Stinchcombe M. B., Notes for a Course in Game Theory.pdf

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gtnotesF02.dvi
Notes for a Course in Game Theory
MaxwellB.Stinchcombe
FallSemester,2002. Unique#29775
Chapter 0.0
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Contents
0 Organizational Stu
7
1 Choice Under Uncertainty 9
1.1 Thebasics modelofchoice under uncertainty.................. 9
1.1.1 Notation.................................. 9
1.1.2 Thebasic modelofchoice underuncertainty .............. 10
1.1.3 Examples ................................. 11
1.2 ThebridgecrossingandrescalingLemmas................... 13
1.3 Behavior...................................... 14
1.4 Problems...................................... 15
2 Correlated Equilibria in Static Games 19
2.1 Generalitiesaboutstaticgames......................... 19
2.2 DominantStrategies ............................... 20
2.3 Twoclassicgames................................. 20
2.4 SignalsandRationalizability ........................... 22
2.5 Twoclassiccoordinationgames......................... 23
2.6 SignalsandCorrelated Equilibria . . . ..................... 24
2.6.1 Thecommonpriorassumption...................... 24
2.6.2 Theoptimizationassumption ...................... 25
2.6.3 Correlatedequilibria ........................... 26
2.6.4 Existence ................................. 27
2.7 Rescaling andequilibrium ............................ 27
2.8 Howcorrelatedequilibria mightarise . ..................... 28
2.9 Problems...................................... 29
3 Nash Equilibria in Static Games 33
3.1 Nashequilibria areuncorrelatedequilibria ................... 33
3.2 2
3
2games.................................... 36
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Chapter 0.0
3.2.1 Threemorestories............................ 36
3.2.2 Rescalingandthestrategicequivalenceofgames............ 39
3.3 Thegapbetween equilibrium andPareto rankings ............... 41
3.3.1 StagHuntreconsidered.......................... 41
3.3.2 Prisoners'Dilemmareconsidered .................... 42
3.3.3 Conclusions aboutEquilibrium andPareto rankings .......... 42
3.3.4 RiskdominanceandParetorankings.................. 43
3.4 Otherstaticgames................................ 44
3.4.1 Innite games ............................... 44
3.4.2 FiniteGames............................... 50
3.5 Harsanyi'sinterpretationofmixedstrategies.................. 52
3.6 Problemsonstaticgames ............................ 53
4 Extensive Form Games: The Basics and Dominance Arguments 55
4.1 Examplesofextensiveformgametrees..................... 55
4.1.1 Simultaneousmovegamesasextensiveformgames .......... 56
4.1.2 Somegameswith\incredible"threats.................. 57
4.1.3 Handling probability0 events . ..................... 58
4.1.4 Signalinggames.............................. 61
4.1.5 Spyinggames............................... 68
4.1.6 OtherextensiveformgamesthatIlike................. 70
4.2 Formalitiesofextensiveformgames....................... 74
4.3 Extensiveformgamesandweakdominancearguments ............ 79
4.3.1 AtomicHandgrenades .......................... 79
4.3.2 Adetourthroughsubgameperfection.................. 80
4.3.3 Arststeptowarddeningequivalenceforgames........... 83
4.4 Weakdominancearguments,plainanditerated ................ 84
4.5 Mechanisms.................................... 87
4.5.1 Hiringamanager............................. 87
4.5.2 Funding apublicgood .......................... 89
4.5.3 Monopolistselling to dierenttypes ................... 92
4.5.4 Eciencyinsalesandtherevelationprinciple............. 94
4.5.5 Shrinkageoftheequilibrium set ..................... 95
4.6 Weakdominancewithrespecttosets...................... 95
4.6.1 Variantsoniterateddeletionofdominatedsets............. 95
4.6.2 Self-referentialtests............................ 96
4.6.3 Ahorsegame............................... 97
4.6.4 Generalitiesaboutsignalinggames(redux)............... 99
4.6.5 Revisitingaspecicentry-deterrencesignalinggame..........100
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Chapter 0.0
4.7 Kuhn'sTheorem . ................................105
4.8 Equivalenceofgames...............................107
4.9 Someotherproblems...............................109
5 Mathematics for Game Theory 113
5.1 Rationalnumbers,sequences,realnumbers...................113
5.2 Limits,completeness,glb'sandlub's ......................116
5.2.1 Limits...................................116
5.2.2 Completeness...............................116
5.2.3 Greatest lower bounds andleast upper bounds . . . ..........117
5.3 Thecontractionmappingtheoremandapplications..............118
5.3.1 StationaryMarkovchains ........................119
5.3.2 Someevolutionary arguments aboutequilibria . . . ..........122
5.3.3 Theexistenceanduniquenessofvaluefunctions............123
5.4 Limitsandclosedsets ..............................125
5.5 Limitsandcontinuity...............................126
5.6 Limitsandcompactness .............................127
5.7 Correspondencesandxedpointtheorem....................127
5.8 Kakutani'sxed point theoremandequilibrium existence results .......128
5.9 Perturbationbased theoriesofequilibrium renement . . . ..........129
5.9.1 Overviewofperturbations........................129
5.9.2 PerfectionbySelten ...........................130
5.9.3 PropernessbyMyerson..........................133
5.9.4 Sequential equilibria ...........................134
5.9.5 Strictperfectionandstability byKohlberg andMertens........135
5.9.6 Stabilityby Hillas.............................136
5.10Signalinggameexercisesinrenement .....................137
6 Repeated Games 143
6.1 TheBasicSet-UpandaPreliminaryResult ..................143
6.2 Prisoners' Dilemma nitely andinnitely ....................145
6.3 Someresultsonniterepetition.........................147
6.4 Threatsinnitelyrepeatedgames........................148
6.5 Threatsininnitely repeated games. . .....................150
6.6 Rubinstein-Stahlbargaining...........................151
6.7 Optimalsimplepenalcodes ...........................152
6.8 Abreu'sexample .................................152
6.9 Harris'formulationofoptimalsimplepenalcodes...............152
6.10 \Shunning," market-place racism, andotherexamples . . . ..........154
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