existence of a Nash equilibrium ........................................ 41 2.5 Strictly competitive games (zero-sum games) ................................. 43 2.6 Existence of a Nash equilibrium: games with discontinuous payoff functions ................ 44 3 Contest 47 4 Bayesian games (strategic games with incomplete information) 49 4.1 Bayes’ rule (Bayes’ theorem) .......................................... 49 4.2 Bayesian games ................................................. 50 4.3 Examples .................................................... 54 4.4 Comments on Bayesian games ......................................... 70 5 Auction 73 i Contents ii 5.1 Preliminary ................................................... 73 5.2 The symmetric model ............................................. 75 5.3 Second-price sealed-bid auction ........................................ 75 5.4 First-price sealed-bid auction ......................................... 76 5.5 Revenue comparison .............................................. 84 5.6 Reserve prices ................................................. 86 5.7 The revenue equivalence principle ....................................... 89 5.8 All-pay auction ................................................. 90 5.9 Third-price auction ............................................... 91 5.10 Uncertain number of bidders ......................................... 92 6 Mixed-strategy Nash equilibrium 95 6.1 Mixed-strategy Nash equilibrium ....................................... 95 6.2 Examples .................................................... 97 6.3 Interpretation of mixed-strategy Nash equilibrium .............................. 100 6.3.1 Purification .............................................. 101 7 Correlated equilibrium 105 7.1 Motivation ................................................... 105 7.2 Correlated equilibrium ............................................. 106 7.3 Examples .................................................... 108 8 Rationalizability 113 8.1 Rationalizability ................................................ 113 8.2 Iterated elimination of never-best response .................................. 117 8.3 Iterated elimination of strictly dominated actions .............................. 118 8.4 Examples .................................................... 120 8.5 Iterated elimination of weakly dominated actions .............................. 124 9 Knowledge model 125 9.1 A model of knowledge ............................................. 125 9.2 Common knowledge .............................................. 130 9.3 Common prior ................................................. 131 9.4 “Agree to disagree” is impossible ........................................ 132 9.5 No-trade theorem ............................................... 134 9.6 Speculation ................................................... 135 Contents iii 9.7 Characterization of the common prior assumption .............................. 137 9.8 Unawareness .................................................. 138 10 Interactive epistemology 141 10.1 Epistemic conditions for Nash equilibrium .................................. 141 10.2 Epistemic foundation of rationalizability ................................... 144 10.3 Epistemic foundation of correlated equilibrium ............................... 145 10.4 The electronic mail game ............................................ 145 11 Extensive games with perfect information 149 11.1 Extensive games with perfect information .................................. 149 11.2 Subgame perfect equilibrium ......................................... 151 11.3 Examples .................................................... 154 11.4 Three notable games .............................................. 161 11.5 Iterated elimination of weakly dominated strategies ............................. 163 11.6 Forward induction ............................................... 164 12 Bargaining games 167 12.1 A bargaining game of alternating offers .................................... 167 12.2 Bargaining games with finite horizon ..................................... 168 12.3 Bargaining games with infinite horizon .................................... 169 12.4 Properties of subgame perfect equilibria in Rubinstein bargaining games .................. 172 12.5 Bargaining games with cost .......................................... 173 12.6 n-person bargaining games .......................................... 173 13 Repeated games 177 13.1 Infinitely repeated games ............................................ 177 13.2 Trigger strategy equilibrium .......................................... 182 13.3 Tit-for-tat strategy equilibrium ........................................ 187 13.4 Folk theorem .................................................. 188 13.5 Nash-threats folk theorem ........................................... 189 13.6 Perfect folk theorem .............................................. 190 13.7 Finitely repeated games ............................................ 193 14 Extensive games with imperfect information 195 14.1 Extensive games with imperfect information ................................. 195 Contents iv 14.2 Mixed and behavioral strategies ........................................ 197 14.3 Subgame perfect equilibrium ......................................... 200 14.4 Perfect Bayesian equilibrium .......................................... 206 14.5 Sequential equilibrium ............................................. 211 14.6 Trembling hand perfect equilibrium ...................................... 214 15 Information economics 217 15.1 Adverse selection ................................................ 218 15.2 Signalling .................................................... 220 15.2.1 The market for “lemons” ....................................... 229 15.2.2 Job-market signaling ......................................... 232 15.2.3 Cheap talk ............................................... 234 15.3 Screening .................................................... 239 15.3.1 Pricing a single indivisible good ................................... 239 15.3.2 Nonlinear pricing ........................................... 240 15.4 Moral hazard and the principle-agent problem ................................ 244 15.4.1 Complete information ......................................... 244 15.4.2 Asymmetric information ....................................... 245 16 Social choice theory 247 16.1 Social choice .................................................. 247 16.2 Arrow’s impossibility theorem ......................................... 248 16.3 Borda count, simple plurality rule, and two-round system .......................... 252 16.4 Gibbard-Satterthwaite theorem ........................................ 255 17 Mechanism design 261 17.1 Envelope theorem ............................................... 262 17.2 A general mechanism design setting ..................................... 263 17.3 Dominant strategy mechanism design .................................... 265 17.3.1 Revelation principle for dominant strategies ............................. 265 17.3.2 Payoff equivalence theorem ...................................... 266 17.3.3 Gibbard-Satterthwaite theorem .................................... 267 17.3.4 VCG mechanism ........................................... 267 17.3.5 Pivot mechanism ........................................... 269 17.3.6 Balancing the budget ......................................... 272 Contents v 17.4 Bayesian mechanism design .......................................... 273 17.5 Characterization of incentive compatibility .................................. 274 17.6 Bilateral trade .................................................. 275 18 Auction: mechanism design approach 279 18.1 The revelation principle for Bayesian equilibrium .............................. 280 18.2 Incentive compatibility and individual rationality .............................. 282 18.3 Optimal auction ................................................ 285 18.4 Maximizing welfare .............................................. 289 18.5 VCG mechanism ................................................ 291 18.6 AGV mechanism ................................................ 293 19 Implementation theory 295 19.1 Implementation ................................................ 295 19.2 Implementation in dominant strategies .................................... 297 19.3 Nash implementation ............................................. 300 20 Coalitional games 305 20.1 Coalitional game ................................................ 305 20.2 Core ....................................................... 306 20.3 Shapley value .................................................
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